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Q: When automatically updating WordPress, how does it grab files from another server and store them on my server? When I automatically update WordPress, how does it grab files from another server and store them on my server? It doesn't have ftp access rights. Does anyone know?
A: According to this thread on the Wordpress forums auto-update works in a variety of ways to try and update itself:
On some hosts (notably ones that run "suPHP", allowing the software to run with the file owner's credentials instead of with the webserver credentials), the software has access to modify its own files directly. In those cases, update simply happens using the 'direct' mode.
On other hosts, it doesn't have the access to modify its own files and maintain the same ownership and permissions and and such. Those latter hosts give the FTP login credentials, and WordPress attempts to log in to itself via FTP, in order to be able to change the files using the proper user credentials.
On a technical level, I believe the actual fetching of files is done through cURL.
A: It can use PHP's fopen function, if Wordpress's directory is writable. Otherwise it will ask for the FTP username and password.
As for actually fetching the file from a remote server I believe it uses curl
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{"url":"https:\/\/www.mersenneforum.org\/showpost.php?s=26a6f0f6c3cfd0d3683d5fabb242882e&p=52853&postcount=9","text":"2005-04-07, 18:01 #9 Templus \u00a0 \u00a0 Jun 2004 2\u00d753 Posts I tried this method for n=81 and 2 < k < 2^32. I made a sieve file which contains about 16000 lines, but I'm not sure about the ABC rule I have to construct. I now have: Code: ABC $a*2^$b+1 & $a*2^$b-1 & $a*2^($b+1)+1 & $a*2^($b+1)-1 & 2^$b+$a & 2^$b-$a & 2^($b+1)+$a & 2^($b+1)-$a Is this definition correct? When I run this code and I check the pfgw.log file I only see lines of the form k*2^n+1, k*2^n-1, k*2^(n+1)+1, k*2^(n+1)-1 but not of the form 2^n+k, 2^n-k, 2^(n+1)+k, 2^(n+1)-k. What am I doing wrong?","date":"2022-12-08 05:43:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6427802443504333, \"perplexity\": 4028.2350681822177}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446711278.74\/warc\/CC-MAIN-20221208050236-20221208080236-00796.warc.gz\"}"}
| null | null |
{"url":"https:\/\/chemistry.stackexchange.com\/questions\/38532\/how-many-atoms-of-helium-are-in-the-chamber","text":"How many atoms of helium are in the chamber?\n\nA $22.7 \\,\\mathrm{L}$ helium-filled chamber at $298\\,\\mathrm{K}$ has a tiny hole of area $1.63 \\times 10^{-5}\\,\\mathrm{mm^2}$ punched in it through which is leaking helium at a rate of $2.25 \\times 10^{15}$ atoms per second. How many atoms of helium are in the chamber?\n\nI'm personally not too sure how to answer this, I'm thinking that it probably uses the root-mean-square speed or collisions with the wall equation ($Z_\\mathrm w$). But I'm sort of lost when it comes to substituting the values and so on. (This is also a question on last years Gas Laws test, and I was planning on solving it for practice questions)\n\nWhat I tried doing was using the Collisions with the wall equation: https:\/\/en.wikipedia.org\/wiki\/Kinetic_theory#Collisions_with_container then solved for $N$, but the answer is supposed to be $N = 9.98 \\times 10^{21}$, and the closest I've gotten was getting $3.38 \\times 10^{22}$, so I'm sort of stuck on the procedure of solving this question.\n\n\u2022 I absolutely hated this topic. Anyway can you identify which equation you need to use? Once you've done that, it's just a matter of making sure your units are consistent. \u2013\u00a0orthocresol Oct 6 '15 at 21:50\n\nYou already mentioned the equation for the rate of collisions of gas molecules with a section of wall:\n\n$$Z_\\mathrm w=\\frac14\\frac NV\\sqrt{\\frac{8RT}{\\pi M}}A$$\n\nWe know the values of\nrate $$Z_\\mathrm w=2.25\\times10^{15}\\ \\mathrm{s^{-1}}$$,\nvolume $$V=22.7\\ \\mathrm l=0.0227\\ \\mathrm{m^3}$$,\nmolar gas constant $$R=8.314462618\\ \\mathrm{J\\ mol^{-1}\\ K^{-1}}$$,\ntemperature $$T=298\\ \\mathrm K$$,\nmolar mass of helium $$M=4.003\\ \\mathrm{g\\ mol^{-1}}=0.004003\\ \\mathrm{kg\\ mol^{-1}}$$, and\narea $$A=1.63\\times10^{-5}\\ \\mathrm{mm^2}=1.63\\times10^{-11}\\ \\mathrm{m^2}$$.\n\nThus, the number $$N$$ of helium atoms is\n\n\\begin{align} N&=\\frac{4Z_\\mathrm wV}{\\sqrt{\\dfrac{8RT}{\\pi M}}A}\\\\[6pt] &=\\frac{4\\times2.25\\times10^{15}\\ \\mathrm{s^{-1}}\\times 0.0227\\ \\mathrm{m^3}}{\\sqrt{\\dfrac{8\\times8.314462618\\ \\mathrm{J\\ mol^{-1}\\ K^{-1}} \\times 298\\ \\mathrm K}{\\pi \\times 0.004003\\ \\mathrm{kg\\ mol^{-1}}}}\\times1.63\\times10^{-11}\\ \\mathrm{m^2}}\\\\[6pt] &=9.98\\times10^{21} \\end{align}\n\n\u2022 Thanks alot, this made sense mainly, the only part from this I didnt understand is, i have the same equation as you provided, but I didnt know that it could be multipled by A (area). Any reason why? \u2013\u00a0Mathquestion-eer Oct 8 '15 at 1:36\n\u2022 @Mathquestion-eer I can only speculate on differences. Maybe, the result $Z_\\mathrm w$ in your equation is defined as \u2018rate per area\u2019. You can always check your calculation with the help of the units. When you use coherent units, conversion factors between units are never required ($1\\ \\mathrm J=1\\ \\mathrm{kg\\ m^2\\ s^{-2}}$). If the units cancel out so that the final result has the correct unit, this is a nice indication, but not a proof, that your calculation is correct. Note that the number $N$ is a quantity of dimension one (a quantity of dimension one is often called dimensionless). \u2013\u00a0Loong Oct 8 '15 at 9:39","date":"2020-04-03 05:10:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 9, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9707013964653015, \"perplexity\": 608.684364135918}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585370510287.30\/warc\/CC-MAIN-20200403030659-20200403060659-00167.warc.gz\"}"}
| null | null |
Q: PHP Get Page Load Stats - How to measure php script execution / load time What I have in the header:
$time = microtime();
$time = explode(' ', $time);
$time = $time[1] + $time[0];
$start = $time;
What I have in the footer:
$time = microtime();
$time = explode(' ', $time);
$time = $time[1] + $time[0];
$finish = $time;
$total_time = round(($finish - $start), 4);
echo 'Page generated in ' . $total_time . ' seconds.';
Output: Page generated in 1292008977.54 seconds.
Can someone please help me figure out why the result is not right?? I am using PHP5.
A: Seeing how this is the first result in Google I thought I'd share my solution to this problem. Put this at the top of your page:
$startScriptTime=microtime(TRUE);
And then put this code at the bottom of your page:
$endScriptTime=microtime(TRUE);
$totalScriptTime=$endScriptTime-$startScriptTime;
echo "\n\r".'<!-- Load time: '.number_format($totalScriptTime, 4).' seconds -->';
When you view source of a page you can see the load time in a comment on the last line of your HTML.
A: microtime() returns the current Unix timestamp with microseconds. i don't see any math there that does the conversion from microseconds to seconds.
microtime(true) returns the time as a float in seconds
A: You can use this simple function to avoid the variable scope issue:
<?php
function timer()
{
static $start;
if (is_null($start))
{
$start = microtime(true);
}
else
{
$diff = round((microtime(true) - $start), 4);
$start = null;
return $diff;
}
}
timer();
echo 'Page generated in ' . timer() . ' seconds.';
A: $page_loadtime_in_millisec = ($page_loadtime / 1000);
echo '<pre>Page Infor:
Page Load Time : ' . $page_loadtime.' <b>Microseconds</b><br/>
Page Load Time : ' . $page_loadtime_in_millisec.' <b>Milliseconds</b><br/>
Page Load Time : ' . number_format(($page_loadtime_in_millisec/1000),18) . ' <b>Seconds</b></pre>';
A: One liner for the footer, which returns seconds as float:
echo number_format(microtime(true) - $_SERVER["REQUEST_TIME_FLOAT"],4);
A: I like to use something like this. Makes it easy to time multiple code blocks without juggling variable names and such. sessions have to be enabled.
function code_timer ($name) {
$mtime = explode(' ',microtime());
$time = $mtime[1] + $mtime[0];
//determine if we're starting the timer or ending it
if ($_SESSION["timer-$name"]) {
$stime=$_SESSION["timer-$name"];
unset($_SESSION["timer-$name"]);
return ($time - $stime);
} else {
$_SESSION["timer-$name"]=$time;
return(true);
}
}
usage:
code_timer ('a');
//do stuff
echo "page generated in " . code_timer('a');
A: Put this in your Header
<?php
$starttime = explode(' ', microtime());
$starttime = $starttime[1] + $starttime[0];
?>
and this in your footer
<html><center>Page generated in <?php $load = microtime();print (number_format($load,2));?> seconds. <?php
$loadtime = explode(' ', microtime()); $loadtime = $loadtime[0]+$loadtime[1]-$starttime; echo 'Peak memory usage: ',round(memory_get_peak_usage()/1048576, 2), 'MB';
?></center></html>
this will tell you how long it took your sites page to generate and how much Memory was used to load the page
A: The issue is with variable scope. You set your $start variable in the header, but in footer, this variable will be empty. So $total_time would just be current time - 0, giving you the current time.
A solution is to use php's GLOBALS. In the header:
$GLOBALS['time_start'] = microtime(true);
And in the footer:
$total_time = round(($finish - $GLOBALS['time_start']), 4);
A: Instead of using microtime in every single page, what i would do is to insert microtime into $_REQUEST and then subtract that time from the current time inside a function and set that function to be called when the script execution is terminated using:
register_shutdown_function ( 'Your_function_name' );
I think that it is useful to use a global script which will be included at the beginning of every script / class throughout the application, it helps me in handling errors, managing sessions, etc...
Adding microtime to $_REQUEST would be the first line in that script and you can include your terminating function there too.
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| 263
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ASP.NET AJAX provides many ways to improve user experience and performance. One of the most common way is undoubtedly the UpdatePanel control. The UpdatePanel control provides partial page rendering capabilities to your web forms. Commonly developers place data bound controls such as GridView and DataList inside an UpdatePanel so that only that area of the web form is refreshed after a post back from within the UpdatePanel. However, there is still better way to use UpdatePanel with grids and lists. Don't agree? Read on to know more!
Often developers use UpdatePanel control to host data bound controls such as GridView and DataList. They place the data bound controls inside the UpdatePanel so that any post back from the data bound control refreshes only the region marked by the UpdatePanel. Though this mechanism works as expected, it is not the most optimized technique. Have a look at the following figure.
The above figure shows a DataList control placed inside an UpdatePanel control. Any post back from the Save button is going to update the region marked as "Area being refreshed". However, in reality only the region marked as "Area being changed" needs some attention. That means for any change in one cell is going to refresh the whole UpdatePanel. Not an optimized way. Isn't it?
How can we avoid the complete UpdatePanel from refreshing? By putting an UpdatePanel inside cells of DataList control. This way only that cell region will be posted back to the server and not the complete DataList.
Let's quickly build an example to illustrate what we just discussed.
Create a new web site using Visual Studio. Drag and drop an SQL data source control on the default web form and configure it to select CustomerID, CompanyName, ContactName and Country columns of Customers table of Northwind database.
Now drag and drop a DataList control on the form and set its DataSourceID property to SqlDataSource1.
The template consists of an UpdatePanel and an UpdateProgress controls.
Arrange a Label control for displaying CustomerID and three TextBox controls for displaying CompanyName, ContactName and Country respectively. Also, add a Button control and set its Text property to Save. Clicking on this button will save the changes made to the data in the database.
The UpdateProgress control contains a Label for displaying progress message during update operation. Also, set its AssociatedUpdatePanelID property to the ID of the UpdatePanel control we added earlier.
Configure the data bindings of the label and textboxes mentioned above using data bindings editor.
The ItemCommand event handler simply obtains the new values entered by the user, fills the UpdateParameters collection of the SDS accordingly and calls Update() method of the SDS. Just for the sake of testing it also introduces a delay of 1 second so that you can see the progress message clearly.
Run the web form test how it updates region of only one cell. The following figure shows a sample run of the web form. Notice how the progress message is being displayed only for the cell being refreshed.
That's it! Now only the region of the cell being changed will refresh and not the whole DataList.
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| 6,593
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Cute pewter charm sets to add help you keep track of what's what when you're knitting. Each set comes in a handy tin and contains 4 unique markers. There is a set for cats, dogs, gardening, positive messages. Pewter charms are made in the USA!
|
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"redpajama_set_name": "RedPajamaC4"
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| 1,862
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filter. Therefore, this converter provides a low ripple output supply over a broad range of input voltage & output current loading.
drops below 3.3V. Normal operation resumes once VCC rises above 3.5V.
affect the ripple current ΔI.
equal to the maximum load current plus half the ripple current to prevent core saturation (3A+0.225A).
capacitor sized for maximum RMS current must be used. A 470µF low ESR capacitor for most applications is sufficient.
ESR value is the most important parameter. The output ripple can be calculated from the following formula.
|
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| 5,603
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Election Do-Over Poll Shows Gains for Gary Johnson, Jill Stein
No sign of third-party regrets
Jesse Walker | 4.24.2017 10:36 AM
(Mike F. Ziethlow)
If the people who participated in last year's election could do it all again, Donald Trump would win the popular vote this time—but he wouldn't actually get more support than before. Instead, according to a new Washington Post/ABC News poll, many Hillary Clinton voters would now stay home or back a third-party candidate.
In the actual election, Clinton bested Trump in the popular vote, 48 percent to 46 percent. In the survey, 46 percent said they voted for Clinton and 43 percent said they voted for Trump—not the same numbers, obviously, but it's a similar margin. When those same people were asked who they'd pick if they could do it again, Trump now won, 43 to 40.
You'll note that Trump hasn't gotten any more popular—he gets 43 percent either way. But Clinton has bled support: Gary Johnson now gets 5 percent of the vote (one point higher than how the respondents said they voted last year), Jill Stein gets 3 percent (another one-point bump), and another 8 percent would either vote for someone else or not vote at all. (The remainder say they have no opinion.) The pollsters note that "nonwhites are 10 points more likely than whites to say they would not support Clinton again, with more than a third of them heading to the Libertarian candidate, Gary Johnson."
It's not all bad news for the Clintonites, though. When you include people who didn't vote in 2016, Clinton comes out ahead in the do-over, 41 percent to 37 percent. (Johnson and Stein are still at 5 and 3 percent, respectively.) So some nonvoters appear to wish they hadn't sat the last election out.
But when it comes to third-party supporters, we don't seem to be seeing anything like the regretful Ralph Nader voters of 2000. If anything, this poll suggests we're witnessing the opposite.
Bonus link: "Again and again this year, Americans looked at the choices before them and said, I'd prefer something else."
NEXT: Trump Beats Clinton in New Poll, White House Pushes Congress to Use Government Shutdown to Secure Border Funding, Le Pen, Macron Headed to Second Round in French Presidential Elections: A.M. Links
Books Editor Jesse Walker is the author of Rebels on the Air and The United States of Paranoia.
Third PartiesPollsGary JohnsonJill SteinDonald TrumpHillary ClintonElection 2016
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{"url":"https:\/\/zbmath.org\/?q=ai%3Aarcondoulis.elias-j-g+se%3A331","text":"# zbMATH \u2014 the first resource for mathematics\n\nImplementation of compressible porous-fluid coupling method in an aerodynamics and aeroacoustics code. II: Turbulent flow. (English) Zbl\u00a01433.76055\nSummary: Due to the potential advantages and applications of porous materials in aerodynamic and aeroacoustic applications, a porous-fluid coupling solver under compressible turbulence circumstances is of significance for CFD researchers. In this paper, a porous-fluid coupling method for compressible turbulence problems based on a $$k - \\omega$$ model is proposed and implemented in the finite-volume-based in-house code Aeroacoustic and Aerodynamic Investigation Simulator (AAISIM). At the porous-fluid interface, the conservation equations of mass, energy, turbulent kinetic energy, specific dissipation rate and isentropic condition are solved to balance the flux through the interface. The stress jump condition and turbulence wall function are also employed for the treatment of the permeable wall. For the Beavers and Joseph\u2019s problem, the current results show good agreement with previous Direct Numerical Simulation results for a Reynolds number of 5500. Flat plates with a porous section in the middle and at the trailing edge are simulated and compared against experimental data thus providing a validation of this porous-fluid coupling method.\nFor Part I of this series, see [the first author et al., ibid. 364, Article ID 124682, 19 p. (2020; Zbl 1433.76152)].\n##### MSC:\n 76F50 Compressibility effects in turbulence 76F60 $$k$$-$$\\varepsilon$$ modeling in turbulence 76G25 General aerodynamics and subsonic flows 76Q05 Hydro- and aero-acoustics 76S05 Flows in porous media; filtration; seepage 80A19 Diffusive and convective heat and mass transfer, heat flow 76D05 Navier-Stokes equations for incompressible viscous fluids 76F10 Shear flows and turbulence\nAUSM; AAISIM\nFull Text:\n##### References:\n [1] Li, Z.; Zhang, H.; Liu, Y.; McDonough, J. M., Implementation of compressible porous-fluid coupling method in an aerodynamics and aeroacoustics code part I: Laminar flow, Appl. Math. Comput., 364, 124682 (2020) \u00b7 Zbl\u00a01433.76152 [2] Vafai, K.; Tien, C., Boundary and inertia effects on flow and heat transfer in porous media, Int. J. Heat Mass Transf., 24, 2, 195-203 (1981) \u00b7 Zbl\u00a00464.76073 [3] Koch, D. L.; Brady, J. F., Dispersion in fixed beds, J. Fluid Mech., 154, 399-427 (1985) \u00b7 Zbl\u00a00592.76130 [4] Gunn, D., Theory of axial and radial dispersion in packed beds, Trans. Inst. Chem. Eng. Chem. Eng., 47, 10, T351 (1969) [5] Hsu, C.; Cheng, P., Thermal dispersion in a porous medium, Int. J. Heat Mass Transf., 33, 8, 1587-1597 (1990) \u00b7 Zbl\u00a00703.76079 [6] Bear, J., Dynamics of Fluids in Porous Media (2013), Courier Corporation [7] Mickley, H.; Smith, K.; Korchak, E., Fluid flow in packed beds, Chem. Eng. Sci., 20, 3, 237-246 (1965) [8] Van der Merwe, D.; Gauvin, W., Velocity and turbulence measurements of air flow through a packed bed, AIChE J., 17, 3, 519-528 (1971) [9] Edwards, M.; Richardson, J., Gas dispersion in packed beds, Chem. Eng. Sci., 23, 2, 109-123 (1968) [10] Gunn, D.; De Souza, J., Heat transfer and axial dispersion in packed beds, Chem. Eng. Sci., 29, 6, 1363-1371 (1974) [11] Levec, J.; Carbonell, R., Longitudinal and lateral thermal dispersion in packed beds. Part I: Theory, AIChE J., 31, 4, 581-590 (1985) [12] Masuoka, T.; Takatsu, Y., Turbulence model for flow through porous media, Int. J. Heat Mass Transf., 39, 13, 2803-2809 (1996) \u00b7 Zbl\u00a00964.76547 [13] Antohe, B.; Lage, J., A general two-equation macroscopic turbulence model for incompressible flow in porous media, Int. J. Heat Mass Transf., 40, 13, 3013-3024 (1997) \u00b7 Zbl\u00a00962.76532 [14] Getachew, D.; Minkowycz, W.; Lage, J., A modified form of the k-\u03b5 model for turbulent flows of an incompressible fluid in porous media, Int. J. Heat Mass Transf., 43, 16, 2909-2915 (2000) \u00b7 Zbl\u00a01094.76529 [15] Pinson, F.; Gr\u00e9goire, O.; Simonin, O., k-\u03b5 macro-scale modeling of turbulence based on a two scale analysis in porous media, Int. J. Heat Fluid Flow, 27, 5, 955-966 (2006) [16] Nakayama, A.; Kuwahara, F., A macroscopic turbulence model for flow in a porous medium, J. Fluids Eng., 121, 2, 427-433 (1999) [17] Braga, E. J.; de Lemos, M. J., Turbulent natural convection in a porous square cavity computed with a macroscopic \u03ba-\u03b5 model, Int. J. Heat Mass Transf., 47, 26, 5639-5650 (2004) \u00b7 Zbl\u00a01121.76339 [18] de Lemos, M. J., Numerical simulation of turbulent combustion in porous materials, Int.s Commun. Heat Mass Transf., 36, 10, 996-1001 (2009) [19] Wang, S.; Mujumdar, A., A comparative study of five low Reynolds number k-\u03b5 models for impingement heat transfer, Appl. Therm. Eng., 25, 1, 31-44 (2005) [20] Yu, P.; Zeng, Y.; Lee, T.; Bai, H.; Low, H., Wake structure for flow past and through a porous square cylinder, Int. J. Heat Fluid Flow, 31, 2, 141-153 (2010) [21] Yu, P.; Lee, T. S.; Zeng, Y.; Low, H. T., A numerical method for flows in porous and homogenous fluid domains coupled at the interface by stress jump, Int. J. Numer. Methods Fluids, 53, 11, 1755-1775 (2007) \u00b7 Zbl\u00a01370.76110 [22] Ochoa-Tapia, J. A.; Whitaker, S., Momentum transfer at the boundary between a porous medium and a homogeneous fluid-I. Theoretical development, Int. J. Heat Mass Transf., 38, 14, 2635-2646 (1995) \u00b7 Zbl\u00a00923.76320 [23] Ochoa-Tapia, J. A.; Whitaker, S., Momentum transfer at the boundary between a porous medium and a homogeneous fluid-II. Comparison with experiment, Int. J. Heat Mass Transf., 38, 14, 2647-2655 (1995) \u00b7 Zbl\u00a00923.76320 [24] Martin, A.; Zhang, H.; Tagavi, K. A., An introduction to the derivation of surface balance equations without the excruciating pain, Int. J. Heat Mass Transf., 115, 992-999 (2017) [25] Weng, H.; Martin, A., Development of a universal solver and its application to ablation problems, Proceedings of the Forty-Seventh AIAA Thermophysics Conference, 3355 (2017) [26] Liu, H.; Wei, J., On the role of surface permeability for the control of flow around a circular cylinder., J. Vibroeng., 18, 8 (2016) [27] Liu, H.; Wei, J.; Qu, Z., Prediction of aerodynamic noise reduction by using open-cell metal foam, J. Sound Vib., 331, 7, 1483-1497 (2012) [28] Arcondoulis, E.; Liu, Y.; Li, Z.; Yang, Y.; Wang, Y.; Li, W., The design and noise characteristics of a structured porous coated cylinder in uniform flow, Proceedings of the AIAA\/CEAS Aeroacoustics Conference, 2963 (2018) [29] Arcondoulis, E. J.; Liu, Y.; Li, Z.; Yang, Y.; Wang, Y., Structured porous material design for passive flow and noise control of cylinders in uniform flow, Materials, 12, 18, 2905 (2019) [30] Rubio Carpio, A.; Avallone, F.; Ragni, D., On the role of the flow permeability of metal foams on trailing edge noise reduction, Proceedings of the AIAA\/CEAS Aeroacoustics Conference, 2964 (2018) [31] Zhou, B. Y.; Gauger, N. R.; Koh, S. R.; Meinke, M. H.; Schroeder, W., On the adjoint-based control of trailing-edge turbulence and noise minimization via porous material, Proceedings of the Twenty-First AIAA\/CEAS Aeroacoustics Conference, 2530 (2015) [32] Geyer, T.; Sarradj, E.; Fritzsche, C., Measurement of the noise generation at the trailing edge of porous airfoils, Exp. Fluids, 48, 2, 291-308 (2010) [33] Revell, J.; Revell, J.; Kuntz, H.; Balena, F.; Horne, C.; Storms, B.; Dougherty, R.; Kuntz, H.; Balena, F.; Horne, C., Trailing-edge flap noise reduction by porous acoustic treatment, Proceedings of the Third AIAA\/CEAS Aeroacoustics Conference, 1646 (1997) [34] Breugem, W. P., The influence of wall permeability on laminar and turbulent flows: theory and simulations (2005.), TU Delft, Delft University of Technology, Ph.D. thesis [35] Showkat Ali, S. A.; Azarpeyvand, M.; Sz\u00f6ke, M.; Il\u00e1rio da Silva, C. R., Boundary layer flow interaction with a permeable wall, Phys. Fluids, 30, 8, 085111 (2018) [36] Showkat Ali, S. A.; Azarpeyvand, M.; da Silva, C. R.I., Trailing-edge flow and noise control using porous treatments, J. Fluid Mech., 850, 83-119 (2018) \u00b7 Zbl\u00a01415.76618 [37] Li, Z., Data-Driven Adaptive Reynolds-Averaged Navier-Stokes k-\u03c9 Models for Turbulent Flow-Field Simulations (2017.), Univeristy of Kentucky, Ph.D. thesis [38] Wilcox, D. C., Formulation of the $$k - \\omega$$ turbulence model revisited, AIAA J., 46, 11, 2823-2838 (2008) [39] M\u00f6\u00dfner, M.; Radespiel, R., Modelling of turbulent flow over porous media using a volume averaging approach and a Reynolds stress model, Comput. Fluids, 108, 25-42 (2015) \u00b7 Zbl\u00a01390.76199 [40] Sondak, D. L., Wall functions for the k-\u03f5 turbulence model in generalized nonorthogonal curvilinear coordinates (1992), Iowa State University, Ph.D. thesis [41] Silva, R. A.; De Lemos, M. J., Turbulent flow in a composite channel, Int. Commun. Heat Mass Transf., 38, 8, 1019-1023 (2011) [42] Liou, M.-S., A sequel to AUSM, part II: AUSM+-up for all speeds, J. Comput. Phys., 214, 1, 137-170 (2006) \u00b7 Zbl\u00a01137.76344 [43] Naito, H.; Fukagata, K., Numerical simulation of flow around a circular cylinder having porous surface, Phys. Fluids, 24, 11, 117102 (2012)\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2021-09-26 22:51:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5908868908882141, \"perplexity\": 10941.289973757157}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057973.90\/warc\/CC-MAIN-20210926205414-20210926235414-00186.warc.gz\"}"}
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{"url":"https:\/\/www.quizover.com\/statistics\/course\/4-2-mean-or-expected-value-and-standard-deviation-by-openstax?page=3","text":"# 4.2 Mean or expected value and standard deviation \u00a0(Page 4\/3)\n\n Page 1 \/ 3\nThis module explores the Law of Large Numbers, the phenomenon where an experiment performed many times will yield cumulative results closer and closer to the theoretical mean over time.\n\nThe expected value is often referred to as the \"long-term\"average or mean . This means that over the long term of doing an experiment over and over, you would expect this average.\n\nThe mean of a random variable $X$ is $\\mu$ . If we do an experiment many times (for instance, flip a fair coin, as Karl Pearson did, 24,000 times and let $X$ = the number of heads) and record the value of $X$ each time, the average is likely to get closer and closer to $\\mu$ as we keep repeating the experiment. This is known as the Law of Large Numbers .\n\nTo find the expected value or long term average, $\\mu$ , simply multiply each value of the random variable by its probability and add the products.\n\n## A step-by-step example\n\nA men's soccer team plays soccer 0, 1, or 2 days a week. The probability that they play 0 days is 0.2, the probability that they play 1 day is 0.5, and the probability that they play 2 days is 0.3. Find the long-term average, $\\mu$ , or expected value of the days per week the men's soccer team plays soccer.\n\nTo do the problem, first let the random variable $X$ = the number of days the men's soccer team plays soccer per week. $X$ takes on the values 0, 1, 2. Construct a $\\mathrm{PDF}$ table, adding a column $\\mathrm{xP\\left(x\\right)}$ . In this column, you will multiply each $x$ value by its probability.\n\nExpected value table\n$x$ $\\text{P(x)}$ $x\\text{P(x)}$\n0 0.2 (0)(0.2) = 0\n1 0.5 (1)(0.5) = 0.5\n2 0.3 (2)(0.3) = 0.6\n\nAdd the last column to find the long term average or expected value: $\\mathrm{\\left(0\\right)\\left(0.2\\right)+\\left(1\\right)\\left(0.5\\right)+\\left(2\\right)\\left(0.3\\right)= 0 + 0.5 + 0.6 = 1.1}$ .\n\nThe expected value is 1.1. The men's soccer team would, on the average, expect to play soccer 1.1 days per week. The number 1.1 is the long term average or expected value if the men's soccer team plays soccer week after week after week. We say $\\mathrm{\\mu =1.1}$\n\nFind the expected value for the example about the number of times a newborn baby's crying wakes its mother after midnight. The expected value is the expected number of times a newborn wakes its mother after midnight.\n\n$x$ $\\text{P(X)}$ $x\\text{P(X)}$\n0 $\\text{P(x=0)}=\\frac{2}{50}$ (0) $\\left(\\frac{2}{50}\\right)$ = 0\n1 $\\text{P(x=1)}=\\frac{11}{50}$ (1) $\\left(\\frac{11}{50}\\right)$ = $\\frac{11}{50}$\n2 $\\text{P(x=2)}=\\frac{23}{50}$ (2) $\\left(\\frac{23}{50}\\right)$ = $\\frac{46}{50}$\n3 $\\text{P(x=3)}=\\frac{9}{50}$ (3) $\\left(\\frac{9}{50}\\right)$ = $\\frac{27}{50}$\n4 $\\text{P(x=4)}=\\frac{4}{50}$ (4) $\\left(\\frac{4}{50}\\right)$ = $\\frac{16}{50}$\n5 $\\text{P(x=5)}=\\frac{1}{50}$ (5) $\\left(\\frac{1}{50}\\right)$ = $\\frac{5}{50}$\n\nAdd the last column to find the expected value. $\\mu$ = Expected Value = $\\frac{105}{50}=2.1$\n\nGo back and calculate the expected value for the number of days Nancy attends classes a week. Construct the third column to do so.\n\n2.74 days a week.\n\nSuppose you play a game of chance in which five numbers are chosen from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. A computer randomly selects five numbers from 0 to 9 with replacement. You pay $2 to play and could profit$100,000 if you match all 5 numbers in order (you get your $2 back plus$100,000). Over the long term, what is your expected profit of playing the game?\n\nTo do this problem, set up an expected value table for the amount of money you can profit.\n\nLet $X$ = the amount of money you profit. The values of $x$ are not 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Since you are interested in your profit (or loss), the values of $x$ are 100,000 dollars and -2 dollars.\n\nTo win, you must get all 5 numbers correct, in order. The probability of choosing one correct number is $\\frac{1}{10}$ because there are 10 numbers. You may choose a number more than once. The probability of choosing all 5 numbers correctly and in order is:\n\n$\\frac{1}{10}*\\frac{1}{10}*\\frac{1}{10}*\\frac{1}{10}*\\frac{1}{10}*=1*{10}^{-5}=0.00001$\n\nTherefore, the probability of winning is 0.00001 and the probability of losing is\n\n$1-0.00001=0.99999$\n\nThe expected value table is as follows.\n\n $x$ $\\text{P(x)}$ $x\\text{P(x)}$ Loss -2 0.99999 (-2)(0.99999)=-1.99998 Profit 100,000 0.00001 (100000)(0.00001)=1\n\nSince $-0.99998$ is about $-1$ , you would, on the average, expect to lose approximately one dollar for each game you play. However, each time you play, you either lose $2 or profit$100,000. The $1 is theaverage or expected LOSS per game after playing this game over and over. Suppose you play a game with a biased coin. You play each game by tossing the coin once. $\\text{P(heads)}=\\frac{2}{3}$ and $\\text{P(tails)}=\\frac{1}{3}$ . If you toss a head, you pay$6. If you toss a tail, you win \\$10. If you play this game many times, will you come out ahead?\n\nDefine a random variable $X$ .\n\n$X$ = amount of profit\n\nComplete the following expected value table.\n\n $x$ ____ ____ WIN 10 $\\frac{1}{3}$ ____ LOSE ____ ____ $\\frac{-12}{3}$\n $x$ $\\mathrm{P\\left(x\\right)}$ $x$ $\\mathrm{P\\left(x\\right)}$ WIN 10 $\\frac{1}{3}$ $\\frac{10}{3}$ LOSE -6 $\\frac{2}{3}$ $\\frac{-12}{3}$\n\nWhat is the expected value, $\\mu$ ? Do you come out ahead?\n\nAdd the last column of the table. The expected value $\\mu =\\frac{-2}{3}$ . You lose, on average, about 67 cents each time you play the game so you do not come out ahead.\n\nLike data, probability distributions have standard deviations. To calculate the standard deviation ( $\\sigma$ ) of a probability distribution, find each deviation from its expected value, square it, multiply it by its probability, add the products, and take the square root . To understand how to do the calculation, look at the table for thenumber of days per week a men's soccer team plays soccer. To find the standard deviation, add the entries in the column labeled ${\\left(x-\\mu \\right)}^{2}\u00b7P\\left(x\\right)$ and take the square root.\n\n $x$ $\\text{P(x)}$ $x\\text{P(x)}$ ${\\text{(x -\u03bc)}}^{2}\\text{P(x)}$ 0 0.2 (0)(0.2) = 0 ${\\left(0-1.1\\right)}^{2}\\left(.2\\right)=0.242$ 1 0.5 (1)(0.5) = 0.5 ${\\left(1-1.1\\right)}^{2}\\left(.5\\right)=0.005$ 2 0.3 (2)(0.3) = 0.6 ${\\left(2-1.1\\right)}^{2}\\left(.3\\right)=0.243$\n\nAdd the last column in the table. $0.242+0.005+0.243=0.490$ . The standard deviation is the square root of $0.49$ . $\\sigma =\\sqrt{0.49}=0.7$\n\nGenerally for probability distributions, we use a calculator or a computer to calculate $\\mu$ and $\\sigma$ to reduce roundoff error. For some probability distributions, there are short-cut formulas that calculate $\\mu$ and $\\sigma$ .\n\nLeo Robinson's definition\nhow is equilibrium defined in financial markets?\nthe concept of it\nDALOM\nCountry A has export sales 20 billion, government purchases 1000billion, business investment is 50 billion, imports are 40billion, and consumption spending is 2000billin. What is the dollar value of GDP ?\nwhat is determination of national income?\neconomic growth\nRukaiya\nstock of capital\nRukaiya\nwe're RBI keep money with them\nAnil\nY =C+l\nFavour\nevaluate the success affirmative action as one of south Africa's redress method\nwhat is market equilibrium\nit is a situation in which the supply of an item is exactly equal to it dd .\nSsmith\ninder wat condition shld a firm stop production in both short n lungrun ?\nSsmith\nwhat is 2nd degree price discrimination?\nSsmith\nwhat is quantity\nTettey\nwhat is quantity2\nAn indefinite amount of something.\nexplorer\nwhat is demand\nin ordinary sense demand means desire\nKhalid\ndemand in economics means both willingness as well as the ability to purchase a commodity by paying a price an also its actuall purchase\nKhalid\nKhalid\ndemand refers to the various quantity of goods and services that consumers are willing and able to purchase at a particular period of time all other things been equal\nDela\nThe amount of a good or service that consumers are willing to buy at a particular price.\nexplorer\nwhat is cost pull inflation?\noru\nwhat is utility\noru\nwhat is cost pull inflation?\noru\ndemand is economic principle referring to a consumer's desire and willingness to pay a price for a specific or service..\nBabakura\nutility is the among of certisfaction driving from using a comundity\nAnas\npull cost of inflation hight population unemployment to some of The country members poor government system\nAnas\nwhat is a buffer scheme\nLukong\nstate the second law of demand and supply\nstate the law of diminishing marginal utility\nRukundo\nmention and explain two Bank I financial institutions and two non baking financial institutions\nwat is demand pull inflation\nDemand-pull inflation\u00a0is asserted to arise when aggregate\u00a0demandin an economy outpaces aggregate supply. It involvesinflation\u00a0rising as real gross domestic product rises and unemployment falls, as the economy moves along the Phillips curve.\nkevin\nPerfectly elastic demand\nthis is a form of demand where goods are demanded at a constant price\nRukundo\nwhat inelastic demanding\nKoire\ndemand of any good demanded more after a certain period. if a commodity prices may high and scarcity of that resources.\nAnil\ncannot demand more\nAnil\nwhat is cross-elasticity of demand\ncross elasticity of demand is an economic concept that measures the responsiveness in the quantity demand of one good when a change in price takes place in other good\nMallekha\nthis is responsiveness quantity demanded keeping other factors constant\nRukundo\nwhat economic growth\nconditions of perfect market\nCONDITIONS OF PERFECT MARKET: 1. Perfect competition(PC): no increasing returns, many buyers and sellers, all are price takers, not prices makers. 2. Perfect Information (PI): buyers and sellers know all they need to know about what they are buying and selling to make the right decisions.\nMallekha\n3. Complete Markets(CM): no externalities or public goods, no transactions costs, \"thick\" markets.\nMallekha\nnice contributor\nMohammed\nA numerous downsized market that does not meet standards.\nLaTasha\nA Perfect Market is a numerous downsized market that does not meet standards.\nLaTasha\nwhat is a market\nis place where buyers and sellers met together for the purpose of buying and selling of good and services\nBabakura\nwhat's the easiest and fastest way to the synthesize AgNP?\nChina\nCied\ntypes of nano material\nI start with an easy one. carbon nanotubes woven into a long filament like a string\nPorter\nmany many of nanotubes\nPorter\nwhat is the k.e before it land\nYasmin\nwhat is the function of carbon nanotubes?\nCesar\nwhat is nanomaterials\u200b and their applications of sensors.\nwhat is nano technology\nwhat is system testing?\npreparation of nanomaterial\nYes, Nanotechnology has a very fast field of applications and their is always something new to do with it...\nwhat is system testing\nwhat is the application of nanotechnology?\nStotaw\nIn this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google\nAzam\nanybody can imagine what will be happen after 100 years from now in nano tech world\nPrasenjit\nafter 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments\nAzam\nname doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world\nPrasenjit\nhow hard could it be to apply nanotechnology against viral infections such HIV or Ebola?\nDamian\nsilver nanoparticles could handle the job?\nDamian\nnot now but maybe in future only AgNP maybe any other nanomaterials\nAzam\ncan nanotechnology change the direction of the face of the world\nAt high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.\nthe Beer law works very well for dilute solutions but fails for very high concentrations. why?\n1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.","date":"2018-05-21 14:51:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 83, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4657007157802582, \"perplexity\": 1126.7103289779304}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-22\/segments\/1526794864405.39\/warc\/CC-MAIN-20180521142238-20180521162238-00555.warc.gz\"}"}
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\section{Introduction}
\label{sec:introduction}
Many relevant applications in the context of routing or logistics call for a temporal component that is part of the input and the actual solution.
In classical flow theory, flow traverses the network in a static fashion, that is, the solutions need to obey capacity restrictions - and possibly additional constraints.
In many real-world applications, however, flow takes some time in order to traverse a network edge.
Hence, a temporal dimension has to be introduced into the models.
Dynamic flow problems that take time into account have been studied for more than half a century.
Dynamic flow problems are also referred to as \emph{flow over time} problems in the literature and in this paper.
Despite the relevance of the topic and the existence of fascinating results, few textbooks cover this topic.
We refer to \cite{skutella2009introduction} for an introduction to flow over time problems.
Furthermore, real-world applications are often also affected by measurement errors as well as by a large degree of uncertainty.
In such situations, the classical models are usually highly inaccurate.
An application may be distribution networks such as gas
\cite{koch2015evaluating} or water networks.
In gas networks, for example, the roughness of the pipes is uncertain due to contamination or aging processes and can only be measured with large effort.
The roughness strongly influences the friction and thus the travel time of gas along a pipe.
As these uncertainties might influence the decision about feasibility or infeasibility of the corresponding complex optimization tasks, a worst-case robust perspective is appropriate.
Although relevant applications exist, little is known about flow over time problems that are protected against uncertain conditions.
In this work, we provide a first step towards studying their properties.
We first define an appropriate model for determining robust optimum flows over time subject to uncertain travel times.
We start from the known classical flow over time theory that ignores uncertainties, that is, the nominal case. Adapting the $\Gamma$-robustness model introduced by
Bertsimas and Sim \cite{BeSi03} that is often applied to combinatorial
optimization problems under uncertainty, we develop a robust flow over time framework.
In this model, the degree of protection against uncertainties can be controlled by a parameter $\Gamma$.
For uncertain objective functions, a robust optimum is a solution with the best guaranteed cost under the assumption that at most $\Gamma$ many objective function coefficients attain their worst-case realization.
Thus, the value of $\Gamma$ determines the conservatism of the solution.
Applying this modeling framework for flows over time, we study the following optimization task.
Let a time horizon $T$ and a network with travel times and potential delays on the edges be given.
Maximize the minimum amount of flow that can be sent through the network within time horizon $T$ under the assumption that at most $\Gamma$ edges are delayed.
In order to study the problem in its most basic version, we restrict ourselves to the case in which flow is not allowed to wait at any intermediate vertex.
For example, this is the case in communication, water, or gas networks without buffer capacities on intermediate vertices.
Although the obtained model seems quite punishing, it is a natural robust counterpart of network flows over time with uncertain travel times.
We provide a formal problem definition in Section~\ref{sec:modelling-techniques}.
\sectionheadline{Related Work}
\label{sec:related-work}
The concept of flows over time was introduced in \cite{ford1958constructing}.
They also showed how to find a maximum $s$-$t$-flow over time using one minimum-cost flow computation, thus generalizing the concept of static maximum $s$-$t$ flows.
Detailed introductions to flows over time and further references can be found in the surveys by Aronson \cite{aronson1989survey} and Skutella \cite{skutella2009introduction}.
In general, the goal of robust optimization is to find a solution that is feasible and as good as possible for any input data in a given uncertainty set.
For a comprehensive introduction to robust optimization, we refer to \cite{ben2009robust}.
In this paper, we consider $\Gamma$-robustness, a concept introduced by Bertsimas and Sim \cite{BeSi03}.
Here, the uncertainty set is determined by $\Gamma$:
Protection is sought against all scenarios in which the input deviates from the nominal input data in at most $\Gamma$ elements simultaneously.
In the context of static network flows, such a scenario could be the failure of at most $\Gamma$ edges in a given network.
Aneja, Chandrasekaran and Nair \cite{aneja2001maximizing} showed how to solve the $\Gamma$-robust maximum $s$-$t$-flow problem in polynomial time for $\Gamma = 1$, even for the case where the flow is required to be integral.
Du and Chandrasekaran \cite{du2007wronghardness} claimed that the $\Gamma$-robust maximum $s$-$t$-flow problem is NP-hard for $\Gamma >1$.
However, Matuschke et al.\ \cite{2015wronghardnessstatic} recently showed that the proof is incorrect.
Personal communication with Disser and Matuschke \cite{Disser2015newhardnessstatic} indicates that this problem is already NP-hard
if $\Gamma$ is not bounded by any constant.
Still, the complexity of the static $\Gamma$-robust flow problem where $\Gamma$ is bounded by any constant bigger than $1$ is open.
Note that in this model, it is not possible to reroute after the edge failure.
In \cite{bertsimas2013robust}, Bertsimas, Nasrabadi and Stiller investigate several variants of robust flow problems.
As far as we know, \cite{bertsimas2013robust} is the only work that considers robustness in a flow over time setting.
In particular, they study a so-called adaptive model where rerouting after edge-failure is allowed within bounds determined by the initial flow and show that this problem is weakly NP-hard.
In contrast, in this paper, rerouting after edge failures is not permitted and we consider a more general robustness scenario:
Besides total edge failures, the travel time can also increase by a finite amount.
Another problem that is highly related to our work is the network interdiction problem.
In contrast to robust maximum flow problems, where the goal is to find a flow that is good no matter which scenario of the uncertainty set is realized, the network interdiction problem takes the opposite perspective:
Here, the goal is to find a set of edges whose deletion minimizes the amount of flow that can be sent in the remaining network.
Wood \cite{W93} showed NP-hardness of this problem.
Köhler and Skutella \cite{koehler2005} consider a flow over time problem where traffic times depend on the actual load of an edge.
While the resulting flow problem is NP-hard, in contrast to our model, temporally repeated flows can be used to obtain a $2$-approximation.
\sectionheadline{Our Contribution}
First, we discuss generalizations of well-studied concepts for the nominal maximum flow over time problem towards robustness.
For uncertain travel times, we point out problems with s straightforward generalization of encoding flow rates on the edges. We show that this concept as well as time-expanded networks are no longer appropriate.
In contrast, we introduce a non-standard, more viable solution encoding in terms of a path decomposition with associated flow rates and dispatch intervals.
We call this concept \emph{general solutions (to the robust maximum flow over time problem)}.
Whereas these two descriptions are equally powerful in the non-robust, i.e.\ \emph{nominal} flow over time case, this is not true for the robust flow over time model.
We also study a robust counterpart of \emph{temporally repeated flows} due to their simple solution encoding, computational complexity and optimality in the nominal case \cite{skutella2009introduction}.
\newcommand{\scriptsize}{\scriptsize}
\newcommand{\footnotesize}{\footnotesize}
\begin{table}[tb]
\begin{center}
\begin{tabular*}{\linewidth}{
@{}l
@{\extracolsep{\fill}} l
@{\extracolsep{3ex}} l
@{\extracolsep{3ex}} l
@{\extracolsep{1ex}} l
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\toprule
\multirow{4}{*}{\normalsize{Task}}& \multirow{4}{*}{\normalsize{\shortstack[l]{static robust\\ max flow}}} & \multicolumn{3}{c}{\normalsize{robust max flow over time}}\\ \cmidrule{3-5}
& & \small{general solutions} & \multicolumn{2}{c}{\small{temporally repeated solutions}} \\ \cmidrule{4-5}
& & & \multicolumn{1}{c}{\scriptsize{arbitrary path length}} & \scriptsize{$T$-bounded path length} \\
\midrule
max, $\Gamma = 1$ & poly time$^1$ \cite{aneja2001maximizing} & ?$^1$ & ? & poly time (Prp. \ref{prop:temp-rep:LP-T-not}) \\[1ex]
\multirow{2}{*}{max, $\Gamma$ bounded} & \multirow{2}{*}{?} & \multirow{2}{*}{\shortstack[l]{at least as hard as\\ static case$^2$ (Prp. \ref{prop:modelling-techniques:robust-flow-np-hard})}} & \multirow{2}{*}{\shortstack[l]{at least as hard as\\ static case (Prp. \ref{prop:temp-rep:complexity})}} & \multirow{2}{*}{poly time (Prp. \ref{prop:temp-rep:LP-T-not})} \\
\\[1ex]
max, $\Gamma$ arb. & NP-hard \cite{Disser2015newhardnessstatic} & NP-hard$^2$ (Prp. \ref{prop:modelling-techniques:robust-flow-np-hard}) & NP-hard (Prp. \ref{prop:temp-rep:complexity}) & poly time (Prp. \ref{prop:temp-rep:LP-T-not}) \\[1ex]
\multirow{2}{*}{\shortstack[l]{feasibility\\ check}} & \multirow{2}{*}{poly time} & \multirow{2}{*}{NP-hard$^2$ (Thm. \ref{thm:modelling-techniques:separation-np-hard})} & \multirow{2}{*}{poly time} & \multirow{2}{*}{poly time}\\
\\
\bottomrule
\end{tabular*}
\end{center}
\caption{An overview of the complexity results from Section \ref{sec:computational-complexity}. \\ \footnotesize{$^1$ If flow rates must be chosen integral, the problem is polynomial time solvable for the static case \cite{aneja2001maximizing} and is inapproximable within any factor for general solutions (see Proposition \ref{prp:SSRFTNE-integral-np-hard-inapproximable}).} \\ \footnotesize{$^2$ These results do not only hold for arbitrary path lengths, but also for instances with the $T$-bounded path length property.}
}
\label{tab:results-computational-complexity}
\end{table}
\begin{table}[tb]
\begin{center}
\begin{tabular*}{\linewidth}{
@{}l
@{\extracolsep{\fill}} l
@{\extracolsep{5ex}} l
}
\toprule
\normalsize{delay restriction} & \normalsize{lower bound} & \normalsize{upper bound} \\
\midrule
$\Delta \in \{0,\infty\}$ & $\max\{T,\Gamma\}$ (Prp. \ref{prop:temp-rep:linear-gap}) & $O(k \log T)$ (Thm. \ref{thm:temp-rep:k-log-T-gap}) \\
$T$-bounded path length & $\max\{\log T,\log \Gamma\}$ (Prp. \ref{prop:temp-rep:log-gap}) & $O(\eta k \log T)$ (Thm. \ref{thm:temp-rep:delta-arbitrary}) \\
none & $\max\{T,\Gamma\}$ (Prp. \ref{prop:temp-rep:linear-gap}) & $O(\eta k \log T)$ (Thm. \ref{thm:temp-rep:delta-arbitrary}) \\
\bottomrule
\end{tabular*}
\end{center}
\caption{Outline of the main results regarding optimality gaps of temporally repeated solutions from Section \ref{sec:bounds}.}
\label{tab:results-bounds}
\end{table}
After introduction and discussion of the models, we address the computational complexity of both solution variants. Our results are summarized in Table \ref{tab:results-computational-complexity}.
The special case of $\Gamma$-robust flow over time with zero travel times and infinite delays on all edges reduces to the \emph{static} $\Gamma$-robust flow problem which searches for an optimum static flow which is robust against up to $\Gamma$ edge failures (cf.\ \emph{related work}).
We show in Proposition \ref{prop:modelling-techniques:robust-flow-np-hard} that our problem contains the static case.
Irrespective of this, our problem for general solutions is considerably more complex.
We show by a reduction from maximum clique that even the verification of feasibility of arbitrary solution candidates is an NP-hard problem (Theorem \ref{thm:modelling-techniques:separation-np-hard}).
By a reduction from the two edge-disjoint paths problem we show that, again in contrast to the static version, the optimization problem is inapproximable for $\Gamma = 1$, if flow rates are required to be integral (Proposition~\ref{prp:SSRFTNE-integral-np-hard-inapproximable}).
For temporally repeated flows, we observe that the computational
complexity depends on the actual edge delays.
In general, they inherit the same complexity status as general solutions (Proposition \ref{prop:temp-rep:complexity}).
However, if the maximum possible path length of each path is bounded by the time horizon, the status changes.
In fact, we show that in this case optimal temporally repeated flows can be computed in polynomial time (Proposition \ref{prop:temp-rep:LP-T-not}).
We formalize this concept in Section \ref{sec:computational-complexity} and call it the \emph{$T$-bounded path length} property.
Note that our hardness results for general solutions also hold for instances with the $T$-bounded path length property.
Subsequently, we study the quality of temporally repeated solutions, when compared to general solutions (Table \ref{tab:results-bounds}).
Temporally repeated solutions do not only have benefits with respect to the computational complexity, but also have a simple solution encoding and are very well studied for the nominal case \cite{skutella2009introduction}.
In the nominal case, it is well known that temporally repeated flows
yield optimal solutions.
Under uncertainty, however, we show that temporally repeated flows can inherit a large optimality gap.
If the delay on all edges is either infinitely large or zero, i.e.\ $\Delta \in \{0,\infty\}$, we show that the optimality gap can be as large as $\max\{T,\Gamma\}$ (Proposition \ref{prop:temp-rep:linear-gap}).
By using a non-trivial primal-dual fitting approach, we show that it is always upper bounded by $O(k \log T)$ (Theorem \ref{thm:temp-rep:k-log-T-gap}).
Here, $k$ is a parameter that is specific for the instance. It does
not depend on the delay but only on the graph structure, travel times
and the time horizon (The definition of $k$ is formalized in Section \ref{sec:bounds}).
Although the instances used in Proposition \ref{prop:temp-rep:linear-gap} have
$k = T$, other classes of instances exist for which the value of $k$
is very small. For example, acyclic digraphs with the $T$-bounded path length property have $k = 1$.
For instances with the $T$-bounded path length property, we provide lower bound examples with an optimality gap of $\max\{\log T,\log \Gamma\}$ (Proposition \ref{prop:temp-rep:log-gap}).
For arbitrary delays, we prove an upper bound of $O(\eta k \log T)$ (Theorem \ref{thm:temp-rep:delta-arbitrary}).
Here, $\eta$ accounts for the relative amount of flow that can be destroyed by scenarios on single paths in the worst case.
Again, this parameter that is again characteristic for an instance is formalized in Section \ref{sec:bounds}.
This bound is tight for the instance from Proposition \ref{prop:temp-rep:log-gap} as, in this case, $k = \eta = 1$.
Finally, if we fix a graph with travel times and finite delays and let the time horizon tend to infinity, we observe that temporally repeated solutions tend to optimality (Proposition \ref{prop:temp-rep:asymptotic-bound}).
\sectionheadline{Outline}
The remainder of this paper consists of three main sections.
Section~\ref{sec:modelling-techniques} is devoted to modelling techniques for the robust maximum flow over time problem.
It introduces a model and two solution concepts which we study subsequently.
Section~\ref{sec:computational-complexity} provides insight in the computational complexity of both concepts under several perspectives.
Finally, Section~\ref{sec:bounds} evaluates the solution quality of optimum temporally repeated flows with respect to their optimality gap to general solutions.
The paper is concluded with final remarks and a discussion of open questions in Section~\ref{sec:conclusions}.
\section{Modeling Techniques for the Robust Maximum Flow over Time Problem}
\label{sec:modelling-techniques}
\sectionheadline{Nominal Maximum Flow over Time Problem}
An instance of the nominal maximum flow over time problem consists of a directed graph $G = (V,E)$ with source and destination vertices $s,d \in V$ and a time horizon $T \in \mathbb{N}$.
Each edge is equipped with a capacity $u: E \rightarrow \mathbb{N}$ and a travel time $\tau: E \rightarrow \mathbb{N}$.
Flow entering edge $e$ at some time $\theta$ leaves the edge at its head at time $\theta + \tau_e$.
We seek to maximize the total amount of flow sent from $s$ to $d$ within the time horizon.
In particular, we use the so-called continuous time model. For further discussion of the relationship between continuous and discrete time models, we refer to \cite{koch2011}.
In classical flow theory, a solution is encoded by a set of Lebesgue-integrable functions $f_e: \mathbb{R} \rightarrow \mathbb{R}_+$ for all $e \in E$
describing the rate of flow entering edge $e$ at time $\theta \in \mathbb{R}$. We assume $f_e(\theta) = 0$ for all $e \in E$ and all $\theta \in \mathbb{R} \setminus [0,T)$.
A feasible solution obeys the capacity limit, that is, $f_e(\theta) \leq u_e$ for all edges and for all $\theta \in [0,T)$.
Depending on the situation to be modeled, waiting at intermediate vertices may or may not be allowed.
For example, if vertices do not have any buffer capacity, it is impossible to store flow at intermediate vertices.
In such situations, \emph{strict} flow conservation is required, that is, the total amount of flow leaving a
vertex up to any point in time $\xi$ is exactly the total amount of flow entering that vertex up to the same time. We have
\[\sum_{e \in \delta^-(v)} \int_0^{\xi - \tau_e} f_e(\theta) d\theta = \sum_{e \in \delta^+(v)} \int_0^{\xi} f_e(\theta) d\theta\]
for all $\xi \in [0,T)$ and for all $v \in V \setminus \{s,d\}$.
The objective function value of a flow $f$ is defined as the total amount of flow reaching vertex $d$ up to time $T$, that is, \smash{$\sum_{e \in \delta^-(d)}\int_0^{T - \tau_e} f_e(\theta) d\theta$}.
Here, we assume that $d$ has no outgoing edges, otherwise, we would have to subtract the amount of flow leaving $d$ up to time $T$.
We make this assumption throughout this paper. Note that flow that
arrives at $d$ after $T$ is allowed but does not give any contribution
to the value of the objective function.
Ford and Fulkerson \cite{ford1958constructing} showed that relaxing strict flow conservation to weak flow conservation, where flow may wait at intermediate vertices, does not change the optimal objective function value in the nominal case.
That is, they proved that there always exists an optimal flow over time which never stores flow at intermediate vertices.
\sectionheadline{$\Gamma$-Robust Flows over Time}
In this paper, we assume that the travel times $\tau_e$ are uncertain and may deviate by a certain delay $\Delta_e \in \mathbb{N}$.
We follow the $\Gamma$-robust approach suggested by Bertsimas and Sim \cite{BeSi03}:
for a given integer $\Gamma \in \mathbb{N}$, we look for a maximum flow over time that is robust against any possible scenario of up to $\Gamma$ edge delays.
This situation can be interpreted as a two-player game in which the first player (the "flow player") decides on a flow over time.
Afterwards, the second player (the "bad adversary") chooses at most $\Gamma$ edges on which she delays the travel times from $\tau_e$ to $\tau_e+\Delta_e$.
The adversary's goal is to minimize the total throughput, or to violate the capacity constraints.
In the robust setting, it might well be that a nominal flow without waiting times has to wait in certain scenarios.
In situations with implied strict flow conservation, such a flow is no longer feasible.
In this paper, we do not consider such situations, and therefore demand strict flow conservation no matter how the adversary chooses edges to be delayed.
We remark that requiring weak flow conservation only introduces additional challenges, as it would have to be decided which flow particles need to wait and which may pass on.
Furthermore, we do not require the network to be empty at time $T$, i.e.\ we allow flow to enter destination $d$ after time $T$. The flow arriving at $d$ after $T$ is not counted in the objective function.
This yields some freedom in reacting to different possible scenarios.
Otherwise, it would be necessary to ensure additionally that - no matter which edges the adversary chooses to delay - all flow can reach $d$ by time $T$.
This would result in a very restrictive model, where we would not be allowed to use any edges whose travel time may exceed $T$. Note that the capacity constraints and the flow conservation constraints are only present up to time $T$ for the same reason. For example, we also allow a feasible solution to violate the edge capacities after $T$.
\sectionheadline{Mathematical Model}
A solution formulated in terms of flow rates $f_e(\theta)$ seems to be unsuitable.
There, the actual flow rate at any intermediate edge depends on the corresponding scenario.
As we are interested in solutions that can be described independently from the scenario, we formulate solutions in terms of $s$-$d$-paths.
A nominal flow over time satisfying strict flow conservation can always be described by a path decomposition $(f_P)_{P\in \mathcal{P}}$, where $\mathcal{P}$ is the set of
$s$-$d$-paths and $f_P: \mathbb{R} \to \mathbb{R}_+$ with $f_P(\theta) = 0$ for all $\theta \in \mathbb{R} \setminus [0,T-\tau(P))$.
For simplicity reasons, in this paper we only allow flow on simple paths.
The function assigns a flow rate $f_P(\theta)$ to each point in time $\theta$ that describes the rate at which flow is sent into path $P$.
Note that this path decomposition is not necessarily unique.
Already in the static robust maximum flow problem, Bertsimas et al.\ have shown that the robust flow value depends on the path decomposition \cite{bertsimas2013robust}.
In this work, however, we do not decompose a flow, but rather define it on the set of all simple paths.
Let $\mathcal{S} = \{z \in \{0,1\}^{|E|} : \sum_{e\in E} z_e \leq \Gamma\}$ denote the set of admissible scenarios, that is, representing all combinations of at most $\Gamma$ delays.
Here, $z_e$ is a decision variable with $z_e = 1$ if and only if $e$ is delayed.
We call a flow over time $(f_P)_{P\in \mathcal{P}}$ \emph{feasible} if the capacity constraints are obeyed under every possible scenario $z\in \mathcal{S}$.
That is, if
\[ \sum_{P\in \mathcal{P}: e\in P} f_P(\theta_{z,e}) \le u_e \quad \forall e\in E, \theta \in [0, T), z\in \mathcal{S}, \]
where
\[\theta_{z,e}=\theta-\sum_{e'\in E: e' <_P e} (\tau_{e'}+\Delta_{e'} z_{e'})\]
denotes the departure time at $s$ of a flow particle on path $P$ which enters edge $e$ at time $\theta$ under scenario $z$.
As usual, $\{e'\in E: e' <_P e\}$ denotes the set of edges that need to be traversed on path $P$ before edge $e$ is reached.
Note that negative values of $\theta_{z,e}$ imply that these flow particles would have started at $s$ before time zero (which is not possible).
The \emph{robust value} of a feasible flow over time $f=(f_P)_{P\in \mathcal{P}}$ is
\[\min_{z\in \mathcal{S}} \sum_{P\in \mathcal{P}} \int_0^{\max\{0,T - \tau(P) - \Delta_z(P)\}} f_P(\nu) d\nu, \]
where $\tau(P) + \Delta_z(P) = \sum_{e\in P} (\tau_e + \Delta_e z_e)$ denotes the travel time on path $P$ in scenario $z$.
\begin{remark}
The nominal maximum flow over time problem, that corresponds to $\Gamma = 0$, can be modeled by an auxiliary time-expanded network.
This graph contains a vertex $(v, t)$ for each relevant time $t \in\{0, \ldots, T - 1\}$.
Edges between the vertices $(v,t)$ and $(w,t + \tau_{vw})$ model edges between vertices $v$ and $w$ with travel time $\tau_{vw}$ in the original graph.
This is not possible in the robust version, since each scenario would induce a different time-expanded network.
Thus, the choice of $z$ defines the topology of the network, and a straight-forward usage of time-expanded networks cannot be applied here.
\end{remark}
By an averaging argument, we show next that it is sufficient to consider functions $f_P$ which are piecewise constant on all integral unit length intervals.
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.32\textwidth} \centering
\resizebox{\textwidth}{!}{
\begin{tikzpicture}
\draw[fill=white,draw=white,use as bounding box] (-0.75,-0.5) rectangle (5,4);
\draw [->,>=latex,very thick] (-0.3,0)--(5,0) node[below left] {$\theta$};
\draw [->,>=latex,very thick] (0,-0.3)--(0,4) node[below left] {$f_P(\theta)$};
\draw[smooth,samples= 200,domain=0.0:2.5]
plot(\x,{8*\x-32.4*\x^2+53.48*\x^3-42.11*\x^4+17.594*\x^5 -3.99*\x^6+0.465713*\x^7-0.0217374*\x^8});
\draw (2.5, 1) -- (3.5, 3);
\draw[bend left] (3.5, 3.5) to (5, 2);
\end{tikzpicture}}
\caption{arbitrary flow}
\label{fig:arbitrary_flow}
\end{subfigure}%
\begin{subfigure}[b]{0.32\textwidth} \centering
\resizebox{\textwidth}{!}{
\begin{tikzpicture}
\draw[fill=white,draw=white,use as bounding box] (-0.75,-0.5) rectangle (5,4);
\draw [->,>=latex,very thick] (-0.3,0)--(5,0) node[below left] {$\theta$};
\draw [->,>=latex,very thick] (0,-0.3)--(0,4) node[below left] {$f_P(\theta)$};
\draw (0,2) to (1,2);
\draw (1,1.5) to (2,1.5);
\draw (2,2) to (3,2);
\draw (3,3.5) to (4,3.5);
\draw (4,2.5) to (5,2.5);
\end{tikzpicture}}
\caption{piecewise constant flow}
\label{fig:triple_flow}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth} \centering
\resizebox{\textwidth}{!}{
\begin{tikzpicture}
\draw[fill=white,draw=white,use as bounding box] (-0.75,-0.5) rectangle (5,4);
\draw [->,>=latex,very thick] (-0.3,0)--(5,0) node[below left] {$\theta$};
\draw [->,>=latex,very thick] (0,-0.3)--(0,4) node[below left] {$f_P(\theta)$};
\draw (0,2.2) to (3.5,2.2);
\end{tikzpicture}}
\caption{temp.\ repeated flow}
\label{fig:temp_rep_flow}
\end{subfigure}
\caption{Comparison of different models for flows over time.}
\label{fig:comparision_between_models}
\end{figure}
\begin{restatable}{proposition}{prpssrftpiecewiseconstant}
\label{prop:modelling-techniques:piecewise-constant}
For every solution to the robust maximum flow over time problem there exists a solution with the same objective function value which consists only of piecewise constant functions $f_P$
whose values change only at integer points.
\end{restatable}
\begin{proof}
Let $\tilde{f}$ be a solution to the robust maximum flow over time problem.
For every $s$-$d$-path $P$, we construct the piecewise constant $f_P: \mathbb{R} \rightarrow \mathbb{R}_+$ as follows.
Intuitively, cut $\tilde{f}_P$ into unit-intervals $[a,a+1) \subseteq [0,T)$.
This is possible due to the fact that $T$ is integral.
Set the flow rate entering path $P$ during each interval to be the average observed in the unit interval, that is,
$$f_P(\theta) = \int\limits_{a}^{a+1}{\tilde{f}_P(\nu) d\nu} \text{ for all } \theta \in [a,a+1),$$
and zero outside of the interval $[0,T)$, see Figures \ref{fig:arbitrary_flow} and \ref{fig:triple_flow} for an illustration.
This ensures that $f$ and $\tilde{f}$ have the same objective function value since all limits of the intervals are integral.
We now argue that all constraints are satisfied.
Suppose there is a scenario $z \in \mathcal{S}$ and a time $\theta \in [a,a+1)$ when $f$ violates the capacity of some edge $e \in E$.
Then the capacity must be violated for all $\theta \in [a,a+1)$ since the solution consists of piecewise constant functions only, and all travel times $\tau_e$ and $\Delta_e z_e$ are integral.
Therefore, the solution sends more than $u_e$ through this edge in the unit interval.
We can conclude that the original solution sends more than $u_e$ through this edge in the unit interval, too.
So there has to be a time $\theta^* \in [a,a+1)$ for which the capacity is violated in the original solution, which contradicts the feasibility of $\tilde{f}$.
\qed
\end{proof}
Hence, it is sufficient to consider solutions that can be described by a family of triples $\left\{\left(P^i,f^i, [a^i,b^i)\right)\right\}_{i = 1,\dots,\omega}$ that describe
the rate $f^i$ at which flow is sent into path $P^i$ during the time interval $[a^i,b^i)$, with $a_i,b_i\in \mathbb{N}$.
We call this interval the \emph{dispatch interval}.
Using Proposition \ref{prop:modelling-techniques:piecewise-constant} we redefine a solution to the robust maximum flow over time problem as follows.
\begin{definition}
\label{def:triples}
A solution to the robust maximum flow over time problem is encoded as a family of triples $\left\{\left(P^i,f^i, [a^i,b^i)\right)\right\}_{i = 1,\dots,\omega}$.
For each $1 \leq i \leq \omega$, $P^i$ is a simple $s$-$d$-path in $G$, $f^i > 0$ and $a^i,b^i \in \{0,\dots,T\}$ with $a^i < b^i$.
\end{definition}
\sectionheadline{Temporally Repeated Flows}
Temporally repeated flows are a classical solution concept used to solve the nominal maximum flow over time problem to optimality.
They are constructed from a path decomposition $x = \sum_{P\in \mathcal{P}} x_P$ of a given static flow $x$ by sending flow at rate $x_P$ along a path $P$ as long as the flow can arrive at the sink $d$ by time $T$.
Therefore, a temporally repeated flow represented in our model has dispatch intervals of the form $[0, T-\tau(P))$ for a path $P$ and flow rates $f_P = x_P$.
A temporally repeated flow is called \emph{feasible}, if
\begin{align}\label{eq:temprep} \sum_{i : e \in P^i} f^i \leq u_e \text { for all edges } e \in E,\end{align}
that is, the capacity constraints are satisfied independent of the actual point in time.
Note that this will always be the case, if the temporally repeated flow was constructed from a \emph{feasible} static flow.
Equivalently, we can state that a flow $\big\{(P^i, f^i, [a^i, b^i))\big \})$ is a feasible temporally repeated flow if and only if $a^i = 0$ and $b^i \ge T - \tau(P^i)$
for all $i$ and \eqref{eq:temprep} holds.
Since the dispatch intervals are fixed by the paths, we will omit them from now on.
See Figure~\ref{fig:comparision_between_models} for a graphical comparison between the different flow models.
\section{Computational Complexity of the Robust Maximum Flow over Time Problem}
\label{sec:computational-complexity}
\sectionheadline{General Solutions}
The following proposition shows that the robust maximum flow over time problem is at least as hard as the static counterpart.
Recently, Disser and Matuschke \cite{Disser2015newhardnessstatic} disclosed that they were able to prove NP-hardness of the static counterpart for unbounded $\Gamma$.
The result, however, is not yet published.
\begin{restatable}{proposition}{SSRFTNNEnphard}
\label{prop:modelling-techniques:robust-flow-np-hard}
The robust maximum flow over time problem is at least as hard as the static robust maximum flow problem.
\end{restatable}
\begin{proof}
We will show that the robust maximum flow over time problem can be
used to solve the static robust maximum flow problem.
Let us assume an instance $(G = (V,E),s,d,u,\Gamma)$ of robust maximum flow is given.
We construct an instance $(G = (V,E),s,d,u,\tau,\Delta,T,\Gamma)$ of robust maximum flow over time as follows.
Let $T = 1, \tau \equiv 0$ and $\Delta \equiv \infty$.
Then any feasible solution to the robust maximum flow problem can be mapped to a solution $\left\{\left(P^i,f^i, [a^i,b^i)\right)\right\}_{i = 1,\dots,\omega}$ of the robust maximum flow over time instance.
For every path $P$ with flow rate $f_P$ in the robust maximum flow
solution, the over-time solution sends the same amount of flow during the dispatch interval $[0,1)$.
The worst-case scenario of the robust maximum flow destroys at most $\Gamma$ edges and decreases the robust flow value - with respect to the nominal flow value - by the total flow rate summed among all
paths that are affected by the set of deleted edges.
The same interference occurs in the constructed instance of robust maximum flow over time.
If an edge was delayed, the travel time increases to an arbitrarily large value.
Hence, any path using such an edge does not reach the destination.
Thus, the objective function values coincide and an optimal solution to the robust maximum flow over time instance is also optimal for the robust maximum flow problem. \qed
\end{proof}
The following two results show that the temporal component introduces additional difficulty to the robust problem, as the static counterparts are both polynomially solvable.
First, we show that computing an optimal integral solution in polynomial time is unlikely and that the problem cannot be approximated.
\begin{proposition}
\label{prp:SSRFTNE-integral-np-hard-inapproximable}
The robust maximum flow over time problem is NP-hard, if we require integral flow rates, that is, $f^i \in \mathbb{Z}$, even if $\Gamma = 1$.
Moreover, the problem is inapproximable within any factor.
\end{proposition}
\begin{proof}
We will show that robust maximum flow over time with integral flow rates solves the two edge-disjoint paths problem.
An instance of two edge-disjoint paths consists of a graph $G = (V,E)$ with two designated pairs of vertices $(s_i,d_i), i = 1,2$ and asks for two edge-disjoint $s_i$-$d_i$-paths in $G$.
We assume that $s_1,s_2,d_1,d_2$ are pairwise disjoint.
The problem was shown to be NP-hard by Fortune et al.\ \cite{fortune1980directed}.
We model this task as a robust maximum flow over time problem as follows (see Figure \ref{fig:SSRFTNE-integral-np-hard}).
We introduce two auxiliary vertices $s$ and $d$ and connect $s$ to $s_i$ and $d$ to $d_i$.
Hence, we end up with the graph $\bar{G} = (V \cup \{s,d\}, E \cup \{s,s_1\} \cup \{s,s_2\} \cup \{d_1,d\} \cup \{d_2,d\})$.
Let $u \equiv 1, \Delta \equiv 2, \tau_e = 0$ for all $e \in E$, $\tau_{\{s,s_1\}} = 0, \tau_{\{s,s_2\}} = 1, \tau_{\{d_1,d\}} = 1, \tau_{\{d_2, d\}} = 0$ and $T = 2$.
\begin{figure}[tb]
\centering
\tikzstyle{vertex}=[circle, draw,
inner sep=1pt, minimum width=10pt]
\tikzstyle{edge} = [draw,very thick]
\resizebox{\textwidth}{!}{
\begin{tikzpicture}[scale = 2]
\node[vertex] (s) at (0, 1){\scriptsize $s$};
\node[vertex] (s1) at (1, 2){\scriptsize $s_1$};
\node[vertex] (s2) at (1, 0){\scriptsize $s_2$};
\node[vertex] (d) at (6, 1){\scriptsize $d$};
\node[vertex] (d1) at (5, 2){\scriptsize $d_1$};
\node[vertex] (d2) at (5, 0){\scriptsize $d_2$};
\draw (s) edge[->, very thick] node[anchor = south] {\scriptsize 0,2} (s1);
\draw (s) edge[->, very thick] node[anchor = south] {\scriptsize 1,2} (s2);
\draw (d1) edge[->, very thick] node[anchor = south] {\scriptsize 1,2} (d);
\draw (d2) edge[->, very thick] node[anchor = south] {\scriptsize 0,2} (d);
\draw (s1) edge[->] node[anchor = south] {\scriptsize 0,2} (2,1.8);
\draw (s1) edge[->] (2,1.6);
\draw (s1) edge[->] (2,1.4);
\draw (s2) edge[->] node[anchor = north] {\scriptsize 0,2} (2,0);
\draw (s2) edge[->] (2,0.2);
\draw (4,0.2) edge[->] node[anchor = north] {\scriptsize 0,2} (d2);
\draw (4,0.4) edge[->] (d2);
\draw (4,1.8) edge[->] node[anchor = south] {\scriptsize 0,2} (d1);
\draw (4,1.6) edge[->] (d1);
\draw (0.8,2.2) -- (5.2,2.2) -- (5.2,-0.2) -- (0.8,-0.2) -- (0.8,2.2);
\node at (3,1) {$G$};
\draw (s1) edge[->,draw=red,ultra thick,bend right=10] node[anchor=south] {\scriptsize \color{red}{$\mathbf{[0,1)}$}} (d1);
\draw (s1) edge[->,draw=red,ultra thick,bend left=10] node[anchor=south,rotate=-25] {\scriptsize \color{red}{$\mathbf{[0,2)}$}} (d2);
\draw (s2) edge[->,draw=red,ultra thick,bend left=10] node[anchor=north] {\scriptsize \color{red}{$\mathbf{[0,1)}$}} (d2);
\end{tikzpicture}}
\caption{Construction for Proposition \ref{prp:SSRFTNE-integral-np-hard-inapproximable}.
Edge labels denote the travel times and possible delays, respectively.
The thick edges depict the possible paths and time intervals a feasible solution may use.}
\label{fig:SSRFTNE-integral-np-hard}
\end{figure}
If we solve the robust maximum $s$-$d$-flow over time problem, the travel times ensure that only the following path types can contribute to the objective function value of any feasible solution:
\begin{enumerate}[1)]
\item $s$-$d$-paths traversing $s_1$ and $d_1$,
\item $s$-$d$-paths traversing $s_1$ and $d_2$,
\item $s$-$d$-paths traversing $s_2$ and $d_2$.
\end{enumerate}
%
Let us assume that there are two edge-disjoint paths $P_1$ and $P_2$ in $G$, that is, a YES-instance.
Then we can construct a solution to robust maximum flow over time which sends flow along ${s,s_1},P_1,{d_1,d}$ in the interval $[0,1)$ and along ${s,s_2},P_2,{d_2,d}$ in the interval $[0,1)$.
Since both paths are disjoint, the total amount of flow reaching the destination is two.
Moreover, no scenario can destroy more than one of the two paths, hence, the robust flow value is equal to one.
Now, let us assume a NO-instance of two edge-disjoint paths.
Due to the integrality of the dispatch intervals and flow rates, a solution to robust maximum flow over time in this network can only consist of at most two different paths.
If it uses a path of type 1) and a path of type 3), both paths have to share some common edge $e^{*} \in E$
as we assumed to have a NO-instance of two edge-disjoint paths.
The scenario delaying exactly this edge reduces the flow value to zero.
In all other cases, the worst case scenario may delay the single utilized edge leaving $s$ (resp.\ entering $t$) in order to decrease the robust flow value to zero.
Hence, the instance is a NO-instance if and only if the robust flow value is zero.
Note that in the construction used above, the objective function value is zero for a NO-instance and one for a YES-instance.
Hence, any approximation algorithm would still distinguish between YES- and NO-instances. \qed
\end{proof}
The following theorem shows that it is strongly NP-hard to verify feasibility
of robust flow over time solutions in general.
\begin{restatable}{theorem}{thmmodellingtechniquesseparationnphard}
\label{thm:modelling-techniques:separation-np-hard}
Deciding feasibility of a given solution $f = \left\{\left(P^i,f^i,
[a^i,b^i)\right)\right\}_{i = 1,\dots,\omega}$ is NP-hard.
\end{restatable}
\begin{proof}
We provide a polynomial-time reduction from the \textsc{clique} decision problem, which is one of Karp's classical NP-hard problems \cite{karp1972reducibility}.
We will show that, given any graph $\bar{G}=(\bar{V},\bar{E})$ and some $r \in \mathbb{N}$, we can construct a maximum robust flow over time instance and a corresponding solution $f$ such that the following holds.
There is a clique of size $r$ in $\bar{G}$ if and only if $f$ is an infeasible solution for the maximum robust flow over time instance.
Without loss of generality, we assume \ $|\bar{E}| \ge |\bar{V}|$, $r\geq 3$ and that the vertices in $\bar{V}$ are numbered from $1, \ldots, n$. Let $m = |\bar{E}|$.
We construct a multigraph $G = (V,E)$, with vertices $V = \{s,d_0,d_1\} \cup \{ v_i^\ell, v_i^r : i \in \bar{V} \}$ as illustrated in Figure \ref{fig:modelling-techniques:separation-np-hard} for the example of $\bar{G}=K_3$.
$V$ consists of two vertices $v_i^\ell$ and $v_i^r$ for each vertex $i \in \bar{V}$, together with three vertices $s$, $d_0$ and $d_1$.
The edge set $E$ consists of five types of edges (omitted values for edges $e \in E$ are $u_e = \infty, \tau_e = 0$ and $\Delta_e = 0$):
\begin{enumerate}[1)]
\item $e = (v_i^\ell,v_i^r)$ for each vertex $i \in \bar{V}$ with $\Delta_e = 2^{i}$,
\item $(v_i^r,d_0)$ for each vertex $i \in \bar{V}$,
\item all ``backward'' edges $(v_i^r,v_j^\ell)$ for $i \neq j\in \bar{V}$,
\item $e = (d_0,d_1)$ with $u_e = \smash{\binom{r}{2}} - 1$,
\item $e = (s,v_i^\ell)$ for each edge $\bar{e} \in \bar{E}$ such that $i \in \bar{e}$.
For $\bar{e} = \{i, j\}$, set $\tau_e = 2^{m+1} - 2^{i} - 2^{j}$. These edges can be parallel.
\end{enumerate}
We set $T = 2^{m+1} + 1$ and $\Gamma = r$. The solution candidate for the feasibility problem is constructed as follows.
We introduce a triple $(P^{\bar{e}}, f^{\bar{e}}, [0,1))$ for each edge $\{i, j\} = \bar{e} \in \bar{E}$,
where $P^{\bar{e}} = (s, v_i^\ell, v_i^r, v_j^\ell, v_j^r, d_0, d_1)$.
The edge $(s, v_i^\ell)$ is chosen to be the designated edge for $\bar{e}$ of type (5) above.
The following two claims conclude the proof as we will show that a $r$-clique in $\bar{G}$ exists if and only if the constructed solution is infeasible due to a capacity violation at time $t = 2^{m+1}$.
\begin{figure}[tb]
\centering
\tikzstyle{vertex}=[circle, draw,
inner sep=1pt, minimum width=13pt,fill=white]
\tikzstyle{edge} = [draw,very thick]
\resizebox{\textwidth}{!}{
\begin{tikzpicture}[scale = 2]
\begin{scope}[xshift=-2cm,yshift=0cm]
\node[vertex] (v1) at (0, 0){\scriptsize $v_1$};
\node[vertex] (v2) at (1.6, 0){\scriptsize $v_2$};
\node[vertex] (v3) at (0.8, 1){\scriptsize $v_3$};
\draw (v1) edge[-] node[anchor=north] {$e_1$} (v2);
\draw (v1) edge[-] node[anchor=south,xshift={-.2cm}] {$e_2$} (v3);
\draw (v2) edge[-] node[anchor=south,xshift={.2cm}] {$e_3$} (v3);
\end{scope}
\draw (-0.2,-0.15) edge[-,thick] (-0.2,1.15);
\node[vertex] (s) at (0, .5){\scriptsize $s$};
\node[vertex] (d0) at (3, .5){\scriptsize $d_0$};
\node[vertex] (d1) at (4, .5){\scriptsize $d_1$};
\node[vertex] (v1l) at (1, 0){\scriptsize $v^{\ell}_1$};
\node[vertex] (v1r) at (2, 0){\scriptsize $v^r_1$};
\node[vertex] (v2l) at (1, 0.5){\scriptsize $v^{\ell}_2$};
\node[vertex] (v2r) at (2, 0.5){\scriptsize $v^r_2$};
\node[vertex] (v3l) at (1, 1){\scriptsize $v^{\ell}_3$};
\node[vertex] (v3r) at (2, 1){\scriptsize $v^r_3$};
\draw (v1l) to node[anchor=south] {\scriptsize $\Delta = 2$} (v1r);
\draw (v2l) to node[anchor=south] {\scriptsize $\Delta = 4$} (v2r);
\draw (v3l) to node[anchor=south] {\scriptsize $\Delta = 8$} (v3r);
\draw (d0) to node[anchor=south,yshift={-.05 cm}] {\scriptsize $u = 2$} (d1);
\draw[draw=red] (s) to[bend right=15] node[anchor=north,rotate=-30] {\scriptsize \color{red}{$P_1$}, $\tau=10$} (v1l);
\draw[draw=red] (v1l) to[bend right=15] (v1r);
\draw[draw=red] (v1r) to[bend right=5] (v2l);
\draw[draw=red] (v2l) to[bend right=15] (v2r);
\draw[draw=red] (v2r) to[bend right=10] (d0);
\draw[draw=red] (d0) to[bend right=40] (d1);
\draw[draw=darkgreen,dashed] (s) to node[anchor=south, rotate=25] {\scriptsize \color{darkgreen}{$P_3$}, $\tau=4$} (v3l);
\draw[draw=darkgreen,dashed] (v3l) to[bend right=15] (v3r);
\draw[draw=darkgreen,dashed] (v3r) to[bend right=5] (v2l);
\draw[draw=darkgreen,dashed] (v2l) to[bend left=30] (v2r);
\draw[draw=darkgreen,dashed] (v2r) to[bend left=10] (d0);
\draw[draw=darkgreen,dashed] (d0) to[bend right=15] (d1);
\draw[draw=blue,dotted,thick] (s) to[bend left=10] node[anchor=south,rotate=-30] {\scriptsize \color{blue}{$P_2$}, $\tau=6$} (v1l);
\draw[draw=blue,dotted,thick] (v1l) to[bend left=30] (v1r);
\draw[draw=blue,dotted,thick] (v1r) to[bend left=15] (v3l);
\draw[draw=blue,dotted,thick] (v3l) to[bend left=55] (v3r);
\draw[draw=blue,dotted,thick] (v3r) to (d0);
\draw[draw=blue,dotted,thick] (d0) to[bend left=45] (d1);
\end{tikzpicture}}
\caption{Construction from Theorem \ref{thm:modelling-techniques:separation-np-hard} for the example of $\bar{G} = K_3$.
The graph $\bar{G}$ is on the left and the corresponding robust flow over time instance with $T=2^{m+1}+1=17$ and $\Gamma = r$ is on the right hand side.}
\label{fig:modelling-techniques:separation-np-hard}
\end{figure}
\omsClaim{The constructed solution obeys the capacity constraints for every point in time $t < 2^{m+1}$.}
For the proof, observe that only the edge $(d_0,d_1)$ has finite capacity and that only edges $(v_i^{\ell}, v_i^r)$ can be delayed.
Hence, it is sufficient to argue that its capacity cannot be exceeded unless $t = 2^{m+1}$.
For any point in time $t < 2^{m+1}$, a path can only contribute to the capacity violation if it was delayed at most once.
If it had been delayed twice, it would have had a total travel time of $2^{m+1}-2^i-2^j+2^i+2^j = 2^{m+1}$ by construction.
Now, let us consider some edge $(v_i^\ell,v_i^r)$ and assume that it was delayed.
Then all paths which cross that edge and have no other delayed edge, will have a total travel time of $2^{m+1} - 2^j$ for some $j \neq i$.
By construction, at most $\Gamma = r$ many edges are delayed.
We conclude that at most $r$ paths can have the same travel time, which in particular, are strictly fewer than $\binom{r}{2}$ paths for $r \geq 3$.
We now observe that for $\{i, j\} \not = \{i', j'\}$, $2^i + 2^j \not = 2^{i'} + 2^{j'}$ and that $2^i \not = 2^{i'} + 2^{j'}$ for all $i, i', j'$.
Therefore, less than $\binom{r}{2}$ paths can arrive at $d_0$ at any point in time before $2^{m+1}$ and thus, the capacity cannot be violated.
\omsClaim{There is a scenario in which the solution violates the capacity constraint for $t = 2^{m+1}$ and edge $e = (d_0,d_1)$ if and only if there is a clique of size $r$ in $\bar{G}$.}
In order to prove the claim, we will show that a clique $C$ of size $r$ in $\bar{G}$ proves that the solution $f$ is infeasible.
Let us select a scenario $z \in \mathcal{S}$ with $z_{(v_i^\ell,v_i^r)} = 1$ if and only if $i \in C$.
Now, for each path $P^{\bar{e}}$ there are three possible situations:
\begin{enumerate}
\item The path contains no delayed edge, then $\tau(P^{\bar{e}}) = 2^{m+1} - 2^u - 2^v < 2^{m+1}$.
Hence, it does not contribute to the capacity of edge $(d_0,d_1)$ at time $2^{m+1}$.
\item If the path is delayed exactly once, the same argument holds and $\tau(P^{\bar{e}}) < 2^{m+1}$.
\item Finally, if the path is delayed exactly twice, $\tau(P^{\bar{e}}) = 2^{m+1}$ and the path contributes to the capacity.
\end{enumerate}
Consequently, a path $P^{\bar{e}}$ contributes to the capacity violation at edge $(d_0,d_1)$ if and only if it was delayed twice.
Due to the construction of $z$, this is the case if and only if both endpoints of $\bar{e}$ and thus $\bar{e}$ itself was part of the clique.
Since the clique consists of $\binom{r}{2}$ edges, the edge capacity is violated.
On the other hand, if the solution is infeasible, there exists a scenario $z \in \mathcal{S}$ such that at least $\binom{r}{2}$ paths are delayed exactly twice.
Since $\Gamma = r$, this is only possible if the delayed paths form a $r$-clique in $\bar{G}$. \qed
\end{proof}
\sectionheadline{Temporally Repeated Flows}
Next, we discuss the computational complexity of temporally repeated flows, which were introduced in Section \ref{sec:modelling-techniques}.
We observe that computing optimal temporally repeated flows is an NP-hard task in general.
\begin{restatable}{proposition}{proptemprepcomplexity}
\label{prop:temp-rep:complexity}
In general, the problem of computing an optimal robust temporally repeated flow is at least as hard as the static robust maximum flow problem.
\end{restatable}
\begin{proof}
We consider again the construction in the proof of Proposition~\ref{prop:modelling-techniques:robust-flow-np-hard}.
Recall that we had $T = 1$. In Proposition~\ref{prop:modelling-techniques:piecewise-constant}, we showed
that it suffices to consider flows with integer dispatch intervals.
Therefore, the dispatch interval for any path is $[0, 1)$ and there is an optimal flow over time in this network that is a temporally repeated flow. \qed
\end{proof}
There are situations in which an optimal temporally repeated flow can be computed efficiently.
The following Proposition \ref{prop:temp-rep:LP-T-not} considers instances whose longest robust path length does not exceed the time horizon for any possible scenario $z \in \mathcal{S}$.
That is, we say an instance has the \emph{$T$-bounded path length property} if and only if
\[\max_{P \in \mathcal{P}, z \in \mathcal{S}} \{ \tau(P) + \Delta_z(P) \} \leq T.\]
This ensures that we can write down an LP model for the problem whose pricing problem is tractable. That means, we can solve the dual separation problem efficiently: given a dual solution,
decide whether the solution is feasible and, if not, return a violated dual inequality. This concludes that the LP can also be solved in polynomial time due to \cite{grotschel1981ellipsoid}.
\begin{restatable}{proposition}{proptemprepTnot}
\label{prop:temp-rep:LP-T-not}
An optimal robust temporally repeated flow can be computed in polynomial time for instances with the $T$-bounded path length property.
\end{restatable}
\begin{proof}
For ease of notation, we denote the set of all feasible temporally repeated flows by $\mathcal{F} = \{ x \in \mathbb{R}_+^{|\mathcal{P}|} : \sum_{P : e \in P} x_P \leq u_e, \forall e \in E \}$.
We can formulate the problem as follows:
\begin{align*}
\max_{x \in \mathcal{F}} \min_{z \in \mathcal{S}} \left\{ \sum_{P \in \mathcal{P}} \left(T - \tau(P) - \Delta_z(P) \right) x_P \right\}
\end{align*}
where, as before, $\mathcal{S}=\{z\in \{0,1\}^E\mid \sum_{e\in E} z_e \le \Gamma\}$, and $\Delta_z(P)=\sum_{e\in P} z_e \Delta_e$.
The variables $x_P$ denote the flow rate at which flow is sent into
path $P \in \mathcal{P}$ and the $z_e$ variables model the decision on
which edge the travel time increases.
Note that the additional assumption on the maximal path length makes sure that the objective coefficients of all paths in all scenarios can never be negative.
Each flow $x\in \mathcal{F}$ induces a load of $x_e=\sum_{P\in \mathcal{P}:e\in P}x_P$ on each edge $e\in E$.
Thus, the term $\sum_{P\in \mathcal{P}} \Delta_z(P) x_P$ can be rewritten as $\sum_{e\in E} \Delta_e z_e x_e$.
As a consequence we can reformulate the objective function of the problem above as:
$$\max_{x\in \mathcal{F}}\left( \sum_{P \in \mathcal{P}} \left(T - \tau(P)\right) x_P - \max_{z\in \mathcal{S}}\sum_{e\in E} \Delta_e z_e x_e \right).$$
Let us consider the inner problem $\max_{z\in \mathcal{S}}\sum_{e\in E} \Delta_e z_e x_e$ for a fixed flow $x\in \mathcal{F}$.
Since the linear inequality system $\{ \sum_{e\in E} z_e \le \Gamma,\ 0\le z_e \le 1 \forall e\in E \}$ is totally unimodular,
we might as well replace $\max\{\sum_{e\in E} \Delta_e z_e x_e : z\in \mathcal{S}\}$ by its linear relaxation
\begin{align*}
\max_{z\in [0,1]^{|E|}} \left\{ \sum_{e\in E} \Delta_e z_e x_e: \sum_{e\in E} z_e \le \Gamma \right\}.
\end{align*}
By strong LP duality (using $\gamma_0$ as dual variable for the GUB constraint and $\gamma_e$ for the upper variable bounds), the objective function value of this inner problem coincides with the objective function value of the dual problem
\begin{align*}
\min_{\gamma \in \mathbb{R}_+^{|E|+1}}\quad & \gamma_0 \Gamma + \sum_{e \in E} \gamma_e \\
\text{s.t.} \quad & \gamma_0 + \gamma_e \geq \Delta_e \sum_{P\in \mathcal{P} : e \in P} x_P\quad && \forall e \in E.
\end{align*}
As a consequence, we can reformulate the problem to compute an optimal temporally repeated flow as:
\begin{align*}
\max_{x \in \mathbb{R}_+^{|\mathcal{P}|},\ \gamma \in \mathbb{R}_+^{|E|+1}} \quad & \sum_{P \in \mathcal{P}} x_P (T - \tau(P)) - \gamma_0 \Gamma - \sum_{e \in E} \gamma_e\\
\text{s.t.} \quad & \sum_{P\in \mathcal{P} : e \in P} x_P \leq u_e \quad && \forall e \in E \\
& \gamma_ 0 + \gamma_e \geq \Delta_e \sum_{P\in \mathcal{P} : e \in P} x_P \quad && \forall e \in E.
\end{align*}
We denote the dual variables for capacity constraints by $\alpha \in \mathbb{R}_+^{|E|}$ and the dual variables for scenarios by $\beta \in \mathbb{R}_+^{|E|}$ and obtain the following dual:
\begin{align*}
\min_{\alpha,\ \beta \in \mathbb{R}_+^{|E|}} \quad & \sum_{e \in E} \alpha_e u_e \\
\text{s.t.}\quad & \sum_{e \in P} \alpha_e + \sum_{e \in P} \beta_e \Delta_e \geq T - \sum_{e \in P} \tau_e\quad && \forall P \in \mathcal{P} \\
& \sum_{e \in E} \beta_e \leq \Gamma \\
& \beta_e \leq 1\quad && \forall e \in E.
\end{align*}
The pricing problem for path variables $x_P$ corresponds to separating the inequalities $\sum_{e \in P}(\alpha_e + \beta_e \Delta_e) \geq T - \sum_{e \in P} \tau_e$ for all paths $P \in \mathcal{P}$.
By moving $\sum_{e \in P} \tau_e$ to the left hand side, it can be solved as a shortest path problem with cost $c_e = \alpha_e + \beta_e \Delta_e + \tau_e$, which can be solved in polynomial time.
Since the ellipsoid method requires only a polynomial time separation oracle in order to run in polynomial time \cite{grotschel1981ellipsoid}, this concludes that the LP can also be solved in polynomial time.
\qed
\end{proof}
Note that, in general, checking whether the longest path length exceeds the time horizon is NP-hard.
Still, whenever the time horizon is sufficiently large, the $T$-bounded path property is certainly fulfilled.
\section{Bounds on the Solution Quality of Temporally Repeated Flows}
\label{sec:bounds}
In the remainder of the paper, we turn our focus on the solution quality of temporally repeated flows compared to a general solution.
For any given instance $\mathcal{I}$ of the robust maximum flow over time problem, let $f_{\text{TR}}^{\text{OPT}}$ be the value of an optimal temporally repeated and $f^{\text{OPT}}$ be the value of an optimal general solution.
We call the ratio ${f^{\text{OPT}}}/{f_{\text{TR}}^{\text{OPT}}}$ the \emph{optimality gap} of $\mathcal{I}$ and provide general upper and lower bounds on the optimality gap.
\subsection{Lower Bounds}
The next two propositions provide lower bounds on the worst case optimality gap and show that the class of temporally repeated flows is not necessarily optimal for the robust maximum flow over time problem, even for $\Gamma = 1$.
Proposition~\ref{prop:temp-rep:linear-gap} presents a family of instances with optimality gap $\Omega(T + \Gamma)$.
For instances with $T$-bounded path length, the following Proposition~\ref{prop:temp-rep:log-gap} yields a gap of $\Omega(\log T + \log \Gamma)$.
Both families of instances are depicted in Figure~\ref{fig:temp-rep:gap-instances}.
\begin{restatable}{proposition}{proptempreploggap}
\label{prop:temp-rep:log-gap}
The optimality gap of temporally repeated flows can be $\mathcal{H}_T$ and $\mathcal{H}_{\Gamma+1}$ for instances satisfying the $T$-bounded path length property.
Here, $\mathcal{H}_r \in \Omega(\log{r})$ is the r-th harmonic number, i.e.\ $\mathcal{H}_r = \sum_{i = 1}^r \frac{1}{i}$.
\end{restatable}
\begin{proof}
For fixed $r \in \mathbb{N}$, we construct an instance $I_r$ with graph $G_r = (V,E_r)$ as follows (see Figure \ref{fig:temp-rep:gap-instances:log-gap}).
The vertex set always consists of three vertices $V = \{s,v,d\}$.
The edge set consists of a designated edge $e^* = (v,d)$ and a set of $r$ parallel edges $e_i = (s,v), i = 0,\dots,r-1$.
We set $\tau_{e^*} = \Delta_{e^*} = 0$, $\tau_{e_i} = i, \Delta_{e_i} = r - i, \Gamma = r - 1, T = r$.
All edges have unit capacity.
For the sake of notation, we use $P^i$ to denote the unique $s$-$d$-path which contains edge $e_{r-i-1}$.
A combination of the two claims below yields an optimality gap of $\mathcal{H}_r = \Omega(\log r)$.
Note that $r = \Gamma + 1 = T$.
\omsClaim{There exists a solution to $I_r$ with objective function value $1$.}
For each $i = 0,\dots,r-1$, we send one unit of flow along path $P^i$ in the time interval $[0,1)$.
The flow of each path reaches the designated edge $e^*$ during the time interval $[i, i+1)$, or if the path was delayed, during ${[r,r+1)}$ which exceeds the time horizon.
Since the time intervals do not overlap during the time horizon, there will be no collisions.
Furthermore, the total flow sent in the nominal setting is equal to the time horizon $T = r$.
Since the total flow sent along each path is equal to $1$, any scenario can destroy at most $\Gamma = (r-1)$ units of flow, thus leaving a remaining flow of $1$.
\omsClaim{An optimal temporally repeated flow can send at most $\mathcal{H}_r^{-1}$ units of flow.}
Let us assume that $x^*$ is an optimal temporally repeated flow with flow rate $x^*_{P^i}$ on path $P^i$.
For ease of notation, we use $x^*_i = x^*_{P^i}$ in the remainder of the proof.
Since the flow is temporally repeated, $\sum x^*_i \leq 1$ holds.
Without loss of generality, we can assume equality.
Otherwise, we could increase some variable $x^*_i$ by a sufficiently small positive amount $\epsilon$ without decreasing the robust flow value (Note that any scenario can destroy at most the additional flow that would be sent by increasing the variable by $\epsilon$).
In the following, we claim that the optimal temporally repeated flow is of the following form:
\[x^*_0 = 2 x^*_1, \quad x^*_1 = \frac{3 x^*_2}{2}, \quad \dots, \quad x^*_{r-2} = \frac{r x^*_{r-1}}{r - 1}. \]
First, let us assume that the solution is of such form.
Then, by substituting, we get $x^*_i = \nicefrac{x^*_0}{i + 1}$ and $\sum x^*_i = 1$ implies that $x^*_0 \leq (\sum_{i=1}^r 1/i)^{-1}$.
Also note that the objective function coefficient of path $P^i$ is $T - \tau_{e_{r - i - 1}} = r - (r - i - 1) = i + 1$ for all $i = 0,\dots,r-1$.
Hence, $x^*$ satisfies for every $i,j$, $$x^*_i (T - \tau_{e_{r - i - 1}}) = \frac{x^*_0 (i+1)}{i+1} = x^*_0 = \frac{x^*_0 (j+1)}{j+1} = x^*_j (T - \tau_{e_{r - j - 1}}).$$
In other words, destroying any set of edges of cardinality $\Gamma$ results in the same objective function value.
Hence, in every scenario, one of the $r$ paths remains unharmed, yielding an objective function value of $x^*_0 \leq (\sum_i 1/i)^{-1}$.
Now, let us assume that the solution is not of such form.
Then, there exist two indices $\ell = \arg\min_{i = 0,\dots,r - 1} x_{i} (T - \tau_{e_{r - i - 1}})$ and $k = \arg\max_{i = 0,\dots,r - 1} x_{i} (T - \tau_{e_{r - i - 1}})$
with $x_{\ell} (T - \tau_{e_{r - \ell - 1}}) < x_k (T - \tau_{e_{r - k - 1}})$.
The worst case scenario will delay all edges except for $e_{\ell}$.
Averaging the flow shows that it was not optimal:
Decreasing the flow on all edges with weighted flow value equal to $x_k(T - \tau_{e_{r - k - 1}})$ and
increasing the flow on all edges with weighted flow value equal to $x_\ell (T - \tau_{e_{r - \ell - 1}})$ would strictly increase the robust flow value. \qed
\end{proof}
\begin{figure}[t]
\begin{center}
\tikzstyle{vertex}=[circle, draw,
inner sep=1pt, minimum width=10pt]
\tikzstyle{edge} = [draw,very thick]
\begin{subfigure}[t]{.49\textwidth} \centering
\begin{tikzpicture}[scale = 1.6]
\draw[fill=white,draw=white,use as bounding box] (-0.2,-0.8) rectangle (3.2,0.8);
\node[vertex] (v0) at (0, 0){ $s$};
\node[vertex] (v1) at (2, 0){ $v$};
\node[vertex] (v2) at (3, 0){ $d$};
\node[anchor=north east] at (3.2, 0.8){$T = 3, \Gamma = 2$};
\draw (v1) edge[->, very thick] node[anchor = south] { 0,0} (v2);
\draw (v0) edge[->, very thick, bend left = 55] node[anchor = south] { 0,3} (v1);
\draw (v0) edge[->, very thick] node[anchor = south] { 1,2} (v1);
\draw (v0) edge[->, very thick, bend right = 55] node[anchor = north] { 2,1} (v1);
\end{tikzpicture}
\caption{\parbox{.9\textwidth}{Instance $I_3$ from Proposition \ref{prop:temp-rep:log-gap}. Edge labels denote the travel times and possible delays, respectively.}}
\label{fig:temp-rep:gap-instances:log-gap}
\end{subfigure}
\begin{subfigure}[t]{.5\textwidth} \centering
\begin{tikzpicture}[scale = 1.6]
\node[vertex] (v0) at (0, 0){ $s$};
\node[vertex] (v1) at (1.2, 0){ $v_1$};
\node[vertex] (v2) at (2.2, 0){ $v_2$};
\node[vertex] (v3) at (3.4, 0){ $d$};
\draw (v0) edge[->, very thick, bend left = 55] node[anchor = south] { $0,\infty$} (v1);
\draw (v0) edge[->, very thick] node[anchor = south] { $1,\infty$} (v1);
\draw (v0) edge[->, very thick, bend right = 55] node[anchor = north] { $2,\infty$} (v1);
\draw (v1) edge[->, very thick] node[anchor = south] { $0,0$} (v2);
\draw (v2) edge[->, very thick, bend left = 55] node[anchor = south] { $0,\infty$} (v3);
\draw (v2) edge[->, very thick] node[anchor = south] { $1,\infty$} (v3);
\draw (v2) edge[->, very thick, bend right = 55] node[anchor = north] { $2,\infty$} (v3);
\node[anchor=north] at (1.7, 0.8){$T = 3, \Gamma = 2$};
\draw[white] (0, -0.7) circle (1mm);
\end{tikzpicture}
\caption{\parbox{.9\textwidth}{Instance $I_3$ from Proposition \ref{prop:temp-rep:linear-gap}. Edge labels denote the travel times and possible delays, respectively.}}
\label{fig:temp-rep:gap-instances:linear-gap}
\end{subfigure}
\end{center}
\caption{Instances showing an optimality gap for temporally repeated flows.}
\label{fig:temp-rep:gap-instances}
\end{figure}
\begin{restatable}{proposition}{proptempreplineargap}
\label{prop:temp-rep:linear-gap}
There are instances for robust maximum flow over time whose gap between an optimal temporally repeated flow and an optimal flow is $T$ and $\Gamma+1$.
\end{restatable}
\begin{proof}
For fixed $r \in \mathbb{Z}_+$, we construct an instance $I_r$ with graph $G_r = (V,E_r)$ as follows (see Figure \ref{fig:temp-rep:gap-instances:linear-gap}).
The vertex set always consists of four vertices $V = \{s,v_1,v_2,d\}$.
The edge set consists of a designated edge $e^* = (v_1,v_2)$ and two sets of $r$ parallel edges $e^1_i = (s,v_1)$ and $e^2_i = (v_2,d), i = 0,\dots,r-1$.
We set $\tau_{e^*} = \Delta_{e^*} = 0$, $\tau_{e^1_i} = \tau_{e^2_i} = i, \Delta_{e^1_i} = \Delta_{e^2_i} = \infty, \Gamma = r - 1, T = r$.
All edges have unit capacity.
Combination of the two claims below yields an optimality gap of $\Omega(r)$.
\omsClaim{There exists a solution to $I_r$ with objective function value $1$.}
We define $r$ paths $P^i = \{e^1_i,e^*,e^2_{r-i-1}\}$ and dispatch a single unit of flow in the interval $[0,1)$ into each.
The flow is clearly feasible and has a nominal objective function value of $r$.
Moreover, any $r-1$ attacked edges can destroy at most $r-1$ units of flow, hence, yielding a flow value of at least $1$.
\omsClaim{An optimal temporally repeated flow can send at most $1 / r$ units of flow.}
We start by defining a temporally repeated flow with robust solution value $1/r$ and show the optimality of this flow. Let $x_i=1/r$ for all paths $P^i, 0\leq i \leq r-1$.
We will argue that $x$ is optimal with robust function value $1/r$.
Let us check the objective value contribution in detail.
Each path $P^i, i= 0,\dots,r - 1$ contributes $$T - \tau_{e^1_i} - \tau_{e^2_{r-i-1}} = r - i - (r - i - 1) = 1$$
to the objective. Hence, sending a flow value $x_i = 1/r$ on all $r$ paths yields an objective function value of $1/r$ independent of which $\Gamma = r - 1$ edges between $s$ and $v_1$ are destroyed.
Note that attacking different edges does not make any sense for this type of solution.
Now, we show that any feasible temporally repeated flow different than $x$ yields a smaller objective function value. Let $x'$ be a different solution, i.e.\ there is some path $P^i$ that has flow rate $x'_i < 1/r$.
We will show that this solution has objective value at most $x'_i$.
More precisely, we will show that there is a scenario which leaves only the path $P^i$ intact.
We will use the scenario $z \in \mathcal{S}$ with $z_{e^1_j} = 1$ for all $0 \leq j < i$, $z_{e^2_j} = 1$ for all $0 \leq j < r - i - 1$ and zero otherwise.
The total number of edges destroyed by $z$ is $r - 1$. In the following, we show that in this scenario only $P^i$ with $x'_i<1/r$ contributes to the objective function value.
If any other path $P \neq P^i$ should survive the scenario, it must avoid all edges affected by $z$. $P$ can neither contain edges from $\{e_1^1,\dots,e_{i-1}^1\}$ nor from $\{e_1^2,\dots,e_{r-i-2}^2\}$. Let us consider the path length of such a path $P$. The only possibility to achieve a path length smaller than $T=r$ is to use edges $e_i^1$ and $e_{r-i}^2$, i.e.\ $P = P^i$. We conclude that the objective value is at most $x'_i < 1/r$, which proves the optimality of $x$ and finishes the proof of the claim.
\qed
\end{proof}
\subsection{Asymptotic Optimality}
Although the gaps seem large, they appear only if the time horizon is
relatively short, when compared to the travel times.
An asymptotic bound shows that the optimality gap diminishes as the time horizon increases.
\begin{restatable}{proposition}{proptemprepasymptoticbound}
For instances of the maximum robust flow over time problem with $\Delta_e < \infty$ for all $e \in E$, temporally repeated flows tend to optimality for $T \rightarrow \infty$, if all other parameters are fixed.
\label{prop:temp-rep:asymptotic-bound}
\end{restatable}
\begin{proof}
Let $(G,s,t,u,\Delta,\Gamma)$ with $\Delta_e < \infty$ for all $e \in E$ be an instance of robust max flow over time, with values that do not depend on the time horizon $T$.
If we denote the optimal objective function value of a temporally repeated flow by $f_{\text{TR}}^{\text{OPT}}(T)$ and the value of a general flow by $f^{\text{OPT}}(T)$ for time horizon $T$,
then for every $\epsilon > 0$ we show that there exists a $T' \in \mathbb{R}$ such that $f^{\text{OPT}}(T)/f_{\text{TR}}^{\text{OPT}}(T) \leq 1 + \epsilon$ for all $T \geq T'$.
Since $\Delta$ is supposed to be a constant, we can assume that the choice of any scenario destroys at most a value of $\lambda^* = \Gamma \max_{e \in E} \Delta_e u_e$.
For sufficiently large $T$, any optimal nominal temporally repeated flow will send a flow value of at least $F^*(T) - \lambda^*$, where $F^*(T)$ is the nominal optimal value for time horizon $T$.
Moreover, as temporally repeated flows are optimal in the nominal case, we can deduce $f^{\text{OPT}} \leq F^*(T)$.
Thus, $$ \frac{f^{\text{OPT}}(T)}{f_{\text{TR}}^{\text{OPT}}(T)} \leq \frac{F^*(T)}{F^*(T) - \lambda^*},$$ which tends to one as $T$ tends to infinity. \qed
\end{proof}
\subsection{Upper Bounds}
In the remainder of this section, we prove an upper bound on the gap between the objective function values of optimal general solutions and optimal temporally repeated flows.
This gap depends on some graph parameter $k$ introduced below.
Note that flow sent along a particular path $P\in \mathcal{P}$ only has a chance to reach the destination if each edge $e\in P$ is reached by the flow within interval
$I_{e,P}:=[\tau^{<e}(P), T - \tau^{\geq e}(P)]$, where $\tau^{<e}(P) = \sum_{e'\in E : e' <_P e}\tau(e')$ is the time required for a flow particle on path $P$ to reach edge $e$.
We call an instance of maximum flow over time \emph{$k$-coverable} if for each edge $e\in E$ it is possible to select $k$ points in time to cover all intervals $\{I_{e,P}\}_{P\in \mathcal{P}: e\in P}$.
The same definition holds for the robust counterpart. Note that this definition only depends on the graph $G$, the vertices $s,d$, the travel times $\tau$ and the time horizon $T$.
Denote these $k$ points in time by $t^e_1, \ldots t^e_k$.
We also call these points \emph{witnesses} of edge $e$.
That is, we define for each edge $e\in E$ the interval graph $H_e$ whose vertex set corresponds to the intervals $\{I_{e,P}\}_{P\in \mathcal{P}: e\in P}$ two of which are linked by an edge if and only if the associated intervals overlap.
Then the instance is $k$-coverable if and only if the vertices of each of the interval graphs $H_e$ can be covered by not more than $k$ cliques.
Since the clique-covering-number equals the maximum cardinality of a stable set by Dilworth's Theorem \cite{dilworth1950decomposition}, we define
\begin{definition}
An instance of maximum robust flow over time is \emph{$k$-coverable} if and only if the maximum size of a stable set in all of the associated interval graphs $H_e, e\in E,$ is at most $k$.
\end{definition}
For example, an instance satisfying $\max_{P \in \mathcal{P}} \tau^{<e}(P) + \max_{P \in \mathcal{P}} \tau^{\geq e}(P) \leq T$ for all $e \in E$ is $1$-coverable:
for an edge $e$, select $t^e_1 \in [\max_{P \in \mathcal{P}} \tau^{<e}(P), T-\max_{P \in \mathcal{P}} \tau^{\geq e}(P)]$.
This inequality is fulfilled in directed acyclic graphs whose longest path length does not exceed the time horizon, i.e.\ where $\max_{P \in \mathcal{P}} \tau(P) \leq T$. Note that maximum robust flow over time instances on a directed acyclic graph with the $T$-bounded path length property, i.e.\ with $\max_{P \in \mathcal{P}, z \in \mathcal{S}} \tau(P)+\Delta_z(P) \leq T$, are also 1-coverable.
More vividly, let us consider the graphs in Figure \ref{fig:temp-rep:gap-instances}.
The graph in (a) has $k=1$, as its longest path length does not exceed the time horizon.
The graph in (b) has $k=2$, as can be seen by considering the edge $e^* = (v_1,v_2)$ and the following three paths.
Let $P_1$ be the path with the top edge from $s$ to $v_1$ and the bottom edge from $v_2$ to $d$.
It has interval $I_{e^*,P_1} = [0,1]$.
$P_2$ with both middle edges has $I_{e^*,P_2} = [1,2]$.
Finally, $P_3$ with the first bottom edge and the last top edge has $I_{e^*,P_3} = [2,3]$.
A minimum covering of these intervals can be achieved by choosing $t^{e^*}_1 = 1, t^{e^*}_2 = 2$.
Hence, $k$ is at least two.
One can easily check that the remaining intervals do not change this.
Moreover, one can observe that all remaining edges are 1-coverable.
It is shown in \cite{gupta1982efficient} that a stable set of maximum cardinality in an interval graph, in general, can be found by a simple greedy approach: \emph{sweep} from right to left through the whole domain, in our case $[0,T]$, and,
iteratively, select the interval with rightmost left endpoint. Remove this interval together with all intersecting intervals from the list, until no intervals are remaining.
In this work, we do not discuss the complexity status of computing $k$ in detail. We only note that the greedy approach described above is, in general, certainly not strongly polynomial as it depends on the time horizon $T$ and on the number of $s$-$d$ paths in the given graph.
Next, we prove upper bounds on the optimality gap of temporally repeated flows.
Since the proofs are rather technical, we distinguish between the case $\Delta_e \in \{0,\infty\}$ for all $e \in E$ and the case $\Delta_e \in \mathbb{Z}_+$.
The former turns out to be simpler and is covered in Theorem \ref{thm:temp-rep:k-log-T-gap}, the latter is covered in Theorem \ref{thm:temp-rep:delta-arbitrary} and builds upon the former.
\begin{theorem}
\label{thm:temp-rep:k-log-T-gap}
Let a $k$-coverable instance with $\Delta_e \in \{0,\infty\}$ for all $e \in E$ be given. Then, an optimal temporally repeated solution is an $O(k \log T)$-approximation for the robust maximum flow over time problem.
\end{theorem}
Additionally, we derive an upper bound for general instances.
Let $\bar{\Delta}_z(P)$ denote the effective amount of flow cut off by scenario $z \in \mathcal{S}$ on path $P$, that is, $\bar{\Delta}_z(P) = \min \{ \Delta_z(P), T - \tau(P) \}$.
Then we define $$\eta = \max_{P \in \mathcal{P},z : \bar{\Delta}_z(P) < T - \tau(P)} \frac{T - \tau(P)}{T - \tau(P) - \bar{\Delta}_z(P)},$$ or $\eta = 1$, if no such path exists.
The value can be interpreted as follows.
If $\eta = 1$, the total contribution of flow on a path $P$ in a temporally repeated solution is either $x_P(T - \tau(P))$, or zero, depending on whether or not the path is attacked by the worst-case scenario.
If $\eta$ is large, the ratio between the contribution of path $P$, if $P$ is not attacked, and the non-zero contribution for some scenarios might be as large as $\eta$.
We will see that this ratio makes it harder to estimate the loss in the proof of Theorem \ref{thm:temp-rep:delta-arbitrary}.
\begin{theorem}
\label{thm:temp-rep:delta-arbitrary}
Let a $k$-coverable instance with $\Delta \in \mathbb{Z}_+$ be given. Then,
an optimal temporally repeated solution is an $O(\eta k \log
T)$-approximation for the robust maximum flow over time problem.
\end{theorem}
The proof strategy of both theorems is the same and can be viewed as a dual fitting approach.
We present primal-dual pairs (\ref{LP:TR-P}), (\ref{LP:TR-D}) modeling robust temporally repeated flows and (\ref{LP:EX-P}), (\ref{LP:EX-D}) modeling general solutions to the robust flow over time problem.
It is clear by strong duality that $opt(\ref{LP:TR-P}) = opt(\ref{LP:TR-D})$ and $opt(\ref{LP:EX-P}) = opt(\ref{LP:EX-D})$.
Hence, in order to prove a bound on the optimality gap, it suffices to show $opt(\ref{LP:EX-D}) \leq \alpha opt(\ref{LP:TR-D})$, where $\alpha \geq 1$ is the upper bound on the optimality gap from Theorem \ref{thm:temp-rep:k-log-T-gap} and \ref{thm:temp-rep:delta-arbitrary}, respectively.
With this factor, we conclude
\begin{align}
opt(\ref{LP:EX-P}) = opt(\ref{LP:EX-D}) \leq \alpha opt(\ref{LP:TR-D}) = \alpha opt(\ref{LP:TR-P})
\Leftrightarrow \frac{f^{\text{OPT}}}{f^{\text{OPT}}_{\text{TR}}} = \frac{opt(\ref{LP:EX-P})}{opt(\ref{LP:TR-P})} \leq \alpha. \label{thm:temp-rep:bound-strategy}
\end{align}
We bound the factor $\alpha$ via a geometric box interpretation of solutions in the dual problems.
From an optimal solution of (\ref{LP:TR-D}), we construct a feasible solution of (\ref{LP:EX-D}), guaranteeing that the objective function values differ by at most the factor of $\alpha$.
In the remainder, we prove Theorems \ref{thm:temp-rep:k-log-T-gap} and \ref{thm:temp-rep:delta-arbitrary}.
We start by introducing the LP models. (\ref{LP:TR-P}) is a model for temporally repeated flows with corresponding dual (\ref{LP:TR-D}).
Without the constraints for scenarios $z$, (\ref{LP:TR-P}) models (nominal) temporally repeated flows.
The variable $\lambda$ corresponds to the amount of flow that is lost due to the actions of the adversary.
\begin{align*}
\max_{x \in \mathbb{R}_+^{|\mathcal{P}|},\ \lambda \in \mathbb{R}_+}\quad & \sum_{P \in \mathcal{P}} (T - \tau(P) )x_P - \lambda \tag{P} \label{LP:TR-P} \\
\text{s.t.} \quad & \sum_{\substack{P \in \mathcal{P} : \\ e \in P}} x_P \leq u_e & \quad && \forall e \in E \\
& \sum_{\substack{P \in \mathcal{P} : \\ z \cap P \neq \emptyset}} \bar{\Delta}_z(P) x_P - \lambda \leq 0 & \quad && \forall z \in \mathcal{S}
\end{align*}
The corresponding dual with variables $\alpha_e$ and $\beta_z$ is:
\begin{align*}
\min_{\alpha \in \mathbb{R}_+^{|E|},\ \beta \in \mathbb{R}_+^{|\mathcal{S}|}}\quad & \sum_{e \in E} u_e \alpha_e \tag{D} \label{LP:TR-D} \\
\text{s.t.} \quad & \sum_{z \in \mathcal{S}} \beta_z \leq 1 \\
& \sum_{e \in P} \alpha_e + \sum_{\substack{z \in \mathcal{S} : \\ z \cap P \neq \emptyset}} \bar{\Delta}_z(P) \beta_z \geq T - \tau(P) & \quad && \forall P \in \mathcal{P}
\end{align*}
Now, we present a model (\ref{LP:EX-P}) for general robust solutions.
We introduce variables $x_P^i$ which correspond to the flow rate sent into path $P$ in the dispatch interval $[i, i+1)$. Here, we implicitly use the fact that we can restrict to
integer dispatch intervals, as shown in Proposition~\ref{prop:modelling-techniques:piecewise-constant}. As before, $\lambda$ is the loss incurred after the adversary acts.
For ease of notation, we assume that all variables with a negative index in (\ref{LP:EX-P}) are omitted, e.g.\ a variable $x_P^{-5}$ is assumed to be excluded from the model.
This can also be seen as having variables $x_P^i$ for all $i \in \mathbb{Z}$ and forcing $x_P^i = 0$ for all $i < 0$ or $i \geq T - \tau(P)$.
We use the notation $\Delta^{<e}_z(P) = \sum_{e' \in P : e' <_P e} \Delta_{e'}= z_{e'}$ to denote the delay on path $P$ induced by scenario $z$ until edge $e$.
\begin{align*}
\max_{x,\ \lambda \geq 0}\quad & \sum_{P \in \mathcal{P}} \sum_{0 \leq i < T - \tau(P)} x_P^i - \lambda \tag{P'} \label{LP:EX-P} \\
\text{s.t.} \quad & \sum_{\substack{P \in \mathcal{P} :\\ e \in P}} x_P^{t - \tau^{<e}(P) - \Delta^{<e}_z(P) } \leq u_e \quad && \forall e \in E, \forall 0 \leq t < T, \forall z \in \mathcal{S} \\
& \sum_{P \in \mathcal{P}} \sum_{i \geq T - \tau(P) - \Delta_z(P)} x_P^i - \lambda \leq 0 \quad && \forall z \in \mathcal{S}
\end{align*}
The corresponding dual with variables $\alpha_e^t(z)$ and $\beta_z$ is:
\begin{align*}
\min_{\alpha,\ \beta \geq 0}\quad & \sum_{e \in E} u_e \sum_{ 0 \leq t < T} \sum_{z \in \mathcal{S}} \alpha_e^t(z) \tag{D'} \label{LP:EX-D} \\
\text{s.t.} \quad & \sum_{z \in \mathcal{S}} \beta_z \leq 1 \\
& \sum_{e \in P} \sum_{z \in \mathcal{S}} \alpha_e^{ i + \tau^{<e}(P) + \Delta^{<e}_z(P) }(z) + \sum_{z \in \scenariosDualPath{i}{P} } \beta_z \geq 1 \quad && \forall P \in \mathcal{P}, \forall 0 \leq i < T - \tau(P).
\end{align*}
By $\scenariosDualPath{i}{P}$ we denote for a path $P$ and time $i$ the set of scenarios for which flow on path $P$ sent into the network at time $i$ will not reach the destination, that is, $\scenariosDualPath{i}{P} = \{ z \in \mathcal{S} : i \geq T - \tau(P) - \Delta_z(P) \}$.
In other words, these are the scenarios for path $P$ which prevent flow sent at time $i$ from contributing to the objective.
\sectionheadline{Graphical Interpretation of (\ref{LP:TR-D})}
\begin{figure}[tb]
\begin{center}
\begin{tikzpicture}[scale=0.7, every node/.style={scale=1.2}]
\draw[fill=white,draw=white,use as bounding box] (0,-1) rectangle (15.5,6.5);
\begin{scope}
\draw[->] (0.5,1) -- (13,1);
\draw[->] (1,0.5) -- (1,6);
\node at (0.74,1.2) {$0$};
\draw[] (0.9,1.1) -- (1,1.1);
\node at (0.6,5.8) {$\alpha^t_{e}$};
\node at (13,0.6) {$t$};
\end{scope}
\begin{scope}
\transparent{0.5}
\draw[pattern=north east lines,pattern color=darkgreen] (3.5,1.1) rectangle (7.5,5.5);
\draw[pattern=north west lines,pattern color=darkblue] (2.0,1.1) rectangle (6.0,3.0);
\draw[pattern=vertical lines,pattern color=darkorange] (2.5,1.1) rectangle (13.0,2.0);
\transparent{1.0}
\draw[-,ultra thick, draw=black] (2.0,1.1) -- (2.0,3.0) -- (3.5,3.0) -- (3.5,5.5) -- (7.5,5.5) -- (7.5,2.0) -- (13.0,2.0) -- (13.0,1.1);
\draw[-,dashed,draw=red] (2.0,1.0) -- (2.0,6.5);
\draw[-,dashed,draw=red] (13.0,1.0) -- (13.0,6.5);
\draw[-,dashed,draw=red] (3.75,1.0) -- (3.75,6.5);
\node[fill=white] at (2.0,6.5) {$t_{min}$};
\node[fill=white] at (13.0,6.5) {$t_{max}$};
\node[fill=white] at (3.75,6.5) {$t^*$};
\node at (3.5,0.6) { $\tau^{<e}(P)$};
\node at (7.5,0.6) { $T - \tau^{\geq e}(P)$};
\draw[thick,black,decorate,decoration={brace,amplitude=8pt}] (7.5,0.4) -- (3.5,0.4) node[midway, below,yshift=-0.3cm]{$T - \tau(P)$};
\draw[thick,black,decorate,decoration={brace,amplitude=12pt}] (7.5,5.5) -- (7.5,1.1) node[midway, right,xshift=0.3cm]{ $\begin{aligned} \Bigl( 1 - \beta^*(P) \Bigr) \frac{\alpha^*_e}{\alpha^*(P)} \end{aligned}$ };
\end{scope}
\end{tikzpicture}
\end{center}
\caption{Illustration of the box model with three boxes. The height profile is a solution constructed for the proof of Theorem \ref{thm:temp-rep:k-log-T-gap}.}
\label{fig:temp-rep:dual-box-model}
\end{figure}
The remaining proofs in this section rely on the following graphical interpretation of the duals (\ref{LP:TR-D}) and (\ref{LP:EX-D}).
We consider an optimal solution $(\alpha^*, \beta^*) \in (\ref{LP:TR-D})$ and interpret the dual constraints and variables of (\ref{LP:TR-D}) as boxes in a two-dimensional space $[0,T] \times [0,1]$ as follows (see Figure \ref{fig:temp-rep:dual-box-model}):
Let us consider the dual constraint for a path $P \in \mathcal{P}$ and assume $\bar{\Delta}_z(P) \in \{0, T - \tau(P)\}$ for all $z \in \mathcal{S}$, that is, $P$ only contains edges that have a very large value for $\Delta$, or zero.
The general case is discussed in the proof of Theorem \ref{thm:temp-rep:delta-arbitrary}.
We can regard the dual constraint of such a path $P$ as a set of boxes
\[ B^e_P = \left[ \tau^{<e}(P), T - \tau^{\geq e}(P) \right] \times \left[ 0, \Bigl( 1 - \beta^*(P) \Bigr) \frac{\alpha_{e}^*}{\alpha^*(P)} \right], e \in P, \]
where $\alpha^*(P) = \sum_{e \in P} \alpha^*_e$ and $\beta^*(P) = \sum_{z \in \mathcal{S} : z \cap P \neq \emptyset} \beta^*_z$.
Each box $B^e_P$ starts at the earliest possible arrival of flow at edge $e$ traveling along path $P$ and ends at the latest reasonable departure time from edge $e$.
As soon as we consider the total area covered by the boxes of a path $P$, the connection between the boxes and their dual constraint becomes evident.
\begin{align*}
\sum_{e \in P} \text{vol}(B^e_P) = \sum_{e \in P} \left(T - \tau(P)\right)\Bigl(1 - \beta^*(P)\Bigr)\frac{\alpha^*_e}{\alpha^*(P)}
= T - \tau(P) - \sum_{\substack{z \in \mathcal{S} : \\ z \cap P \neq \emptyset}} \beta^*_z \bar{\Delta}_z(P)
\end{align*}
Here we used that $\bar{\Delta}_z(P) = T -\tau(P)$ for $z \cap P \neq \emptyset$ as $\Delta_e \in \{0, \infty\}$.
The sum of volumes of all boxes of a path is equal to the corresponding right hand side value of its dual constraint in (\ref{LP:TR-D}) (assuming all terms involving $\beta^*$ are moved to the right hand side).
Furthermore, the volume of $B^e_P$ is a lower bound for the value of $\alpha^*_e$ for all edges $e \in P$.
This can be seen by considering the dual constraint for path $P$ in (\ref{LP:TR-D}), multiplying it by $\alpha^*_e$ and rearranging the terms to have $vol(B^e_P)$ as the right hand side value.
\begin{align}
\alpha^*(P) = \sum_{e' \in P} \alpha^*_{e'} \geq \left( 1- \beta^*(P) \right) \left(T - \tau(P) \right) \notag \\
\Leftrightarrow \alpha^*_e \alpha^*(P) \geq \alpha^*_e \left( 1- \beta^*(P) \right) \left( T - \tau(P) \right) \notag \\
\Leftrightarrow \alpha^*_e \geq \frac{\alpha^*_e}{\alpha^*(P)} \left(1- \beta^*(P) \right) \left(T - \tau(P) \right) = vol(B^e_P). \label{bounds:alpha-volume-bound}
\end{align}
In the following, we will use the \emph{area covered} by boxes in order to construct a feasible solution for (\ref{LP:EX-D}) and use the \emph{volume} of the boxes in order to estimate the total cost of the solution.
In a $k$-coverable instance, every box $B^e_P$ intersects at least one of the lines $t^e_i \times [0, 1]$.
Recall that $t^e_i$ are the witnesses of edge $e$, that is, the points used in order to cover the interval graph in the definition of $k$-coverability.
In the interval graph $H_e$, we had intervals $I_{e,P}:=[\tau^{<e}(P), T - \tau^{\geq e}(P)]$ for path $P$.
Note that this is exactly the shadow of box $B^e_P$ on the horizontal $t$-axis.
For the remainder of this part, we will restrict all arguments to boxes which intersect a specific line $t^* = t^e_i \times [0, 1]$.
That is, we will consider only boxes $\{B^e_P : \tau^{<e}(P) \leq t^* \leq T - \tau^{\geq e}(P) \}$.
Afterwards, in the proofs of the theorems, we will do a summation over all lines $t^e_i$.
A box may intersect multiple lines - and thus be considered multiple times - but this does not harm the approximation guarantee.
In particular, we assume that no box starts after, respectively ends before, $t^*$.
By $t_{min}$ and $t_{max}$ we denote the earliest and latest points on $[0,T]$ covered by any of the considered boxes (see Figure \ref{fig:temp-rep:dual-box-model}).
That is, $t_{min} = \min \{ \tau^{<e}(P) : \tau^{<e}(P) \leq t^* \leq T - \tau^{\geq e}(P) \}$ and $t_{max}$ is defined analogously.
\sectionheadline{Construction of a feasible solution for (\ref{LP:EX-D})}
Let us examine feasible values for the dual variables $\alpha_e^t(z)$ in (\ref{LP:EX-D}) which can be constructed from the graphical interpretation of $(\alpha^*,\beta^*)$.
First, let us see how $\alpha_e^t(z)$ can be set in order to become feasible for the constraints induced by a single path $P$.
We set $\alpha_e^t(z) = 0$ for all $z \neq \emptyset$.
Setting these variables to a positive value would be hard to analyze.
But in order to become feasible, we set $\alpha_e^t(\emptyset)$ to the height of box $B^e_P$ in each slice $(t,t+1)$.
This results in $\alpha_e^t(\emptyset)$ being the height of $B^e_P$, if $\tau^{<e}(P) \leq t < T - \tau^{\geq e}(P)$, or zero otherwise.
Then
\begin{align*}
\sum_{e \in P} \sum_{z \in \mathcal{S}} \alpha_e^{i + \tau^{<e}(P) + \Delta_z^{<e}(P)}(z)
= \sum_{e \in P} \left( 1 - \sum_{z \in \scenariosDualPath{i}{P}} \beta^*_z \right)\frac{\alpha_e^*}{\alpha^*(P)}
= 1 - \sum_{\substack{z \in \scenariosDualPath{i}{P}}} \beta^*_z.
\end{align*}
That means, if we had only a constraint for a single path, the solution would be feasible.
But we have many paths and the height of their boxes $B^e_P$ differs.
Hence, if we set $\alpha_e^t(\emptyset)$ to the maximal height of all boxes $B^e_P$ intersecting slice $(t,t+1)$, we obtain a feasible solution $(\alpha,\beta^*) \in (\ref{LP:EX-D})$.
Formally, the solution can be described as
\[ \alpha_e^t(\emptyset) = \max_{P \in \mathcal{P}_e^t} \left\{ \left( 1 - \sum_{z \in \scenariosDualPath{t}{P} } \beta^*_z \right)\frac{\alpha_e^*}{\alpha^*(P)} \right\}, \]
where $\mathcal{P}_{e}^t = \{P : e \in P \wedge \tau^{<e}(P) \leq t < T - \tau^{\geq e}(P) \}$ describes the set of paths which can utilize edge $e$ at time $t$.
And $\alpha_e^t(z) = 0$ for all other variables.
The height profile in Figure \ref{fig:temp-rep:dual-box-model} depicts this solution.
The constructed solution is feasible for (\ref{LP:EX-D}).
We will now analyze its total cost.
Intuitively, we argue as follows:
The sum of dual variables, $\sum_{0 \leq t < T} \alpha_{e}^t(\emptyset)$ for any fixed edge $e$ is equal to the area covered by the union of all boxes $B^e_P$ induced by paths $P \in \mathcal{P}$ such that $e \in P$.
Thus, in order to prove the $O(k \log T)$-approximation factor for Theorem~\ref{thm:temp-rep:k-log-T-gap}, we will show that, for any fixed edge $e \in E$, the total area covered by the union of all boxes is bounded by $O(k \log T)$ times the area of the largest box.
Recall that (\ref{bounds:alpha-volume-bound}) implies that the value of $\alpha^*_e$ in the temporally repeated solution is lower bounded by the area of the largest box.
With this, we can estimate the total solution cost of $(\alpha,\beta^*)$ in terms of $(\alpha^*,\beta^*)$.
\begin{figure}[tb]
\begin{center}
\begin{tikzpicture}[scale=0.7, every node/.style={scale=1.2}]
\draw[fill=white,draw=none,use as bounding box] (0,0.25) rectangle (15.5,8.5);
\begin{scope}
\draw[->] (0.5,1) -- (13,1);
\draw[->] (1,0.5) -- (1,7);
\node at (0.74,1.2) {$0$};
\draw[] (0.9,1.1) -- (1,1.1);
\node at (0.74,6.2) {$1$};
\draw[] (0.9,6.25) -- (1,6.25);
\node at (0.6,6.8) {$\alpha^t_{e}$};
\node at (13,0.6) {$t$};
\end{scope}
\begin{scope}
\transparent{0.5}
\draw[pattern=north east lines,pattern color=darkgreen] (3.5,1.1) rectangle (7.5,5.5);
\draw[pattern=north west lines,pattern color=darkblue] (2.0,1.1) rectangle (6.0,3.0);
\draw[pattern=vertical lines,pattern color=darkorange] (2.5,1.1) rectangle (13.0,2.0);
\transparent{1.0}
\draw[-,dashed,draw=red] (13,1.0) -- (13,7);
\draw[-,dashed,draw=red] (3.75,1.0) -- (3.75,7);
\node[fill=white] at (13,6.75) {$t_{max}$};
\node[fill=none] at (3.55,6.75) {$t^*$};
\draw[-,dashed,draw=red] (8.375,1.0) -- (8.375,6.75);
\node[fill=none,rotate = 45] at (9.5,7.5) {$t^* + \frac{t_{max} - t^*}{2}$};
\node at (12.5,1.5) {$P_1$};
\draw[draw=black,thick] (8.375,1.2) rectangle (15,2.1);
\draw[draw=white,fill=white,thick] (14.9,1.2) rectangle (15.1,2.1);
\draw[-,dashed,draw=red] (6.0625,1.0) -- (6.0625,6.75);
\node[fill=none, rotate = 45] at (7.25,7.5) {$t^* + \frac{t_{max} - t^*}{4}$};
\node at (7.0,5.0) {$P_{2,3}$};
\draw[draw=black,thick] (6.0625,1.1) rectangle (10.0625,5.6);
\draw[-,dashed,draw=red] (4.90625,1.0) -- (4.90625,6.75);
\node[fill=none, rotate = 45] at (5.95,7.5) {$t^* + \frac{t_{max} - t^*}{8}$};
\draw[draw=black,thick] (4.90625,1.2) rectangle (8.90625,5.7);
\draw[draw=black,thick] (3.75,1.1) rectangle (4.90625,6.25);
\draw[draw=red,thick] (3.75,5.5) -- (7.5,5.5) -- (7.5,2.0) -- (13.0,2.0) -- (13.0,1.1);
\end{scope}
\end{tikzpicture}
\end{center}
\caption{Illustration of the proof of Theorem \ref{thm:temp-rep:k-log-T-gap}.
The interval $[t^*,t_{max}]$ contains three boxes (orange,blue,green).
The total area is covered by copies of $P_i, i \geq 1$, the copies are black (slightly moved up/down to distinguish different copies).
The box used in iteration $2$ and $3$ is the same, moved to a different offset.
We also assume that $\frac{t_{max} - t^*}{8} \leq \alpha^*_e$, hence, the final box is just the slice of height $[0,1]$.
The height profile of the solution is depicted in red.}
\label{fig:temp-rep:dual-covering}
\end{figure}
\begin{proof}[Proof of Theorem \ref{thm:temp-rep:k-log-T-gap}]
In order to prove the theorem, we will show $opt(\ref{LP:EX-D}) \leq O(k \log T) opt(\ref{LP:TR-D})$, which is sufficient in combination with (\ref{thm:temp-rep:bound-strategy}).
Therefore, we will use the argumentation from the preceding paragraphs, in particular, we will use the constructed solution $\alpha_e^t(z)$.
We will show
\begin{align}
\sum_{ 0 \leq t < T} \sum_{z \in \mathcal{S}} \alpha_e^t(z)
= vol \left(\bigcup_{\substack{P \in \mathcal{P} :\\ e \in P}} B^e_P \right)
\leq \sum_{i=1}^{k} vol \left(\bigcup_{\substack{P \in \mathcal{P}_e^{t_i}}} B^e_P \right)
\leq O(k \log T) \alpha^*_e \quad \forall e \in E. \label{proof:temp-rep:l-log-T-gap:pointwise}
\end{align}
Recall that $(\alpha^*, \beta^*)$ was an optimum solution to (\ref{LP:TR-D}).
Thus, result (\ref{proof:temp-rep:l-log-T-gap:pointwise}) then will prove
$$ opt(\ref{LP:EX-D}) \leq \sum_{e \in E} u_e \sum_{ 0 \leq t < T} \sum_{z \in \mathcal{S}} \alpha_e^t(z) \leq O(k \log T) \sum_{e \in E} u_e \alpha^*_e = O(k \log T) opt(\ref{LP:TR-D}). $$
Without loss of generality, we can assume $\sum_{z \in \mathcal{S}}\beta^*_z = 1$.
For the remainder of the proof, we will consider the graphical interpretation of the duals from the preceding paragraphs and fix an edge $e \in E$ and witness at $t^* = t^e_i$.
For ease of notation, we use $\mathcal{P}$ to denote the set $\mathcal{P}_e^{t^*}$ of all paths whose boxes intersect $t^*$.
If a path intersects multiple witnesses, we consider it multiple times.
We will show that the total volume of the union of the boxes intersecting $t^*$ can be bounded by $O(\log T)\alpha_e^*$.
Summation over $t_i^e$ with $i \le k$ yields the result.
Since every box $B^e_P$ for path $P \in \mathcal{P}$ intersects the point $t^*$, it is easy to see that the sets of paths $\mathcal{P}^\ell = \{P \in \mathcal{P} : T - \tau^{\geq e}(P) \geq t^* + (t_{max} - t^*) 2^{-\ell} \}$ are monotonically increasing, that is, $\mathcal{P}^i \subseteq \mathcal{P}^{i+1}$.
The set $\mathcal{P}^\ell$ contains all rectangles covering some area to the right of the line $t^* + (t_{max} - t^*) 2^{-\ell}$.
Now, let us consider the set $\mathcal{P}^1$ and pick the path $P_1 \in \mathcal{P}^1$ whose box $B_1$ is the highest among all candidates, that is,
\[ P_1 = \arg\max_{P \in \mathcal{P}^1} \left( 1 - \sum_{z \in \scenariosDualPath{t^*}{P}} \beta^*_z \right)\frac{\alpha_e^*}{\alpha^*(P)}. \]
$B_1$ has the largest height among all boxes which intersect the interval $[t^* + (t_{max} - t^*)\frac{1}{2}, t_{max}]$.
Furthermore, it has a total width exceeding the width of the interval, as it contains both, $t^*$ and $t^* + (t_{max} - t^*)\frac{1}{2}$.
Hence, we can take a copy of $B_1$, shift it such that it starts at $t^* + (t_{max} - t^*)/2$ and cover the total area of the interval $[t^* + (t_{max} - t^*)/2, t_{max}]$.
An example is given in Figure \ref{fig:temp-rep:dual-covering}.
Analogously, we can proceed with increasing $\ell$ in order to pick a path $P_\ell \in \mathcal{P}^\ell$ whose box $B_\ell$ covers the interval $[t^* + (t_{max} - t^*) 2^{-\ell}, t^* + (t_{max} - t^*) 2^{-\ell+1}]$.
As soon as $t^* + (t_{max} - t^*) 2^{-\ell} \leq \alpha^*_e$, we can stop as the remaining area in the interval $[t^*, t^* + \alpha^*_e]$ is at most $\alpha^*_{e}$.
The total area consumption of chosen boxes is bounded by $O(\log T) \alpha^*_e$ as the length of the remaining interval to be covered is divided by two in each step.
In summary, we have just shown that the total area covered by boxes in the interval $[t^*, t_{max}]$ can be bounded.
By symmetry, the same arguments also hold for the area covered by boxes in the interval $[t_{min}, t^*]$.
Here, we shift boxes to the left instead of to the right.
So the total area covered by boxes that intersect $t^*$ can be bounded by $2 \cdot O(\log T) \alpha^*_e$.
Recall $\mathcal{P} = \cup_{1 \leq i \leq k} \{ P \in \mathcal{P} : P \cap t^e_i \neq \emptyset \}$, hence,
$$vol \left(\bigcup_{\substack{P \in \mathcal{P} :\\ e \in P}} B^e_P \right)
\leq \sum_{i=1}^{k} vol \left(\bigcup_{\substack{P \in \mathcal{P}_e^{t_i}}} B^e_P \right) \leq \sum_{i=1}^k 2 \cdot O(\log T) \alpha^*_e = O(k \log T) \alpha^*_e$$
holds, which concludes the proof. \qed
\end{proof}
Since we restricted the preceding argumentation only to the case when $\Delta \in \{0,\infty\}$, we will now generalize it towards arbitrary delays in the proof of Theorem \ref{thm:temp-rep:delta-arbitrary}.
\begin{proof}[Proof of Theorem \ref{thm:temp-rep:delta-arbitrary}]
We begin the proof by generalizing the box model to general scenarios.
Let us fix a path $P$.
Again, we want to construct geometric objects $B^e_P$ such that $\sum_{e \in P} vol(B^e_P)$ equals the right hand side value of the dual constraint of path $P$ in (\ref{LP:TR-D}) (assuming that all terms involving $\beta$ are moved to the right hand side).
Therefore, let us start with the following boxes
\[ \hat{B}^e_P = \left[ \tau^{<e}(P), T - \tau^{\geq e}(P) \right] \times \left[ 0, \frac{\alpha^*_e}{\alpha^*(P)} \right]. \]
Let us also fix an edge $e \in P$ and order the scenarios $z_i, 1 \leq i \leq \ell$ with $\beta^*_{z_i} > 0$ such that $\bar{\Delta}_{z_i}(P) \geq \bar{\Delta}_{z_{i+1}}(P)$ for all $1 \leq i < \ell$.
For each $1 \leq i \leq \ell$, we cut out a section of box $B^e_P$ of height $\beta^*_{z_i}$ starting at the top right boundary of the box.
That is, for each scenario, define the following rectangles
\[ A^e_{z_i} = \left[ T - \tau^{\geq e}(P) - \bar{\Delta}_{z_i}(P), T - \tau^{\geq e}(P) \right] \times \left[ \left( 1 - \sum_{j=1}^i \beta^*_{z_j} \right) \frac{\alpha^*_e}{\alpha^*(P)}, \left( 1 - \sum_{j=1}^{i-1} \beta^*_{z_j} \right) \frac{\alpha^*_e}{\alpha^*(P)} \right] \]
and cut these out of $\hat{B}^e_P$.
So we end up with a polygon $B^e_P = \hat{B^e_P} \setminus \left( \bigcup_{i=1}^\ell A^e_{z_i} \right)$ (see the box shaped as a staircase in Figure \ref{fig:temp-rep:dual-box-model-delta-arb}).
Also note that the boxes $A$ do not overlap - except for their boundaries - and are always contained in $\hat{B^e_P}$.
Hence, it is simple to compute the total volume of the polygons of any path.
\begin{align*}
\sum_{e \in P} vol(B^e_P)
&= \sum_{e \in P} \left( \frac{\alpha^*_e}{\alpha^*(P)} \left( T - \tau(P) \right) - \sum_{i=1}^{\ell} vol(A_{z_i}) \right) \\
&= \sum_{e \in P} \left( \frac{\alpha^*_e}{\alpha^*(P)} \left( T - \tau(P) \right) - \sum_{i=1}^{\ell} \bar{\Delta}_{z_i}(P) \beta^*_{z_i} \frac{\alpha^*_e}{\alpha^*(P)} \right) \\
&= T - \tau(P) - \sum_{\substack{z \in \mathcal{S} : \\ z \cap P \neq \emptyset}} \bar{\Delta}_z(P) \beta^*_z.
\end{align*}
Analogously to (\ref{bounds:alpha-volume-bound}) we can deduce that $\alpha^*_e \geq vol(B^e_P)$ holds for all $P$ and $e \in P$, in other words, the volume of polygons is again a lower bound on the variables of the dual temporally repeated solution.
\begin{figure}[tb]
\centering
\resizebox{0.7\textwidth}{!}{
\begin{tikzpicture}[scale=1, every node/.style={scale=2}]
\draw[fill=white,draw=none,use as bounding box] (0,0.25) rectangle (12.5,6.85);
\begin{scope}
\draw[->] (0.5,1) -- (11,1);
\draw[->] (1,0.5) -- (1,6);
\node at (0.74,1.2) {$0$};
\draw[] (0.9,1.1) -- (1,1.1);
\node at (0.6,5.8) {$\alpha^t_{e}$};
\node at (11,0.6) {$t$};
\end{scope}
\begin{scope}
\draw[draw=darkblue,ultra thick] (2,1.1) -- (2,5) -- (4,5) -- (4,4) -- (5,4) -- (5,2) -- (7,2) -- (7,1.1) -- (2,1.1);
\draw[draw=red,dashed, ultra thick] (1.95,1.05) rectangle (7.05,5.05);
\node at (2.5,5.5) {$A_{z_1}$};
\node at (4.5,4.5) {$A_{z_2}$};
\node at (6,3) {$A_{z_3}$};
\transparent{0.4}
\draw[pattern=north east lines,pattern color=red] (5,2) rectangle (7,4);
\draw[pattern=north west lines,pattern color=darkblue] (4,4) rectangle (7,5);
\draw[pattern=vertical lines,pattern color=darkorange] (2,5) rectangle (7,6);
\transparent{1}
\draw[thick,black,decorate,decoration={brace,amplitude=8pt}] (7,6) -- (7,5) node[midway, right,xshift=+0.25cm]{$\beta_{z_1}(P) \frac{\alpha^*_e}{\alpha(P)}$};
\draw[thick,black,decorate,decoration={brace,amplitude=8pt}] (2,6) -- (7,6) node[midway, above,yshift=0cm]{$\bar{\Delta}_{z_1}(P)$};
\draw[thick,black,decorate,decoration={brace,amplitude=8pt}] (7,5) -- (7,4) node[midway, right,xshift=+0.25cm]{$\beta_{z_2}(P) \frac{\alpha^*_e}{\alpha(P)}$};
\draw[thick,black,decorate,decoration={brace,amplitude=8pt}] (4,5) -- (7,5) node[midway, above,yshift=0cm]{$\bar{\Delta}_{z_2}(P)$};
\draw[thick,black,decorate,decoration={brace,amplitude=8pt}] (7,4) -- (7,2) node[midway, right,xshift=+0.25cm]{$\beta_{z_3}(P) \frac{\alpha^*_e}{\alpha(P)}$};
\draw[thick,black,decorate,decoration={brace,amplitude=8pt}] (5,4) -- (7,4) node[midway, above,yshift=0cm]{$\bar{\Delta}_{z_3}(P)$};
\end{scope}
\end{tikzpicture}}
\caption{Illustration of the box model for arbitrary values of $\Delta$.
The boundary of the polygon $B^e_P$ is drawn in solid blue.
The corresponding rectangular box $\bar{B}^e_P$ from the proof of Theorem \ref{thm:temp-rep:delta-arbitrary} is drawn red, dashed.}
\label{fig:temp-rep:dual-box-model-delta-arb}
\end{figure}
Note that the preceding \emph{construction of a feasible solution for (\ref{LP:EX-D})} used only variables $\alpha_e^t(\emptyset)$.
Hence, the solution constructed therein is not only feasible if $\Delta_e \in \{0,\infty\}$, but also for general $\Delta_e \in \mathbb{Z}_+$ by the same arguments.
Fix a path $P$ and set $\alpha_e^t(\emptyset)$ to the height of polygon $B^e_P$ in the slice $(t,t+1)$.
This time, note that the height of a polygon corresponding to a specific path $P$ and an edge $e \in P$ may differ depending on the actual slice.
Then the solution is feasible as the height of polygons in slice $(t,t+1)$ is, by construction, equal to $\frac{\alpha^*_e}{\alpha^*(P)} \left( 1 - \sum_{z \in \scenariosDualPath{t}{P}} \beta^*_z \right)$.
It remains to prove that the total cost of the constructed solution $\alpha$ is bounded by the total cost of the temporally repeated solution times at most a factor $O(\eta k \log T)$.
Unfortunately, the staircase structure of $B^e_P$ prevents us from covering the area as in the proof of Theorem \ref{thm:temp-rep:k-log-T-gap}.
If we take a copy and shift it to the left, the area to the left of $t^*$ will not necessarily be covered.
Therefore, we will use the following strategy.
In a first step, we will replace all polygons $B^e_P$ from the temporally repeated solution by rectangles $\bar{B}^e_P$.
This will cost at most a factor $\eta$.
The polygons will be replaced in such a way that the height profile with respect to $B^e_P$ is contained in the height profile with respect to $\bar{B}^e_P$.
Afterwards, we can apply exactly the same proof as in Theorem \ref{thm:temp-rep:k-log-T-gap}.
The proof will be concluded.
The replacement is done as follows.
Fix a path $P$ and $e \in P$ and consider the polygon $B^e_P$.
We want to replace it by a rectangle $\bar{B}^e_P$ that contains the polygon.
Therefore, let $h = \frac{\alpha^*_e}{\alpha^*(P)} \left( 1 - \sum_{z \in \mathcal{S} : \bar{\Delta}_z(P) = T - \tau(P)} \beta^*_z \right)$ be the highest point of the polygon.
We will replace $B^e_P$ by the rectangle
$\bar{B}^e_P = \left[ \tau^{<e}(P), T - \tau^{\geq e}(P) \right] \times \left[ 0, h \right]$.
Clearly, it contains the polygon.
The factor we lose can be bounded as follows.
We use $\mathcal{S}_1 = \{z \in \mathcal{S} : \bar{\Delta}_z(P) = T - \tau(P) \}$ and $\mathcal{S}_2 = \mathcal{S} \setminus \mathcal{S}_1$ to partition the scenarios.
The first set contains all scenarios that destroy the entire flow on path $P$, the latter contains scenarios that destroy only a smaller amount.
\begin{align*}
\frac{vol(\bar{B}^e_P)}{vol(B^e_P)}
&= \frac{\frac{\alpha^*_e}{\alpha^*(P)} \left( T - \tau(P) \right) \left( 1 - \sum_{z \in \mathcal{S}_1} \beta^*_z \right)}{\frac{\alpha^*_e}{\alpha^*(P)} \left(T - \tau(P) - \sum_{z \in \mathcal{S}} \beta^*_z \bar{\Delta}_z(P) \right)} \\
&= \frac{\left( T - \tau(P) \right) \left( 1 - \sum_{z \in \mathcal{S}_1} \beta^*_z \right)}{ T - \tau(P) - \sum_{z \in \mathcal{S}_1} \beta^*_z \left(T - \tau(P) \right) - \sum_{z \in \mathcal{S}_2} \beta^*_z \bar{\Delta}_z(P)} \\
&= \frac{\left( T - \tau(P) \right) \left( 1 - \sum_{z \in \mathcal{S}_1} \beta^*_z \right)}{ \left(T - \tau(P) \right) \left( 1 - \sum_{z \in \mathcal{S}_1} \beta^*_z \right) - \sum_{z \in \mathcal{S}_2} \beta^*_z \bar{\Delta}_z(P)} \\
&\leq \frac{\left( T - \tau(P) \right) \left( 1 - \sum_{z \in \mathcal{S}_1} \beta^*_z \right)}{ \left(T - \tau(P) \right) \left( 1 - \sum_{z \in \mathcal{S}_1} \beta^*_z \right) - \max_{z \in \mathcal{S}_2}\{ \bar{\Delta}_z(P)\} \sum_{z \in \mathcal{S}_2} \beta^*_z} \\
&= \frac{T - \tau(P)}{ T - \tau(P) - \max_{z \in \mathcal{S}_2}\{ \bar{\Delta}_z(P)\} } \leq \eta.
\end{align*}
The last equality is due to $\sum_{z \in \mathcal{S}_2} \beta^*_z = 1 - \sum_{z \in \mathcal{S}_1} \beta^*_z$.
The proof is concluded since
\[opt(\ref{LP:EX-D}) \leq \sum_{e \in E}{u_e\sum_{0\leq t < T}{\sum_{z \in \mathcal{S}}{\alpha_e^t(z)}}} \leq \eta \sum_{e \in E}{u_e\sum_{0\leq t < T}{\sum_{z \in \mathcal{S}}{\bar{\alpha}_e^t(z)}}}\leq O(\eta k \log T) \cdot opt(\ref{LP:TR-D}).\] \qed
\end{proof}
\section{Open Problems}
\label{sec:conclusions}
In this work, we provided a first step towards modeling and solving robust flow over time problems.
We have shown that temporally repeated flows under the presence of uncertainty are no longer optimal.
We provided lower and upper bounds on the optimality gap.
Moreover, we have shown that the relation between delays, the time horizon and the longest path length has a strong impact on the complexity status.
We have shown that, for instances with $T$-bounded path length, an optimum temporally repeated solution can be computed in polynomial time.
Clearly, many interesting questions remain open.
We want to point out a few of these explicitly which may inspire follow-up work.
In our opinion, one of the biggest questions is clearly the complexity status of robust maximum flow over time if $\Gamma$ is bounded by a constant, even for $\Gamma = 1$.
Since the static counterpart is solvable in polynomial time for $\Gamma = 1$, it is interesting to see if the complexity status changes already due to the introduction of travel times.
The same question could be asked for temporally repeated solutions.
Although we were able to show that temporally repeated solutions can
be computed in polynomial time if the instance has $T$-bounded path
length, we were not able to provide insight for the case in which this
setting is not true.
A related question is the following.
Note that Proposition \ref{prop:temp-rep:LP-T-not} implies that, for $T$-bounded instances, there always exists an optimum temporally repeated solution which utilizes at most $2 |E|$ paths.
This is due to the number of constraints in the reformulated LP which we solve.
It would be interesting to understand if, for general instances or even for general solutions, there always exists an optimum solution which utilizes only a polynomial number of paths.
The LP formulations used in Section \ref{sec:bounds} contain an exponential number of variables and constraints.
Hence, it is unclear whether there always exists a basic feasible solution which is \emph{not} of exponential size.
If one could show that there are instances for which every optimum solution is of exponential size, one should also ask if the gap between polynomial size solutions and exponential size solutions can be bounded and if such solutions can also be computed in polynomial time.
The optimality gaps provided in Section \ref{sec:bounds} are not necessarily tight.
We have seen that for $k = \eta = 1$ in $T$-bounded instances, the lower bound from Proposition \ref{prop:temp-rep:log-gap} and the upper bound from Theorem \ref{thm:temp-rep:delta-arbitrary} coincide up to constant factors.
Apart from that, larger gaps remain.
It would be interesting to close these gaps, for example by constructing instances for $k$-coverable instances with $k > 1$ matching the upper bound from the Theorem.
This is especially interesting for instances with $\Delta \in \{0,\infty\}$ as discussed in Theorem \ref{thm:temp-rep:k-log-T-gap}.
Here, the gap between our lower and upper bounds is even bigger.
Moreover, it would be interesting to see if the dependency on $\eta$ in Theorem \ref{thm:temp-rep:delta-arbitrary} is necessary.
In the proof strategy we used, we were not able to remove it, although our lower bound instances always satisfied $\eta = 1$ and we were not able to construct gap instances with $\eta > 1$.
As mentioned before, the model studied here is a natural formulation for robust flow over time problems under uncertain travel times, using the well-established $\Gamma$-robustness model.
Due to the worst-case nature of the models introduced here, the resulting robust counterpart is quite restrictive and in general yields conservative solutions.
However, its study is interesting in order to understand robust flow over time problems.
Based upon the results obtained here, more advanced and potentially less conservative models could also be studied in the future.
\section*{Acknowledgments}
We thank the reviewers for their very careful reading of the manuscript and their valuable comments.
We thank the DFG for their support within Project B06 in CRC TRR 154.
\Urlmuskip=0mu plus 1mu
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\section{Introduction}
A useful formulation of gauge theories, both from the
conceptual and methodological point of view, is the one in terms
of gauge invariant excitations or string-like objects.
The so-called {\em P-representation} \cite{fg}, consisting
of a Hilbert space of path labeled states, has been used on the lattice
to perform analytical Hamiltonian
calculations. A cluster approximation allowed to provide
qualitatively good results for the $(2+1)$ QED \cite{af2d}
and the $(3+1)$ QED \cite{af3d} with staggered fermions.
A description in terms
of paths or strings, besides the general advantage of
only involving gauge invariant excitations, is appealing because
all the gauge invariant operators have a
simple geometrical meaning when
realized in the path space.
However, the computational method implemented, up to now, on a formally
{\em infinite} lattice, has the serious
drawback of the explosive proliferation of
clusters with the order of the approximation.
In order to tackle this difficulty we propose in this
paper to explore the previous method implemented now on a
{\em finite lattice}.
As a first test, we choose the simplest lattice
gauge theory with dynamical
fermions, the Schwinger model or (1+1) QED.
This massless model can be exactly solved in the continuum
and it is rich enough to share relevant features
with 4-dimensional QCD as confinement or
chiral symmetry breaking with an axial anomaly \cite{col76}.
For this reason it has been extensively used as a laboratory to
analyze the previous phenomena.
The lattice Schwinger model also become a popular
benchmark to test different techniques to handle
dynamical fermions \cite{hamer80}-\cite{hamer97}.
This article is organized into four sections. In section 2
we show the formulation of the model in the P-representation.
The electric and interaction components of the Hamiltonian
operator are realized in this basis of ``electromeson" states.
In section 3, first, we describe the finite lattice Hamiltonian
approach. Second, we show the calculation of
the ground-state energy, the
mass gap and chiral condensate.
These results are discussed in the concluding section.
\section{Schwinger Model in the Lattice P-Representation}
The P-representation offers a
gauge invariant description of physical
states in terms of kets $\mid P >$,
where $P$ labels a set of connected paths
$P_x^y$ with ends $x$ and $y$
in a lattice of spacing $a$.
In the continuum, the connection
between the P-representation and the ordinary
representation (``configuration" representation),
in terms of the fermion fields $\psi$ and the gauge
fields $U_\mu(x)=\exp [iea A_\mu(x)]$, can be performed
considering the natural gauge invariant object constructed
from them:
\begin{equation}
\Phi (P_x^y) = {\psi}^{\dagger} (x) U(P_x^y)\psi (y),
\end{equation}
where $U(P_x^y)=\exp [ie \prod_{\ell \in P} A_\ell]$
( $\ell \equiv (x, \mu)$ denote the links ).
The immediate problem we face is that $\Phi$
is not purely an object belonging
to the ``configuration'' basis because it
includes the canonical conjugate
momentum of $\psi$, ${\psi}^{\dagger}$.
The lattice offers a solution to this problem consisting in
the decomposition of the fermionic degrees
of freedom. Let us consider the
Hilbert space of kets $\mid {\psi}_u^{\dagger}
,{\psi}_d , A_{\mu} >$, where
$u$ corresponds to the $up$ part of the Dirac
spinor and $d$ to the $down$
part. Those kets are well defined in terms
of ``configuration'' variables (the
canonical conjugate momenta of ${\psi}_d$ and
${\psi}_u^\dagger$ are
${\psi}_d^\dagger$ and ${\psi}_u$ respectively.)
Then, the internal product of
one of such kets with one of the path dependent
representation (characterized
by a lattice path $P_x^y$ with ends $x$ and $y$) is given by
$$
\Phi (P_x^y) \equiv <P_{x;i}^{y;j} \mid {\psi}_u^{\dagger} ,
{\psi}_d, A_{\mu} >
$$
\begin{equation}
= {\psi}_{u;i}^{\dagger} (x) U(P_x^y) {\psi}_{d;j}(y),
\label{eq:Phi}
\end{equation}
where $i$ and $j$ denote a component of the
spinor $u$ and $d$ respectively.
Thus, it seems that the choice
of staggered fermions is the natural one
in order to build the lattice P-representation.
Therefore, the lattice
paths $P_x^y$ start in
sites $x$ of a given parity and end in sites
$y$ with opposite parity.
The one spinor component at each site
can be described in terms of the Susskind's
$\chi (x)$ single Grassmann fields \cite{sus}.
The path creation
operator $\hat{\Phi}_Q$ in the space of kets $\{ |\, P> \}$
of a path with ends $x$ and $y$ is defined as
\begin{equation}
\hat{\Phi}_Q= \hat{\chi}^{\dagger} (x) \hat{U}(Q_x^y)
\hat{\chi}(y).
\label{eq:Phiop}
\end{equation}
Its adjoint operator $\hat{\Phi}_Q^{\dagger}$ acts
in two possible ways \cite{fg}: annhilating the path $Q_x^y$
or joining two existing paths in $|\, P>$ one ending at
$x$ and the other starting at $y$.
The Schwinger Hamiltonian is given by
$$
\hat{H}=\frac{ae^2}{2}\hat{W}
$$
$$
\hat{W}=\hat{W}_E +\lambda \hat{W}_I \, , \;\;\;
\lambda =\frac{1}{a^2e^2} \, ,
$$
\begin{equation}
\hat{W}_E=\sum_\ell \hat{E}_\ell^2 \, ,
\label{eq:H}
\end{equation}
$$
\hat{W}_I=\sum_{\ell} (\hat{\Phi}_\ell +
\hat{\Phi}_\ell^{\dagger}) \hspace{3mm}
(\hat{\Phi}_\ell=\hat{\chi}^{\dagger}(x)
\hat{U}_n(x)\hat{\chi} ( x+ n) )
$$
where $x$ labels sites, $\ell \equiv (x, n)$ the spatial
links pointing along the spatial unit vector $n$,
$\hat{E}_\ell$ is the electric field operator,
the kets $|\, P \, >$ are eigenvectors of this operator
\begin{equation}
\hat{E}_\ell |\, P \, > = N_\ell (P) |\, P \, >,
\label{eq:E}
\end{equation}
where the eigenvalue $N_\ell(P)$ is the number of times that the
link $\ell$ appears in the set of paths $P$.
The $\hat{\Phi}_\ell$ are
``displacement" operators corresponding to the quantity
defined in (\ref{eq:Phi}) for the case of a one-link
path i.e. $P\equiv \ell$. The realization of both
Hamiltonian terms in this representation is
as follows \cite{fg}:
\vspace{3mm}
By (\ref{eq:E})
the action of the electric Hamiltonian is given by
\begin{equation}
\hat{W}_E\mid P >=\sum_\ell N_\ell^2(P)\mid P >.
\label{eq:We}
\end{equation}
The interaction term $\hat{W}_I$
is realized in the loop space as
\begin{equation}
\hat{W}_I\mid P >=\sum_{\ell}\epsilon (P,\ell)
\mid P\cdot \ell>
\label{eq:Wi}
\end{equation}
where
the factor $ \epsilon (P,\ell )$
is 0 or $\pm 1$ dictated by the algebra of the operators.
The different actions of operators
$\hat{\Phi}_\ell$ over path-states $|P(t)>$
and their corresponding $ \epsilon (P,\ell )$
are schematically summarized in FIG.1.
\begin{center}
\begin{figure}[t]
\hskip 1cm \psfig{figure=WId1.eps,height=5cm}
\vskip -6mm
\caption{A summary of the different actions of
operators $\Phi_\ell$, $\ell$ from $x$ to $y$,
applied over path-states $|\,P \,>$
with their respective $\epsilon$.
The original path is on the top.
The resulting paths $|P'>$ are plotted below: (a) represent
the addition of a disconnected link, (b) the union of 2 disconnected
pieces, (c) the separation of a connected piece into 2 pieces
and (d) the annihilation of a one-link path.}
\label{fig1}
\vskip -5mm
\end{figure}
\end{center}
\section{Finite Lattice Hamiltonian Method and Results}
Our method of calculation works assuming a lattice of
some fixed even number of sites $N$ and periodic boundary
conditions (PBC). Starting with the zero-path
state $|\, \emptyset \, >$ (infinite coupling vacuum), then
a collection of new states
$|\, P_i \, >$
are generated by applying successively the non-diagonal $W_I$
interaction Hamiltonian operator
-- whose action is to add or to eliminate links to to the path
$P_{i-1}$ as it was
described in the previous section --
up to order $K$.
The traslational symmetry can be exploited
in order to reduce the dimension of the space
tacking only one representative
$\bar{P}_\alpha$ of
each class of translationally equivalent paths $\{ P_i \}$.
The Hamiltonian matrix, with all the
transitions between the different states
$|\, \bar{P}_\alpha \, >$, is then built
for the scalar and vector sectors
and their eigenvalues $\omega_i$ are numerically evaluated.
In order to perform
the generation and recognition of
diagrams (the elementary lattice paths) as well as
the computation of transitions
between them,
we resorted to the PROLOG language which is very suitable
to carry out the symbolic manipulations.
\vspace{1mm}
The calculations of the ground-state energy, mass gap and
chiral order parameter were performed on lattices ranging from
size $N=2$ to $N=16$
and at least up to order $K=N$ in each case.
Our aim is to extrapolate these results to the continuum
limit: $N\rightarrow \infty$, $a \rightarrow 0$
($\lambda \rightarrow \infty$.)
It is clear from the plots (FIGS. 2 to 5) that the lattice
results show
convergence to the expected continuum values. This convergence
is, however, non-uniform and for $\lambda$ large enough
the plots show deviation from the continuum values
although the region of assimptotic regime
becomes larger when the size is increased. It is patent
that for a fixed lattice size $N$ the best results for the vacuum energy
and the chiral condensate are achieved for order $N-2$.
This appears to be the order at which the finite size effects
are minimized. This is not the case with the mass gap which
always gets closer to the continumm value when the order
increases.
\vspace{2mm}
{\em GROUND STATE ENERGY}
\vspace{2mm}
In the continuum limit the ground-state energy density
is known
exactly \cite{hamer80}:
\begin{equation}
\lim \frac{\omega_0}{2N\lambda}=-\frac{1}{\pi}=-0.3183.
\label{eq:E-exact}
\end{equation}
When the order increases $\omega_0/(2N\lambda)$ tends to a
fixed value. For a fixed size $N$ the closer value to
(\ref{eq:E-exact}) is given by order $N-2$. The value
for size $N=16$ and order $K=14$ at $\lambda =1000$ is
$\omega_0/(2N\lambda) =0.31844$,
so the discrepancy from the exact value is less than 0.05 \%.
The approximations converge with considerable rapidity.
FIG. 2 shows $\frac{\omega_0}{2N\lambda}$
for orders $K=1,2,\ldots ,14$ for $\lambda$ ranging from
0 to 100
on a lattice of size $N=14$.
\begin{center}
\begin{figure}[t]
\psfig{figure=e14.eps ,height=4.5cm ,width=8.6cm}
\vskip -13mm
\caption{The ground-state energy density
over $2\lambda^{1/2}$ for orders $K=1,2,\ldots ,14$
on a lattice of size $N=14$. }
\label{fig2}
\end{figure}
\end{center}
\vskip -10mm
In order to obtain a result in a consistent way we compute
the energy
for two large values of $\lambda$, for three correlative
large orders and for three correlative large sizes.
Then, for fixed size and order we first extrapolate
to $\lambda =\infty$ assuming
the behaviour $a+b/\lambda$. Second, for fixed size
we extrapolate to infinite order
assuming exponential dependence. Finally we extrapolate to
infinite size assuming exponential behaviour. The results
are given in TABLE 1.
The error using lattice sizes up to $N=16$ is 0.17\% .
\begin{table*}[hbt]
\setlength{\tabcolsep}{1.5pc}
\newlength{\digitwidth} \settowidth{\digitwidth}{\rm 0}
\catcode`?=\active \def?{\kern\digitwidth}
\caption{Ground state energy
for different lattice sizes and orders
at $\lambda =\infty$.}
\label{tab:gse}
\begin{tabular*}{\textwidth}{@{}l@{\extracolsep{\fill}}rrrr}
\hline
& \multicolumn{1}{r}{$N=12$}
& \multicolumn{1}{r}{$N=14$}
& \multicolumn{1}{r}{$N=16$}
& \multicolumn{1}{r}{$N=\infty$} \\
\hline
$K=N-4$ & $-0.317075$ & $-0.317723$ & $-0.318023$ & \\
\hline
$K=N-3$ & $-0.318010$ & $-0.318249$ & $-0.318348$ & \\
\hline
$K=N-2$ & $-0.318697$ & $-0.318656$ & $-0.318608$ & \\
\hline
$K=\infty$ & $-0.320611$ & $-0.320045$ & $-0.319660$ & $-0.318847$ \\
\hline
\multicolumn{5}{@{}p{120mm}}
{
\end{tabular*}
\end{table*}
\vspace{2mm}
{\em MASS GAP}
\vspace{2mm}
The mass gap for the massless continuum Schwinger model
can be computed exactly \cite{s62}:
\begin{equation}
\frac{M^c}{e^c}=\frac{1}{\pi^{1/2}}=0.564,
\label{eq:mg_c}
\end{equation}
The lattice mass gap is computed as:
\begin{equation}
\frac{M}{e}=\frac{\omega_1 -\omega_0}{2\sqrt{\lambda}}
\end{equation}
Comparing our results with those
of Crewther and Hamer \cite{hamer80} obtained by a similar
method, although they use a different representation (Jordan-Wigner
transformation), we find complete agreement for given values
of $N$ and $K$. When we reach larger $N$ we observe that the value
of the mass gap improves substantially. For instance, in FIG. 3
we show a plot of the mass gap for $N=10$ for several orders.
As it can be seen in the region $10<\lambda < 30$
, the mass gap values decrease
with the order and the size approaching the continuum result.
Given the non-uniformity of the convergence it is more
difficult to extrapolate to the limit although values
$\approx 0.579$ are obtained at the modest size of $N=8$.
\begin{center}
\begin{figure}[t]
\hskip 1cm \psfig{figure=m10.eps,width=8.6cm}
\caption{The mass gap over $2\lambda^{1/2}$ vs. $\lambda$
for orders $K=1,2,..,10$
on a lattice of size $N=10$ .}
\label{fig3}
\end{figure}
\end{center}
\vspace{3mm}
{\em CHIRAL ORDER PARAMETER}
\vspace{2mm}
An interesting quantity to compute is the vacuum expectation
of the chiral condensate per-lattice-site
$<\bar{\chi}\chi >$, defined as
\begin{equation}
\bar{\chi}\chi =\frac{1}{2N_s}\sum_x (-1)^{x}
[\hat{\chi}^\dagger
(\mbox{\bf x}),\hat{\chi}(\mbox{\bf x})],
\label{eq:chir-cond}
\end{equation}
where $N_s$ is the number of lattice sites.
The corresponding operator is realized in the
P-representation and thus we get for
the chiral condensate:
\begin{equation}
\bar{\chi}\chi |P>=
(\frac{1}{2}-\frac{2{\cal N}_P}{N_s})|P>,
\label{eq:chir-cond2}
\end{equation}
where ${\cal N}_P$ is the number of connected paths in $P$.
To compute
$<\bar{\chi}\chi >$ the $\hat{W}$ Hamiltonian is modifyed as
\begin{equation}
\hat{W}'=\hat{W}+\frac{\alpha}{2} \sum_x (-1)^{x}
[\hat{\chi}^\dagger
(\mbox{\bf x}),\hat{\chi}(\mbox{\bf x})],
\label{eq:W'}
\end{equation}
where $\alpha$ is an arbitrary parameter.
Thus, $<\bar{\chi}\chi >$ is obtained in the standard way as
\begin{equation}
<\bar{\chi}\chi >= \frac{\partial \omega '}
{\partial \alpha} |_{\alpha =0}.
\end{equation}
The massless continuum Schwinger model
undergoes a breaking of chiral symmetry with
\begin{equation}
<\bar{\psi} \psi>/e = \frac{e^\gamma}{2\pi^{3/2}}=0.15995,
\label{eq:chiral-cont}
\end{equation}
where $\gamma$ is the Euler constant.
This non-zero value of the chiral condensate is one of the main efects
of the axial anomaly.
In FIG. 4 we report the value of the chiral condensate
per-lattice-site for
lattice sizes ranging from $N=2$ to $N=10$.
FIG. 5 shows this chiral order parameter for different lattice
sizes up to order $K=N-2$ for each size.
\begin{center}
\begin{figure}[t]
\hskip 1cm \psfig{figure=q10.eps
,width=8.6cm}
\caption{The chiral condensate per-lattice-site
times $\lambda^{1/2}$ vs. $\lambda$
for orders $K=1,2,\ldots,10$
on a lattice of size $N=10$ .
}
\label{fig4}
\end{figure}
\end{center}
\begin{center}
\begin{figure}[t]
\hskip 1cm \psfig{figure=qtot.eps,width=8.6cm}
\caption{The chiral condensate per-lattice-site
times $\lambda^{1/2}$ vs. $\lambda$
on lattices of size $N=4,6,8,10$
for order $K=N-2$ respectively.}
\label{fig5}
\end{figure}
\end{center}
Notice that the results
in the weak coupling region converge to the corresponding continuum
value (\ref{eq:chiral-cont})
as long as $N$
increases while for a fixed $N$ the value improves with the
order $K$ till the value $K=N-2$ is reached.
\section{Conclusions and Final Remarks}
Our general proposal is to
to show that the P-representation
is a valuable and alternative computational tool
for gauge theories
with dynamical fermions. In particular, in this work,
we wanted to test the Hamiltonian approach on finite lattices.
With tis aim, we chose the simplest model: (1+1) QED.
This also enables us to compare with the corresponding numerical
simulations \cite{f97} using the Lagrangian counterpart of the
P-representation or the socalled
{\em worldsheet formulation} \cite{afg}. This comparison shows that
, for this case of one spatial dimension, the Hamiltonian method is less
time consuming.
The results are very good and confirm the belief of Hamer et al
\cite{hamer97} in obtaining with considerable
accuracy the observables working on
lattices of moderate size. Consequently, this procedure is appealing
because one can run all the needed computations in small machines
obtaining quite fair results.
Finally, we would like to stress (once more)
that our aim was not to present another
solution to the Schwinger model, but, to test an alternative general
approach to tackle dynamical fermions.
\vspace{15mm}
{\large \bf Acknowledgements}
|
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}
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A Site of Beautiful Resistance
GODS & RADICALS
Write With Us!
Editorial: Post-Truth Capitalism and Pre-Truth Revolution
January 13, 2017 February 5, 2017 by Rhyd Wildermuth, posted in Capitalism, Climate Change, Editorial, Uncategorized
You can't miss either of the two operative buzz-words being bandied around like so much meaningless noise in the last few months. The media rails about it, politicians whine over it, activists shout it: we're now in a 'post-truth' world, drowning in 'fake news.' The election of Trump was blamed on it, the rise of the alt/new/fascist right is a sign of it, and the Brexit vote in the United Kingdom happened because of it.
Really, though? Does no one care about truth any longer? Are lies suddenly masquerading as journalism? Is there some new scourge of deception and delusion sweeping across the Western world, making it impossible to tell what's really going on around us?
Nah. This isn't new. And it's not what we're told to think is happening, either.
Let's look at some news stories of this last week in the United States, shall we?
A few nights ago, Meryl Streep criticized Trump and called herself and other Hollywood actors 'the most vilified segment in America.'
Two days later, the new president of the United States engaged in this exchange with a reporter for a news company owned by the third largest media conglomerate in the world:
This was a day after it was reported there is video footage of Donald Trump paying Russian prostitutes to urinate on a bed where the Obama's liked to stay in Moscow.
There was also a report about US tanks entering Germany and a brief story about Russia's propaganda network interrupting live footage of US congressional speeches.
All these examples I mention were news stories, reporting on actual events which occurred. It's true that Meryl Streep gave that speech, it's true there are claims about that video, that Trump argued with that reporter. Tanks moved into Germany to protect against Russian invasion, and also C-Span's live footage was interrupted by Russian Television.
But in each case, truth was utterly irrelevant to the stories. Let's look at them all again.
Meryl Streep is currently worth $45 million dollars. While there are certainly some who probably think Hollywood is full of degenerate reprobates, unless she meant that rich people are the most vilified people, it's hard to imagine she wasn't just engaging in one of her award-winning performances.
The second of these stories is a bit more complex. Watch Trump's exchange with the reporter again if you can (I've watched in over thirty times now, it's so fascinating).
The reporter is the chief White House correspondent for CNN. A couple of things you probably already know about CNN: they're owned by the third largest media conglomerate in the world, are worth $10 billion dollars, likely turned a 1 billion dollar profit in 2016, and were the first media outlet to break a certain story Trump was understandably upset about.
That story, of course, was of the piss-video. Or, less sensational but potentially more-damning (c'mon, it's just piss) is the dossier which claims such a video exists. That document, by the way, is available to you on Buzzfeed, the same quality news site that keeps you up-to-date on J.K. Rowling's twitter feuds and quizzes about the Kardashians and whether or not you are devoted enough to chocolate. (I'm not, apparently).
Read the dossier if you like. You won't find the video (and probably wouldn't want to see it anyway) because no one's certain there actually is one. The dossier suggests it, but before you go trusting that, there's some stuff you should know about who wrote it.
It was written for a private intelligence firm by a private investigator originally commissioned by a rich Republican customer who wanted to stop Trump. But then, according to the New York Times:
the Republican interest in financing the effort ended. But Democratic supporters of Hillary Clinton were very interested, and Fusion GPS kept doing the same deep dives, but on behalf of new clients.
And from Glenn Greenwald at The Intercept (a reporter hated by both Democrats and Republicans alike for his whistleblowing activities:)
ALMOST IMMEDIATELY AFTER it was published, the farcical nature of the "dossier" manifested. Not only was its author anonymous, but he was paid by Democrats (and, before that, by Trump's GOP adversaries) to dig up dirt on Trump. Worse, he himself cited no evidence of any kind but instead relied on a string of other anonymous people in Russia he claims told him these things. Worse still, the document was filled with amateur errors.
So the dossier exists, but the tapes probably don't, and the whole thing is likely false and was paid for by people who wanted to prove that Trump is being manipulated by Russia.
Which brings us to the last two news stories. The report of the tanks rolling into Bremerhaven, Germany was short but chilling:
The deployment — which also includes 3,500 U.S. Troops — is to protect Eastern Europe against a potential Russian invasion.
In the dock area of the German city of Bremerhaven all around is American military hardware just off the boat — everything from Humvees to tanks. The official name for this display of military muscle is Operation Atlantic Resolve.
Its purpose is to reassure America's nervous European allies that the U.S. military will stand with them against any aggressive moves by Russia.
Sounds scary, huh? And it should be a bit scary. But what the report doesn't mention is that Operation Atlantic Resolve was initiated in 2014 and started its primary roll-out in April 2015. That is, the tanks rolling in to Germany from the US are definitely an escalation in military tensions, but not a spontaneous one. In fact, they happened before the Russians were accused of meddling in the US election, and might even help explain a Russian motive for hacking the pro-war Democratic campaign of Hillary Clinton.
The last story is the easiest to resolve. RT (which is, again, a Russian-funded media organisation fully doing the government's bidding, just like American news companies usually do) didn't hijack C-Span. According to C-Span, they themselves made the error.
Guardians at the Gate of Truth
IF YOU'RE feeling a bit dizzy with all this, don't worry. I'm done deconstructing news stories. But it's worth returning Meryl Streep's acceptance speech. After the bit about being vilified, just before talking about what a privilege it is to be the voice of empathy to the world, she urged everyone to support the Committee to Protect Journalists because, as she said,
"they'll need us to safeguard the truth."
Need who, though? Not actors. Don't get me wrong, I actually really like Meryl Streep. But I don't get my truth from her. And anyway, she was talking about the media.
But what truth is possible in such a world where both political parties pay private investigators to come up with a story about piss-play to stop Trump? What truth is possible in a world where a company worth $10 billion dollars is seen as a victim against another billionaire? That same news company, by the way, who made $1 billion partially due to election coverage and campaign advertisements? Truth probably isn't going to come from Buzzfeed either, though according to Dan Rather, Teen Vogue seems to be doing some cutting-edge reporting of late. (omg #couplegoals!)
All this is to suggest that yeah, we are in a fake-news, post-truth world. But the problem isn't Trump or the rise of the alt/new/fascist-right, or Russian meddling in elections. If anything, they're symptoms, and the real problem's not even new.
We've mostly been taught to think of news companies as some sort of independent check upon the government and corporations. They're supposed to investigate things, to bring stuff to light that the powerful don't want to see, report things to us, inform us.
It's a pretty story, sure, and it happens that way sometimes. And perhaps it happened more like that in the past, though with the Protocols of the Elders of Zion and the legacy of 'yellow journalism,' it's a bit hard to prove such a nostalgic idea is any less propagandistic than Trump's "Make America Great Again."
News shapes the way we see the world. We call it 'the media' as a shorthand (the news 'media': that is, newspapers, television, etc.) but it's more accurate to employ the other meaning of that world, 'middle.' News is the mediator between the world and our understanding of it, the narrative which shapes how we view politics and power. What it tells us about a story determines how we understand not just that story, but the sorts of people it reports on and what is relevant.
Need an example? Consider the relationship to race and crime in news reporting. Black suspects are almost always described by their race; white suspects very rarely are. Decades of news stories where a murder or rape suspect's race is only included in a story if they happen not to be white has the obvious affect of associating Blackness with criminality. Worse, because we are told to think of journalists as 'objective,' we tend to see the facts they report as objectively-selected facts. It's easy to forget that it's actually the reporter, and the editor, and the publisher who decide what's relevant to a story, not the story itself.
We naturally omit details we think are irrelevant and emphasize things we think are important. If you ask me what I'm doing right at this moment, I'll tell you that I'm writing an essay for God&Radicals. I wouldn't mention that I'm also waiting for tea water to boil or happen to be shirtless, because that seems irrelevant. But now that I've mentioned I'm shirtless, you might have just envisioned me as such while reading this.
The point, then, is that narrative is selective, and what gets included or excluded shapes the experience of truth. I'm shirtless, waiting for tea water to boil while writing an essay. I've just shaped how you experience me.
Expand that on a large scale, and throw in two things we very often forget about news. The first? Well, capitalism. CNN, Buzzfeed, the New York Times, etc. etc., they're capitalist enterprises. They need to make money. They are in the business of shaping narrative, telling you stories, giving you 'news' (or telling you how many times you'll get married according to your choices in cheese–in my case, three). To make money, they need your attention–they need you checking back, seeing them as reliable or entertaining, the place you look to when you want to find out about the world.
Capitalism isn't the whole story, though. Because news shapes how you see the world, because media outlets are the fastest way to get a narration out into the world, and because we have a desire to understand things, the media is in a position of immense power over our behavior. Advertising is an obvious example, but every facet of our relationships to government and each other is an open playground to their whims. As in the example of racialized crime reporting, journalists shape the way we see Black folk, or Muslims, or immigrants. But more so, they shape the way we relate to the government and to other countries. They often act in the service of the government, but always act in their own interest.
Whether or not Russia is really actively meddling in the political affairs of the United States is quite impossible to tell. What's more important is whether or not we think they are, and some political powers have more interest in us believing this than others. For a different example, consider the lead-up to the war in Iraq: there were no Weapons of Mass Destruction, but every capitalist media company in the United States reported the government's story as if there were.
Were they then and are they now merely trying to keep our attention? Or did the rich who ran those media companies have an interest in seeing the US go to war then and possibly go to war now? Are they being manipulated by the government, or are they trying to manipulate the government? Do some want us to go to war with Russia, or are they trying to fight off foreign intrusion?
These are questions we can't really answer, but that brings us anyway to an even more important question:
Why are we letting capitalists decide what's true for us?
Post-Truth Capitalism and Pre-Truth Revolution
MAYBE you're feeling what I've been feeling. It's like quakes shuddering through our pysches, the ground slipping beneath us. A friend described it as 'giants throwing rocks at each other.' When I was a kid, I watched my baby-sitter's boyfriend fight with another guy, and I hid with my sisters behind the couch until the fighting was over. It feels like that.
Something does seem to have happened to the truth, but it's not that it disappeared. The truth was never actually there in the first place, and we're only now just starting to see this. Everything we thought was solid seems to be melting into air. Everything we held sacred seems like it is being profaned.
There's a war for truth being fought, the same war that has always occurred between priests and kings. Who gets to decide what the people believe, who gets to hold ultimate power over the minds and souls of millions?
If it seems like this is a new war, it's probably that one side won for awhile. The truth was occupied, colonized, an imperial subject too beaten down to throw off its oppressors. But now? Now the empire's starting to crumble. The capitalists are fighting each other, political alliance against political alliance, media conglomerate against media conglomerate, government against government. Liberals or Conservatives, Russia or US, CNN or Breitbart, it's impossible to tell who's going to win, who will capture the throne of meaning and truth.
Maybe they'll all lose, and that's actually the best thing we could possibly hope for. In fact, this is the opening we need, the opportunity we've been waiting for, the potential for a revolutionary change in the entire realm of truth-creation.
While they fight each other for dominance over the truth, the rest of us can see more clearly how subjective truth really is. When news companies publish fake news and teen style magazines publish in-depth analysis, everything's gone into flux, the truth is slipping, going where it wants to go, and might just escape back into our hands.
Because in all those battles, certain things aren't said. None talk about the environment, climate collapse, extinction. There are natural limits to capitalism, and we've probably hit them. Dwindling resources, melting ice-caps, degraded soil, economic collapse–these are the truths we see in front of us, things those closest to the earth don't need a screen or smartphone to tell them. The truth is in front of us, under our feet, in the eyes of the panicked people around us.
Everything else is just distraction for the profit of the rich, the same people causing this crisis in the first place. Fortunately, they're pretty distracted themselves at the moment. They won't be for long, and they might even try their greatest weapon against us to hold onto truth–an actual war.
In capitalism's post-truth moment, our chance has arrived. The revolution is not yet a truth, but it can be. The same media who tells us it's impossible told us there were weapons of mass destruction in Iraq and a video of Trump covered in urine: they're losing their grip on our narrative. The same politicians who assure us that empire will last forever are fighting as we speak to keep their thrones from toppling under the weight of capitalist in-fighting.
Anarchists and Marxists both insist on seizing the means of production back from the capitalists. It's time to expand this: we must seize the production of meaning back from them, too.
By no longer believing their stories about the world while also creating our own. By ignoring their narrative while crafting a new one. Be it newspapers or books, radio shows or podcasts, we can must tell our stories against theirs, make ours more beautiful, more compelling, more intoxicating than their flashy yet shallow truths.
Most of all, we must refuse to take either side in the war the rich are fighting against each other. Neither Liberals nor Conservatives, neither the media nor the president, neither Russia or the United States. They depend on us to fight these wars for them, to take one side or another.
If we withdraw, they will have to fight these wars themselves, and while they're distracted, we'll make our own truth and build our own world without them.
Rhyd's a co-founder and the managing editor of Gods&Radicals. He writes here and at Paganarch, or you can also read about his sex life on Fur/Sweat/Flesh, or read his near-daily "Anarchist Thought of the Day" on Facebook. He lives nomadically, likes tea, and probably really likes you, too.
Like this piece? You will probably love our print and digital publications, including our journal A Beautiful Resistance and Christopher Scott Thompson's new book, Pagan Anarchism! Find out more here.
Tagged CNN, dossier, media, Meryl Streep, Piss-tape, Post-Truth, propaganda, Rhyd Wildermuth, RT, Russia, Trump
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10 thoughts on "Editorial: Post-Truth Capitalism and Pre-Truth Revolution"
whaleoilbeefoct says:
Reblogged this on whaleoilbeefoct and commented:
Better get the Ben Hur video out again Keith.
Prometheus Jones says:
Good Godds! I got seven times married on my cheese choices. What stats are they using for this analysis, this hardly seems legit. LOL. Great article in any case. It argues an excellent point about how propaganda often masquerades as truth and media ultimately have to be put to the bullshit test as there needs to be a questioning of the underlying assumptions and biases that modern media are subject to.
Patacelsus says:
An ounce of planning is worth a pound of cure, especially in such times of flux. Did anyone plan ahead, does anyone have a plan? I know I do, but my plans may not be a proper PLAN in the sense of ceasing the day and all that. There still seems to be an ennui in the populace, they still feel powerless. My plan is to replace that feeling with a drive to reclaim that power. It's slowly working, I can see unused wheels turning in people around me, in my life, but I am only one human.
Dr. Bones says:
That's how it starts.
I have my concerns for the larger movement as a whole. But since I have no aspirations to be a political commissar, I leave the matter to Eris and Fortuna.
Alley Valkyrie says:
Shit. I'm not devoted enough to chocolate either. 😦
lornasmithers says:
'We must seize the production of meaning back from them, too… By no longer believing their stories about the world while also creating our own. By ignoring their narrative while crafting a new one.' – YES!
TheMysticalGoddess says:
Reblogged this on The Mystical Goddess and commented:
Great read!! ❤
Vince Clortho says:
Beautifully written! It's like he was reading my mind, but using a much more eloquent vocabulary than my crass mind would have. It makes me feel all warm and shitty to know that there really are others out there who seem to really understand the depths to which the human race has truly sunk.
Igal says:
I am sorry, any grown up can understand and agree with you that the capitalist drama you describes is not ideal, that people who call Maryl Strip's speech as something admirable, all the while she is rich and has an interest, is of course obvious and part of the rightful criticism we should all have about the world, yet unfortunately there is no other way at this moment then democracy together with capitalism, which had brought a huge prosperity and age of relative peace to the world. I think China is key here, cause the Chinese system is creating a model, not yet there, that can be equivalent to democracy. They put a bigger emphasis on meritocracy and polls to see what the people need, something that exists in democracies, yet not enough .. we should learn from them and they should learn from us slowly and gradually, while countries like Russia need to be isolated as much as possible cause Russia is nothing but a totalitarian fascistic society with few things to offer to the society. Same goes for Islamic theocratic dogmatic brainwashing countries – these entities should be fought against, fought culturally, and be checked military if needed – cause their thinking poses a real threat to the world, while the democratic – Chinese ways, pose a gradual development of salvation to this world. A revolution which you "seek" will always be bloody and that is something that is worse then what we have now and should be avoided, though again I agree we need to teach people from all sides, liberal and conservative more and more septicemic and criticism and also activism.
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\section{Introduction}
Over the past few years, deep learning based approaches are became popular in video coding research \cite{lei2021deep,li2021high,liu2021fast}, among which convolution neural network-based methods have demonstrated impressive performance on enhancing quality of HEVC coded frames. A variable-filter-size CNN with residual learning (VR-CNN) is proposed in \cite{dai2017vrcnn} to remove HEVC compression artifacts. Frame-enhancement CNN (FE-CNN) proposed in \cite{li2018fecnn} employs long and short skip connections to improve frame quality. Multi-frame guided attention network (MGANet) \cite{meng2021robust} enhances video quality based on long-short-term frame dependency and coding unit boundaries. Content-aware CNN (CA-CNN) \cite{jia2019contentaware} contains multiple models and selects the most appropriate one to enhance each coding tree unit. Adaptive-switching network (ASN) \cite{lin2020partitionaware} switches to different enhancement models according to frame content. A network adopting recursive design and residual learning (RR-CNN) is proposed in \cite{zhang2020rrcnn} as HEVC intra frame post-processing filter. Video coding prediction modes and units partition map are utilized as side information in frame-wise quality enhancement CNN (FQE-CNN) \cite{huang2021fqecnn} to boost quality enhancement of HEVC coded frames.
Different from other image restoration tasks, the raw data is available at the encoder side in image and video coding task. However, these existing CNN-based methods neglect employing it to enhance coded frame quality. In this work, we use the raw frame at encoder as ground-truth to online train OL-DN to overfit to this frame. Then, we transmit the new parameters to update the OL-DN at decoder. Therefore, the model gains high enhancement performance, as well as being adoptable to various coding scenarios. Moreover, we design a lightweight adaptive layers (AL) as the only online trainable part to reduce online training complexity. AL implements channel attention similarly to SE block \cite{hu2019senet}, but with less computation complexity. In OL-DN, we combine wide activation \cite{fan2018wdsr} and channel attention mechanism to design the wide block (WB) as an effective feature extraction block. And furthermore, we use AL to implement channel attention in WB to obtain online learning based wide block (OL-WB).
Besides, since quantization error in discrete cosine transform (DCT) coefficients is the root cause of various quality degradation in HEVC, Guo et al. \cite{guo2016ddcn} and Zhang et al. \cite{zhang2018dmcnn} introduce the priors of DCT coefficients as side information to remove the JPEG artifacts. Nevertheless, the strength of CNN lies in learning the inter dependencies between adjacent elements, while DCT coefficients have weak correlation with adjacent ones. Therefore the frequency coefficients are sub-optimal for CNN learning and frequency priors are not efficiently extracted in those methods. Therefore, we propose a DCT-based convolution layer (DCT-conv), which clusters DCT coefficients of the same frequency spectrum into one channel, and remains their relative positional relationship in spatial domain consistent with that in frequency domain. DCT-conv strengthens the frequency coefficients correlation and makes them more compatible with CNN learning. Consequently, our method can effectively extract frequency priors to improve quality enhancement performance.
The coding information was also adopted in the CNN-based quality enhancement solution. The coding unit mask was defined in \cite{huang2021fqecnn} as the prior information for the CNN-based post-processing. Also, the transform unit partition map was employed in \cite{meng2021robust} to improve the quality of video frames. Besides the coding information, the channel correlation between the luma image (Y) and the chrom images (U and V) may also be utilized to improve the quality of compressed images.However, most of the existing CNN-based methods process each channel independently, which ignores the channel correlation and cannot achieve a high efficiency. In this work, we design the deep network in which we use the luma component as the guidance to restore the chroma component in both spatial and frequency domains.
\begin{figure*}[t]
\vspace{-1em}
\centering
\includegraphics[width=0.7\textheight]{figures/network.pdf}
{\small{\caption{The framework of our proposed OL-DN. Green layer indicates 3$\times$3 convolution layer with $n$ output channels.}\label{f:network}}}
\vspace{-.5em}
\end{figure*}
\begin{figure}[h]
\vspace{-.em}
\centering
\includegraphics[width=0.35\textheight]{figures/olearn.pdf}
\vspace{-1.em}
{\small{\caption{The flow chart of our online learning method.}\label{f:olearn}}}
\end{figure}
\section{Proposed Method}
\subsection{Online Learning}
\label{sec:ol}
Our proposed online learning method updates the OL-DN to gain performance improvement, and the its procedure is summarized in Fig. \ref{f:olearn}. Specifically, an OL-DN model is offline trained as the baseline model and it is arranged at both encoder and decoder sides. During the practical coding, OL-DN is online trained at the encoder side with raw data as the ground-truth image. Then, the online trained model parameters are coded via Huffman coding and transmitted to decoder side. Finally, the received parameters update the baseline model at decoder side. Note that to limit the extra encoder complexity and bit-rate for online learning, only the parameters of the designed adaptive layer (AL) are updated during online learning, and only the residual of parameters between the baseline model and online trained model is transmitted.
Online learning enables the model to learn the mapping from input to desired output directly, bringing considerable performance improvement to OL-DN. Therefore, OL-DN does not require either extremely deep network or multiple offline trained models to can outperform previous methods.
\subsection{Network Architecture of OL-DN}
\noindent
\textbf{Overview:} The architecture of OL-DN is illustrated in Fig. \ref{f:network}. It takes luma and chroma inputs, and outputs enhanced chroma image. OL-DN mainly consists of wide block (WB), online learning-based wide block (OL-WB), and DCT-based convolution layer (DCT-conv). We employ WB to extract input features and reconstruct output image.
In reconstruction process, we insert adaptive layer (AL) in WB to obtain OL-WB, allowing the network online learnable. The DCT-conv converts image from pixel domain into frequency domain, and IDCT-conv conducts the inverse process. In OL-DN, the pixel shuffle and unshuffle layers are utilized to reduce computation complexity\cite{shi2016pixelshuffle}. Moreover, to avoid the exploding and vanishing gradient problems we apply skip connections \cite{he2016residual} between inputs and outputs of OL-DN, WB, and OL-DB.
\noindent
\textbf{Wide block:} WB contains two wide-activated convolution layers and a channel attention block (CAB). We expand WB input channels before ReLU activation, to achieve better model performance for given computation complexity\cite{fan2018wdsr}.
Then, CAB implements channel attention mechanism to boost the learning ability of network \cite{hu2019senet}. A CAB input $\in \mathbb{R} {^{N \times H \times W}}$ is globally averaged to a 1-D vector and passes through two fully connected layers and sigmoid activation function, then it re-calibrates the channel weights of CAB input via multiplication, producing the CAB output.
Let ${conv}^k_n$ denote the convolutional layer with $k \times k$ kernel and $n$ output channels. With input $x$, the output of WB is obtained by
\begin{equation}\label{Eq.wb}
y = x + CAB\left( {conv_{64}^3({\mathop{\rm Re}\nolimits} LU(conv_{256}^3(x)))} \right)
\end{equation}
\noindent
\textbf{Online learning-based wide block:}
We substitute AL for CAB in WB to obtain OL-WB. Our proposed AL consists of $N$ weighting parameters $\{w_1,\ldots,w_N\}$, which are corresponding to $N$ channels of the input $X$, where $X=\{x_1,\ldots,x_N\}$ and $x_i$ is the $i$-th channel of $X$. The output of AL is obtained by $AL(X) = \{ x_1 \cdot w_1,\ldots,x_N \cdot w_N \}$. Its lightweight structure guarantees low time complexity for online learning, and low bit-rate for transmitting the parameter difference. Since the neural network channel weights might be similar at near depths, we employ OL-WB at intervals to maximize the efficiency of online learning.
Compared to CAB, AL also implements channel attention via channel-wise multiplication but its re-calibration parameters are directly online learned from the current input. Thus, it can achieve more accurate re-calibration for channels to improve the learning ability of the network. In addition, it can reduce computational complexity for the model.
\begin{figure}[h]
\vspace{-.em}
\centering
\includegraphics[width=0.35\textheight]{figures/DCT-conv.pdf}
\vspace{-1.em}
{\small{\caption{The process of DCT-conv converting image into frequency domain.}\label{f:dct}}}
\vspace{-.5em}
\end{figure}
\noindent
\textbf{DCT-based convolution layer:} DCT-conv converts image from pixel domain to frequency domain and IDCT-conv accomplishes the inverse process. More specifically, DCT-conv consists $N^2$ kernels sliding over input image in a non-overlapped manner, with each kernel corresponding to a frequency spectrum. To obtain a DCT coefficient $y_{(u,v)}$, the corresponding DCT-conv kernel $F_{(u,v)}$ contains $N^2$ weights
\begin{equation}\label{Eq.dct}\scalebox{1.0}
{$
{w_{(i,j)}} = c(u)c(v)\cos \left( {\frac{{\left( {i + 0.5} \right)\pi }}{N}u} \right)\cos \left( {\frac{{\left( {j + 0.5} \right)\pi }}{N}v} \right)
$}
\end{equation}
\noindent
where $u,v,i,j \in \{0,...,N-1\}$, and $c(u)={\sqrt {1/N}}$ if $u=0$ otherwise $c(u)={\sqrt {2/N}}$. Moreover, $N=8$ in this work. The process of DCT-conv is illustrated in Fig. \ref{f:dct}. For a given image $x\in \mathbb{R} {^{1 \times H \times W}}$, the DCT-conv outputs $y\in \mathbb{R} {^{N^2 \times (H/N) \times (W/N)}}$. Each $N\times N$ block in $x$ is converted into $N^2$ DCT coefficients in output channels from $y_{(0,0)}$ to $y_{(N-1,N-1)}$, representing frequencies from low to high. The corresponding pixel blocks and frequency coefficients are marked with the same color in Fig. \ref{f:dct}.
In conventional block-wise DCT, adjacent coefficients represent amplitudes of different frequency spectrum, so they are weakly correlated especially at block boundaries. In contrast, our proposed DCT-conv clusters coefficients of the same frequency spectrum into one channel, and adjacent coefficients also correspond to block neighbors in spatial domain. Therefore, we have strengthen elements correlation to match CNN's ability in learning local relationship, allowing the model to effectively extract frequency priors to improve quality enhancement performance.
\noindent
\textbf{Y guidance:}In color video compression, the input frame is firstly converted from RGB to YUV domain before encoding, The conversion matrix under YUV-420 format is as below
\begin{equation}\label{Eq.yuv}
\scalebox{.95}{$
\!\begin{bmatrix}
Y\\U\\V
\end{bmatrix}\!=\!
\begin{bmatrix}\!
0.2126&0.7152&0.0722\\-0.1146&-0.3854&0.5000\\0.5000&-0.4542&-0.0458
\!\end{bmatrix}\! \!
\begin{bmatrix}\!
R\\G\\B
\!\end{bmatrix}\!+\begin{bmatrix}\!
16\\128\\128
\!\end{bmatrix}\!.$}
\end{equation}
\noindent
It is observed that although luma and chroma components are obtained via different weighting factors, they are still highly correlated. Moreover, luma contains clearer textures compared with the down-sampled chroma component, so luma can potentially provide high-quality structure information to improve the chroma quality. Consequently, we extract its features to guide chroma quality enhancement.
In both spatial and frequency domain, we employ 3$\times$3 convolution layers and WBs to extract luma features and aggregate with chroma features via element-wise addition. Then, the aggregated spatial and frequency domain features are concatenated in channel-wise to produce the dual-domain Y-guided chroma feature, which is used to reconstruct the final output.
\section{Experimental Results}
\subsection{Implementation Details}
To form the training set, we compress Flickr2K \cite{timofte2017ntire} images by HM-16.7 with YUV-420 format under the all intra (AI) configuration, where the quantization parameter (QP) is 27. Then, we crop Y and U components into 64$\times$64 and 32$\times$32 small patches, respectively. Moreover, 15 videos from Common-Test Dataset for HEVC \cite{bossen2013common} are compressed under AI configuration with QP = [22,27,32,37] to generate the test set. We only train one baseline model and apply it on both U/V components and 4 QPs. Our method is implemented with PyTorch and NVIDIA GTX 2080Ti GPU. The baseline network is offline trained for 20 epochs with the batch size of 64 and learning rate of 1e-4.
\begin{table*}[t]
\vspace{-3em}
\renewcommand\arraystretch{1.5}
\centering
\fontsize{7}{7}\selectfont
\caption{BD-rate (\%) savings for HEVC intra chrominance frames (U/V)}
\label{table:bdrate}
\vspace{-1em}
{
\begin{tabular} {c | l | cc | cc | cc | cc | cc}
\Xhline{1.2pt}
\hline\hline
Resolution
&Sequence
&\multicolumn{2}{c|}{VR-CNN\cite{dai2017vrcnn}}
&\multicolumn{2}{c|}{FE-CNN\cite{li2018fecnn}}
&\multicolumn{2}{c|}{RR-CNN\cite{zhang2020rrcnn}}
&\multicolumn{2}{c|}{FQE-CNN\cite{huang2021fqecnn}}
&\multicolumn{2}{c}{OL-DN (Ours)}\\
\hline
\multirow{2}*{2560$\times$1600}
&A1 Traffic& -3.5& -4.1& -4.2& -5.8& -11.4& -15.4& -14.1& -17.1& \textbf{-28.3}& \textbf{-38.6}\\
&A2 PeopleOnStreet& -5.9& -5.7& -8.2& -8.6& -33.3& -29.8& -30.3& -29.5& \textbf{-46.2}& \textbf{-38.5}\\
\hline
\multirow{5}*{1920$\times$1080}
&B1 Kimono& -1.5& -1.4& -5.0& -5.2& -20.2& -8.0& -24.0& -11.6& \textbf{-32.5}& \textbf{-38.5}\\
&B2 ParkScene& -3.3& -2.5& -4.4& -4.1& -30.9& -6.9& -27.2& -11.1& \textbf{-40.1}& \textbf{-46.1}\\
&B3 Cactus& -3.9& -6.3& -5.5& -10.7& -9.2& -11.7& -14.6& -22.5& \textbf{-27.4}& \textbf{-37.1}\\
&B4 BQTerrace& -3.7& -3.0& -5.3& -6.4& -15.9& -16.4& -21.6& -33.0& \textbf{-34.2}& \textbf{-41.5}\\
&B5 BasketballDrive& -3.3& -5.3& -10.8& -12.6& -23.3& -26.7& -25.8& -30.4& \textbf{-38.4}& \textbf{-38.3}\\
\hline
\multirow{4}*{832$\times$480}
&C1 RaceHorses& -6.7& -11.0& -8.4& -12.5& -16.5& -23.2& -15.2& -23.4& \textbf{-21.1}& \textbf{-31.3}\\
&C2 BQMall& -5.3& -5.3& -6.9& -7.6& \textbf{-29.3}& -32.3& -28.8& \textbf{-33.0}& -27.7& -27.7\\
&C3 PartyScene& -4.4& -4.4& -5.4& -5.7& -17.8& -21.4& -18.9& -23.2& \textbf{-26.4}& \textbf{-24.1}\\
&C4 BasketballDrill& -5.8& -6.8& -12.2& -14.9& -26.7& -31.8& -34.6& -41.4& \textbf{-41.4}& \textbf{-44.6}\\
\hline
\multirow{4}*{416$\times$240}
&D1 RaceHorses& -8.5& -11.5& -9.8& -12.8& \textbf{-28.9}& \textbf{-33.7}& -26.9& -32.9& -27.0& -31.0\\
&D2 BQSquare& -4.2& -6.4& -3.8& -6.8& -26.6& -26.5& -22.1& -29.2& \textbf{-26.9}& \textbf{-37.7}\\
&D3 BlowingBubbles& -8.4& -7.9& -8.5& -9.0& -19.7& -23.6& -25.2& -27.7& \textbf{-25.5}& \textbf{-30.5}\\
&D4 BasketballPass& -4.4& -6.5& -8.2& -10.3& -27.1& -29.7& \textbf{-31.9}& \textbf{-30.1}& -31.1& -28.3\\
\hline
2560$\times$1600 &class A &-4.7& -4.9 &-6.2& -7.2 &-22.3& -22.6 &-22.2& -23.3 &\textbf{-37.3}& \textbf{-38.6}\\
1920$\times$1080 &class B &-3.1& -3.7 &-6.2& -7.8 &-19.9& -13.9 &-22.6& -21.7 &\textbf{-34.7}& \textbf{-40.3}\\
832$\times$480 &class C &-5.6& -6.9 &-8.2& -10.2 &-22.6& -27.2 &-24.4& -30.2 &\textbf{-29.1}& \textbf{-31.9}\\
416$\times$240 &class D &-6.4& -8.1 &-7.6& -9.7 &-25.6& -28.4 &-26.5& -30.0 &\textbf{-27.7}& \textbf{-31.9}\\
\hline
\multicolumn{2}{c|}{\textbf{Overall Average}} &-5.0& -5.9 &-7.1& -8.7 &22.6& -23.0 &-23.9& -26.3 &\textbf{-32.2}& \textbf{-35.7}\\
\hline\hline
\Xhline{1.2pt}
\end{tabular}}
\vspace{-1em}
\end{table*}
\begin{table}[h]
\renewcommand\arraystretch{1.4}
\centering
\fontsize{7}{7}\selectfont
{\caption{BD-rate(\%) results of ablation study on U/V components}\label{table:ablation}}
\vspace{-1em}
{
\begin{tabular} {c c c c c}
\Xhline{1.2pt}
\hline\hline
\multicolumn{1}{c}{Class}
&OL-DN(w/o O)
&OL-DN(w/o F)
&OL-DN(w/o Y)
&OL-DN \\
\hline
A & -21.5 / -25.0 & -34.2 / -34.7 & -11.8 / -11.9 & -37.3 / -38.6 \\
B & -16.9 / -16.2 & -32.0 / -34.2 & -11.0 / -16.3 & -34.7 / -40.3 \\
C & -20.4 / -19.3 & -27.8 / -30.0 & -13.1 / -16.3 & -29.1 / -31.9 \\
D & -18.7 / -17.9 & -26.6 / -30.9 & -13.0 / -15.0 & -27.7 / -31.9 \\
\hline
Average & -19.4 / -19.6 & -30.2 / -32.5 & -12.2 / -14.9 & -32.2 / -35.7 \\
\hline\hline
\Xhline{1.2pt}
\end{tabular}}
\vspace{-1em}
\end{table}
\begin{table}[h]
\renewcommand\arraystretch{1.4}
\centering
\fontsize{7}{7}\selectfont
{\caption{Comparison of Increased Time $\Delta t$ at Encoder Side}\label{t.oltime}}
\vspace{-1em}{
\begin{tabular} {c c c c}
\Xhline{1.2pt}
\hline\hline
Class
&RR-CNN\cite{zhang2020rrcnn}
&FQE-CNN\cite{huang2021fqecnn}
&OL-DN\\
\hline
A &18.00\%&22.57\%&5.21\%\\
B &19.00\%&23.48\% &3.60\% \\
C &18.00\%&19.77\% &2.80\% \\
D &39.00\% &19.91\% &4.90\% \\
\hline
\textbf{Average} &23.50\% &21.43\% &4.13\% \\
\hline\hline
\Xhline{1.2pt} \end{tabular}}
\vspace{-1em}
\end{table}
\begin{table}[h]
\renewcommand\arraystretch{1.5}
\centering
\fontsize{7}{7}\selectfont
{\caption{Comparison of Complexity at Decoder Side}\label{t.dectime}}
\vspace{-1em}
{
\begin{tabular} {c c c | c c}
\Xhline{1.2pt}
\hline\hline
&\multicolumn{2}{c|}{Decoding Time (ms)}&\multicolumn{2}{c}{Increased Decoding Time Ratio $\Delta t$}\\
\hline
class
&FQE-CNN\cite{huang2021fqecnn}
&OL-DN
&RR-CNN\cite{zhang2020rrcnn}
&OL-DN
\\
\hline
A &565&337&2947\% &24\%\\
B &285&177&2947\% &29\%\\
C &55&42&1667\% &22\%\\
D &15&28&1478\% &56\%\\
\hline
Average&230&146&1825\% &33\%\\
\hline\hline
\Xhline{1.2pt}
\end{tabular}}
\vspace{-1em}
\end{table}
\subsection{Comparison of R-D performance}
Compared with other methods, the R-D performance in terms of BD-rate (\%) is given Table \ref{table:bdrate}. According Table \ref{table:bdrate}, it is found that our method outperforms the other methods for each video class, and achieves -32.2\% and -35.7\% (on average) for U and V components. Moreover, one can see that OL-DN brings more coding gains at high-resolution classes. Because high-resolution frames offer more pixel information for online training, thus improving model performance. Also, they are at larger bit rate so OL-DN extra bits occupy less portion.
\begin{figure}[h]
\vspace{-.5em}
\centering
\includegraphics[width=0.35\textheight]{figures/visualResults.pdf}
\vspace{-.5em}
{\small{\caption{Visual results of \textit{Cactus\_1920x1080\_50} at QP=37. The first and second rows show Cb and Cr components, respectively. (a) raw frame. (b): compressed frame. (c): our results.}\label{f:visualresults}}}
\vspace{-2em}
\end{figure}
\subsection{Visual Results Comparison}
Fig. \ref{f:visualresults} shows the visual results of our method enhancing HEVC compressed frames, and it is seen that our method can offer authentic textures after quality enhancement.
\subsection{Ablation Study}
In this ablation study, we verify the effectiveness of proposed method via removing correspondent module of OL-DN, including removing online learning (w/o O), frequency domain priors (w/o F), and Y component guidance (w/o Y). The results are listed in Table \ref{table:ablation}, and they demonstrate that our methods are effective in improving quality enhancement performance.
\subsection{Comparison of Complexity}
Table \ref{t.oltime} and Table \ref{t.dectime} show the running time at encoder side and decoder side, respectively. The relative time is defined as $\Delta t = t'/t$, where $t'$ is neural network running time and $t$ is HEVC running time. The results indicate that OL-DN has lower time complexity compared with other methods.
\section{Conclusion}
In this paper we propose a quality enhancement network for HEVC chroma component termed as OL-DN. It online updates the model parameters to gain performance improvement. Also, we design a DCT-conv to efficiently utilize frequency priors to assist quality enhancement. Furthermore, we extract Y component feature and aggregate it with chroma feature to guide chroma component reconstruction.
Experimental results demonstrate that this network has achieved superior performance compared with other state-of-the-art methods. Moreover, running time analysis has verified that OL-DN has acceptable complexity at both encoder and decoder sides.
\bibliographystyle{IEEEbib}
|
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Părul reprezintă totalitatea firelor subțiri, producții de origine epidermică și de natură cornoasă care cresc pe tegumentul mamiferelor și denumite științific foliculi piloși. Ele au rol protector. Firul de păr, ca și unghia, are o compoziție dură de proteine numit cheratină.
Legături externe
Misterioasele bucle ale evolutiei, 20 iunie 2007, Cecilia Stroe, Descoperă
Cauzele cele mai comune ale caderii parului, 21 aprilie 2013, Iulian Voicu, Ziare.com
Păr
|
{
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Q: unordered_set: is pointer address a good hash? i want to store a set of (smart) pointers in a hash set, either <boost/unordered_set>. After 10 seconds of thought, i came up with this hash function:
typedef boost::shared_ptr<myType> ref_t;
struct SharedPtrHash : public std::unary_function<ref_t, std::size_t> {
std::size_t operator()(ref_t const& obj) const {
return reinterpret_cast<std::size_t>( obj.get() );
}
};
My question is: is this hash a good idea? i'm entertaining the thought that this hash will have zero or very few collisions (maybe there is some prime-number modulus under the hood spoiling all my fun).
Further Details on purpose: The purpose of the hash is for recycling storage of big objects, so i need a fast way to detect if a big object is already in the bin.
in case it is not, what would be an ideal hash for pointers, either smart or dumb ones?
A: If you want to detect objects that are not identical even though their contents might be equal, you have no choice but to use the address of the object in the hash. The only question is whether to use the address directly or to run it through a formula. Dividing by sizeof(mytype) would tighten up the holes in the distribution.
Edit: Here's an untested template implementation that should work with all shared_ptr types, along with an equal_to function to complete the requirements for std::unordered_set. Don't use this generic implementation if you have other objects that require a hash based on the value instead of the pointer.
template<typename T>
size_t hash(const std::shared_ptr<T> & ptr)
{
return ((size_t) ptr.get()) / sizeof(T);
}
template<typename T>
bool equal_to(const std::shared_ptr<T> & left, const std::shared_ptr<T> & right)
{
return left.get() == right.get();
}
A: The following code compiles perfectly (GCC 4.7, Boost 1.47):
#include <boost/unordered_set.hpp>
#include <boost/shared_ptr.hpp>
struct Foo { };
int main()
{
boost::unordered_set<boost::shared_ptr<int>> s;
boost::shared_ptr<int> pi(new int);
s.insert(pi);
boost::unordered_set<boost::shared_ptr<Foo>> t;
boost::shared_ptr<Foo> pf(new Foo);
t.insert(pf);
}
A: The default Boost.Hash hash function for integral types is the identity function, so I don't think doing the same for pointers is a bad idea. It would have the same collision ratio.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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Up close and personal in travel retail – Sponsored by Victorinox
The Moodie Davitt Report
Hitting new heights in Seoul at Lotte World Tower
by Martin Moodie | August 27, 2015 August 27, 2015
Martin Moodie
Martin Moodie is the Founder & Chairman of The Moodie Report.
Latest posts by Martin Moodie (see all)
How Caelumest Ginis and Ginger Vitis got Blitzed then Bloomed - January 16, 2021
Catching up with old friends for a last tango and the perfect re-torte - January 13, 2021
Marking a grim anniversary and rediscovering the industry of beautiful memories - January 11, 2021
"Are you afraid of heights?" Lotte Duty Free's delightful Communication Team Senior Assistant Sunju Park (above right) asked me with a smile as wide as the Hangang river as we clambered into the steel cage lift.
"No, I'm fine," I lied, deciding that any mention of the fact that I get vertigo from standing on a kitchen stool would not be helpful to her or me at this point.
We're not talking just any old heights here – we're talking 85 floors or 356 metres above ground in the South Korean capital of Seoul.
Sunju had kindly arranged a VIP tour for me of the Lotte World Tower, still under construction and set to become one of the world's tallest buildings (with the second-highest viewing point) and greatest landmarks when it opens late next year. When completed it will be 123 floors or 555 metres high, towering above this sprawling city like a giant rocket in launch mode.
As the cage door closed, I realised we would be ascending up the side of the building with an eagle's view of the ground below and the city beyond through the gaps in the tightly meshed steel construction. Quickly, very quickly, I began to see Seoul in an entirely different light from anything I had experienced in 25 years of coming here.
I suspect the meshed construction of the cage has been carefully thought out. The dreaded feelings of spinning, swaying and wooziness that you get with vertigo seemed to be absent. I tried to only look out but then couldn't help but look down…………
Wow! Down and down and then down again. I felt like I was on a Korean Air flight. My goodness, we were high and moving higher. It was only when I looked over at the Tower building itself that I noticed how fast we were rising, the various storeys rushing by. The cars on the streets below had suddenly transformed into tiny model toys, while the vastness of the city stretched out all around us into what felt like an infinite beyond.
Sunju, who hadn't experienced this trip before, was laughing with the exhiliration of it all. And despite myself, so was I. This was unforgettable.
The lift stopped. We had reached level 79. "We're going higher," said Sunju as we walked across a metal plate and into the building itself. In future you'll be able to see the same amazing views as a Lotte Hotel will operate on floors 72 through 101. We got onto another lift, which dropped us off on the 85th floor, 356 metres up. It's incredible to think another 38 floors have yet to be added, (though that's as they say, another storey).
"There's the swimming pool," said Sunju, pointing to a long oblong construction structure. I imagined what it will be like to take a dip here, an infinity pool like no other, I suspect, like swimming out into the clouds.
"It's very rare to see it so clear," said our excellent guide Woo Seuk Choi (pictured top), Lotte World Tower General Manager Experience Center Team. So clear in fact that we could easily make out the distinctive high-rise skyline of Incheon City, 34km away. Over 55,000 people will live, work or stay in this building once it is completed. "It's more like a city, not just a building," he added.
Just look at the almost humbling power of these images, these extraordinary vistas. The age-old power of the Hangjang river winding its way through the capital and on to the Yellow Sea. The contemporary might of another kind of sea, one comprised of thousands of high-rise buildings. The mountains looming regally, eternally in the distance.
When it is complete, the US$3.8 billion Lotte World Tower will have 21 cinema halls (the largest in Asia), a hotel, offices, residences, duty free and other shopping and numerous other world-class facilities. But most of all, it will offer the most extraordinary 360 degree views of the majesty that is Seoul, capital of the 'Land of the Morning Calm'.
As we walked around the bare concrete floors of the 85th floor, Sunju and I stopped time and again to gasp at the magnificence beyond. The outside of the building now bears a giant Korean flag, a reminder of just how important this building has already become to Korean national identify. As the promotional video we watched later proclaimed, "Korea's Dream becomes a reality high up in the sky." Few would argue with that.
I felt privileged to have been given the honour of a preview visit to this amazing facility. Vertigo or no vertigo I'm coming back when the building is finished and going right to the top. Just like that. 1, 2, 3.
Footnote: Look out for a forthcoming Blog and story about Lotte's adjacent and highly impressive World Mall duty free store (which will be extended into the World Tower when the latter is completed).
["It's time to get back into your cage Mr Moodie."]
Martyn Westbury says:
Hi Martin
I too struggle with heights and had sweaty palms just reading the story – will be happy to go up in an enclosed lift when open though!!
Amazing story and views.
How Caelumest Ginis and Ginger Vitis got Blitzed then Bloomed
Catching up with old friends for a last tango and the perfect re-torte
Marking a grim anniversary and rediscovering the industry of beautiful memories
TVA in place but now free of duty – thank you and farewell Mr Danzka
Your flight time will be 8,760 hours… you may experience some turbulence
Copyright 2021 The Moodie Blog
|
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| 144
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{"url":"https:\/\/www.quantumstudy.com\/two-wires-are-made-of-the-same-material-and-have-the-same-volume-however-wire-1-has-cross-sectional-area-a\/","text":"# Two wires are made of the same material and have the same volume. However, wire 1 has cross-sectional area A\u2026..\n\nQ: Two wires are made of the same material and have the same volume. However, wire 1 has cross-sectional area A and wire 2 has cross-sectional area 3A. If the length of wire 1 increases by \u0394x on applying force F, then how much force is needed to stretch wire 2 by the same amount?\n\n(a) F\n\n(b) 4F\n\n(c) 6F\n\n(d) 9F\n\nAns: (d)","date":"2021-09-25 06:37:14","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8489459156990051, \"perplexity\": 615.9718853143844}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057598.98\/warc\/CC-MAIN-20210925052020-20210925082020-00394.warc.gz\"}"}
| null | null |
Clinical guidelines are used by the health professionals at Great Ormond Street Hospital for a wide range of conditions, treatments and tests.
Search below for clinical guidelines, or filter by speciality.
The purpose of the guideline is to provide all staff who insert and care for CVAD's the evidence and knowledge of when to use Biopatch®, how to use Biopatch® and also to be aware of any exclusion criteria.
The aim of this guideline is to support staff in ensuring the prompt assessment, recognition and treatment of jaundice, while minimizing the risks of unintended harm such as parental anxiety, decreased breastfeeding, and unnecessary costs or treatment.
The aim of these guidelines is to facilitate the safe administration of injections (non-intra venous) for patients at Great Ormond Street Hospital for Children NHS Foundation Trust (GOSH).
The purpose of this guideline is to guide healthcare staff in the appropriate management and on-going care of neonatal longlines (peripherally inserted central venous catheters (PICCs).
The purpose of this guideline is to provide guidance about blood tests at Great Ormond Street Hospital (GOSH).
Assessment and measurement of growth is vital and must be done with precision and accuracy to be meaningful (Voss, 2000).
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| 8,892
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{"url":"https:\/\/www.math24.net\/related-rates\/","text":"# Related Rates\n\nSuppose we have two quantities, which are connected to each other and both changing with time. A related rates problem is a problem in which we know the rate of change of one of the quantities and want to find the rate of change of the other quantity.\n\nLet the two variables be $$x$$ and $$y.$$ The relationship between them is expressed by a function $$y = f\\left( x \\right).$$ The rates of change of the variables $$x$$ and $$y$$ are defined in terms of their derivatives $$\\large{\\frac{{dx}}{{dt}}}\\normalsize$$ and $$\\large{\\frac{{dy}}{{dt}}}\\normalsize.$$ If $$\\large{\\frac{{dx}}{{dt}}}\\normalsize$$ is known, we can determine $$\\large{\\frac{{dy}}{{dt}}}\\normalsize$$ (and vice versa).\n\nAny related rates problem can be solved as follows:\n\n1. Decide what are the two variables describing your system or process.\n2. Drawing a picture can often be useful.\n3. Write an equation that relates the variables. To derive the equation you may use a geometric fact (like the Pythagorean theorem or similar triangles), a trigonometric identity, or a physical law.\n4. Take the derivative $$\\large{\\frac{d}{{dt}}}\\normalsize$$ of both sides of the equation.\n5. Solve for the unknown rate of change.\n6. Substitute all known values to get the final answer.\n\nAs an example, let\u2019s consider the well-known sliding ladder problem.\n\nA ladder $$13$$ feet long leans against a wall. The top of the ladder slides down at a constant rate of $$12\\,\\large{\\frac{\\text{ft}}{\\text{sec}}}\\normalsize.$$ Find the velocity of the ladder when $$h = 5\\,\\text{ft}.$$\n\nBy the Pythagorean equation:\n\n${{x^2} + {h^2} = {13^2}.}$\n\nDifferentiate both sides with respect to time $$t.$$\n\n${\\frac{d}{{dt}}\\left( {{x^2} + {h^2}} \\right) = \\frac{d}{{dt}}\\left( {{{13}^2}} \\right),\\;\\;} \\Rightarrow {2x\\frac{{dx}}{{dt}} + 2h\\frac{{dh}}{{dt}} = 0,\\;\\;} \\Rightarrow {x\\frac{{dx}}{{dt}} = \u2013 h\\frac{{dh}}{{dt}},\\;\\;} \\Rightarrow {\\frac{{dx}}{{dt}} = \u2013 \\frac{h}{x}\\frac{{dh}}{{dt}}.}$\n\nAs $$x = \\sqrt {{{13}^2} \u2013 {h^2}} ,$$ we have\n\n${\\frac{{dx}}{{dt}} = \u2013 \\frac{h}{{\\sqrt {{{13}^2} \u2013 {h^2}} }}\\frac{{dh}}{{dt}}.}$\n\nSubstitute the known values:\n\n${\\frac{{dh}}{{dt}} = \u2013 12\\,\\frac{\\text{ft}}{\\text{sec}}}$\n\n(the minus sign denotes that the ladder is sliding down),\n\n${h = 5\\,\\text{ft}.}$\n\nThis yields:\n\n${\\frac{{dx}}{{dt}} = \u2013 \\frac{h}{{\\sqrt {{{13}^2} \u2013 {h^2}} }}\\frac{{dh}}{{dt}} }={ \u2013 \\frac{5}{{\\sqrt {{{13}^2} \u2013 {5^2}} }} \\cdot \\left( { \u2013 12} \\right) }={ \\frac{5}{{12}} \\cdot 12 }={ 5\\,\\frac{\\text{ft}}{\\text{sec}}}.$\n\n## Solved Problems\n\nClick or tap a problem to see the solution.\n\n### Example 1\n\nOil from an uncapped well is radiating outward in the form of a circular film on the surface of the water. If the radius of the circle is increasing at the rate of $$0.5$$ meters per minute, how fast is the area of the oil film growing at the instant when the radius is $$100\\,\\text{m}?$$\n\n### Example 2\n\nA triangle has two sides $$a = 1\\,\\text{cm}$$ and $$b = 2\\,\\text{cm}.$$ How fast is the third side $$c$$ increasing when the angle $$\\alpha$$ between the given sides is $$60^\\circ$$ and is increasing at the rate of $$3^\\circ$$ per second (Figure $$2$$)?\n\n### Example 3\n\nA light is at the top of a $$16$$ ft pole. A man $$6$$ ft tall walks away from the pole at a rate of $$5\\,\\large{\\frac{\\text{ft}}{\\text{sec}}}\\normalsize$$ (Figure $$3$$). How fast is the tip of his shadow moving when he is $$20$$ ft from the pole?\n\n### Example 4\n\nThe volume of a cube is increasing at the rate of $$2$$ cubic inches per minute. How fast is the surface of the cube increasing when the side is $$8$$ inches?\n\n### Example 5\n\nAn object is moving along the curve $$y = {x^2} \u2013 2x + 3.$$ Find a point on the curve at which the $$y-$$coordinate is changing $$4$$ times faster than the $$x-$$coordinate.\n\n### Example 6\n\nA water tank in the form of an inverted cone is being emptied at the rate of $$2$$ cubic feet per second. The height of the cone is $$8$$ feet and the radius is $$4$$ feet (Figures $$5, 6$$). Find the rate of change of the water level when the depth is 6 feet.\n\n### Example 7\n\nA kite $$160$$ feet above the ground moves in a direction parallel to the ground at the rate of $$5$$ feet per second (Figure $$7$$). How fast is the string unwinding when the length of string already let out is $$200$$ feet?\n\n### Example 8\n\nShip A is 60 miles north of point O and moving in the north direction at $$20$$ miles per hour. Ship B is 80 miles east of point O and moving west at 25 miles per hour (Figure $$8$$). How fast is the distance between the ships changing at this moment?\n\n### Example 9\n\nThe legs of an isosceles triangle are increasing at the rate of $$0.5$$ feet per minute, and the vertical angle $$\\alpha$$ is increasing at the rate of $$1$$ radian per minute (Figure $$9$$). At what rate is the area $$A$$ of the triangle changing when the leg $$b$$ is 2 feet and the angle $$\\alpha$$ is $$\\large{\\frac{{2\\pi }}{3}}\\normalsize?$$\n\n### Example 10\n\nFind all points on the ellipse $$9{x^2} + 16{y^2} = 400,$$ at which the $$y-$$coordinate is decreasing and the $$x-$$coordinate is increasing at the same rate (Figure $$10$$).\n\n### Example 1.\n\nOil from an uncapped well is radiating outward in the form of a circular film on the surface of the water. If the radius of the circle is increasing at the rate of $$0.5$$ meters per minute, how fast is the area of the oil film growing at the instant when the radius is $$100\\,\\text{m}?$$\n\nSolution.\n\nSuppose that t is time in minutes, R and A are the radius and area of the circle, respectively.\n\nThe rate of change of the area is giveb by the derivative $$\\large{\\frac{{dA}}{{dt}}}\\normalsize,$$ where\n\n$A = \\pi {R^2}.$\n\nDifferentiating the right-hand side of the relation by the chain rule, we get\n\n${\\frac{{dA}}{{dt}} = \\frac{d}{{dt}}\\left( {\\pi {R^2}} \\right) }={ 2\\pi R\\frac{{dR}}{{dt}}.}$\n\nIt is known that $$\\large{\\frac{{dR}}{{dt}}}\\normalsize = 0.5\\,\\large{\\frac{\\text{m}}{\\text{min}}}\\normalsize.$$ Therefore, the oil spot is growing at the rate\n\n${\\frac{{dA}}{{dt}} = 2\\pi R\\frac{{dR}}{{dt}} }={ 2\\pi R \\cdot 0.5 }={ \\pi R.}$\n\nFor $$R = 100\\,\\text{m},$$ we have\n\n$\\frac{{dA}}{{dt}} = 100\\pi \\approx 314\\,\\frac{{{\\text{m}^2}}}{\\text{min}}.$\n\n### Example 2.\n\nA triangle has two sides $$a = 1\\,\\text{cm}$$ and $$b = 2\\,\\text{cm}.$$ How fast is the third side $$c$$ increasing when the angle $$\\alpha$$ between the given sides is $$60^\\circ$$ and is increasing at the rate of $$3^\\circ$$ per second (Figure $$2$$)?\n\nSolution.\n\nAccording to the law of cosines,\n\n${c^2} = {a^2} + {b^2} \u2013 2ab\\cos \\alpha .$\n\nWe differentiate both sides of this equation with respect to time t:\n\n${\\frac{d}{{dt}}\\left( {{c^2}} \\right) }={ \\frac{d}{{dt}}\\left( {{a^2} + {b^2} \u2013 2ab\\cos \\alpha } \\right),}$\n\n${2c\\frac{{dc}}{{dt}} }={ \u2013 2ab\\left( { \u2013 \\sin \\alpha } \\right)\\frac{{d\\alpha }}{{dt}},}$\n\nor\n\n$\\frac{{dc}}{{dt}} = \\frac{{ab\\sin \\alpha }}{c}\\frac{{d\\alpha }}{{dt}}.$\n\nCalculate the length of the side $$c:$$\n\n${c = \\sqrt {{a^2} + {b^2} \u2013 2ab\\cos \\alpha } }={ \\sqrt {{1^2} + {2^2} \u2013 2 \\cdot 1 \\cdot 2 \\cdot \\cos 60^\\circ } }={ \\sqrt {1 + 4 \u2013 2} }={ \\sqrt 3 .}$\n\nNow we know all quantities to determine the rate of change $$\\large{\\frac{{dc}}{{dt}}}\\normalsize:$$\n\n${\\frac{{dc}}{{dt}} = \\frac{{ab\\sin \\alpha }}{c}\\frac{{d\\alpha }}{{dt}} }={ \\frac{{1 \\cdot 2 \\cdot \\sin 60^\\circ }}{{\\sqrt 3 }}\\frac{{d\\alpha }}{{dt}} }={ \\frac{{2\\frac{{\\sqrt 3 }}{2}}}{{\\sqrt 3 }} \\cdot 3 }={ 3\\,\\frac{\\text{cm}}{\\text{sec}}.}$\n\n### Example 3.\n\nA light is at the top of a $$16$$ ft pole. A man $$6$$ ft tall walks away from the pole at a rate of $$5\\,\\large{\\frac{\\text{ft}}{\\text{sec}}}\\normalsize$$ (Figure $$3$$). How fast is the tip of his shadow moving when he is $$20$$ ft from the pole?\n\nSolution.\n\nBy similar triangles,\n\n$\\frac{L}{{16}} = \\frac{{L \u2013 x}}{6}.$\n\nSolving for $$L,$$ we have:\n\n${6L = 16\\left( {L \u2013 x} \\right),\\;\\;} \\Rightarrow {6L = 16L \u2013 16x,\\;\\;} \\Rightarrow {L = \\frac{8}{5}x.}$\n\nDifferentiating both sides with respect to time $$t$$ yields:\n\n$\\frac{{dL}}{{dt}} = \\frac{8}{5}\\frac{{dx}}{{dt}}.$\n\nWe are told that the rate $$\\frac{{dx}}{{dt}} = 5\\,\\large{\\frac{\\text{ft}}{\\text{sec}}}\\normalsize.$$ Therefore\n\n$\\frac{{dL}}{{dt}} = \\frac{8}{5} \\cdot 5 = 8\\,\\frac{\\text{ft}}{\\text{sec}}.$\n\nThus, the tip of the shadow is moving at the rate of $$8\\,\\large{\\frac{\\text{ft}}{\\text{sec}}}\\normalsize.$$\n\n### Example 4.\n\nThe volume of a cube is increasing at the rate of $$2$$ cubic inches per minute. How fast is the surface of the cube increasing when the side is $$8$$ inches?\n\nSolution.\n\nThe volume of the cube of side $$a$$ is given by\n\n$V = {a^3}.$\n\nWe differentiate both sides of the equation to find the relation between the rates of change:\n\n$\\frac{{dV}}{{dt}} = 3{a^2}\\frac{{da}}{{dt}}.$\n\nSolve for $$\\large{\\frac{{da}}{{dt}}}\\normalsize:$$\n\n$\\frac{{da}}{{dt}} = \\frac{1}{{3{a^2}}}\\frac{{dV}}{{dt}}.$\n\nRecall that the surface area of the cube is\n\n$A = 6{a^2}.$\n\nSimilarly, take the derivatives of both sides:\n\n$\\frac{{dA}}{{dt}} = 12a\\frac{{da}}{{dt}}.$\n\nNow we substitute the expression for $$\\large{\\frac{{da}}{{dt}}}\\normalsize$$ from the first equation. This yields\n\n${\\frac{{dA}}{{dt}} = 12a\\frac{{da}}{{dt}} }={ 12a \\cdot \\frac{1}{{3{a^2}}}\\frac{{dV}}{{dt}} }={ \\frac{4}{a}\\frac{{dV}}{{dt}}.}$\n\nAs the volume of the cube is increasing at the rate of $$2\\,\\large{\\frac{{{\\text{in}^3}}}{\\text{min}}}\\normalsize,$$ we can write\n\n${\\frac{{dA}}{{dt}} = \\frac{4}{a}\\frac{{dV}}{{dt}} }={ \\frac{4}{a} \\cdot 2 }={ \\frac{8}{a}.}$\n\nFinally, we substitute the side length of the cube, which is equal to $$8$$ inches. As a result, we find the rate of change of the area:\n\n${\\frac{{dA}}{{dt}} = \\frac{8}{a} }={ \\frac{8}{8} }={ 1\\,\\frac{{{\\text{in}^2}}}{\\text{min}}.}$\n\n### Example 5.\n\nAn object is moving along the curve $$y = {x^2} \u2013 2x + 3.$$ Find a point on the curve at which the $$y-$$coordinate is changing $$4$$ times faster than the $$x-$$coordinate.\n\nSolution.\n\nFirst, we differentiate both sides of the curve equation with respect to time $$t:$$\n\n${\\frac{{dy}}{{dt}} = \\frac{d}{{dt}}\\left( {{x^2} \u2013 2x + 3} \\right),\\;\\;} \\Rightarrow {\\frac{{dy}}{{dt}} = \\left( {2x \u2013 2} \\right)\\frac{{dx}}{{dt}}.}$\n\nIn the problem, we are told that\n\n$\\frac{{dy}}{{dt}} = 4\\frac{{dx}}{{dt}}.$\n\nTherefore, we get the relationship\n\n$2x \u2013 2 = 4,$\n\nso $$x=3.$$\n\nThe $$y-$$coordinate is given by\n\n$y = {3^3} \u2013 2 \\cdot 3 + 3 = 6.$\n\nThus, the answer is $$\\left( {x,y} \\right) = \\left( {3,6} \\right).$$\n\n### Example 6.\n\nA water tank in the form of an inverted cone is being emptied at the rate of $$2$$ cubic feet per second. The height of the cone is $$8$$ feet and the radius is $$4$$ feet (Figures $$5, 6$$). Find the rate of change of the water level when the depth is 6 feet.\n\nSolution.\n\nFigure 5.\nFigure 6.\n\nLet $$h$$ and $$r$$ be the height and the radius of the water. The volume of the water (in the form of right circular cone) is given by\n\n$V = \\frac{1}{3}\\pi {r^2}h.$\n\nWe can use similar triangles to get a relationship between $$r$$ and $$h:$$\n\n$\\frac{h}{r} = \\frac{8}{4} = 2.$\n\nHence,\n\n${r = \\frac{h}{2}.}$\n\nPlug it into the formula for the volume:\n\n${V = \\frac{1}{3}\\pi {r^2}h }={ \\frac{1}{3}\\pi {\\left( {\\frac{h}{2}} \\right)^2}h }={ \\frac{1}{{12}}\\pi {h^3}.}$\n\nDifferentiate this formula with respect to time $$t:$$\n\n${\\frac{{dV}}{{dt}} = \\frac{d}{{dt}}\\left( {\\frac{1}{{12}}\\pi {h^3}} \\right) }={ \\frac{1}{{12}}\\pi \\cdot 3{h^2}\\frac{{dh}}{{dt}} }={ \\frac{1}{4}\\pi {h^2}\\frac{{dh}}{{dt}}.}$\n\nSolve the last equation for $$\\large{\\frac{{dh}}{{dt}}}\\normalsize :$$\n\n$\\frac{{dh}}{{dt}} = \\frac{4}{{\\pi {h^2}}}\\frac{{dV}}{{dt}}.$\n\nNow we can substitute known values to compute the rate of change of the water level:\n\n${\\frac{{dh}}{{dt}} = \\frac{4}{{\\pi {h^2}}}\\frac{{dV}}{{dt}} }={ \\frac{4}{{\\pi \\cdot {6^2}}} \\cdot 2 }={ \\frac{2}{{9\\pi }}\\frac{\\text{ft}}{\\text{sec}}.}$\n\n### Example 7.\n\nA kite $$160$$ feet above the ground moves in a direction parallel to the ground at the rate of $$5$$ feet per second (Figure $$7$$). How fast is the string unwinding when the length of string already let out is $$200$$ feet?\n\nSolution.\n\nWe denote the length of string by $$s.$$ By the Pythagorean theorem,\n\n${s^2} = {x^2} + {h^2}.$\n\nTake the derivatives of both sides with respect to time $$t$$ assuming the height $$h$$ remains constant.\n\n${\\frac{d}{{dt}}\\left( {{s^2}} \\right) = \\frac{d}{{dt}}\\left( {{x^2} + {h^2}} \\right),\\;\\;} \\Rightarrow {2s\\frac{{ds}}{{dt}} = 2x\\frac{{dx}}{{dt}},\\;\\;} \\Rightarrow {\\frac{{ds}}{{dt}} = \\frac{x}{s}\\frac{{dx}}{{dt}}.}$\n\nWrite $$x$$ in terms of $$s$$ and $$h:$$\n\n$x = \\sqrt {{s^2} \u2013 {h^2}} .$\n\nHence,\n\n$\\frac{{ds}}{{dt}} = \\frac{{\\sqrt {{s^2} \u2013 {h^2}} }}{s}\\frac{{dx}}{{dt}}.$\n\nThe speed of the kite $$\\large{\\frac{{dx}}{{dt}}}\\normalsize$$ is known (it is equal to $$5\\,\\large{\\frac{\\text{ft}}{\\text{sec}}}\\normalsize$$). Substituting the values of $$s$$ and $$h,$$ we find the rate $$\\large{\\frac{{ds}}{{dt}}}\\normalsize:$$\n\n${\\frac{{ds}}{{dt}} = \\frac{{\\sqrt {{{200}^2} \u2013 {{160}^2}} }}{{200}} \\cdot 5 }={ 3\\,\\frac{\\text{ft}}{\\text{sec}}.}$\n\n### Example 8.\n\nShip A is 60 miles north of point O and moving in the north direction at $$20$$ miles per hour. Ship B is 80 miles east of point O and moving west at 25 miles per hour (Figure $$8$$). How fast is the distance between the ships changing at this moment?\n\nSolution.\n\nLet us denote the distance between the two ships as $$s.$$ By the Pythagorean theorem,\n\n${s^2} = {x^2} + {y^2},$\n\nwhere $$y$$ is the coordinate of point $$A$$ and $$x$$ is the coordinate of point $$B.$$ Take the derivatives of both sides:\n\n${2s\\frac{{ds}}{{dt}} = 2x\\frac{{dx}}{{dt}} + 2y\\frac{{dy}}{{dt}},\\;\\;} \\kern0pt {\\frac{{ds}}{{dt}} = \\frac{{x\\frac{{dx}}{{dt}} + y\\frac{{dy}}{{dt}}}}{s} }={ \\frac{{x\\frac{{dx}}{{dt}} + y\\frac{{dy}}{{dt}}}}{{\\sqrt {{x^2} + {y^2}} }}.}$\n\nSubstitute the known values to get the rate of change of the distance between the ships:\n\n${\\frac{{ds}}{{dt}} = \\frac{{80 \\cdot \\left( { \u2013 25} \\right) + 60 \\cdot 20}}{{\\sqrt {{{80}^2} + {{60}^2}} }} }={ \\frac{{ \u2013 800}}{{100}} }={ \u2013 8\\frac{\\text{mi}}{\\text{h}}.}$\n\nThe minus sign means that the distance is decreasing at this moment.\n\n### Example 9.\n\nThe legs of an isosceles triangle are increasing at the rate of $$0.5$$ feet per minute, and the vertical angle $$\\alpha$$ is increasing at the rate of $$1$$ radian per minute (Figure $$9$$). At what rate is the area $$A$$ of the triangle changing when the leg $$b$$ is 2 feet and the angle $$\\alpha$$ is $$\\large{\\frac{{2\\pi }}{3}}\\normalsize?$$\n\nSolution.\n\nThe area of an isosceles triangle is given by\n\n$A = \\frac{{{b^2}}}{2}\\sin \\alpha .$\n\nDifferentiate both sides of the equation with respect to time $$t.$$ The right-hand side contains two variables that change with time: $$b$$ and $$\\alpha.$$ Therefore, we differentiate the right-hand side combining the product and chain rules.\n\n${\\frac{{dA}}{{dt}} = \\frac{d}{{dt}}\\left( {\\frac{{{b^2}}}{2}\\sin \\alpha } \\right) }={ \\frac{{\\sin \\alpha }}{2} \\cdot 2b\\frac{{db}}{{dt}} + \\frac{{{b^2}}}{2} \\cdot \\cos \\alpha \\frac{{d\\alpha }}{{dt}} }={ b\\sin \\alpha \\frac{{db}}{{dt}} + \\frac{1}{2}{b^2}\\cos \\alpha \\frac{{d\\alpha }}{{dt}}.}$\n\nSubstitute the known values to calculate $$\\large{\\frac{{dA}}{{dt}}}\\normalsize:$$\n\n${\\frac{{dA}}{{dt}} }={ 2 \\cdot \\sin \\frac{{2\\pi }}{3} \\cdot 0,5 }\\kern0pt{+ \\frac{1}{2} \\cdot {2^2} \\cdot \\cos \\frac{{2\\pi }}{3} \\cdot 1 }={ \\frac{{\\sqrt 3 }}{2} \u2013 1 \\approx \u2013 0.134\\frac{{{\\text{ft}^2}}}{\\text{min}}}.$\n\nThe area of the triangle is decreasing at the approximate rate of $$0.134\\,\\large{\\frac{{{\\text{ft}^2}}}{\\text{min}}}\\normalsize.$$\n\n### Example 10.\n\nFind all points on the ellipse $$9{x^2} + 16{y^2} = 400,$$ at which the $$y-$$coordinate is decreasing and the $$x-$$coordinate is increasing at the same rate (Figure $$10$$).\n\nSolution.\n\nWe differentiate both sides of the ellipse equation with respect to $$t:$$\n\n$\\frac{d}{{dt}}\\left( {9{x^2} + 16{y^2}} \\right) = \\frac{d}{{dt}}\\left( {400} \\right),$\n\n$18x\\frac{{dx}}{{dt}} + 32y\\frac{{dy}}{{dt}} = 0,$\n\nor\n\n$9x\\frac{{dx}}{{dt}} + 16y\\frac{{dy}}{{dt}} = 0.$\n\nWe are asked to find the points where\n\n$\\frac{{dx}}{{dt}} = \u2013 \\frac{{dy}}{{dt}},$\n\nso we can write the system of equations:\n\n$\\left\\{ \\begin{array}{l} 9x\\frac{{dx}}{{dt}} + 16y\\frac{{dy}}{{dt}} = 0\\\\ \\frac{{dx}}{{dt}} = \u2013 \\frac{{dy}}{{dt}} \\end{array} \\right..$\n\nHence, the points on the ellipse must satisfy the condition\n\n${9x = 16y,\\;\\;}\\kern0pt{\\text{or}\\;\\;x = \\frac{{16}}{9}y.}$\n\nSubstitute this in the ellipse equation to calculate the $$y-$$coordinates:\n\n${9{x^2} + 16{y^2} = 400,\\;\\;} \\Rightarrow {9 \\cdot {\\left( {\\frac{{16}}{9}y} \\right)^2} + 16{y^2} = 400,\\;\\;} \\Rightarrow {\\frac{{256}}{9}{y^2} + 16{y^2} = 400,\\;\\;} \\Rightarrow {\\frac{{400}}{9}{y^2} = 400,\\;\\;} \\Rightarrow {{y^2} = 9,\\;\\;} \\Rightarrow {{y_1} = 3,{y_2} = \u2013 3.}$\n\nThe $$x-$$coordinates are given by\n\n${{x_1} = \\frac{{16}}{9}{y_1} }={ \\frac{{16}}{9} \\cdot 3 }={ \\frac{{16}}{3},}$\n\n${{x_2} = \\frac{{16}}{9}{y_2} }={ \\frac{{16}}{9} \\cdot \\left( { \u2013 3} \\right) }={ \u2013 \\frac{{16}}{3}.}$\n\nWe\u2019ve got two points on the ellipse:\n\n${A\\left( {\\frac{{16}}{3},3} \\right),\\;}\\kern0pt{B\\left( { \u2013 \\frac{{16}}{3}, \u2013 3} \\right)}.$","date":"2020-11-26 06:04:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.876785933971405, \"perplexity\": 158.91223123988172}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141186761.30\/warc\/CC-MAIN-20201126055652-20201126085652-00397.warc.gz\"}"}
| null | null |
/**
* @since 2018-07-08 15:58:13
* @author vivaxy
*/
const PriorityQueue = require('../../../priority-queue/implements/index.js');
module.exports = function dijkstra(graph, startVertex) {
const distances = {};
const visitedVertices = {};
const previousVertices = {};
const queue = new PriorityQueue();
graph.getAllVertices().forEach((vertex) => {
distances[vertex.getKey()] = Infinity;
previousVertices[vertex.getKey()] = null;
});
distances[startVertex.getKey()] = 0;
queue.add(startVertex, distances[startVertex.getKey()]);
while (!queue.isEmpty()) {
const currentVertex = queue.poll();
currentVertex.getNeighbors().forEach((neighborVertex) => {
if (!visitedVertices[neighborVertex.getKey()]) {
const edge = graph.findEdge(currentVertex, neighborVertex);
const existingDistanceToNeighbor = distances[neighborVertex.getKey()];
const distanceToNeighborFromCurrent = distances[currentVertex.getKey()] + edge.weight;
if (distanceToNeighborFromCurrent < existingDistanceToNeighbor) {
distances[neighborVertex.getKey()] = distanceToNeighborFromCurrent;
if (queue.hasValue(neighborVertex)) {
queue.changePriority(neighborVertex, distanceToNeighborFromCurrent);
}
previousVertices[neighborVertex.getKey()] = currentVertex;
}
if (!queue.hasValue(neighborVertex)) {
queue.add(neighborVertex, distances[neighborVertex.getKey()]);
}
}
});
visitedVertices[currentVertex.getKey()] = currentVertex;
}
return {
distances,
previousVertices,
};
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,618
|
Base58 è un insieme di schemi di codifica da binario a testo, specificati da Satoshi Nakamoto per la rete Bitcoin, al fine di rappresentare numeri interi grandi come testo alfanumerico.
Da allora, è stato applicato ad altre crittovalute e applicazioni. È simile al Base64, ma è stato modificato per eliminare sia i caratteri non alfanumerici che quelle lettere che potrebbero essere confuse con altre, quando stampate. È quindi progettato per gli utenti umani che inseriscono manualmente i dati, copiandoli da una qualche sorgente visiva, ma consente anche una facile copia e incolla, perché un doppio clic di solito seleziona l'intera stringa.
Rispetto al Base64, sono state eliminate le seguenti lettere tra loro simili e in grado generare errati riconoscimenti: 0 (zero) e O ("o" maiuscola), così come I ("i" maiuscola) e l (lettere L minuscola), oltre ai caratteri non alfanumerici + (più) e / (slash).
Diversamente da Base64, le cifre della codifica non si allineano bene con i limiti di bit dei dati originali. Per questo motivo, il metodo è adatto per codificare interi di grandi dimensioni, ma non è progettato per codificare porzioni più lunghe di dati binari. L'ordine effettivo delle lettere nell'alfabeto varia a seconda dell'applicazione, motivo per il quale il solo termine "Base58" non è sufficiente per descrivere completamente il formato. Base58Check è uno dei formati di codifica Base58, che inizia con l'identificatore del tipo di dati e termina con un codice di controllo nei rimanenti caratteri.
La variante Base56 differisce dalla Base58 per l'esclusione dei caratteri 1 (uno) e o (o minuscola).
Esempi
Note
Voci correlate
Bitcoin
Algoritmo di Hash
Repertori di caratteri
Formati di file
Crittografia
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,059
|
Donald Walsh was an American professional basketball player. He played in the National Basketball League for the Kankakee Gallagher Trojans in four games during the 1937–38 season and averaged 1.0 point per game.
References
Year of birth uncertain
Year of death uncertain
American men's basketball players
Basketball players from Illinois
Forwards (basketball)
Kankakee Gallagher Trojans players
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,459
|
Q: Add third party library in chromium project I'm working on an Android project that extends chromium project.
I want to add a third party library (www.intercom.com). The problem is that the intercom documentation is just for gradle projects, and chromium is not using gradle so I have to add the library manual.
I've tried to copy all the .aar and .jar files in third_party/intercom and create a gn file, but I get an error:
Class "kotlin/coroutines/Continuation" not found on any classpath. Used by class "io/intercom/retrofit2/RequestFactory$Builder"
I've include org_jetbrains_kotlin_kotlin_stdlib_java, but still get the same error.
Questions:
*
*Should I include anything else to get rid of this error?
*Is there any automated tool/script, that generates a gn file for a library?
Thanks!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,147
|
package com.linkedin.pinot.integration.tests;
import java.io.File;
import java.io.FileInputStream;
import java.sql.DriverManager;
import java.sql.ResultSet;
import java.sql.Statement;
import java.util.ArrayList;
import java.util.Collections;
import java.util.List;
import java.util.Map;
import java.util.Set;
import java.util.concurrent.CountDownLatch;
import java.util.concurrent.ExecutorService;
import java.util.concurrent.Executors;
import java.util.concurrent.TimeUnit;
import kafka.server.KafkaServerStartable;
import org.apache.commons.io.FileUtils;
import org.apache.helix.ExternalViewChangeListener;
import org.apache.helix.HelixManager;
import org.apache.helix.HelixManagerFactory;
import org.apache.helix.InstanceType;
import org.apache.helix.NotificationContext;
import org.apache.helix.model.ExternalView;
import org.json.JSONArray;
import org.json.JSONObject;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import org.testng.Assert;
import org.testng.annotations.AfterClass;
import org.testng.annotations.BeforeClass;
import org.testng.annotations.Test;
import com.linkedin.pinot.common.data.Schema;
import com.linkedin.pinot.common.utils.FileUploadUtils;
import com.linkedin.pinot.common.utils.KafkaStarterUtils;
import com.linkedin.pinot.common.utils.TarGzCompressionUtils;
import com.linkedin.pinot.common.utils.ZkStarter;
import com.linkedin.pinot.util.TestUtils;
/**
* Hybrid cluster integration test that uploads 8 months of data as offline and 6 months of data as realtime (with a
* two month overlap).
*
*/
public class HybridClusterIntegrationTest extends BaseClusterIntegrationTest {
private static final Logger LOGGER = LoggerFactory.getLogger(RealtimeClusterIntegrationTest.class);
private final File _tmpDir = new File("/tmp/HybridClusterIntegrationTest");
private final File _segmentDir = new File("/tmp/HybridClusterIntegrationTest/segmentDir");
private final File _tarDir = new File("/tmp/HybridClusterIntegrationTest/tarDir");
private static final String KAFKA_TOPIC = "hybrid-integration-test";
private static final int SEGMENT_COUNT = 12;
private static final int QUERY_COUNT = 1000;
private static final int OFFLINE_SEGMENT_COUNT = 8;
private static final int REALTIME_SEGMENT_COUNT = 6;
private KafkaServerStartable kafkaStarter;
@Override
protected String getHelixClusterName() {
return "HybridClusterIntegrationTest";
}
protected void setUpTable(String tableName, String timeColumnName, String timeColumnType, String kafkaZkUrl,
String kafkaTopic, File schemaFile, File avroFile) throws Exception {
Schema schema = Schema.fromFile(schemaFile);
addSchema(schemaFile, schema.getSchemaName());
addHybridTable(tableName, timeColumnName, timeColumnType, kafkaZkUrl, kafkaTopic, schema.getSchemaName(),
"TestTenant", "TestTenant", avroFile);
}
@BeforeClass
public void setUp() throws Exception {
//Clean up
ensureDirectoryExistsAndIsEmpty(_tmpDir);
ensureDirectoryExistsAndIsEmpty(_segmentDir);
ensureDirectoryExistsAndIsEmpty(_tarDir);
// Start Zk and Kafka
startZk();
kafkaStarter =
KafkaStarterUtils.startServer(KafkaStarterUtils.DEFAULT_KAFKA_PORT, KafkaStarterUtils.DEFAULT_BROKER_ID,
KafkaStarterUtils.DEFAULT_ZK_STR, KafkaStarterUtils.getDefaultKafkaConfiguration());
// Create Kafka topic
KafkaStarterUtils.createTopic(KAFKA_TOPIC, KafkaStarterUtils.DEFAULT_ZK_STR);
// Start the Pinot cluster
startController(true);
startBroker();
startServers(2);
// Create tenants
createBrokerTenant("TestTenant", 1);
createServerTenant("TestTenant", 1, 1);
// Unpack the Avro files
TarGzCompressionUtils.unTar(
new File(TestUtils.getFileFromResourceUrl(OfflineClusterIntegrationTest.class.getClassLoader().getResource(
"On_Time_On_Time_Performance_2014_100k_subset_nonulls.tar.gz"))), _tmpDir);
_tmpDir.mkdirs();
final List<File> avroFiles = new ArrayList<File>(SEGMENT_COUNT);
for (int segmentNumber = 1; segmentNumber <= SEGMENT_COUNT; ++segmentNumber) {
avroFiles.add(new File(_tmpDir.getPath() + "/On_Time_On_Time_Performance_2014_" + segmentNumber + ".avro"));
}
File schemaFile = getSchemaFile();
// Create Pinot table
setUpTable("mytable", "DaysSinceEpoch", "daysSinceEpoch", KafkaStarterUtils.DEFAULT_ZK_STR, KAFKA_TOPIC, schemaFile,
avroFiles.get(0));
// Create a subset of the first 8 segments (for offline) and the last 6 segments (for realtime)
final List<File> offlineAvroFiles = new ArrayList<File>(OFFLINE_SEGMENT_COUNT);
for (int i = 0; i < OFFLINE_SEGMENT_COUNT; i++) {
offlineAvroFiles.add(avroFiles.get(i));
}
final List<File> realtimeAvroFiles = new ArrayList<File>(REALTIME_SEGMENT_COUNT);
for (int i = SEGMENT_COUNT - REALTIME_SEGMENT_COUNT; i < SEGMENT_COUNT; i++) {
realtimeAvroFiles.add(avroFiles.get(i));
}
// Load data into H2
ExecutorService executor = Executors.newCachedThreadPool();
setupH2AndInsertAvro(avroFiles, executor);
// Create segments from Avro data
buildSegmentsFromAvro(offlineAvroFiles, executor, 0, _segmentDir, _tarDir, "mytable");
// Initialize query generator
setupQueryGenerator(avroFiles, executor);
executor.shutdown();
executor.awaitTermination(10, TimeUnit.MINUTES);
// Set up a Helix spectator to count the number of segments that are uploaded and unlock the latch once 12 segments are online
final CountDownLatch latch = new CountDownLatch(1);
HelixManager manager =
HelixManagerFactory.getZKHelixManager(getHelixClusterName(), "test_instance", InstanceType.SPECTATOR,
ZkStarter.DEFAULT_ZK_STR);
manager.connect();
manager.addExternalViewChangeListener(new ExternalViewChangeListener() {
@Override
public void onExternalViewChange(List<ExternalView> externalViewList, NotificationContext changeContext) {
for (ExternalView externalView : externalViewList) {
if (externalView.getId().contains("mytable")) {
Set<String> partitionSet = externalView.getPartitionSet();
if (partitionSet.size() == OFFLINE_SEGMENT_COUNT) {
int onlinePartitionCount = 0;
for (String partitionId : partitionSet) {
Map<String, String> partitionStateMap = externalView.getStateMap(partitionId);
if (partitionStateMap.containsValue("ONLINE")) {
onlinePartitionCount++;
}
}
if (onlinePartitionCount == OFFLINE_SEGMENT_COUNT) {
System.out.println("Got " + OFFLINE_SEGMENT_COUNT + " online tables, unlatching the main thread");
latch.countDown();
}
}
}
}
}
});
// Upload the segments
int i = 0;
for (String segmentName : _tarDir.list()) {
System.out.println("Uploading segment " + (i++) + " : " + segmentName);
File file = new File(_tarDir, segmentName);
FileUploadUtils.sendSegmentFile("localhost", "8998", segmentName, new FileInputStream(file), file.length());
}
// Wait for all offline segments to be online
latch.await();
// Load realtime data into Kafka
pushAvroIntoKafka(realtimeAvroFiles, KafkaStarterUtils.DEFAULT_KAFKA_BROKER, KAFKA_TOPIC);
// Wait until the Pinot event count matches with the number of events in the Avro files
int pinotRecordCount, h2RecordCount;
long timeInTwoMinutes = System.currentTimeMillis() + 2 * 60 * 1000L;
Statement statement = _connection.createStatement(ResultSet.TYPE_FORWARD_ONLY, ResultSet.CONCUR_READ_ONLY);
do {
Thread.sleep(5000L);
// Run the query
JSONObject response = postQuery("select count(*) from 'mytable'");
JSONArray aggregationResultsArray = response.getJSONArray("aggregationResults");
JSONObject firstAggregationResult = aggregationResultsArray.getJSONObject(0);
String pinotValue = firstAggregationResult.getString("value");
pinotRecordCount = Integer.parseInt(pinotValue);
statement.execute("select count(*) from mytable");
ResultSet rs = statement.getResultSet();
rs.first();
h2RecordCount = rs.getInt(1);
rs.close();
LOGGER.info("H2 record count: " + h2RecordCount + "\tPinot record count: " + pinotRecordCount);
Assert.assertTrue(System.currentTimeMillis() < timeInTwoMinutes, "Failed to read all records within two minutes");
TOTAL_DOCS = response.getLong("totalDocs");
} while (h2RecordCount != pinotRecordCount);
}
@AfterClass
public void tearDown() throws Exception {
stopBroker();
stopController();
stopServer();
KafkaStarterUtils.stopServer(kafkaStarter);
try {
stopZk();
} catch (Exception e) {
// Swallow ZK Exceptions.
}
FileUtils.deleteDirectory(_tmpDir);
}
@Override
protected int getGeneratedQueryCount() {
return QUERY_COUNT;
}
@Override
@Test
public void testMultipleQueries() throws Exception {
super.testMultipleQueries();
}
@Test
public void testSingleQuery() throws Exception {
String query;
query = "select count(*) from 'mytable' where DaysSinceEpoch >= 16312";
super.runQuery(query, Collections.singletonList(query.replace("'mytable'", "mytable")));
query = "select count(*) from 'mytable' where DaysSinceEpoch < 16312";
super.runQuery(query, Collections.singletonList(query.replace("'mytable'", "mytable")));
query = "select count(*) from 'mytable' where DaysSinceEpoch <= 16312";
super.runQuery(query, Collections.singletonList(query.replace("'mytable'", "mytable")));
query = "select count(*) from 'mytable' where DaysSinceEpoch > 16312";
super.runQuery(query, Collections.singletonList(query.replace("'mytable'", "mytable")));
}
@Override
@Test
public void testHardcodedQuerySet() throws Exception {
super.testHardcodedQuerySet();
}
@Override
@Test
public void testGeneratedQueries() throws Exception {
super.testGeneratedQueries();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,167
|
Bookends — четвертый студийный альбом американского дуэта Simon & Garfunkel, выпущенный 3 апреля 1968 года на лейбле Columbia.
В 2003 году журнал Rolling Stone поместил альбом Bookends на 233 место своего списка «500 величайших альбомов всех времён». В списке 2012 года альбом находится на 234 месте.
Список композиций
Чарты
Сертификации
Примечания
Ссылки
Альбомы 1968 года
Альбомы Simon & Garfunkel
Альбомы Columbia Records
Концептуальные альбомы
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,580
|
LHaladás-Viktória Football Club, o più semplicemente Haladás-Viktória, è una squadra di calcio femminile ungherese con sede nella città di Szombathely. Nata nel 2017 dalla collaborazione del con il precedente club interamente femminile Viktória Football Club Szombathely, e iscritta alla Női Nemzeti Bajnokság I, la massima serie del campionato ungherese di calcio femminile, dalla stagione 2017-2018 con una denominazione congiunta dei due club, utilizza colori sociali e tenute di gioco del Szombathelyi Haladás.
Storia
Fondata nel 1993, ha vinto la Női Nemzeti Bajnokság I per due volte, nella stagione 2003-2004 e nella stagione 2008-2009. Nel 2009 ha vinto anche la Női Magyar Kupa (la coppa nazionale ungherese), competizione vinta complessivamente tre volte. Grazie ai due campionati vinti, ha partecipato due volte alla UEFA Women's Champions League. Ha fatto il suo esordio nelle competizioni UEFA nell'edizione 2004-2005 accedendo alla prima fase a gironi: ha concluso il girone al secondo posto a pari merito con il FC Codru Anenii Noi, venendo eliminato dalla competizione. Con la vittoria del secondo campionato ha partecipato alla prima edizione della UEFA Women's Champions League nell'edizione 2009-2010, accedendo direttamente alla fase ad eliminazione diretta. Nei sedicesimi di finale ha affrontato la squadra tedesca del , perdendo sia l'andata sia il ritorno per un complessivo di 2-9.
Il club ha cambiato denominazione diverse volte nel corso della sua esistenza. Queste erano:
1993-?: Viktória FC
?-2006: WHC Viktória FC
2006-2017: Viktória FC-Szombathely
2017-: Haladás-Viktória FC
Calciatrici
Palmarès
2003-2004, 2008-2009
2007-2008, 2008-2009, 2010-2011
Statistiche
Risultati nelle competizioni UEFA
Note
Collegamenti esterni
Haladás-Viktória F.C.
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,688
|
Phyllodiaptomus sasikumari is een eenoogkreeftjessoort uit de familie van de Diaptomidae. De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1989 door Reddy & Venkateswarlu.
Diaptomidae
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,708
|
// This placeholder is referenced by all the TS projects to allow generating an empty dummy DLL
// so we can build and resolve the projects in the correct order
// This placeholder is used for Visual Studio to track the build output of specific projects and not trigger rebuilds.
public static class Placeholder { }
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,026
|
{"url":"http:\/\/mymathforum.com\/economics\/20864-value-annuity-any-given-month.html","text":"My Math Forum Value of an annuity in any given month\n\n Economics Economics Forum - Financial Mathematics, Econometrics, Operations Research, Mathematical Finance, Computational Finance\n\n September 6th, 2011, 09:15 AM #1 Newbie \u00a0 Joined: Sep 2011 Posts: 4 Thanks: 0 Value of an annuity in any given month I have an annuity where $2,000 is being invested at the beginning of every year for 20 years and the interest is compounded monthly. I want to find out for any given month what the balance of the annuity is. Is there a formula that will allow this? Example: I invest$2,000 at the start of every year for 20 years. The annual interest rate over the life of the annuity is 9%, compounded monthly. I want to be able to find out the balance at month 15, 30, 45, 60, etc. Thank you very much in advance for any help you can offer. It is much appreciated!\n September 6th, 2011, 09:49 AM #2 Global Moderator \u00a0 \u00a0 Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Value of an annuity in any given month You can use this in reverse, with some care: http:\/\/en.wikipedia.org\/wiki\/Mortgage_c ... nt_formula I'm sure you can find the formula written \"forward\" if you search for it.\n September 6th, 2011, 09:53 AM #3 Newbie \u00a0 Joined: Sep 2011 Posts: 4 Thanks: 0 Re: Value of an annuity in any given month This is a great start. I'll give it a try. Thanks!\n September 6th, 2011, 10:05 AM #4 Global Moderator \u00a0 \u00a0 Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Value of an annuity in any given month The formula can also be derived as a geometric series, but that may be more difficult. (It's probably the easiest way for me, but ymmv.)\n September 7th, 2011, 12:39 PM #5 Newbie \u00a0 Joined: Sep 2011 Posts: 4 Thanks: 0 Re: Value of an annuity in any given month I came across this link and the \"Calculating the Future Value of an Annuity Due\" sections seems to outline what I am looking to do. The only issue is that I do not know how to modify the formula to account for different payment and compounding periods (annual payments, monthly compounding). Every formula I find assumes the payment and compounding periods are the same. http:\/\/www.investopedia.com\/articles...#ixzz1XIgPHZ6Y I took your geometric series suggestion and found the link below. At the very end under the \"A Common Application\" section they given an example of how to calculate the annuity value assuming the deposit cycle and the compounding cycle are the same. http:\/\/fym.la.asu.edu\/~tturner\/MAT_1..._Sequences.htm How can I modify these formulas to account for different payment and compounding periods?\nSeptember 7th, 2011, 01:43 PM \u00a0 #6\nGlobal Moderator\n\nJoined: Nov 2006\nFrom: UTC -5\n\nPosts: 16,046\nThanks: 938\n\nMath Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms\nRe: Value of an annuity in any given month\n\nQuote:\n Originally Posted by Business School Guy How can I modify these formulas to account for different payment and compounding periods?\nYou're letting a fixed (integer) number of compounding periods pass between investing, right? Then just find the equivalent interest over that whole period and you can use the effective interest rate instead. If over 10 years you go from $P to$P * k, then the interest rate per ten-year period is k-1.\n\n September 8th, 2011, 07:33 PM #7 Senior Member \u00a0 Joined: Apr 2011 From: USA Posts: 782 Thanks: 1 Re: Value of an annuity in any given month The problem with the future value annuity equation is that this situation isn't the type of annuity for which this equation applies. It only works when the payments match the compounding periods, so you can't use it, nor am I aware of a way to \"adjust\" it to work. My degree's in accounting and we did a fair amount of time value of money, but I've only had one finance class (and no banking\/investment classes or anything of the sort), so there may be some special way to do this that I'm not aware of. We did problems like this, but we used a combination of equations, and just good old addition\/subtraction. If there were another equation, not sure why we wouldn't have learned it, unless the math behind it was considered too advanced. The following is how I would solve something like this, and if it were personal for me, I'd stick it into Excel and make some type of input section so that I could take it to any time period. The equation for just a future value (no payments) is: $\\text{FV= PV(1 + i)^n}$ I noticed on the one site they quote i to be the \"interest rate.\" Actually, i is the interest per compounding period, which is an important point to note. That is, i in this case is .09\/12, or .0075. This equation does not account for making more payments. It's a lump sum equation; that is, you stick $2000 in at the beginning of the year, and what's it worth after n periods, but without adding any more payments to it. So I would use this to get the value of the thing after one year, since you'd be changing it after a year: $\\text{FV= 2000 (1 + \\frac{.09}{12})^{12}}$ $\\text{FV= 2187.61}$ That gives you the value after the first year. At that point, you're adding another$2000 to it. So I would just add it: $\\text{2187.61 + 2000= 4187.61}$ Now you have a value for the beginning of the second year. And you can repeat this with your new number: $\\text{FV= 4187.61 (1 + \\frac{.09}{12})^{12}}$ $\\text{FV= 4580.44}$ Repeat ad nauseum. If you want to know what it is after, say, 15 months, just adjust the n in the second one: $\\text{FV= 4187.61 (1 + \\frac{.09}{12})^3}$ Because I can handle Excel fairly well, I'm sure I could find a way to put it in there. I'm also sure there's some fancy dancy equation that could take care of this, without manually adding the new payment each year. As I said, this is how we would do problems like this, and perhaps the math to do it as one equation was beyond our level. Perhaps someone here can take what I've done and turn it into one nice equation. It's also possible that CRGreathouse's method would work, but I don't see how. The effective rate stays the same on an annual basis. But I don't know how you can do this between years, like at 15 months. Perhaps if he used these numbers as example to demonstrate that?\nSeptember 9th, 2011, 07:55 AM \u00a0 #8\nGlobal Moderator\n\nJoined: Nov 2006\nFrom: UTC -5\n\nPosts: 16,046\nThanks: 938\n\nMath Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms\nRe: Value of an annuity in any given month\n\nQuote:\n Originally Posted by Erimess It's also possible that CRGreathouse's method would work, but I don't see how. The effective rate stays the same on an annual basis. But I don't know how you can do this between years, like at 15 months. Perhaps if he used these numbers as example to demonstrate that?\nUsing BSG's example, the yearly interest is (1 + .09\/12)^12 = 9.38...%, so use that rate and a period of 1 year with the standard formula. That only tells you what happens at the end of each year, so if you want, e.g. 15 months, use the formula for one year and then do a standard interest calculation for 3 months.\n\nSeptember 9th, 2011, 08:49 PM \u00a0 #9\nSenior Member\n\nJoined: Apr 2011\nFrom: USA\n\nPosts: 782\nThanks: 1\n\nRe: Value of an annuity in any given month\n\nQuote:\n Originally Posted by CRGreathouse Using BSG's example, the yearly interest is (1 + .09\/12)^12 = 9.38...%, so use that rate and a period of 1 year with the standard formula.\nI had to play with this for about 20 minutes to figure this out. When I said using his example, I kind of meant the whole thing. I know how to do the effective interest rate. I wasn't even sure which equation you were referring to.\n\nFuture value annuity equation:\n\n$\\text{FV=Pmt \\left(\\frac{(1+i)^n-1}i\\right)}$\n\nThen plug in the effective interest rate (the 9.38%) where the i is. This can only work if you do it for a full year. Also use it for the i in the denominator as well:\n\n$\\text{FV=2000\\,\\left(\\frac{(1+.093^2-1}{.0938}\\right)}\" \/>\n\nI'm going to jump straight to 2 periods, the value of which was 4580.44 as we saw.\n\nIf you work that out, it comes out to 4187.60. You still have to do the annuity due adjustment, still using the effective interest rate: 4187.60 (1.093 = 4580.40.\n\nIf you wanted the value in the middle of a year, you have to remember to add the next payment on first, and then proceed how I said in the other post, with the other equation for however many months you want.\n\nI also worked it with 3 periods to check it. That comes out to 6580.40 * 1.0938 for the adjustment = 7197.64.\n\nEtc.\n\nVery interesting. I wonder why we never learned this in finance. I don't know how the annuity equations are derived, but don't need to know, so I wouldn't consider that the math was beyond us. I'll have to play with that and figure out how to use it for various combinations of compounding and payments.\n\nI still don't get the k - 1 thing. k in finance stands for interest rate, so that's like saying 9% - 1, or 9.38% - 1. Don't know where that fits in.\n\n September 10th, 2011, 04:18 PM #10 Newbie \u00a0 Joined: Sep 2011 Posts: 4 Thanks: 0 Re: Value of an annuity in any given month CRG and Erimess, I was thinking about this in the shower and I thought of the exact same approach that you guys are coming up with. I am glad to see that you guys thought it would work also! Basic formula outline will be something like this: IF Months Since Start of Extra Payments <= 12 THEN (Annual Pmt) + (Interest Earned Over X Months) ELSE IF Months Since Start of Extra Payments > 12 THEN 'To figure out the # of years passed as of month X - for FV formula Excel formula FLOOR(Months Since Start of Extra Payments\/12,1) 'To figure out the # of months passed from the new year as of month X - for FV formula Months - Excel formula FLOOR(Months Since Start of Extra Payments\/12,1) ELSE 0 END IF So if the month was 34 the FLOOR formula would tell me that 2 full years have passed (34\/12 = 2.8333 with the number rounded down to the nearest number). 34 - (12*2) = 10 months of compounding in the current year. I could then use these figures in the FV formulas to calculate the value of the payments. I could do this very easily in Excel, but our teacher has challenged us to do this without using many of the Excel financial functions. He has also placed certain other restrictions in place, such as not making a straight monthly payment schedule and manually adding in the interest and annual payments. I am forced to take a different approach and I think this will work. I am going to try it out and I'll let you know how it goes. Thanks a ton for the help!\n\n Tags annuity, month\n\n### annuities at the start of the month math\n\nClick on a term to search for related topics.\n Thread Tools Display Modes Linear Mode\n\n Similar Threads Thread Thread Starter Forum Replies Last Post Bom55 Algebra 2 April 10th, 2013 08:10 PM Brandonlive Elementary Math 1 September 14th, 2012 06:29 AM warqzaka Algebra 12 July 5th, 2011 09:47 AM alexis2 Advanced Statistics 0 November 20th, 2010 06:45 PM Bom55 Calculus 0 December 31st, 1969 04:00 PM\n\n Contact - Home - Forums - Cryptocurrency Forum - Top","date":"2019-09-16 16:26:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 9, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6186578273773193, \"perplexity\": 891.5896008197045}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514572879.28\/warc\/CC-MAIN-20190916155946-20190916181946-00367.warc.gz\"}"}
| null | null |
Juan Jacobo Arbenz Guzmán (Quetzaltenango, 14 de setembro de 1913 – Cidade do México, 27 de janeiro de 1971) foi um político, militar e presidente da Guatemala de 1951 a 1954. Por ser descendente de suíços (seu pai, Hans-Jakob, era imigrante do país europeu), era conhecido como "El Suizo" e "El Chelón" ().
Tentou realizar uma reforma agrária, entrando em choque com o monopólio das empresas dos Estados Unidos nas terras da Guatemala, sobretudo a United Fruit Company. Em resposta seu governo foi alvo de golpe de Estado organizado pela CIA que instalou uma ditadura militar no país. Este foi o primeiro golpe de Estado promovido pela CIA na América Latina, durante a Guerra Fria.
Após o afastamento de Arbenz, sua família refugiou-se na embaixada do México, onde permaneceu por 73 dias. A morte de sua filha, Arabella Arbenz, em outubro de 1965, deixou-o profundamente abalado. Em 27 de janeiro de 1971, um acidente incomum foi fatal para o ex-presidente: durante o banho, o rádio caiu na banheira em que Arbenz se encontrava. Ele sofreu um infarto e faleceu aos 57 anos. Seu corpo foi enterrado ao lado de Arabella.
Era casado com María Cristina Vilanova, com quem tinha outros 2 filhos: Leonora e Jacobo Arbenz Jr.
Ver também
Guerra Civil da Guatemala
Revolução Guatemalteca
Nascidos em 1913
Naturais de Quetzaltenango (Guatemala)
Militares da Guatemala
Políticos da Guatemala
Governantes depostos por golpe de Estado
Mortos em 1971
|
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Q: Panel.Controls.Remove not deleting all the controls It should(.NET) I'm making a database manager program and the program creates only as much textbox
as needed (counts the columns in the table). But for some reason the program is not removing all the textboxes I want.
New textboxes should be created and old ones should be removed on every "ComboBoxSelectedIndexChange".
Here is my code snippet:
var elm = panel20.Controls.OfType<System.Windows.Forms.TextBox>();
foreach (var item in elm)
{
DeleteControlsWithTag(item.Tag.ToString());
}
button19.Enabled = true;
one_cond = string.Empty;
if (comboBox19.Text != string.Empty)
{
one_cond = comboBox19.Text.ToUpper();
if (one_cond == "ÖSSZES")
{
string allcolumn = "(";
colnames.Remove("Összes");
for (int i = 0; i < colnames.Count - 1; i++)
{
allcolumn += colnames.ElementAt(i) + ",";
}
allcolumn += colnames.Last() + ")";
button19.Enabled = false;
textBox16.Text = ManualQuery + " " + from + " " + allcolumn + " " + "VALUES" + " ";
if (3 < colnames.Count)
{
for (int i = 0; i < colnames.Count; i++)
{
System.Windows.Forms.TextBox textbox = new System.Windows.Forms.TextBox();
textbox.Name = $"textbox_{i}";
textbox.AccessibleName = $"textbox_{i}";
textbox.Tag = $"textbox_{i}";
panel20.Controls.Add(textbox);
textbox.Parent = panel20;
//"textbox_" + colnames[i] + i
if (2<i)
{
if (i == 3)
{
textbox.Location = new System.Drawing.Point(489, 49);
}
else
{
textbox.Location = new System.Drawing.Point(489 + ((i - 3) * 71), 49);
}
}
else
{
if (0 < i)
{
textbox.Location = new System.Drawing.Point(467 + (i * 71), 17);
}
else
{
textbox.Location = new System.Drawing.Point(467, 17);
}
}
textbox.Size = new System.Drawing.Size(64, 20);
}
}
else
{
for (int i = 0; i < colnames.Count; i++)
{
System.Windows.Forms.TextBox textbox = new System.Windows.Forms.TextBox();
textbox.Name = $"textbox_{i}";
textbox.AccessibleName = $"textbox_{i}";
textbox.Parent = panel20;
panel20.Controls.Add(textbox);
textbox.Tag = $"textbox_{i}";
if (0<i)
{
textbox.Location = new System.Drawing.Point(467 + (i*71), 17);
}
else
{
textbox.Location = new System.Drawing.Point(467, 17);
}
textbox.Size = new System.Drawing.Size(64, 20);
}
}
}
else
{
onecondinsert = "(" + one_cond + ")";
textBox16.Text = ManualQuery + " " + from + " " + onecondinsert + " " + "VALUES" + " ";
System.Windows.Forms.TextBox textbox = new System.Windows.Forms.TextBox();
textbox.Name = $"textbox";
textbox.AccessibleName = $"textbox";
textbox.Location = new System.Drawing.Point(464, 20);
textbox.Size = new System.Drawing.Size(100, 20);
textbox.Parent = panel20;
textbox.Tag = "textbox";
panel20.Controls.Add(textbox);
//tbox.TextChanged += tbox_TextChanged;
}
}
MessageBox.Show(elm.Count().ToString());
}
and here is some screenshot of how the problem looks like
on the first select it works
on this picture you can see the program did not delete all the text box.
longer textbox is the new one
On the third picture I selected another column (only one textbox should be visible) but the program deletes one textbox
as you can see here
After the third selection it deletes the last extra textbox too
3. try it "works"
I have tried different ways to delete from the panel20.controls
but all produced the same bug
Hope someone can help me, Thanks
Update:
I tried another method from here:
C#, deleting all the controls with the same .Tag
private List<Control> GetTaggedControls(string tag, Control parent)
{
var taggedControls = new List<Control>();
foreach (Control control in parent.Controls)
{
if (control.Tag?.ToString() == tag)
{
taggedControls.Add(control);
}
// Recursively call this method in case this is a container
taggedControls.AddRange(GetTaggedControls(tag, control));
}
return taggedControls;
}
// Deletes all controls with the specified tag
private void DeleteControlsWithTag(string tag)
{
foreach (Control control in GetTaggedControls(tag, this))
{
panel20.Controls.Remove(control);
}
}
and I tried to delete the textboxes like this:
var elm = panel20.Controls.OfType<System.Windows.Forms.TextBox>();
foreach (var item in elm)
{
DeleteControlsWithTag(item.Tag.ToString());
}
but it is still not working
A: Your GetTaggedControls() function is finding ANY type of control, RECURSIVELY starting at the main Form container level, that has a specific tag. So returned controls might not be from many containers, but your DeleteControlsWithTag() method tries to remove all matches only from panel20. That may or may not be a problem; it's unclear from the pictures and possibly incomplete posted code.
When you say remove the old textboxes, can we simply remove ALL existing TextBoxes, or do you really need to match tags?
From your original attempt:
var elm = panel20.Controls.OfType<System.Windows.Forms.TextBox>();
foreach (var item in elm)
{
DeleteControlsWithTag(item.Tag.ToString());
}
This is finding ALL textboxes in panel20, then trying to delete them based on matching tags with the helper function. So the tag doesn't even matter since you are finding all textboxes to begin with...
Just try removing all TextBoxes from panel20?
var elm = panel20.Controls.OfType<System.Windows.Forms.TextBox>().ToList();
elm.ForEach(tb => panel20.Controls.Remove(tb));
A: Okay, now I tried to tie together Idle_Mind's and LarsTech's recommendation
and finally it's working fine with "Dispose".
Thanks guys for your quick help.
here is the previous code:
var elm = panel20.Controls.OfType<System.Windows.Forms.TextBox>();
foreach (var item in elm)
{
DeleteControlsWithTag(item.Tag.ToString());
}
and here is the new one:
var elm = panel20.Controls.OfType<System.Windows.Forms.TextBox>().ToList();
elm.ForEach(tb => tb.Dispose());
|
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{"url":"https:\/\/www.shaalaa.com\/question-bank-solutions\/monochromatic-light-wavelength-589-nm-incident-air-water-surface-if-water-133-find-wavelength-frequency-speed-refracted-light-refraction_16383","text":"# Monochromatic Light of Wavelength 589 nm is Incident from Air on a Water Surface. If \u00b5 for Water is 1.33, Find the Wavelength, Frequency and Speed of the Refracted Light - Physics\n\nMonochromatic light of wavelength 589 nm is incident from air on a water surface. If \u00b5 for water is 1.33, find the wavelength, frequency and speed of the refracted light.\n\n#### Solution\n\nWavelength of light in air, \u03bb = 589 nm\nTherefore,\nFrequency of light in air is given by\n\nf=c\/lambda=(3xx10^8)\/(589xx10^-9)=5.09xx10^14 Hz\n\n\u200bSince, frequency of light does not depend on the medium. Thus, the frequency of the refracted light in water will be equal to the frequency of the incident or reflected light in air.\n\nHence, the frequency of refracted light = 5.09xx10^14 Hz\n\nNow\n\nSpeed of light in water, v=c\/mu=(3xx10^8)\/(1.33)=2.26xx10^8 m\/s\n\nWavelength of light in water,lambda=v\/f=(2.26xx10^8)\/(5.09xx10^14)=444.007xx10^-9 \"m or \" 444.007 \"nm\"\n\nTherefore the speed, frequency and wavelength of refracted light are 2.26 \u00d7 108 m\/s, 5.09\u00d71014 Hz and 444.007 nm, respectively.\n\nConcept: Refraction\nIs there an error in this question or solution?","date":"2021-03-03 15:07:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6993670463562012, \"perplexity\": 1290.33846838824}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178366969.45\/warc\/CC-MAIN-20210303134756-20210303164756-00371.warc.gz\"}"}
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\section{Introduction}
The search for alternatives to the Higgs mechanism generating the masses of gauge bosons and fermions in the electroweak theory is motivated by a feeling that this mechanism is hand-made, and some dynamical explanation might be actually possible (for a recent review see ref.~\cite{Ba93}).
A fundamental Higgs field is considered to provide a comfortable description of the spontaneous symmetry breaking, allowing a perturbative approach, but its use might not be really necessary.
Therefore most alternative attempts try to avoid the introduction of a fundamental scalar field.
The obligatory Goldstone bosons, as well as the Higgs boson, should it exist, are interpreted as composite states, formed by some strong dynamics beyond the standard model.
However, one should distinguish between a fundamental scalar field whose selfinteraction triggers the symmetry breaking already on the tree level, as in the conventional Higgs mechanism, and such a field playing some different important role in the dynamical symmetry breaking occuring only beyond the perturbative expansion.
In this second role a fundamental scalar field is acceptable provided its particle excitations are massive or confined.
Already known examples are strongly coupled Yukawa models, which can exhibit spontaneous chiral symmetry breaking (S$\chi$SB) even if the bare scalar potential in the action does not have the form of the classical Mexican hat~\cite{KoTa90,Ca91,ClRo91}.
On the lattice the S$\chi$SB\ can occur in such models even if the nearest neighbour coupling of the scalar field is antiferromagnetic, i.e.~competing against the S$\chi$SB\ generated dynamically by the Yukawa interaction term (see e.g.~\cite{LeShi90a,BoDe92b}, for reviews see refs.~\cite{Shr92,DeJe92}).
The scalar boson can be made heavy within the joint upper bounds on the Higgs
and fermion masses~\cite{BoFr93a,BoFr93b,FrLi93,LiMu93}.
In this paper we would like to point out that the experience accumulated on the lattice with strongly coupled vectorlike gauge theories with matter fields suggests the existence of still another alternative of dynamically generated S$\chi$SB.
The mechanism we want to describe assumes some new confining gauge field~$A$ of a compact gauge group~$G$, and makes use of a fundamental scalar field~$\phi$ which is coupled to this gauge field and, consequentely, confined.
The scalar field, however, does not generate the S$\chi$SB, in fact it is crucial that~$\phi$ acts against it.
The S$\chi$SB\ is generated dynamically by the interaction between the gauge field~$A$ and some fermion fields~$\chi$, making the gauge invariant condensate~$\langle\overline{\chi} \chi \rangle$ nonzero, in an analogy to the S$\chi$SB\ in QCD.
The role of the scalar field~$\phi$ is twofold.
First, it shields the $G$-charge of the fermion $\chi$, so that composite $G$-neutral physical fermion states of the form~$F = \phi^{\dagger}\chi$ can exist asymptotically in spite of the confinement of the $G$-charge.
This is why we call our proposal the {\em shielded gauge mechanism} (SGM).
In the phase with S$\chi$SB\ ({\em Nambu phase}) the fermion mass~$m_{\rm F}$ is nonzero,
so we have the case of dynamical mass generation (DMG).
The Goldstone bosons are composed of $\overline{\chi}$ and $\chi$.
We use the terms ``composite'' and ``dynamical'', whose meaning is somewhat obscure in strongly coupled field theories \cite{HaHa91}, essentially in the same sense as they are used in QCD.
Second, to make the model applicable in the continuum physics, we have to make the lattice constant small and thus approach a phase transition of 2$^{\rm nd}$ order.
Usually phase transitions in the lattice gauge theories without scalars occur at couplings of order one (lattice QED) or even at zero coupling (lattice QCD).
But the scalar field has the tendency to suppress the S$\chi$SB\ and induces for very strong gauge coupling a new second order phase transition, at which the chiral symmetry is smoothly restored.
In the scaling region of this phase transition in the Nambu phase, the mechanism might be applicable to the continuum physics if the corresponding field theory is nonperturbatively renormalizable.
What kind of continuum physics?
If e.g. the broken global chiral symmetry is SU(2) with one fermion field doublet $\chi$ coupled to~$A$, we find a massive fermion doublet $F= \phi^{\dagger}\chi$ and three massless Goldstone bosons $\pi$, all being gauge invariant with respect to~$G$.
Because of confinement due to the new gauge interaction there are no physical states corresponding to the field~$\phi$, but bosonic gauge invariant states consisting of~$\chi$,~$\phi$ and~$A$, e.g. of the type~$\phi^{\dagger}\phi$ and~$\overline{\chi} \chi$, should be expected, some of them possibly looking like the Higgs boson.
The situation is then quite similar to the standard model with the standard gauge fields switched off, symmetry breaking present, and one degenerated weak isospin fermion doublet heavy.
The other fermions, not coupling to $A$, are massless.
If the standard SU(2)$\otimes$U(1) gauge fields are then switched on, the broken global SU(2) changes into the local one, and the $\pi$'s lead to the massive vector bosons (see e.g. \cite{LiRo92,Li93}).
The Higgs mechanism has been replaced by the SGM but, of course, the fermion spectrum is quite nonrealistic, except possible speculations about a very heavy fourth family with small mass differences in weak isospin doublets.
A more realistic application is suggested by ideas ascribing the top quark a special role in the symmetry breaking.
In fact, the mechanism we are describing might be related to the top quark condensate models based on new strong gauge interactions at some energies beyond the electroweak scale, as suggested by various authors~\cite{LiRo92,Li93,Ho87,Hi91,Boe91,Li92,Ma92
\footnote{We thank M. Lindner for many elucidating discussions on this idea},
and our suggestion is inspired by their work.
Here it is e.g. assumed that the top-bottom doublet couples strongly to~$A$
whereas the other fermions couple only weakly or not at all to it, the interaction being flavour diagonal.
The problem is that these models are chiral gauge theories which we still cannot simulate on the lattice.
Thus for this kind of physical application of the SGM\ we have to assume that the dynamics of strongly coupled chiral gauge theories with scalars is similar to the vectorlike lattice models.
We point out that the dynamical scenario we consider is different from the hypothetical strongly coupled standard model~\cite{AbFa81a,AbFa81b,ClFa86} searched for in vain some years ago within the same class of lattice models.
There a confining phase with shielded fermions but without the chiral condensate has been assumed, whereas we are looking for a confining phase with both the shielded fermions and the nonvanishing chiral condensate.
On the lattice, such a phase is known since long, and the question is what are the properties of the continuum limit in this phase.
This is a fundamental question concerning the strongly coupled, not asymptotic free gauge theories.
The long term aim is to find out whether and how far the physical content of the SGM is different from the usual Higgs mechanism.
We do not insist on the existence of some new nontrivial fixed point, accepting the possibility that this mechanism could operate only in some energy range in analogy to the ``trivial'' Higgs-Yukawa sector of the standard model.
Even in such a case significant differences from the usual Higgs mechanism might be possible, as for example higher upper bounds for its validity.
To discuss the SGM\ we consider a specific chirally symmetric lattice model with a staggered fermion field~$\chi$, the gauge field described by the group elements~$U$, and the complex scalar field~$\phi$ of fixed absolute value ($\chi U\phi$\ {\em model}).
As the gauge group we choose the compact~$G$ = U(1) group with both scalar and fermion fields having ``charge'' one with respect to it.
This lattice model can be considered as a generic case for the models we have in mind, since it is a confining theory at strong coupling and thus similar also to nonabelian theories.
We note, however, that some similar extensions of the standard model based on the U(1) gauge group with strong coupling exist also in continuum \cite{LiRo92,Ho87,Boe91}.
In spite of the existing experience with this and similar lattice models, the proposed SGM\ is yet quite speculative because the critical line, which should be used for an approach to the continuum limit, is at present only poorly understood.
The scaling behaviour and the properties of the continuum limit at this transition are not yet known.
However, we argue that at least in the limit of infinite gauge coupling the SGM makes sense.
Here we recover a lattice transcription of the
Nambu--Jona-Lasinio~(NJL)~model which has been used by various authors
as an effective theory for some strong dynamics with S$\chi$SB\ beyond the standard~model~\cite{Na89,MiTa89,BaHi90}.
This observation is based on the Lee-Shrock~transformation~\cite{LeShr87a}
relating the lattice theories with fermions, gauge and scalar fields
at infinite gauge coupling to a pure four fermion theory on the lattice (see sec.~2.3).
Thus it is possible that the SGM can be described by the NJL~model in the low~energy limit and that the $\chi U\phi$\ model is a generalization of the NJL model.
In this paper we report results of a nonperturbative investigation of the SGM by means of numerical simulations of the 4D $\chi U\phi$~models on the lattice in the quenched approximation.
Several $\chi U\phi$~models in~4D have already been studied numerically some time ago, both in the quenched approximation (e.g.~\cite{LeShi86d,LeShr87b,DeShi88,LeShr88b,Shr89}), and with dynamical fermions~\cite{DaKo88,AoLe88,Ku89a,DaMe89,MePe91}.
The aim was either to clarify the general features of the models or to search for the strongly coupled standard model.
But the questions we are asking now require a more systematic investigation of several relevant physical quantities in the Nambu phase,
in particular the chiral condensate, the fermion mass and the bosonic spectrum.
The reason is that we want to look for lines of constant mass ratios, which can best elucidate the physical content in the continuum limit.
Up to now we obtained quenched numerical results for the chiral condensate,
the fermion mass and the pseudoscalar Goldstone boson mass, and have got an insight about the behaviour of these quantities in the vicinity of the critical line, where the approach to the continuum limit should be performed.
We also gained first experience with the determination of scalar and vector
bound states.
These numerical results confirm the existence of the Nambu phase and the
expectations about the scaling of some fermionic observables.
They support the hope that a sensible continuum limit can be found.
For the long-term~goal to investigate the renormalizability properties
of the $\chi U\phi$~models in 4D further simulations with dynamical
fermions~\cite{FrFr95} are necessary.
We want to remark that the SGM can be considered also in lower dimensional models.
As will be explained, the relationship between the $\chi U\phi$\ models and four fermion theories at strong gauge coupling \cite{LeShr87a} is independent of the dimension.
Therefore both in 3D and 2D the SGM exists as a renormalizable quantum field theory at least in the strong coupling limit of the $\chi U\phi$\ model.
Namely, in 3D we recover in this limit the nonperturbatively renormalizable~\cite{Gr75,RoWa89b,RoWa91} 3D~Gross-Neveu model.
However, we do not yet know whether in 3D the SGM for a large, but finite, coupling is in the same universality class as this model.
In 2D, we find at infinite gauge coupling the cherished asymptotically free
chiral Gross-Neveu model \cite{GrNe74} which is thus related to the SGM in 2D at large coupling.
This 2D~model exhibits DMG without S$\chi$SB~\cite{Wi78b,DeFo93b}
due to the Mermin-Wagner-Coleman~theorem.
This shows that the DMG is a little bit more general phenomenon
than the S$\chi$SB\, and we therefore prefer to use the term DMG in this paper.
The outline of the paper is as follows: In the following section we review the already known relevant properties of the lattice $\chi U\phi$\ model we are investigating.
In section 3 we describe the properties of the Nambu phase and of its critical boundary.
Then we discuss the question of renormalizability.
In section 4 the numerical data are presented.
In section 5 we give a summary and conclude with several speculative remarks.
\section{Lattice $\chi U\phi$~model at strong coupling}
Here we summarize the previous relevant knowledge about the lattice $\chi U\phi$\ models with coupled fermion field~$\chi$, gauge field variable~$U$ and scalar field~$\phi$.
\subsection{The $\chi U\phi$~model with U(1) gauge symmetry}
For definiteness we make the following choices on the euclidean hypercubic lattice in~$d$ dimensions:
\begin{itemize}
\item~$\chi$ is one staggered fermion field. The model has the continuous global U(1) chiral symmetry with respect to the transformations
\begin{equation}
\chi_x \rightarrow e^{i \alpha \varepsilon_x} \chi_x \;, \hspace*{1.5cm}
\overline{\chi}_x \rightarrow e^{i \alpha \varepsilon_x} \overline{\chi}_x \;,
\label{chtrafo}
\end{equation}
with the standard~$\varepsilon_x = (-1)^{x_1+ \cdots +x_d}$.
This is, however, only a residual symmetry, achieved on the lattice, whereas in the continuum limit the expected global chiral symmetry is U($N_f$)$\otimes$U($N_f$), $N_f$ being the number of fermion species found in this limit.
The charge of~$\chi$, determining its U(1) gauge transformation properties, is one.
\item $U$ is the gauge field defined on the lattice links.
The link variables are elements of the compact gauge group U(1).
They can be imagined as
\begin{equation}
U_{x,\mu} = e^{i g a A_{\mu} (x)} \; ,
\end{equation}
with~$A_\mu(x)$ being the abelian gauge field,~$g$ the gauge coupling constant and~$a$ the lattice constant.
\item $\phi$ is a complex scalar field of charge one.
It is convenient to impose the constraint~$|\phi |$=1, which corresponds to the choice of infinite quartic scalar selfcoupling.
This constraint is known not to restrict the physical content of the scalar sector.
For example block spin transformations set this constraint off.
\end{itemize}
The action is
\begin{equation}
S_{\chi U \phi} = S_\chi + S_U + S_\phi \; ,
\label{action}
\end{equation}
where
\begin{eqnarray}
S_\chi &=& {\textstyle \frac{1}{2}} \sum_x \overline{\chi}_x \sum_{\mu = 1}^d
\eta_{\mu x} \left[ U_{x,\mu} \chi_{x + \mu} -
U_{x-\mu,\mu}^\dagger \chi_{x - \mu} \right]
\, + \, a m_0 \sum_x \overline{\chi}_x \chi_x \; ,
\label{SCH} \\
S_U &=& \beta \, \sum_{\rm P} \left[ 1 - \mbox{Re} \{ U_{\rm P} \} \right] \; ,
\label{SU} \\
S_\phi &=& - \kappa \, \sum_x \sum_{\mu=1}^d \left[ \phi_x^\dagger
U_{x,\mu} \phi_{x + \mu} \,+\, {\rm h.c.} \right] \; .
\label{SPH}
\end{eqnarray}
Here~$\beta = a^{d-4}/g^2$, $U_{\rm P}$ is the plaquette product of link variables $U_{x,\mu}$ and $\eta_{\mu x} = (-1)^{x_1 + \cdots + x_{\mu - 1}}$.
All the fields and coupling parameters are dimensionless and therefore the calculated masses and expectation values are always in
the lattice units, e.g.\ the masses are of the form~$am$,~$m$ being
the mass in physical units.
The hopping parameter~$\kappa$ vanishes (is infinite) when the squared bare scalar mass is positive (negative) infinite.
The bare fermion~mass~$m_0$ is introduced for technical reasons, and the model is meant in the limit~$m_0 \!=\! 0$.
\subsection{Properties of various subsystems in 4D} \label{subsystems}
First we shall describe those properties of various subsystems which are relevant for the understanding of the whole $\chi U\phi$\ model in 4D at strong gauge coupling and of its phase diagram shown schematically in fig.~\ref{PD}.
\begin{figure
\centerline{
\fpsxsize=13.0cm
\fpsbox{figures/dmg94pd.ps}
}
\vspace{-1.cm}
\captio
[...caption in list of figures...
{
The schematic phase diagram of the~$\chi U\phi$~model~({\protect\ref{action}}) in~4D at $m_0 = 0$.
The emphasized points are: \\*[3mm]
\phantom{~~~}{\bf N}:~ critical point of the {\bf N}JL~model, which is the limit
of the~$\chi U\phi$\ model at~$\beta \!=\! 0$, \\
\phantom{~~~}{\bf E}:~ critical {\bf E}ndpoint of the
Higgs phase transition line~ETS, \\
\phantom{~~~}{\bf T}:~ {\bf T}riple point, \\
\phantom{~~~}{\bf C}:~ phase transition from the {\bf C}onfinement (at strong gauge coupling)
to \\
\phantom{~~~{\bf C}:}~ the {\bf C}oulomb phase (at weak gauge coupling)
in the model without the scalar field. \\
\phantom{~~~}{\bf S}:~ critical point of the {\bf S}pin model.
\\*[3mm]
The Nambu, Higgs and Coulomb phases are described in the text.
The dashed line corresponds to a 2$^{\rm nd}$ order phase~transition,
full lines to 1$^{\rm st}$ order transitions.
\label{PD}
\end{figure}
\begin{itemize}
\item \underline{$S_\chi$ at~$g\!=\!0$:} Setting~$U_{x,\mu}\!=\!1$, this action describes a free staggered fermion field.
Because of the lattice fermion doubling
the model in 4D describes $N_f \!=\! 4$ fermions in the scaling region.
For $m_0 = 0$, the global chiral symmetry is then U(4)$\otimes$U(4).
\item \underline{$S_U$:} This is the pure gauge theory on the lattice.
For a compact gauge group it is confining at small~$\beta$ (strong coupling) in the sense of the Wilson~loop criterion.
The chosen U(1) gauge theory with action (\ref{SU}) is a generic example.
It has a phase transition around~$\beta \!=\! \beta_c \simeq 1.0$, separating the confinement and Coulomb phases.
The spectrum in the confinement phase consists of glueball-like states (``gaugeballs'').
\item \underline{$(S_\chi + S_U)$:} The coupled fermion-gauge system exhibits for small~$\beta$ the S$\chi$SB\ with a nonvanishing chiral condensate~$\langle\overline{\chi} \chi \rangle$ and massless Goldstone~bosons.
The system is also confining in the sense that only gauge invariant states like~$\overline{\chi} \chi$ and gaugeballs are present in the spectrum.
The situation is quite similar to QCD.
The confinement-Coulomb phase transition is of first order~\cite{KoDa87} and occurs at~$\beta \!= \beta_c \simeq 0.89$ for~$N_f \!=\!4$ (point~C in fig.~\ref{PD}).
\item \underline{$S_\phi$ at~$g\!=\!0$:} We restrict our description to~$\kappa \ge 0$.
The pure scalar field theory, selfinteracting in consequence of the constraint~$|\phi| = 1$, has the global U(1) symmetry, i.e.\ it is the XY~model in 4D.
This symmetry is gauged for~$g > 0$.
At $g = 0$ it is spontaneously broken for~$\kappa > \kappa_c$ with~$\kappa_c \simeq 0.15$.
Below~$\kappa_c$ the symmetry is restored.
The phase transition at~$\kappa_c$ (point~S in fig.~\ref{PD}) is of magnetic type, the order parameter being~$\langle\phi\rangle$ as in spin models of ferromagnetism.
The mass~$am_\phi$ of the lowest mass excitation of the~$\phi$ field below~$\kappa_c$, the~$\phi$ boson doublet, diverges as~$\kappa \rightarrow 0$ and vanishes as~$\kappa \rightarrow \kappa_c$.
\item \underline{$(S_U + S_\phi)$:} This ``$U\phi$ {\em sector}'' is a typical example of lattice Higgs models, studied intensively nearly 10 years ago (for a review see e.g. ref.~\cite{Je86}).
Its limit cases at~$\beta \!=\! \infty$ and~$\kappa \!=\! 0$ are the pure scalar theory~$S_\phi$ at~$g\!=\!0$, and pure gauge theory~$S_U$, respectively.
The limit cases~$\beta \!=\! 0$ and~$\kappa \!=\! \infty$ have no phase transitions~\cite{OsSe78,FrShe79}, as can be seen by choosing the unitary gauge.
The magnetic pure scalar field transition point~S extends for~$\beta < \infty$ into a line of Higgs phase transitions, the line~ETS in fig.~\ref{PD}, whereas the confinement-Coulomb transition~C is the basis of the line~CT, T being a triple~point.
The ETS~line is weakly first order, E being a 2$^{\rm nd}$~order critical endpoint.
At this point the correlation length, corresponding to the inverse scalar boson mass in lattice units, diverges.
The Higgs mechanism operates above the ETS~line, whereas below the ET~line confinement takes place, and the physical states are gauge invariant composites of~$\phi^\dagger_x$ and~$\phi_y$ or gaugeballs.
The most important are the scalar and vector bosons.
Above the ETS line they correspond to the Higgs and gauge bosons in the gauge invariant formulation of the Higgs mechanism.
Below the ET line they are massive bound states of two massive scalars.
The analytic connection between both regions~\cite{OsSe78,FrShe79} (the so-called {\em complementarity} occuring in the absence of fermions) gave rise to speculations about the strongly coupled standard model~\cite{AbFa81a,AbFa81b,ClFa86}.
However, the transition region to the left of the point~E is difficult to understand in the continuum field theory language (for a recent attempt to explain complementarity see ref.~\cite{Hs93}).
Just this is the region where we expect the SGM\ to operate when the fermions are included.
There is no local order parameter distinguishing the regions above and below the line~ETS, as~$\langle\phi\rangle \!=\! 0$ identically for any~$g > 0$.
Nevertheless, the expectation value of the link product in eq.~(\ref{SPH}), the so-called link energy~$E_{\rm L}$ (see eq.~(\ref{EL})),
is very small below the ETS~line and starts to rise with increasing~$\kappa$ above this line, in some reminiscence to the behaviour of~$\langle\phi\rangle^2$ in the~$g\!=\!0$ case.
This can be used for a localization of the Higgs phase transition line~\cite{JaJe86,AlAz92,AlAz93}.
Another, more expensive possibility is to calculate the masses (in lattice units) $am_{\rm S}$ and~$am_{\rm V}$ of the scalar and vector bosons.
They have minima on the ETS~line when considered as functions of~$\kappa$ at fixed~$\beta$~\cite{EvJa87b}.
We note that the $U\phi$ sector describes the behaviour of all observables constructed from the $U$ and $\phi$ fields in the $\chi U\phi$\ model in the quenched approximation.
This approximation, neglecting virtual fermion loops, is frequently used in numerical simulations of theories with fermion fields on the lattice.
If the constraint $|\phi |=1$ is not imposed, the endpoint E moves to the left as the quartic scalar selfcoupling decreases, eventually reaching $\beta = 0$.
\end{itemize}
\subsection{The relation of the $\chi U \phi$ model to
four fermion theories in the strong coupling limit}
At~$\beta \!=\! 0$, the $\chi U\phi$~model in~$d$ dimensions can be rewritten exactly
as a lattice four fermion model~\cite{LeShr87a}.
In the path integral
\begin{equation}
Z = \int \prod_{x, \mu} {\rm d} \chi_x {\rm d} \overline{\chi}_x
{\rm d} \phi_x {\rm d} \phi_x^\dagger
{\rm d} U_{x,\mu}
\exp \{ \,-\, S_{\chi U \phi} \, \} \; ,
\label{PI}
\end{equation}
with~$S_{\chi U \phi}$ given in~(\ref{action}), at~$\beta \!=\! 0$ the scalar and gauge fields can be integrated out exactly.
This results in
\begin{equation}
Z = r^N J_1^{N_l} \, \int \, \prod_x {\rm d} \chi_x {\rm d} \overline{\chi}_x
\exp \{ \,-\, S_{\rm 4f} \, \} \; ,
\label{PI4f}
\end{equation}
where~$N$ and~$N_l$ is the number of lattice sites and links, respectively, and
\begin{equation}
S_{\rm 4f} = - \sum_x \sum_{\mu = 1}^d \left[ \, G \;
\overline{\chi}_x \chi_x \overline{\chi}_{x+\mu} \chi_{x+\mu} \, - \,
{\textstyle \frac{1}{2}} \eta_{\mu x}
\left( \overline{\chi}_x \chi_{x+\mu} - \overline{\chi}_{x+\mu} \chi_x \right)
\right] \, + \, \frac{am_0}{r} \sum_x \overline{\chi}_x \chi_x \; ,
\label{action4f}
\end{equation}
with
\begin{equation}
\renewcommand{\arraystretch}{1.4}
r = r(\kappa) = \frac{J_U}{J_1} \;\;\;\;\;\; \mbox{ and } \;\;\;
\begin{array}{rclll}
J_U &=& \int {\rm d} U e^{2 \kappa {\rm Re} \{ U \} } U &=&
I_1(2\kappa) \;, \\
J_1 &=& \int {\rm d} U e^{2 \kappa {\rm Re} \{ U \} } &=&
I_0(2\kappa) \;.
\end{array}
\renewcommand{\arraystretch}{1.}
\label{rofkappa}
\end{equation}
The fermion field has been rescaled by~$\sqrt{r}$.
The parameter $r$ is an analytic function of~$\kappa$ increasing monotonically
from~$r(0) \!=\! 0$ to~$r(\infty) \!=\! 1$.
The action (\ref{action4f}) obviously describes a lattice version of the four-fermion theory. The coupling term contains the nearest neighbour variables because the Grassmann variable $\chi$ has only one component.
The four~fermion coupling parameter~$G$ is related to~$\kappa$ via~$r$:
\begin{equation}
G := \frac{1-r^2}{4 r^2} \; .
\label{GNJL}
\end{equation}
From (\ref{rofkappa}) one sees that~$G$ is decreasing monotonically
with increasing~$\kappa$;~$G \!=\! \infty$ at~$\kappa \!=\! 0$,
and~$G \!=\! 0$ at~$\kappa \!=\! \infty$.
As discussed in~\cite{LeShr87a},
in~$S_{\rm 4f}$~(\ref{action4f}) the first and third terms are
explicitly gauge invariant.
For the second term it should be noted that
the only contributions to the expansion of the exponential
which survive the Grassmann integration are of the form~$\prod_{x \in C} \overline{\chi}_x \chi_x$, $C$ being closed loops, hence also the
second term is gauge invariant.
The Lee-Shrock transformation~(\ref{PI}) and~(\ref{PI4f}) is a gauge invariant analogy to the Hub\-bard-Stratonovich transformation used to bilinearize the four fermion actions by introducing an auxiliary scalar field.
The Lee-Shrock transformation introduces for the same purpose both an auxiliary gauge field (there is no pure gauge field term in the action, which corresponds to~$g \!=\! \infty$) and a dynamical scalar field.
The scalar field can be avoided, but the field~$U$ is then a vector auxiliary field without being gauge invariant~\cite{BoKe89}.
The transformation (\ref{PI}) and (\ref{PI4f}) holds only for one staggered fermion field, it can be generalized, however, to the case with several fermion fields.
The resulting pure fermionic theory then contains multifermion couplings of higher degree.
The constraint $|\phi |=1$ is important for the possibility to integrate out the scalar field.
In 4D, the action~(\ref{action4f}) describes the lattice NJL~model~\cite{BoKe89,Ho89,AlGo94}, having S$\chi$SB\ and DMG for~$G > G_c \approx 0.28$ corresponding to $\kappa < \kappa_c \simeq 0.97$.
The character of the phase transition is chiral, i.e. the order parameter is~$\langle\overline{\chi} \chi \rangle$.
In lower dimensions the lattice Gross-Neveu models are obtained.
\subsection{Phases of the complete $\chi U\phi$~system in 4D}
The phase diagram of the $\chi U\phi$~model with U(1) gauge symmetry in 4D is shown schematically in fig.~\ref{PD}.
It has been obtained by studying various limiting cases and by numerical simulations with quenched (e.g.~\cite{LeShi86d,LeShr87b,DeShi88,LeShr88b,Shr89,Ku89a}) and unquenched fermions~\cite{DaKo88,AoLe88,Ku89a,DaMe89,MePe91}.
The chiral phase transition of the NJL~model at~$\beta\!=\!0$ (point~N) extends to nonzero values of~$\beta$.
As pointed out in ref.~\cite{LeShr87b}, this can be derived by means of a convergent expansion around $\beta = 0$.
Therefore, this transition must remain to be of second order at least at small nonzero $\beta$ and cannot end in the interior of the phase diagram.
Furthermore, some properties of the NJL model will persist also at small nonzero $\beta$.
However, the striking observation is that, within the numerical precision, the chiral phase transition joins the Higgs phase transition line~ET at point~E, forming a smooth line~NET.
This is so both in the quenched and unquenched case, though the fermion feedback changes somewhat the position of the line~ET with respect to the quenched approximation.
The position of the chiral phase transition line~NE changes correspondingly.
In the vicinity of the point E new critical phenomena, different from the NJL model, might occur.
To our knowledge the interweaving of the chiral and Higgs phase transition is not understood theoretically.
(When the charges of the scalar and fermion fields are different, it does not occur \cite{LeShi86d,LeShr87b,LeShr88b}.)
Assuming that the transition lines join exactly, there are just two lines of phase transitions, the NETS~line and the CT~line, which separate the~$g \ge 0, \kappa \ge 0$ area into three phases:
\begin{itemize}
\item \underline{Nambu phase.}
This is the area below the NET~line.
Here the chiral condensate~$\langle\overline{\chi} \chi \rangle$ is nonvanishing.
Both~$\chi$ and~$\phi$ fields are confined, in analogy to the quark confinement in QCD.
The mass~$am_{\rm F}$ of the fermion state~$F = \phi^{\dagger}\chi$ is nonzero, and thus the DMG occurs.
The justification for the name of this phase is provided by the~$\beta\!=\!0$~limit.
In this limit the $F$ and $\chi$ fermions are identical.
As explained above, here one recovers the four~fermion~model with the mechanism of mass generation, which is traditionally called~``dynamical''.
In this model the ideas of the top quark condensate have been originally formulated \cite{Na89,MiTa89,BaHi90}.
\item \underline{Higgs phase.}
This is the area above the NETS~line.
Here~$\langle\overline{\chi} \chi \rangle\!=\! am_{\rm F} \!=\! 0$ and the Higgs mechanism operates.
All physical states are gauge invariant and the U(1) charge is screened by the scalar condensate~\cite{EvGr87,EvJa87b}.
Choosing the unitary gauge, one recovers for weak~$g$ by standard perturbative methods the massive U(1) vector boson and the Higgs boson.
\item \underline{Coulomb phase.}
This is the area below the TS~line.
The gauge boson is massless and~$\langle\overline{\chi} \chi \rangle\!=\! am_{\rm F} \!=\! 0$.
\end{itemize}
A very important observation in 4D is that the line~NE is of second order, whereas the ETS~line is most probably also in the unquenched case weakly first order~\cite{DaKo88,DaMe89,FrFr95}.
In this case the point~E is a tricritical~point, where the order of the phase transition changes from the second order (the NE~line) to the weakly first order (the ETS~line).
\section{Dynamical mass generation and chiral phase transition in strongly coupled gauge theories}
Numerous investigations of various strongly coupled gauge theories with fermions on the lattice revealed that the DMG is a generic property of these theories.
It occurs not only in QCD and similar nonabelian theories, but even in the lattice QED at strong coupling, in both its compact~\cite{KoDa87} and noncompact~\cite{KoHa90,GoHo92} formulations.
Can such a naturally occuring dynamical S$\chi$SB\ and DMG be of use for the symmetry breaking in the standard model or beyond it, e.g. replacing the Higgs mechanism?
What is required is a mechanism generating fermion masses, breaking chiral symmetry and providing 3 Goldstone bosons, which is the role of the Higgs-Yukawa sector of the standard model.
The large top quark mass makes acceptable an approximation to the fermion spectrum, in which only the top quark is massive, the other fermions of the standard model being massless.
In this simplified situation we need to break dynamically a global chiral SU(2) symmetry and ensure that there will be a massive fermion in the spectrum.
This can be the top quark, as suggested in the top condensate models~\cite{Na89,MiTa89,BaHi90,Hi91,Boe91,LiRo92,Ma92}, or possibly some fermion(s) in a new generation.
Except for being nonchiral, the lattice $\chi U\phi$~model with the compact U(1) gauge group is a prototype of the lattice models with such properties.
The DMG occurs naturally at strong coupling and not too large~$\kappa$ (the Nambu phase in fig.~\ref{PD}) due to the gauge interaction, i.e. without any help of the charged scalar field~$\phi$.
The question is how to approach the continuum limit and what are its properties, in particular what spectrum is to be expected.
\subsection{Confinement and spectrum in the Nambu phase}
In order to find the most suitable way to the continuum, let us first discuss the spectrum within the Nambu phase.
The fundamental fermions are confined, but the scalar field allows us to construct gauge invariant composite fermions of the form~$F = \phi^{\dagger}\chi$, which should correspond to the physical fermions (here we neglect the quark confinement due to QCD).
For an illustration of what can happen in the Nambu phase nonperturbatively, one can think of QCD with a scalar ``quark''~$\varphi$ in addition to the standard quarks~$q$.
We would then expect the fermionic ``mesons'' of the form~$\varphi^\dagger q$, analogous to the above fermion $F$, the lowest one being stable with respect to the interaction mediated by $A$.
Then there would be many other quark-antiquark~states.
This analogy suggests the existence of massive scalar and vector states like~$\phi^{\dagger}\phi$ and~$\overline{\chi} \chi$ in the $\chi U\phi$~model.
Thus an occurrence of a Higgs-like bound state is probable.
The ratio of its mass to the fermion mass will most probably depend on the way the continuum limit is approached.
Possibly also ``gaugeballs'', states similar to glueballs in the QCD, etc., might exist.
In any case the massless pseudoscalar Goldstone bosons composed of~$\overline{\chi}$ and~$\chi$ must be present.
On the other hand, the scalar field~$\phi$ itself is confined.
It is important to realize that for small~$\kappa$ all states containing~$\phi$ get much heavier than the~$\overline{\chi} \chi$~states or gaugeballs, because the constituent~$\phi$ gets infinitely heavy as~$\kappa \rightarrow 0$.
As seen from eq.~(\ref{SPH}), the scalar field loses its kinetic term in this limit and simultaneously decouples, as there is no Yukawa coupling.
At~$g \!=\! 0$, when there is no confinement, one can define the physical mass~$am_\phi$ of the scalar~$\phi$ as the inverse correlation length in the XY~model, which is known to be infinite at~$\kappa \!=\! 0$.
The mass~$am_\phi$ is small only in the vicinity of the point~S and rapidly increases with the distance from this point.
For~$g > 0$ one can think of the ``constituent'' mass of~$\phi$ being roughly equal to $am_\phi$ at the same distance from the line NETS as in the $g = 0$ limit.
That this picture is appropriate has been indicated by the quenched calculations of the masses of the~$\phi^\dagger \phi$~states in the present and similar models~\cite{EvJa87b,EvJe86}.
The masses of various physical states containing~$\phi$ in the $\chi U\phi$\ model can be understood qualitatively by assuming that the constituent mass of~$\phi$ is small along the line~NETS and grows below this line with increasing distance from it, getting infinite at~$\kappa \!=\! 0$.
Thus also the fermion mass~$am_F$ in lattice units is large at small~$\kappa$ and infinite at~$\kappa \!=\! 0$, in agreement with fermion confinement, if the shielding scalar is absent.
\subsection{Where to approach the continuum limit}
As lattice models are applicable to continuum physics only in the vicinity of critical points, where the masses~$am$ in lattice units scale to zero, we have to approach the boundary~NETC of the Nambu phase.
The phase transition along the segment~ET is most probably of first~order and thus unsuitable.
The phase transition along the CT~line is probably of first~order, too~\cite{KoDa87}.
One can think of changing the gauge part of the action~(\ref{SU}) in such a way that this transition gets continuous~\cite{EvJe85,Ok89}.
However, as discussed above, the scalar constituent of the fermion~$F$ would be heavy at a larger distance from the ETS~line, and the fermion mass~$m_{\rm F}$ would be therefore much larger than that of the various~$\overline{\chi} \chi$ and gaugeball states.
This is undesirable if we look for an alternative to the Higgs mechanism.
What remains is the NE~phase transition line.
Its most important property is a smooth vanishing of~$\langle\overline{\chi} \chi \rangle$
and of~$am_{\rm F}$, like in the NJL model at $\beta = 0$.
However, also other observables, constructed by means of the~$U$
and~$\phi$~fields, can be considered, though they, being of the~$\phi^\dagger \phi$ or gaugeball type, have no natural counterpart in the NJL~model.
The prominent examples are the masses $am_{\rm S}$ and~$am_{\rm V}$ of the scalar and vector bosons occuring already in the spectrum of the $U\phi$ sector.
They are important for the understanding of the Higgs~phase transition at point E and to the right of it.
In particular,~$am_{\rm S}$ vanishes at the point~E.
Here also the order of the phase transition changes, which in the statistical mechanics is usually associated with a tricritical behaviour (values of critical exponents, etc.)
different from the behaviour at ordinary critical points.
Around the point~E the character of the phase transition changes from
the chiral to the mixed chiral-Higgs phase transition. Therefore, when
considering the approach to the continuum limit we should distinguish
between the vicinity of the point~E and the rest of the NE~line.
For example, quenched calculations suggest that along the NE line, except the point E, no state of the type $\phi^{\dagger}\phi$ scales and thus would not be present in the continuum spectrum.
Gaugeballs are not expected to scale if the NE line is approached.
It is instructive to elucidate the importance of the scalar field
for the chiral phase transition.
The DMG at small~$\kappa$ is understood as a consequence of the
strong gauge coupling, like in QCD at large distances.
The required fluctuations of the gauge field are, however, gradually
suppressed by the scalar field when~$\kappa$ increases.
In the limit~$\kappa \rightarrow \infty$, which corresponds to large negative squared bare scalar mass, the gauge field freezes.
This is seen from~(\ref{SPH}) in the unitary gauge,~$\phi_x \!=\! 1$,
when at~$\kappa \!=\! \infty$ all the gauge variables are frozen
at the value~$U_{x,\mu} \!=\! 1$, and consequently the chiral condensate is zero.
So the chiral transition on the line~NET takes place when the
suppression of the gauge field fluctuations by the scalar field is
sufficiently strong.
This is consistent with the picture in the~NJL~model
at~$\beta\!=\!0$:
here the DMG ends when the four~fermion coupling,
which according to~(\ref{GNJL}) decreases with increasing~$\kappa$,
is sufficiently small.
\subsection{The shielded gauge mechanism in continuum}
We are now in a position to formulate the idea of the SGM in the $\chi U\phi$\ model.
Essentially it is a possibility that this model, or some suitable generalization of it, is renormalizable in the vicinity of the NE line and thus can be used for the continuum physics in a large range of energies.
This is meant in the following nonperturbative sense:
On the lattice the concept of renormalizability is conveniently formulated in terms of the lines of constant physics.
These are the lines in the bare parameter space along which the dimensionless ratios of physical observables stay approximately constant as the cutoff is varied.
Such lines, if at all, are found in the vicinity of critical manifolds in their scaling regions, where the lattice artefacts are negligible.
Lines of constant physics can hit the critical manifold, i.e.\ the masses~$am$ in lattice units get zero, as they are inverse correlation lengths.
Then these lines can be used for continuum physics without any intrinsic energy restriction, because the lattice cutoff can be completely removed, as for example in the asymptotically free theories.
It can also happen that lines of constant physics approach the critical manifold, yet remain in some small but finite distance from it.
The cutoff can then be made large, but has to be kept finite, and the lines can be used for continuum physics only in some energy range substantially smaller than the cutoff.
This is the situation in the trivial theories like the Higgs-Yukawa sector of the standard model.
Both types of theories can be considered as renormalizable, in the second case with an energy restriction.
For the SGM to operate it is necessary that the $\chi U\phi$\ model with the U(1) gauge symmetry, or some similar model, have lines of constant physics in the Nambu phase in the vicinity of the NE~line.
The NJL~model~points at~$\beta \!=\! 0$ can be special points of these lines, but we cannot exclude that these lines completely avoid the~$\beta \!=\! 0$ line in fig.~\ref{PD}.
We do not insist that the cutoff can be removed completely, i.e. that a nontrivial fixed point exists.
This would be nice and is not impossible but, as the success of the standard model shows, not necessary.
It can happen that in a given model the lines of constant physics do not exist.
This is apparently the case in the lattice NJL~model~\cite{AlGo94}.
In such a situation one may try to generalize the model by introducing new fields and/or couplings, so that the parameter space is enlarged by a few relevant parameters, and the lines of constant physics can be found.
One well known example is the nonlinear~$\sigma$ model in 4D with its generalization to the full~$\phi^4$ theory.
Also the NJL~model can be generalized to the Yukawa model by changing the auxiliary character of the scalar field into a dynamical one~\cite{HaHa91,Zi91,BoDe92b,DeJe92}.
The strongly coupled QED on the lattice is another example of a model in which the existence of the lines of constant physics is at least disputed~\cite{KoHa90,GoHo92}, but its appropriate generalization has not yet been found.
We have two qualitative arguments for the expectation that the SGM\ could operate in the $\chi U\phi$\ model as defined in sec. 2.1.
First, the generalization from the NJL~model to the $\chi U\phi$\ model is similar to the step from the auxiliary to the dynamical scalar field generalizing successfully the NJL to the Yukawa model.
Second, as~$\beta$ increases, the character of the NE~line gets more complicated than in the NJL case.
As we have discussed above, the chiral character gets mixed with the Higgs type of phase transition, in particular around the point~E.
This means that new relevant parameter(s) may come into game.
This should be expected in particular if the point~E is a tricritical~point.
However, it could be necessary to generalize the model by adding one or two more couplings, possibly in a similar way as the four fermion theories are generalized to Yukawa theories \cite{HaHa91,Zi91,BoDe92b}.
To explain this generalization we first recall that the four fermion theories are usually bilinearized by introducing an scalar auxiliary field, which we call~$\Omega$.
Its symmetry properties are those of the global chiral symmetry group.
Such a transformation of a four fermion theory opens space for a generalization by including also the kinetic and selfcoupling term of the~$\Omega$ field into the action, obtaining in this way a Yukawa model~\cite{HaHa91,Zi91,BoDe92b}.
In 4D this generalization makes from the NJL~model a renormalizable theory.
For a strong Yukawa coupling the field~$\Omega$ can be understood as a composite of~$\overline{\chi} \chi$.
An analogous generalization of the $\chi U\phi$\ model might thus consist in introducing a composite scalar field~$\Omega = \overline{\chi} \chi$ with charge zero and including into the action (\ref{action}) also the kinetic and selfcoupling terms of this field.
One might hope that the model has similar renormalizability properties as strongly coupled Yukawa models.
These models have been found to have DMG and fit well into scenarios with SGM.
We note that some gauged Yukawa models have been considered \cite{Ho91,Ko93,Kr93,KoShi94} for similar purposes.
Finally, it could be also important to relax the constraint $|\phi |$=1.
Unfortunately, with such amendments the models with SGM would get quite complex.
The idea of the SGM\ is profoundly nonperturbative and requires numerical verification in a very difficult regime of the lattice field theories around the point E: the gauge fields are strongly coupled, i.e. strongly fluctuating, and ultimately the SGM should be investigated with dynamical fermions.
However, as in QCD, it could be that many essential features of the $\chi U\phi$\ models, in particular the DMG, can be studied in the quenched approximation, which is the aim of the next section.
\section{Some numerical results in the quenched approximation}
Our nonperturbative numerical results in the strong coupling region, obtained up to now, confirm some of the anticipated features of the Nambu phase, which are necessary for the SGM\ to work:
In the phase with S$\chi$SB, where~$\langle\overline{\chi} \chi \rangle$ is nonzero, also the fermion~mass in lattice units~$a m_{\rm F}$ is nonzero, implying DMG.
Both observables seem to scale to zero when the transition line~NE is approached. This is so even in the vicinity of the point E, whereas along the ET line the transition is apparently of first order also in fermionic observables.
The results for~$\langle\overline{\chi} \chi \rangle$ and~$a m_{\rm F}$ are also consistent with
the prediction of the gap~equation relating these two quantities and obtained from the spectral function of the free fermion with the physical mass $am_F$.
The pseudo-scalar fermion-antifermion state $\pi$ behaves in the Nambu phase like a Goldstone~boson, i.e.~$(a m_\pi)^2$ is a linear function of the bare fermion~mass~$a m_0$.
This behaviour is unique for the Nambu phase.
We have initiated the study of further states in the spectrum
which we will use in our forthcoming work \cite{FrFr95} in the search for the lines of constant mass ratios.
The mass of the scalar fermion-antifermion state $\sigma$ has been determined only very roughly, however.
This state might be a candidate for the effective (composite) Higgs boson.
At the moment we can only say that the staggered fermions can be used in this kind of models as at least in the pseudoscalar and vector channels no problems with the flavour symmetry restoration have been detected.
These results are described in some detail in this rather technical section.
\subsection{Definitions of the observables}
For the localization of the phase transition lines and the
determination of the particle spectrum the following
observables were used:
The normalized plaquette and link energies, defined as
\begin{eqnarray}
E_{\rm P} &=& \frac{1}{6V} \sum_{\rm P}
{\rm Re} \{ U_{\rm P} \} \;, \label{EP} \\
E_{\rm L} &=& \frac{1}{4V} \sum_{x,\mu}
{\rm Re} \{ \phi^\dagger_x U_{x,\mu} \phi_{x+\mu} \} \;,
\label{EL}
\end{eqnarray}
where~$V\!=\!L^3 T$ is the lattice volume.
These observables have been used for the localization of
the Higgs phase transition in the $U\phi$ sector
(see section~\ref{subsystems})~\cite{JaJe86,AlAz92,AlAz93},
i.e.\ in the quenched approximation of the $\chi U\phi$~model.
In particular, in ref.~\cite{AlAz93}~$E_{\rm P}$ and~$E_{\rm L}$
where used in the framework of the multihistogram method
for a high-precision determination of the position of
the endpoint~E in this approximation.
For the localization of the chiral phase transition line,
in particular the segment~NE,
we must use fermionic observables. In the quenched approximation
this line can only be seen in fermionic quantities because
of the missing feedback of the fermions to the bosonic fields.
We measure the chiral~condensate with the stochastic~estimator~method:
\begin{equation}
\langle\overline{\chi} \chi \rangle = \left\langle \frac{1}{V} {\rm Tr} \{ M^{-1} \} \right\rangle
\approx \left\langle \vec{\eta}^{\,\dagger} M^{-1} \vec{\eta} \right\rangle \;,
\label{cbc}
\end{equation}
where~$\vec{\eta}$ is a vector of dimension~$V$ filled with
gaussian random numbers (see,~e.g.,~\cite{BiKe89}).
It should be noted that with fixed-length scalar field~$|\phi_x| \!=\! 1$
this condensate~$\langle\overline{\chi} \chi \rangle$ coincides with the
condensate~$\langle \overline{F} F \rangle$ constructed from the
neutral fermionic fields~$F$ defined below.
The next important fermionic quantity is the mass of the physical fermion.
We expect in the spectrum of the Nambu phase a neutral fermionic state whose mass~$a m_{\rm F}$ is nonzero and scaling to zero when the chiral~transition line~NE is approached.
We have considered the gauge invariant fermionic field
\begin{equation}
F_x := \phi^\dagger_x \chi_x \;, \hspace*{2cm}
\overline{F}_x = \phi_x \overline{\chi}_x \; ,
\label{FFbar}
\end{equation}
and determined numerically the corresponding fermion propagator
\begin{equation}
\renewcommand{\arraystretch}{0.5}
G_{\rm F}(t) = \frac{2^3}{V} \sum_{\begin{array}{c} \scriptstyle \vec{x} \\*[0.5mm]
\scriptstyle x_1,x_2,x_3 \\
\scriptstyle {\rm even}
\end{array}}
\sum_{\begin{array}{c} \scriptstyle \vec{y} \\*[0.5mm]
\scriptstyle y_1,y_2,y_3 \\
\scriptstyle {\rm even}
\end{array}}
\left\langle F_{\vec{x},t} \overline{F}_{\vec{y},0}
\right\rangle \;.
\label{GF}
\renewcommand{\arraystretch}{1}
\end{equation}
The sums mean that we use a special wall source
where only one point per hypercube is set to~1, the remaining ones to~0,
as usual for staggered fermions (see e.g. \cite{GuGu91}).
The numerical data for~$G_{\rm F}(t)$
are fitted with MINUIT to the Ansatz
\begin{equation}
G_{\rm F}(t) = A_{\rm F}
\left( e^{- E_{\rm F} t} - (-1)^t e^{- E_{\rm F}(T-t)}
\right) \;.
\label{GFfit}
\end{equation}
The mass $a m_{\rm F}$ of the gauge invariant fermion and the corresponding wave function renormalization constant~$Z_{\rm F}$
are then
\begin{eqnarray}
a m_{\rm F} &=& \sinh E_{\rm F} \;, \label{mFfromEF} \\
Z_{\rm F} &=& A_{\rm F} \left( 1 + e^{- E_{\rm F} T} \right) \cosh E_{\rm F} \;. \label{ZFfromAFEF}
\end{eqnarray}
Having in mind our long-term aim to determine lines of constant
physics, i.e.\ lines of constant mass ratios, we have started to look
at further states in the spectrum.
First we define the fermion-antifermion composite states,
the ``mesons''.
Table~\ref{tabmesons} shows a list of the operators in the
more familiar continuum notation and the corresponding
translation to the formulation with staggered fermions.
Lattice experts will notice that we use the four simplest local operators from the Golterman~tables~\cite{Go86}.
We note that it is irrelevant whether these operators are
constructed with the $\chi$~field or with the neutral~$F$~field~(\ref{FFbar})
because of the fixed length of the~$\phi$~field.
\begin{table}
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{|r|c|r|c|l|l|} \hline
\vphantom{$\displaystyle \sum^a$} $i$ & continuum & staggered fermions & $s^{ik}_x$ & $J^{PC}$ & particle \\*[1mm] \hline\hline
1&$\overline{\ps} \psi$&$\overline{\chi}_x \chi_x$&1&$\begin{array}{l}
0^{++}_s \\ 0^{-+}_a
\end{array}$ &
$\begin{array}{l}
\sigma \; {\rm (f}_0) \\ \pi^{(1)}
\end{array}$ \\ \hline
2&$\overline{\ps} \gamma_5 \psi$&$\eta_{4x}\xi_{4x} \overline{\chi}_x \chi_x$&$\eta_{4x} \xi_{4x}$&
$\begin{array}{l}
0^{+-}_a \\ 0^{-+}_a
\end{array}$ &
$\begin{array}{l}
- \\ \pi^{(2)}
\end{array}$ \\ \hline
3&$\overline{\ps} \gamma_k \psi$&$\eta_{kx} \varepsilon_x \xi_{kx} \overline{\chi}_x \chi_x$&$\eta_{kx} \varepsilon_x \xi_{kx}$&
$\begin{array}{l}
1^{++}_a \\ 1^{--}_a
\end{array}$ &
$\begin{array}{l}
a \\ \rho^{(3)}
\end{array}$ \\ \hline
4&$\overline{\ps} \gamma_k \gamma_5 \psi$&$\eta_{4x} \xi_{4x} \eta_{kx} \varepsilon_x \xi_{kx} \overline{\chi}_x \chi_x$&$\eta_{4x} \xi_{4x} \eta_{kx} \varepsilon_x \xi_{kx}$&
$\begin{array}{l}
1^{+-}_a \\ 1^{--}_a
\end{array}$ &
$\begin{array}{l}
b \\ \rho^{(4)}
\end{array}$ \\ \hline\hline
\end{tabular}
\renewcommand{\arraystretch}{1}
\caption%
[Mesonic operators]%
{
List of mesonic operators in familiar continuum notation
and in the staggered fermion formulation, together with
the continuum quantum number assignement~$J^{PC}$ and the names
of the corresponding QCD particles.
The sign factors~$s^{ik}_x$ are composed of the standard
staggered phase factors~$\eta_{\mu x} = (-1)^{x_1+ \cdots +x_{\mu-1}}$,
$\xi_{\mu x} = (-1)^{x_{\mu+1}+ \cdots +x_4}$
and~$\varepsilon_x = (-1)^{x_1 + \cdots + x_4}$.
}
\label{tabmesons}
\end{center}
\end{table
The timeslice operators for the mesons are given by
\begin{equation}
{\cal O}^{ik} (t) = \sum_{\vec{x}} s^{ik}_{\vec{x},t} \overline{\chi}_{\vec{x},t}
\chi_{\vec{x},t} \;,
\label{Omesons}
\end{equation}
with~$s^{ik}_x$ given in table~\ref{tabmesons}.
We measure the correlation functions
\begin{equation}
G^{(i)}_{\rm mesons} (t) = \frac{1}{N^{(i)}_k} \sum_k^{N^{(i)}_k}
\left\langle {\cal O}^{ik} (t) {\cal O}^{ik} (0)
\right\rangle \;, \;\;\; i = 1,\ldots,4 \;,
\label{Gmesons}
\end{equation}
where~$N_k^{(i)} \!=\! 1$ for~$i \!=\! 1$,~2 and~$N_k^{(i)} \!=\! 3$
for~$i \!=\! 3$,~4, i.e.\
for~$i\!=\!3$,~4 we average
over the space directions~$k\!=\!1,\ldots,3$.
For the actual measurement~$G^{(i)}_{\rm mesons}$ has to be expressed
in terms of the inverse fermion matrix~$M^{-1}$:
\begin{eqnarray}
G^{(i)}_{\rm mesons} (t) &=& \frac{1}{N_k^{(i)}} \sum_k^{N_k^{(i)}}
\left\langle \sum_{\vec{x} \vec{y}}
s^{ik}_{\vec{x},t} \overline{\chi}_{\vec{x},t} \chi_{\vec{x},t}
s^{ik}_{\vec{y},0} \overline{\chi}_{\vec{y},0} \chi_{\vec{y},0}
\right\rangle \nonumber \\
&=& - \frac{1}{N_k^{(i)}} \sum_k^{N_k^{(i)}}
\left\langle \sum_{\vec{x} \vec{y}}
s^{ik}_{\vec{x}} s^{ik}_{\vec{y}}
(M^{-1})_{(\vec{x},t),(\vec{y},0)}
(M^{-1})_{(\vec{y},0),(\vec{x},t)}
\right\rangle \;
\label{Cm2}
\end{eqnarray}
up to a constant disconnected term.
In the second line we used the fact that the~$s^{ik}$ factorize according to:
\begin{equation}
s^{ik}_x = s^{ik}_{\vec{x},t} = s^{ik}_t s^{ik}_{\vec{x}} =
s^{ik}_{\vec{x}} \;,
\label{sik}
\end{equation}
because~$s^{ik}_t \!=\! 1$ for all~$(ik)$.
The relation
\begin{equation}
(M^{-1})_{(\vec{x},t),(\vec{y},0)}
(M^{-1})_{(\vec{y},0),(\vec{x},t)} =
\varepsilon_{\vec{y},0} \varepsilon_{\vec{x},t}
(M^{-1})_{(\vec{x},t),(\vec{y},0)}
\left[ (M^{-1})_{(\vec{x},t),(\vec{y},0)} \right]^*
\label{MMeeMM}
\end{equation}
shows that actually only one fermion matrix inversion
is needed per source~point.
With
\begin{equation}
\varepsilon_{\vec{x},t} = (-1)^{t} \, \eta_{4 \vec{x}}
\end{equation}
we finally have
\begin{equation}
G^{(i)}_{\rm mesons} = - (-1)^t \frac{1}{N_k^{(i)}} \sum_k^{N_k^{(i)}}
\left\langle \sum_{\vec{x},\vec{y}}
s^{ik}_{\vec{x}} s^{ik}_{\vec{y}}
\eta_{4 \vec{x}} \eta_{4 \vec{y}}
(M^{-1})_{(\vec{x},t)(\vec{y},0)}
\left[ (M^{-1})_{(\vec{x},t)(\vec{y},0)}
\right]^* \right\rangle \;.
\end{equation}
In our simulations we use point~sources for this measurement.
We fit these~$G^{(i)}_{\rm mesons}$ to the Ansatz~\cite{AlGo94}
\begin{eqnarray}
G_{\rm mesons} (t) &=& \phantom{(-1)^t} \sum_n A^+_n
\left( e^{-E^+_n t} + e^{-E^+_n (T-t)} \right) \nonumber \\
&+& (-1)^t \sum_n A^-_n
\left( e^{-E^-_n t} + e^{-E^-_n (T-t)} \right) \nonumber \\
&+& \phantom{(-1)^t} B^+ \nonumber \\
&+& (-1)^t B^- \;.
\label{Gmesonsfit}
\end{eqnarray}
Here $E^+$ is the energy of the s-wave states whereas $E^-$ that of the parity partners, which are p-waves.
The masses of the mesons are then computed from these energies by
\begin{equation}
a m = 2 \sinh ( E / 2) \;.
\label{mfromE}
\end{equation}
Note that already this minimal set of~$4$~mesonic operators
given in table~\ref{tabmesons} allows us to check
for flavour symmetry restoration, because there are
two states having overlap with
two different operators: the~$\pi$ shows up in~${\cal O}^{(1),(2)}$
and the~$\rho$ in~${\cal O}^{(3),(4)}$.
We are further interested in the scalar and vector bosons, which
are present also in the~$U \phi$~sector without fermions.
The corresponding operators are
\begin{eqnarray}
{\cal O}^{({\rm S})} (t) &=& \frac{1}{L^3} \sum_{\vec{x}} {\rm Re}
\left\{
\sum_{i=1}^{3} \phi^\dagger_{\vec{x},t} U_{(\vec{x},t), i}
\phi_{\vec{x}+\vec{i},t} \right\} \;,
\label{OS} \\
{\cal O}^{({\rm V})}_{i} (t) &=& \frac{1}{L^3} \sum_{\vec{x}} {\rm Im}
\left\{ \phi^\dagger_{\vec{x},t} U_{(\vec{x},t), i}
\phi_{\vec{x}+\vec{i},t} \right\} \;, \;\;\;\;\;\;
i = 1,\;2,\;3 \;,
\label{OV}
\end{eqnarray}
and the correlation functions are
\begin{eqnarray}
G^{({\rm S})} (t) &=& \left\langle {\cal O}^{({\rm S})} (t)
{\cal O}^{({\rm S})} (0)
\right\rangle \;, \label{GS} \\
G^{({\rm V})} (t) &=& \frac{1}{3} \sum_{i=1}^3 \left\langle
{\cal O}^{({\rm V})}_i (t)
{\cal O}^{({\rm V})}_i (0)
\right\rangle \;. \label{GV}
\end{eqnarray}
We fit them to a simple~$\cosh$-Ansatz:
\begin{equation}
G (t) = A \left( e^{-E t} + e^{-E (T-t)} \right) \;.
\label{GSVfit}
\end{equation}
The masses of the scalar boson~$a m_{\rm S}$ and the vector boson~$a m_{\rm V}$
are then computed from the energies~$E_{\rm S,V}$ by means of the formula~(\ref{mfromE}).
\subsection{Numerical investigation of the NE~line
in the quenched approximation}
As already stated above, the existing lattice results for the
magnetic transitions in the~$U \phi$~sector~\cite{JaJe86,AlAz92,AlAz93}
are also relevant for the $\chi U\phi$~model in the quenched approximation,
and thus the only new feature of the phase diagram with quenched fermions
is the chiral~transition.
The Lee-Shrock transformation~(\ref{PI}),~(\ref{PI4f}) in combination
with mean field theory predicts a
critical point at~$\beta\!=\!0$ and~$\kappa_c \approx 1.15$
for~$N_f \!=\! 4$,~\cite{LeShr87a} with
S$\chi$SB\ below~$\kappa_c$ and chiral~symmetry restauration above~$\kappa_c$.
The first question is whether this chiral phase transition
continues to finite~$\beta$ and where it runs to.
Earlier quenched investigations~\cite{LeShi86d,LeShr87b,DeShi88} already come to the result
that the chiral~phase~transition joins the line~ET.
Our comprehensive results for~$\langle\overline{\chi} \chi \rangle$ and~$a m_{\rm F}$ confirm
this observation.
In particular on the line~ET, where the bosonic observables indicate
a~$1^{\rm st}$~order transition, we also observe jumps in~$\langle\overline{\chi} \chi \rangle$
at the same places.
On the other hand, along the line~NE, even quite close to point~E,
we do not observe any indication for a discontinuous phase transition.
This is demonstrated
in figure~\ref{figcbc}, where we compare the numerical results for~$\langle\overline{\chi} \chi \rangle$
as a function of~$\kappa$ at~$\beta$~values slightly below and
above~$\beta_{\rm E} \!=\! 0.8485(8)$~\cite{AlAz93}.
\begin{figure
\centerline{
\fpsxsize=9.0cm
\fpsbox{figures/cbgb83k.ps}
}
\centerline{
\fpsxsize=9.0cm
\fpsbox{figures/cbgb90k.ps}
}
\captio
[...caption in list of figures...
{
The chiral condensate as a function of~$\kappa$ at two~$\beta$~values:
one slightly below~($\beta = 0.83$, upper figure) and one slightly
above~$\beta_{\rm E} \approx 0.85$~($\beta = 0.90$, lower figure).
The dashed lines are included to guide the eye.
\label{figcbc}
\end{figure}
The fermion~mass~$a m_{\rm F}$ shows a similar behaviour, as is shown
in figure~\ref{figmF}.
From these numerical results we conclude that in the quenched approximation
the chiral phase transition line joins the line~ET, as shown in
the schematic phase diagram in figure~\ref{PD},
changing at the point~E
from~$2^{\rm nd}$~order on the~NE~line to~$1^{\rm st}$~order on the~ET~line.
\begin{figure
\centerline{
\fpsxsize=9.0cm
\fpsbox{figures/mbgb83k.ps}
}
\centerline{
\fpsxsize=9.0cm
\fpsbox{figures/mbgb90k.ps}
}
\captio
[...caption in list of figures...
{
The fermion mass~$a m_{\rm F}$ as a function of~$\kappa$ at two~$\beta$~values:
one slightly below ($\beta = 0.83$, upper figure) and one slightly
above~$\beta_{\rm E} \approx 0.85$~($\beta = 0.90$, lower figure).
The dashed lines are included to guide the eye.
\label{figmF}
\end{figure}
Figure \ref{figmF} also demonstrates the DMG below the line~NE, where the fermion mass is clearly nonzero and seems to scale to zero when this line is approached.
The finite~$a m_{\rm F}$ values in the chirally symmetric phase are probably
an artefact of the small lattice volume and the finite bare~mass~$m_0$,
but of course this conjecture has to be confirmed by some theoretically
well-funded extrapolation scheme to the thermodynamic limit and
to~$m_0 \rightarrow 0$, e.g. by methods developed in refs.~\cite{LeShr87b,LeShr88a,Ho89}.
In figure~\ref{figmFcbc} we plot our numerical results for~$a m_{\rm F}$
as a function of~$\langle\overline{\chi} \chi \rangle$ together with the relation between them, which follows from the free fermion propagator with the observed fermion mass (the gap equation):
\begin{equation}
\langle\overline{\chi} \chi \rangle = \frac{1}{V} \sum_{p_\mu}
\frac{a m_{\rm F}}{(a m_{\rm F})^2 + \sum_\mu \sin^2 p_\mu } \;.
\label{gapeq}
\end{equation}
Here the sum runs over the set of momenta corresponding to
a finite lattice with periodic (antiperiodic) boundary~conditions
in the space (time) directions, as we have it in our simulations.
Nearly all the data (except at~$\beta\!=\!0$) lie in a narrow band
around the curve (\ref{gapeq}), indicating that the fermion wave function renormalization constant is close to one.
Such a behaviour was previously observed also in
strongly coupled noncompact~QED on the lattice~\cite{GoHo92}
and in the NJL~model~\cite{AlGo94}, both with dynamical fermions.
\begin{figure
\centerline{
\fpsxsize=12.0cm
\fpsbox{figures/mFcbc.ps}
}
\captio
[...caption in list of figures...
{
All our quenched results for~$a m_{\rm F}$ as a function of~$\langle\overline{\chi} \chi \rangle$.
The dotted line is the prediction of the gap~equation~({\protect\ref{gapeq}}).
\label{figmFcbc}
\end{figure}
An indicator for S$\chi$SB\ is the
pseudo-scalar~mass~$a m_\pi$ as a function of~$a m_0$:
in the chirally broken phase the ``$\pi$ meson'' should behave
like a Goldstone~boson, i.e.~$(a m_\pi)^2$ should go to zero linearly
as a function of~$a m_0$.
Conversely, one should see deviations
from such a behaviour in the chirally symmetric phase.
In our numerical results for~$a m_\pi$ we observe both these
possibilities; figure~\ref{figmpi2} is an example.
\begin{figure
\centerline{
\fpsxsize=12.0cm
\fpsbox{figures/mpi2.ps}
}
\captio
[...caption in list of figures...
{
The square of the pseudoscalar~mass~$(a m_\pi)^2$ as a function of~$a m_0$
(it is denoted~$a m_\pi^{(2)}$ because it was measured with the second operator in table~{\protect\ref{tabmesons}};~$a m_\pi^{(1)}$, measured with the first operator, is consistent with~$a m_\pi^{(2)}$ but is much more noisy).
For~$\kappa$ within the Nambu phase~$(a m_\pi)^2$ is linear in~$a m_0$,
for the higher~$\kappa$ deviations are observed.
\label{figmpi2}
\end{figure}
The resulting estimation of the position of the chiral~phase~transition
agrees well with those from~$\langle\overline{\chi} \chi \rangle$ and~$a m_{\rm F}$.
We have also measured the other mesonic states listed in table~\ref{tabmesons}
as well as the bosonic masses~$a m_{\rm V}$~(\ref{OV}) and~$a m_{\rm S}$~(\ref{OS}).
However, these states would require much higher statistics than
we could achieve in this first explorative stage of the investigations
we are reporting here.
The mass of the composite Higgs boson $\sigma$, $am_\sigma$, is quite difficult to determine, which indicates that it might be rather large.
Furthermore, we can make the statement that within the (large) error bars
we do not see from this side any indications for problems e.g.\ with
flavour symmetry restauration, i.e.\ within the error bars~$a m_\pi^{(1)}$ agrees with~$a m_\pi^{(2)}$
and~$a m_\rho^{(3)}$ with~$a m_\rho^{(4)}$, respectively.
\section{Summary and discussion}
The SGM, which we are considering as a possible substitute for the Higgs mechanism, is based on the observation that strongly coupled lattice gauge theories as a rule break dynamically a global chiral symmetry.
The model we have discussed demonstrates that, at least on the lattice, an arrangement is possible in which the Goldstone bosons and the mass of some heavy fermion arise, in a qualitative analogy to the Higgs-Yukawa sector of the standard model with a heavy quark.
The role of the scalar field is crucial, though completely different from that in the standard Higgs mechanism.
Whether a useful approach to the continuum limit can be achieved is still an open question.
However, at least in the strong coupling limit the model we have considered is promising:
In 4D it reduces to the lattice NJL~model which, though not renormalizable, has many attributes required for a viable theory of symmetry breaking in the standard model.
In 3D and even more in 2D the lattice four fermion theories found in this limit are established as renormalizable field theories and their continuum limit is no principal obstacle.
The question whether, and exactly how, the SGM operates beyond the strong coupling limit is a difficult nonperturbative problem requiring a substantial effort.
The difficulty is mainly due to the lack of analytic understanding of the transition between the Higgs and confinement regions.
We see essentially three possibilities.
One of them is extremely optimistic, namely that the $\chi U\phi$\ models could have a nontrivial fixed point, at which the cutoff can be removed completely without losing the interaction and thus the SGM.
The most obvious candidate is the point E.
This point has been up to now investigated only in the quenched approximation.
In the SU(2) model \cite{Bo90} some indication that the point might be nontrivial has been found, whereas in the U(1) model \cite{AlAz92,AlAz93} the data are consistent with the mean field like behaviour.
In any case an investigation with unquenched fermions is required, which, as the experience with the strongly coupled QED shows, might be a very difficult task.
However, also if the $\chi U\phi$\ models would turn out to be trivial, their quantitative properties might still differ significantly from the normal Higgs-Yukawa models.
For example the upper bounds could be larger than the unitarity bounds in these models.
In contrast to the Higgs-Yukawa models there might exist a strongly interacting sector.
A somewhat disappointing possibility would be that the SGM might turn out to be equivalent to the Yukawa theory, in a similar way as the generalized four fermi theory is \cite{HaHa91,Zi91}.
Then the SGM would be only an alternative formulation of the Higgs mechanism, a particularly difficult one.
This outcome would mean that it is very difficult to substantiate the top condensate idea by means of strongly coupled gauge theories.
In any case it seems to us that an investigation of the shielded gauge mechanism of dynamical mass generation might shed new light on the little understood properties of (chiral?) gauge theories at strong coupling.
How far the SGM might be viable phenomenologically is difficult to judge at this stage.
An important question of the models replacing the Higgs mechanism is the occurence of some scalar, which might look like the Higgs boson.
Within the SGM there are two obvious candidates: the scalar bound states of
the type $\phi^{\dagger}\phi$ and~$\overline{\chi} \chi$.
In the quenched approximation the first of these states scales only at the point E.
As the fermion mass scales in the vicinity of the whole NE line, one might think that the ratio of their masses might be chosen quite arbitrarily by approaching the point E in different ways.
But this must be studied in a simulation with dynamical fermions, as well as the mass of the scalar $\overline{\chi} \chi$ state.
So at the moment we can say little about a possible occurence and mass of the Higgs-like state in the SGM.
Of course, from the phenomenological point of view, the considered lattice model is not specific enough.
It is much simpler than the existing more elaborate attempts in the continuum field theory \cite{Ho87,Hi91,Boe91,LiRo92,Ma92}.
We agree with those models only in some basic ideas.
It could be that the existing models are not yet good enough, e.g. giving incorrect top quark mass.
The contribution of the lattice methods to the further search for better models might consist in a more reliable control of the dynamical problems, treated otherwise in a rather qualitative way.
\vspace{2cm}
{\large\bf Acknowledgements}\\
We thank W. Bock, W. Franzki, P. Hasenfratz, C.B. Lang, M. Lindner, X.-Q. Luo, R.E. Shrock and M.A. Stephanov for discussions and various suggestions, and H.A. Kastrup for continuous support.
\clearpage
\bibliographystyle{wunsnot}
|
{
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(English)Zbl\u00a01483.65076\n\nMSC:\u00a0 65F99 15A83 90B06\nFull Text:\n\n### Multiobjective approximate gradient projection method for constrained vector optimization: sequential optimality conditions without constraint qualifications. (English)Zbl\u00a007503423\n\nMSC:\u00a0 90C29 90C46 90C30\nFull Text:\n\nFull Text:\n\nFull Text:\n\n### A matrix-free trust-region Newton algorithm for convex-constrained optimization. (English)Zbl\u00a007500342\n\nMSC:\u00a0 90C26 90C53\nFull Text:\n\nFull Text:\n\n### Vibration analysis of a strain gradient plate model via a mesh-free moving kriging interpolation method. (English)Zbl\u00a007496061\n\nMSC:\u00a0 74-XX 76-XX\nFull Text:\n\n### Accelerated proximal envelopes: application to componentwise methods. (English. Russian original)Zbl\u00a007492860\n\nComput. Math. Math. Phys. 62, No. 2, 336-345 (2022); translation from Zh. Vychisl. Mat. Mat. Fiz. 62, No. 2, 342-352 (2022).\nMSC:\u00a0 90C30 90C52\nFull Text:\n\n### Stable solution of a quadratic minimization problem with a nonuniformly perturbed operator by applying a regularized gradient method. (English. Russian original)Zbl\u00a01484.65115\n\nComput. Math. Math. Phys. 62, No. 1, 10-19 (2022); translation from Zh. Vychisl. Mat. Mat. Fiz. 62, No. 1, 12-22 (2022).\nMSC:\u00a0 65J20\nFull Text:\n\nFull Text:\n\n### Perturbation techniques for convergence analysis of proximal gradient method and other first-order algorithms via variational analysis. (English)Zbl\u00a007490849\n\nMSC:\u00a0 90C52 49J52 49J53\nFull Text:\n\n### An efficient three-term conjugate gradient-based algorithm involving spectral quotient for solving convex constrained monotone nonlinear equations with applications. (English)Zbl\u00a007490257\n\nMSC:\u00a0 65Kxx 65Nxx\nFull Text:\n\n### On the acceleration of the Barzilai-Borwein method. (English)Zbl\u00a007490087\n\nMSC:\u00a0 90C20 90C25 90C30\nFull Text:\n\nFull Text:\n\nMSC:\u00a0 90C25\nFull Text:\n\n### Linear convergence of a nonmonotone projected gradient method for multiobjective optimization. (English)Zbl\u00a007489940\n\nMSC:\u00a0 90C29 90C30 65K05\nFull Text:\n\n### Four algorithms to solve symmetric multi-type non-negative matrix tri-factorization problem. (English)Zbl\u00a007489927\n\nMSC:\u00a0 90C26 90C90\nFull Text:\n\n### Restrictive preconditioning for convection-diffusion distributed control problems. (English)Zbl\u00a01482.65051\n\nMSC:\u00a0 65F10 65F08 65N22\nFull Text:\n\n### Generalized fractional algebraic linear system solvers. (English)Zbl\u00a007488735\n\nMSC:\u00a0 65F10 65N22 35R11\nFull Text:\n\nFull Text:\n\nFull Text:\n\n### Transcranial stimulation analysis using the smoothed finite element method. (English)Zbl\u00a007487684\n\nMSC:\u00a0 65-XX 76-XX\nFull Text:\n\n### A Hermite interpolation element-free Galerkin method for functionally graded structures. (English)Zbl\u00a007483694\n\nMSC:\u00a0 74Sxx 74Kxx 65Nxx\nFull Text:\n\nFull Text:\n\n### Inexact variable metric method for convex-constrained optimization problems. (English)Zbl\u00a007477380\n\nMSC:\u00a0 90C30 90C59\nFull Text:\n\n### A proximal gradient splitting method for solving convex vector optimization problems. (English)Zbl\u00a007477374\n\nMSC:\u00a0 90C25 90C52 65K05\nFull Text:\n\n### Modified spectral PRP conjugate gradient method for solving tensor eigenvalue complementarity problems. (English)Zbl\u00a007475186\n\nMSC:\u00a0 90C30 65K05 15A18\nFull Text:\n\nFull Text:\n\nFull Text:\n\n### Tensorial conditional gradient method for solving multidimensional ill-posed problems. (English)Zbl\u00a01484.65074\n\nMSC:\u00a0 65F22 65F10 68U10\nFull Text:\n\nall top 5\n\nall top 5\n\nall top 5\n\nall top 3\n\nall top 3","date":"2022-07-07 16:03:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6964088082313538, \"perplexity\": 11216.165227358082}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104495692.77\/warc\/CC-MAIN-20220707154329-20220707184329-00698.warc.gz\"}"}
| null | null |
Q: PHP how to hide some of page numbers? I want to ask how i can hide some of the page numbers on the result page so they show like this:
[First] [Back] [1] [2] [3] [4] [5] ..... [10] [11] [12] [13] [14] [15] [Next] [Show Last]
Now it's showing like this:
[1] [2] [3] [4] [5] [6] [7] [8] [8] [10] [11] [12] [13] [14] [15]
But i don't want that. I want to make them show like in the first example.
Here is the code that i use for making the results in pages :
<?PHP
$requested_page = isset($_GET['page']) ? intval($_GET['page']) : 1;
// Get the product count
$r = mysql_query("SELECT COUNT(*) FROM `oferts`");
$d = mysql_fetch_row($r);
$product_count = $d[0];
$products_per_page = 20;
// 55 products => $page_count = 3
$page_count = ceil($product_count / $products_per_page);
// You can check if $requested_page is > to $page_count OR < 1,
// and redirect to the page one.
$first_product_shown = ($requested_page - 1) * $products_per_page;
// Then we retrieve the data for this requested page
$r = mysql_query("SELECT * FROM `oferts` ORDER BY id DESC LIMIT $first_product_shown, $products_per_page ");
while($rowi = mysql_fetch_array($r))
{
$title = addslashes($rowi['title']);
$fromweb = addslashes($rowi['fromweb']);
$image = addslashes($rowi['image']);
$link = addslashes($rowi['link']);
$price = addslashes($rowi['price']);
$realprice = addslashes($rowi['realprice']);
$endDate = addslashes($rowi['endDate']);
$calc = $realprice / $price;
if ($realprice == 0)
{
$procent = 0;
}
if ($realprice != 0)
{
$percent = 100/$calc;
$reshenie = 100-$percent;
$procent = substr($reshenie,0,2);
}
if ($fromweb == '1') {
$divname = "ozo";
}
if ($fromweb == '2') {
$divname = "vip";
}
$seconds = strtotime("$endDate") - time();
$days = floor($seconds / 86400);
$seconds %= 86400;
$hours = floor($seconds / 3600);
$seconds %= 3600;
$minutes = floor($seconds / 60);
$seconds %= 60;
$position=65;
$titleNumbered = mb_substr($title, 0, $position, "utf-8");
$healthy = array('\"');
$yummy = array('');
$TitleText = str_replace($healthy, $yummy, $titleNumbered);
?>
<div class="OfferBox">
<center>
<div class="<?PHP echo $divname;?>"></div>
<a href="<?PHP echo $link;?>" target="_blank"><img src="<?PHP echo $image;?>" style="width:190px;border:1px solid #E5E5E5;"/></a>
<div class="OfferText"><?PHP echo "$TitleText...";?></div>
<div class="p1"><b><?PHP echo $realprice;?>лв.</b></div><div class="p2"><b>-<?PHP echo $procent;?>%</b></div><div class="p3"><b><?PHP echo $price;?>лв.</b></div>
<a href="<?PHP echo $link;?>" class="OfferButton" target="_blank"></a>
<div class="timeleft">Изтича след:<br><?PHP echo "$days дни | $hours часа | $minutes мин.";?></div>
</center>
</div>
<?PHP
}
?>
</div>
<?PHP
echo '<div style="clear:both;display:block;bottom:0;float:right;margin-right:20px;">';
for($i=1; $i<=$page_count; $i++) {
if($i == $requested_page) {
echo "<span class='pagenumberSelected'><b>$i</b></span> ";
} else {
echo '<a href="index.php?page='.$i.'" class="pagenumber"><b>'.$i.'</b></a> ';
}
}
echo '</div><br>';
?>
So how i can make it?
How i can hide some of the page numbers when they are more?
A: This is the relevant peace of code:
<?php
for($i=1; $i<=$page_count; $i++) {
if($i == $requested_page) {
echo "<span class='pagenumberSelected'><b>$i</b></span> ";
} else {
echo '<a href="index.php?page='.$i.'" class="pagenumber"><b>'.$i.'</b></a> ';
}
}
?>
There are 3 variables which matter:
$i, $page_count and $requested_page
*
*$i = iterator
*$page_count = total pages
*$requested_page = page which is loaded
in PSEUDO CODE
if $page_count > 10
show if $i < 3
show if $i = $requested_page -1
show if $i = $requested_page
show if $i = $requested_page +1
show if $i > $page_count - 3
A: Here an example how to show only the last 10 pages instead of every page before the current page:
for($i=$page_count-10; $i<=$page_count; $i++) {
if($i == $requested_page) {
echo "<span class='pagenumberSelected'><b>$i</b></span> ";
} else {
echo '<a href="index.php?page='.$i.'" class="pagenumber"><b>'.$i.'</b></a> ';
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,910
|
\section{Introduction}
Orthogonal time frequency space (OTFS) modulation has received considerable attention in the past few years since its introduction in~\cite{Hadani2017orthogonal}, thanks to its capability of enabling highly reliable communication over high-mobility channels~\cite{Zhiqiang_magzine}.
The most important new feature of OTFS modulation compared to conventional orthogonal frequency-division multiplexing (OFDM) modulation is the delay-Doppler (DD) domain information multiplexing, which motivates OTFS transceiver design based on the DD domain channel response. Consequently, conventional transceiver designs for OFDM systems optimized based on the time-frequency (TF) domain channel characteristics cannot be directly applied in OTFS systems as they are not able to harvest the full benefits of DD domain information multiplexing.
In Part II of this three-part tutorial, we aim to provide an in-depth discussion on OTFS transceiver design. Specifically, we study the key elements of the transceiver, including cyclic prefix (CP) insertion, pulse shaping, channel estimation, and signal detection. In particular, the commonly used message passing algorithm (MPA) for OTFS detection is explained based on the \emph{maximum a posteriori} (MAP) criterion, and simulation results are presented to evaluate the error performance of various detection schemes.
Furthermore, we compare the performances of OTFS and OFDM in terms of diversity gain and achievable rate, where we also numerically verify the advantages of coded OTFS modulation over coded OFDM.
\section{Transmitter Design}
The transmitter design is of great importance for practical application of OTFS. As explained in Part I, there are two common implementations of OTFS, namely, symplectic finite Fourier transform (SFFT)-based OTFS and discrete Zak transform (DZT)-based OTFS. In this section, we will provide further details on the transmitter design for both SFFT-based and DZT-based OTFS, respectively.
Similar to the Part I, we assume that one OTFS frame occupies a bandwidth of $B_{\rm OTFS}$ and a time duration of $T_{\rm OTFS}$, which accommodates $M$ subcarriers with subcarrier spacing $\Delta f = \frac{{{B_{{\rm{OTFS}}}}}}{M}$ and $N$ time slots with slot duration $T = \frac{{{T_{{\rm{OTFS}}}}}}{N}$.
\subsection{Cyclic Prefix Design for SFFT-based OTFS}
The SFFT-based implementation was proposed in the first OTFS paper~\cite{Hadani2017orthogonal}. In particular, the SFFT-based implementation can be viewed as the concatenation of an inverse SFFT (ISFFT) module and the Heisenberg transform, where the latter one can be realized with an inverse fast Fourier transform (IFFT) module followed by a transmit pulse shaping filter~\cite{Hadani2017orthogonal}.
The details of SFFT-based OTFS have been covered in Part~I. Here,
we focus on CP design. Specifically, there are two commonly used options for inserting the CP into SFFT-based OTFS, i.e., \textit{full-CP} OTFS and \textit{reduced-CP} OTFS.
In the full-CP scheme, a CP is inserted in each time slot to combat the delay spread of the channel, similar to what is done in conventional OFDM~\cite{RezazadehReyhani2018analysis}. On the other hand,
in the reduced-CP scheme, only one CP is appended at the start of the frame
with a duration longer than the maximum delay spread of the channel. Reduced-CP OTFS has been officially introduced in the literature in~\cite{Raviteja2019practical}.
A key property of full-CP OTFS is that intersymbol interference (ISI)-free transmission can be guaranteed after CP removal at the receiver side, similar to conventional OFDM~\cite{RezazadehReyhani2018analysis}. As a result,
signal detection can be performed in the TF domain, where only the impact of the Doppler shifts of the channel has to be considered. Therefore, full-CP OTFS transmissions may enable reduced-complexity signal detection.
On the other hand, reduced-CP OTFS may be the more attractive option. In contrast to full-CP OTFS, the reduced-CP scheme does not guarantee ISI-free transmission, but it generally requires a much smaller signaling overhead.
In fact, the purpose of the reduced-CP scheme is to ensure that the received sequence is $MN$-periodic ($MN$ is the frame length) after CP removal, such that DZT can be employed for receiver processing, yielding an effective DD domain channel matrix with block diagonal structure~\cite{Raviteja2019practical}. There are some interesting variations of reduced-CP OTFS. For example, it is reported in~\cite{Raviteja2018interference,pandey2021low} that padding zeros instead of adding a CP results in a more structured effective DD domain channel matrix, at the cost of a small power loss.
\subsection{Window Design for SFFT-based OTFS}
An appealing advantage of SFFT-based OTFS is that it facilitates TF domain window design~\cite{wei2021transmitter}, which introduces additional DoFs for further improvements of the channel estimation and data detection performance compared to the commonly used rectangular window.
The windowing at the transmitter can be interpreted as power allocation in the TF domain, while the windowing at the receiver causes colored noise\cite{wei2021transmitter}.
If channel state information (CSI) is available at both transmitter and receiver, the transmitter window can be optimized for minimization of the detection mean squared error (MSE). The obtained solution can be interpreted as a mercury/water-filling power allocation, where the mercury is filled first, before water is poured to pre-equalize the doubly selective TF domain channel\cite{wei2021transmitter}.
If CSI is not available at the transmitter, fixed window designs, such as the Dolph-Chebyshev (DC) window\cite{wei2021transmitter}, in the TF domain can enhance channel sparsity and thus improve channel estimation performance, enabling a smaller guard space overhead.
We refer interested readers to \cite{wei2021transmitter} for a more detailed discussion of window designs for OTFS modulation.
\subsection{Pulse Shaping for DZT-based OTFS}
Different from SFFT-based OTFS, DZT-based OTFS directly converts the DD domain signal into the time-delay (TD) domain without converting the signal first into the TF domain. DZT-based OTFS transmitters generally comprise an IDZT module and a pulse shaping filter ${g_{{\rm{tx}}}}\left( t \right)$. According to (15) in Part I and~\cite{lampel2021orthogonal},
the discrete DD domain equivalent transmitted symbols can be obtained via the DZT of the samples of the TD domain transmit signal $s\left( t \right)$,
such that
\begin{align}
{\cal D}{{\cal Z}_{{s}}}\left[ {l,k} \right] = \sqrt {MN} {X_{{\rm{DD}}}}\left[ {l,k} \right]{\cal D}{{\cal Z}_{g_{\rm tx}}}\left[ {l,k} \right],\label{DD_equivalent_with_pulse}
\end{align}
where information symbol ${X_{{\rm{DD}}}}\left[ {l,k} \right]$ is the $(l,k)$-th element of the DD domain information matrix ${\bf X}_{\rm DD}$ of size $M \times N$, with $l \in \left\{ {0,...,M - 1} \right\}$ and $k \in \left\{ {0,...,N - 1} \right\}$. In~\eqref{DD_equivalent_with_pulse}, ${{\cal DZ}_{g_{\rm tx}}}$ denotes the DZT of
vector ${\bf g}_{\rm tx}$ containing the \emph{periodically extended} pulse samples, i.e., for the $k$-th element of ${\bf g}_{\rm tx}$, we have $g_{\rm tx}\left[ k \right] \buildrel \Delta \over = {g_{{\rm{tx}}}}\left( {\frac{{{{\left[ k \right]}_{MN}}}}{M}T} \right)$, $k \in {\mathbb Z}$, where ${\left[ \cdot \right]}_{N}$ denotes the modulo operation with respect to (w.r.t.) $N$.
\begin{figure}
\centering
\includegraphics[width=3.5in]{Fig/Equivalent_model.eps}
\caption{Block diagram of the equivalent model for DZT-based OTFS.}\vspace{-5mm}
\label{Zak_equivalent_model}
\centering
\end{figure}
The literature on OTFS pulse shape design is not mature yet, however, we may still provide some intuition for pulse shape design in the DD domain.
In fact,~\eqref{DD_equivalent_with_pulse} suggests an interesting interpretation of pulse shape design in the DD domain, where the pulse shaping can be viewed as a point-wise multiplication.
Notice that the DZT is defined for $MN$-periodic sequences. With the reduced-CP scheme mentioned in the previous subsection, the overall DZT-based OTFS communication system can be equivalently modelled as in Fig.~\ref{Zak_equivalent_model}.
In Fig.~\ref{Zak_equivalent_model}, we assume that the same pulse is employed for
both transmit pulse shaping and receive matched-filtering, and its \emph{periodically extended} sample vector is given by $\bf g$, i.e., ${{\bf g}_{{\rm{tx}}}}={{\bf g}_{{\rm{rx}}}}={{\bf g}}$.
Thus, the pulse shape design for OTFS may be formulated as an optimization problem that aims to optimize the effective DD domain channel ${{\cal DZ}_g}\left[ {l,k} \right]h_{\rm DD}\left[ {l,k } \right]{\cal DZ}_g^*\left[ {l,k } \right]$, $l \in \left\{ {0,...,M - 1} \right\}$ and $k\in \left\{ {0,...,N - 1} \right\}$, where $h_{\rm DD}\left[ {l,k } \right]$ denotes the samples of the continuous DD domain channel response $h_{\rm DD}\left( {\tau ,\nu } \right)$.
An effective tool used for pulse shape design is the \emph{cross ambiguity function}. The ambiguity function characterizes the correlation between two time domain signals w.r.t. delay variable $\tau$ and Doppler variable $\nu$, and is defined as follows~\cite{Raviteja2018interference}
\begin{align}
{A_{x,y}}\left( {\tau ,\nu } \right) \buildrel \Delta \over = \int_{ - \infty }^\infty {x\left( t \right)} {y^*}\left( {t - \tau } \right){{\rm{e}}^{ - j2\pi \nu \left( {t - \tau } \right)}}{\rm{d}}t. \label{AF}
\end{align}
In the literature, a pulse is referred to as an \emph{ideal} pulse, if it satisfies the bi-orthogonality condition~\cite{Hadani2017orthogonal}, i.e.,
\begin{align}
{A_{{g_{\rm tx}},g_{\rm rx}}}\left( {nT,\frac{m}{T}} \right) = \delta \left[ n \right]\delta \left[ m \right],
\end{align}
where $\delta \left[ {\cdot} \right]$ is the Dirac delta function.
Note that the ideal pulse is defined on a TF domain grid, which implies two-dimensional (2D) orthogonality between TF domain transmitted symbols. However, a pulse satisfying the bi-orthogonality condition in the TF domain may not have ideal properties in the DD domain, where the grid (corresponding to the DD resolution) is defined differently.
For the design of the pulse shape in the DD domain, we may exploit the relation between a product of DZTs and the ambiguity function. In particular, as shown in~\cite{Bolcskei1994Gabor}, the product of two DZTs can be expanded into a 2D Fourier series w.r.t. the sampled cross ambiguity function, which could be exploited for pulse shape design.
More specifically, ${{\cal DZ}_g}\left[ {l,k} \right]h_{\rm DD}\left[ {l,k } \right]{\cal DZ}_g^*\left[ {l,k } \right]$ can be optimized by leveraging the cross ambiguity function with the objective to promote certain properties, such as improved channel sparsity and larger Euclidean distance.
\section{Receiver Design}
In this section, we consider the receiver design for OTFS systems. Due to the space limitation, we focus on DZT-based OTFS.
\subsection{Channel Estimation}
Different from its OFDM counterpart, OTFS channel estimation is usually performed in the DD domain rather than the TF domain as this allows the exploitation of
the appealing properties of DD domain channel responses, such as sparsity, compactness, separability, and quasi-static behaviour~\cite{Zhiqiang_magzine}.
A commonly used channel estimation approach for OTFS may be the one published in~\cite{Raviteja2019embedded}, which only requires one embedded pilot symbol in the DD domain.
Specifically, a sufficiently large guard interval is applied around the pilot to facilitate the acquisition of the delay and Doppler responses. As the DD domain relationship between the transmitted signal and the channel response corresponds to a 2D circular convolution as discussed in Part I, the embedded pilot is smeared over several DD grid points around the original location.
Therefore, the channel can be estimated by simply checking the received signal's values around the DD grid point where the pilot was embedded.
Channel estimation based on compressed sensing methods has also been considered for OTFS systems. Compressed sensing is suitable for sparse signal recovery, where the number of measurements is much smaller than the number of unknown parameters. Therefore, compressed sensing-based channel estimation is well-suited for OTFS with fractional delays or/and fractional Doppler shifts~\cite{Wei2022off}.
For instance, the
authors in~\cite{Wenqian2019channel} proposed a three-dimensional (3D) structured orthogonal matching pursuit (OMP) algorithm to estimate the delay-Doppler-angle domain channel by exploiting the underlying 3D structured sparsity.
A 3D Newtonized OMP (NOMP) algorithm was proposed in~\cite{Muye2021new_path}, which exploits the fractional components in the Doppler and angle domains via Newton's method.
Furthermore, channel estimation based on sparse Bayesian learning (SBL) techniques has been recently proposed~\cite{Wei2022off}, and was shown to achieve a better error performance compared to OMP-based schemes.
\begin{figure}
\centering
\includegraphics[width=2.8in]{Fig/Channel_Estimation_Results.eps}\vspace{-2mm}
\caption{Channel estimation performance comparison for SBL~\cite{Wei2022off}, OMP~\cite{Wenqian2019channel}, NOMP~\cite{Muye2021new_path}, and the conventional embedded pilot approach~\cite{Raviteja2019embedded}.}\vspace{-5mm}
\label{CS_performance}
\centering
\end{figure}
We present a performance comparison of the above-mentioned channel estimation schemes w.r.t. the signal-to-noise ratio (SNR) in Fig.~\ref{CS_performance}, where $M=N=32$, $P=5$, $l_{\rm max}=4$, and $k_{\rm max}=3$, respectively, and the fading coefficients are generated according to the exponential power delay profile with exponent $0.1$. As can be observed, both on-grid SBL and off-grid SBL~\cite{Wei2022off} outperform OMP~\cite{Wenqian2019channel}, NOMP~\cite{Muye2021new_path}, and the conventional embedded pilot approach~\cite{Raviteja2019embedded} in terms of the normalized mean squared error (NMSE). Furthermore, off-grid SBL achieves roughly $1$ dB NMSE gain over on-grid SBL in the high SNR regime because off-grid SBL can model the effects of fractional Doppler components. Meanwhile, we also observe that NOMP~\cite{Muye2021new_path} achieves roughly $2$ dB NMSE gain over OMP~\cite{Wenqian2019channel}, since NOMP can refine the delay and Doppler shift estimation via Newton's method.
\subsection{Signal Detection}
For OTFS, conventional detectors can be used, such as the minimum mean square error (MMSE) detector. Thanks to the properties of the effective DD domain channel matrix, MMSE detection can be implemented with linear time complexity~\cite{Chockalingam2020low_comp}.
Apart from MMSE detection, MPA~\cite{Raviteja2018interference,li2021hybrid,Yuan2020simple} has also been widely applied for OTFS detection.
Let us briefly introduce MPA from an MAP detection point of view~\cite{li2021hybrid}.
In the case of integer delay and Doppler shifts, assuming an ideal pulse, the $(l,k)$-th element $Y_{\rm DD}\left[ {l,k} \right]$ of the DD domain received symbol matrix ${\bf Y}_{\rm DD}$, for $l \in \left\{ {0,...,M - 1} \right\}$ and $k\in \left\{ {0,...,N - 1} \right\}$, is given as follows~\cite{Raviteja2018interference}
\begin{align}
Y_{\rm DD}\left[ {l,k} \right] &\!=\! \sum\nolimits_{i = 1}^P {{h_i}{e^{ - j2\pi {\nu _i}{\tau _i}}}X_{\rm DD}\!\!\left[ {{{\left[ {l - l_i} \right]}_M},{{\left[ {k - k_i} \right]}_N}} \right]}\notag\\
& + Z_{\rm DD}\left[ {l,k} \right], \label{IO_ideal_integer}
\end{align}
where $P$ is the number of resolvable paths, $h_i$, ${\tau _i}$, and ${\nu _i}$ are the fading coefficient, the delay, and the Doppler shift associated with the $i$-th path, respectively, while ${l _i}$ and ${k _i}$ are the corresponding delay and Doppler indices, as defined in Part I.
For ease of presentation, let us define the following sets
\begin{align}
\mathbb{H}^{\left( i \right)}&\buildrel \Delta \over = \left\{ {{h_j}\left| {1 \le j \le P,j \ne i} \right.} \right\},\notag\\
\mathbb{Y}_{l,k} &\buildrel \Delta \over = \left\{ {Y_{\rm DD}\left[{{\left[ {l + l_i} \right]}_M},{{\left[ {k + k_i} \right]}_N}\right]\big| {1 \le i \le P} } \right\}, \text{and}\notag\\
\mathbb{X}_{l,k}^{\left( i \right)}\!&\buildrel \Delta \over = \!\!
\left\{ {{X_{{\rm{DD}}}}\!\!\left[ {\left[ {l\! +\! {l_i} \!- \!{l_j}} \right]_M\!,\!\left[ {k \!+\! {k_i}\! - \!{k_j}} \right]_N} \right]\!\left| {1 \le j \le P,j \ne i} \right.} \right\}\notag,
\end{align}
where the $j$-th element of $\mathbb{H}^{\left( i \right)}$, $\mathbb{Y}_{l,k}$, and $\mathbb{X}_{l,k}^{\left( i \right)}$, are denoted by $\mathbb{H}^{\left( i \right)}[j]$, $\mathbb{Y}_{l,k}[j]$, and $\mathbb{X}_{l,k}^{\left( i \right)}[j]$, respectively.
According to~\eqref{IO_ideal_integer}, it can be shown that set $\mathbb{Y}_{l,k}$ contains $P$ received symbols that are associated with DD domain transmitted symbol $X_{\rm DD}\left[ {l,k} \right]$,
while set $\mathbb{X}_{l,k}^{\left( i \right)}$ contains $P-1$ DD domain transmitted symbols that are related to received symbol $\mathbb{Y}_{l,k}\left[ i \right]$, i.e., ${Y_{\rm DD}\left[{{\left[ {l + l_i} \right]}_M},{{\left[ {k + k_i} \right]}_N}\right]}$.
In fact, the \emph{a posteriori} probability $\Pr \left\{ {{X_{{\rm{DD}}}}\left[ {l,k} \right]|{{\bf{Y}}_{{\rm{DD}}}}} \right\}$ can be factorized based on a graphical model, where the nodes and calculations can be characterized by $\mathbb{H}^{\left( i \right)}$, $\mathbb{Y}_{l,k}$, and $\mathbb{X}_{l,k}^{\left( i \right)}$, respectively, as shown in~\cite{li2021hybrid}. Due to the page limitation, we cannot provide the implementation details for MPA. However, we note that the main idea of MPA is to pass messages among the connected nodes iteratively in a graphical model, such that the target probability, e.g., the \emph{a posteriori} probability, is approximately calculated after a sufficient number of iterations.
Note that the MPA designed based on~\eqref{IO_ideal_integer} assumes integer delays and Doppler shifts. In the fractional Doppler case, cross domain iterative detection (CDID) proposed in~\cite{li2021cross} has been shown to achieve a near-optimal performance with reduced complexity{\footnote{We note that detection algorithms for OTFS are generally designed specifically with different channel conditions in mind. For example, the MPA algorithm reported in~\cite{li2021hybrid} is not suitable for fractional Doppler shifts, as its detection complexity would become prohibitively high in this case.}}. CDID employs simple estimation/detection schemes in both the TD and DD domains and iteratively updates the extrinsic information via the unitary transformations between the TD and DD domains. Fig.~\ref{Detection_P4} shows the bit error ratio (BER) performance of OTFS transmission for conventional MMSE detection, MPA in~\cite{li2021hybrid}, MPA in~\cite{Raviteja2018interference}, and CDID, where we adopted $M=32$, $N=16$, $P=4$ and the fading coefficients are generated based on a uniform power delay profile with $l_{\rm max}=10$ and $k_{\rm max}=5$. The MPA in~\cite{Raviteja2018interference} and conventional MMSE detection achieve roughly the same BER, which also coincides with that of the first iteration of CDID. Furthermore, as the number of iterations increases, CDID gradually approaches the performance of the MPA in~\cite{li2021hybrid} with integer Doppler shifts, which is approximately the MAP detection performance. This observation suggests that CDID bridges the performance gap between MMSE and MAP as the number of iterations increase, which indicates that CDID achieves a favorable performance-complexity tradeoff. For more details regarding the performance analysis of CDID, we refer to~\cite{li2021cross}.
\begin{figure}
\centering\vspace{-3mm}
\includegraphics[width=2.8in]{Fig/detection_P4.eps}\vspace{-3mm}
\caption{BER comparison of MMSE detection, MPA~\cite{li2021hybrid}, MPA~\cite{Raviteja2018interference}, and CDID.}\vspace{-5mm}
\label{Detection_P4}
\centering
\end{figure}
\vspace{-2mm}
\section{Performance Analysis of OTFS Modulation}
In this section, we analyze the performance of OTFS modulation and draw comparisons with OFDM.
To this end, we consider system representative parameters to facilitate our discussion. In practice, all parameters have to be selected carefully according to the underlying channel conditions, of course.
\vspace{-2mm}
\subsection{Diversity Gain vs. Coding Gain}
The diversity gain characterizes the exponential scaling of the error performance w.r.t. the SNR in the high SNR regime.
OFDM requires channel coding to extract the diversity gain offered by multipath channels. In contrast, OTFS has the potential to exploit the full channel diversity without channel coding~\cite{Surabhi2019on,Raviteja2019effective,li2020performance}.
Nevertheless, channel coding will further improve the error performance of OTFS.
In particular, it is shown in~\cite{li2020performance} that the unconditional pair-wise error probability (PEP) of coded OTFS modulation over Rayleigh fading channels can be approximately upper-bounded by
\begin{equation}
\Pr\left( { {{\bf{x}},{\bf{x'}}} } \right)\mathbin{\lower.3ex\hbox{$\buildrel<\over
{\smash{\scriptstyle\sim}\vphantom{_x}}$}}{\left( {\frac{{d_{\rm{E}}^2\left( {\bf{e}} \right)}}{P}} \right)^{ - P}}{\left( {\frac{{{E_s}}}{{4{N_0}}}} \right)^{ - P}},\label{diversity_coding_tradeoff}
\end{equation}
where $\Pr\left( { {{\bf{x}},{\bf{x'}}} } \right)$ denotes the probability that DD domain transmitted sequence $\bf x$ is mistakenly detected as $\bf x'$, and the Euclidean distance between $\bf x$ and $\bf x'$ is ${d_{\rm{E}}^2\left( {\bf{e}} \right)}$. In~\eqref{diversity_coding_tradeoff}, the SNR exponent, i.e., the diversity gain, is equal to the number of resolvable paths of the underlying wireless channel $P$. On the other hand, the term ${{d_{\rm{E}}^2\left( {\bf{e}} \right)} \mathord{\left/
{\vphantom {{d_{\rm{E}}^2\left( {\bf{e}} \right)} P}} \right.
\kern-\nulldelimiterspace} P}$ is referred to as the coding gain, indicating the SNR gain achieved with channel coding~\cite{li2020performance}.
Two interesting observations can be obtained from~\eqref{diversity_coding_tradeoff}. Firstly, the PEP upper-bound does not depend on the delays and Doppler shifts, which implies that OTFS modulation causes ``channel hardening''. This is because each DD domain transmitted symbol experiences the fluctuation of the entire TF domain channel response thanks to the employed ISFFT.
Secondly, there is a tradeoff between diversity gain and coding gain for OTFS. In particular,~\eqref{diversity_coding_tradeoff} indicates that the diversity gain of OTFS improves with the number of resolvable paths $P$, while the coding gain declines. This observation suggests a rule-of-thumb for code design, i.e., the
Euclidean distance between transmitted sequences should be maximized, which actually aligns with the code design criterion for the additive white Gaussian noise (AWGN) channel as a consequence of the ``channel hardening'' effect.
\begin{figure}
\centering
\includegraphics[width=2.8in]{Fig/diversity_coding_tradeoff.eps}\vspace{-3mm}
\caption{Comparison of FER performances of coded and uncoded OTFS and OFDM.}\vspace{-5mm}
\label{div_coding_tradeoff}
\centering
\end{figure}
Fig.~\ref{div_coding_tradeoff} depicts the frame error rate (FER) performances of coded and uncoded OTFS and OFDM modulation with maximum-likelihood (ML) detection, where we apply the half-rate (3,1) feedforward convolutional code and binary phase shift keying (BPSK). From the figure, we observe that for a larger number of resolvable paths, the coding gain for OTFS decreases, e.g., from $2.0$ dB to $1.7$ dB, while the diversity gain increases, which is consistent with our discussions based on~\eqref{diversity_coding_tradeoff}. Furthermore, we also notice that the diversity gain of coded OTFS with $P = 8$ is larger than that of coded OFDM, which suggests that coded OTFS is a more attractive option for reliable communication over multipath fading channels than coded OFDM.
\vspace{-3mm}
\subsection{Achievable Rate Performance}
The achievable rate is an important performance metric characterizing how much information can be reliably transmitted over a channel with given resources. The achievable rates of OTFS and OFDM have been compared in~\cite{RezazadehReyhani2018analysis,Chong2022achievable}.
We present the achievable rate performance for both OTFS and OFDM in Fig.~\ref{Achievable_rate}, where we assume that perfect CSI is available at the receiver side and the achievable rate is calculated based on
\begin{align}
\setcounter{equation}{5}
R = \frac{1}{{MN}}{\log _2}\det \left( {{{\bf{I}}_{MN}} + {\rm{SNR}}{{\bf{H}}^{\rm{H}}}{\bf{H}}} \right).\label{achievable_rate_formula}
\end{align}
In~\eqref{achievable_rate_formula}, ${{\bf{I}}_{MN}}$ denotes the identity matrix of size $MN$, ${\rm{SNR}}$ denotes the operating SNR, ${\rm det}(\cdot)$ denotes the determinant, $(\cdot)^{\rm H}$ denotes the Hermitian conjugate, and ${\bf{H}}$ stands for the \emph{effective channel matrix} for reduced-CP OTFS or OFDM with and without CP (the CP length equals $l_{\rm max}$), as given in~\cite{Raviteja2019practical} and~\cite{RezazadehReyhani2018analysis}, respectively.
We set $M=32$ and $N=16$, and assume $P=4$ independent resolvable paths with maximum delay and Doppler indices given by $l_{\rm max}=5$ and $k_{\rm max}=5$, respectively.
As can be observed from Fig.~\ref{Achievable_rate}, reduced-CP OTFS and OFDM provide almost the same achievable rate, while OFDM with CP clearly suffers from a rate degradation due to the CP insertion. The intuition behind this observation is that the transformation between the TF domain and the DD domain is unitary, and thus, does not affect matrix determinants, leading to the same achievable rate.
However, the unitary property of the domain transformation may not hold in the multiuser case, where only a limited number of resource blocks can be allocated to each user. In this case, it has been shown that OTFS yields an overall achievable rate gain compared to OFDM if practical successive interference cancelation (SIC) detection is employed at the receiver~\cite{Chong2022achievable}.
An alternative, more practical performance metric is the pragmatic capacity, defined as the achievable rate of the channel
induced by the signal constellation and the soft-output of the detector~\cite{Lorenzo2021otfs}. The authors in~\cite{Lorenzo2021otfs} showed that OTFS transmission enjoys a better pragmatic capacity performance compared to OFDM over static channels with practical channel estimation and detection schemes, thanks to the smaller signaling overhead. Furthermore, the pragmatic capacity of OFDM is
very sensitive to the Doppler effect~\cite{Lorenzo2021otfs}, such that OTFS has a clear advantage in high-mobility channels.
\begin{figure}
\centering
\includegraphics[width=2.8in]{Fig/achievable_rate_P4.eps}\vspace{-3mm}
\caption{Comparison of the achievable rates of OTFS, OFDM without CP, and OFDM with CP, where perfect CSI is assumed and $P=4$.}\vspace{-4mm}
\label{Achievable_rate}
\centering
\end{figure}
\vspace{-2mm}
\section{Conclusions and Future Research Directions}
In this letter, we reviewed OTFS transceiver design principles, including CP insertion, pulse shaping, channel estimation, and signal detection. We also discussed the diversity gain and achievable rate of OTFS systems.
It is worth pointing out that OTFS transceiver design still faces many practical issues. For example, OTFS receivers may induce a long latency as the demodulation can only be carried out once the whole block of TF symbols is received due to the symbol spreading from the DD domain to the TF domain. Furthermore, without a carefully designed pulse shape, OTFS may cause high out-of-band emissions and other practical issues. Therefore, low latency receiver and pulse designs are important research topics for facilitating practical OTFS implementation.
\vspace{-3mm}
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 33
|
{"url":"https:\/\/math.stackexchange.com\/questions\/832536\/in-signal-processing-every-where-you-see-infinity-why","text":"# In signal processing, every where you see infinity. Why?\n\nEverywhere, in signal processing you see infinity. For example, in Fouriers, correlations. But no body would live to see infinity. Why do we aritificially talk about infinite time signals and then backtrack the thing using windows. Is inifinity necessary? Or can we do processing without this do-undo process?\n\n\u2022 The fast Fourier transform deals with finite samples. No infinities there. \u2013\u00a0copper.hat Jun 13 '14 at 5:54\n\u2022 @copper.hat FFT is in discrete or digital space. Do we equivalent in analog domain without windows? How about correlations and other operations that talk about infinite duration signals? \u2013\u00a0Seetha Rama Raju Sanapala Jun 13 '14 at 6:01\n\u2022 Well, its a model of reality that simplifies some aspects. Like using reals to do financial calculations, most of us will never see infinities. You could analyse using a finite time span, but will hit mostly irrelevant complexities just dealing with boundary conditions. \u2013\u00a0copper.hat Jun 13 '14 at 6:04\n\u2022 @copper.hat Thank you. \u2013\u00a0Seetha Rama Raju Sanapala Jun 13 '14 at 6:08\n\nFourier analysis requires a function to be defined on a (locally compact, abelian) group. The set of real numbers is a group under addition; this is the setting of Fourier transform. So is the circle $\\mathbb R\/(a\\mathbb Z)$ for some $a>0$; this is the setting of Fourier series. So is the cyclic group $\\mathbb Z\/(n\\mathbb Z)$, which is the setting of the discrete Fourier transform.\nAn interval $[a,b]$ is not a group. So, even though we may only have a function defined on such an interval, from the mathematical point of view the Fourier transform is taken on the real line. The function can be set to $0$ outside of $[a,b]$.","date":"2020-05-28 05:57:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7042306661605835, \"perplexity\": 653.0577910129653}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347396495.25\/warc\/CC-MAIN-20200528030851-20200528060851-00315.warc.gz\"}"}
| null | null |
package org.zalando.nakadi.webservice;
import com.fasterxml.jackson.core.JsonProcessingException;
import com.jayway.restassured.http.ContentType;
import com.jayway.restassured.response.Response;
import org.apache.http.HttpStatus;
import org.hamcrest.Matchers;
import org.junit.Before;
import org.junit.Test;
import org.zalando.nakadi.domain.EventType;
import org.zalando.nakadi.service.BlacklistService;
import org.zalando.nakadi.utils.EventTypeTestBuilder;
import org.zalando.nakadi.webservice.utils.NakadiTestUtils;
import java.io.IOException;
import java.text.MessageFormat;
import static com.jayway.restassured.RestAssured.given;
import static org.zalando.nakadi.utils.TestUtils.waitFor;
public class BlockEventPublishingAT extends BaseAT {
private EventType eventType;
@Before
public void setUp() throws JsonProcessingException {
eventType = EventTypeTestBuilder.builder().build();
NakadiTestUtils.createEventTypeInNakadi(eventType);
}
@Test
public void whenPublishingToBlockedEventTypeThen403() throws IOException {
publishEvent(eventType)
.then()
.statusCode(HttpStatus.SC_OK);
SettingsControllerAT.blacklist(eventType.getName(), BlacklistService.Type.PRODUCER_ET);
waitFor(() -> publishEvent(eventType)
.then()
.statusCode(403)
.body("detail", Matchers.equalTo("Application or event type is blocked")));
SettingsControllerAT.whitelist(eventType.getName(), BlacklistService.Type.PRODUCER_ET);
waitFor(() -> publishEvent(eventType)
.then()
.statusCode(HttpStatus.SC_OK));
}
private Response publishEvent(final EventType eventType) {
return given()
.body("[{\"foo\":\"bar\"}]")
.contentType(ContentType.JSON)
.post(MessageFormat.format("/event-types/{0}/events", eventType.getName()));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 50
|
{"url":"https:\/\/stats.stackexchange.com\/questions\/504018\/right-side-t-test-reject-null-but-sample-confidence-interval-includes-hypothet","text":"# Right Side T-Test - Reject Null but Sample Confidence Interval Includes Hypothetical Value\n\nThis has me puzzled. I am doing a right sided t-test where the null hypothesis is rejected based on the critical t value and the p value but the sample confidence interval includes the expected value. There is something missing in my understanding because I thought if the confidence interval included the value you could conclude there is no difference between the sample and hypothesized value.\n\nThe null hypothesis is rejected for the following reasons:\n\n\u2022 Reject the null hypothesis because the t-statistic, 1.54, was greater than the critical t value, 1.383.\n\n\u2022 Reject the null hypothesis because the p-value, 0.079, was less than the significance level, 0.1.\n\nHere are the values used in the calculation:\n\nHo: mean <= 50.5\nHa: mean > 50.5\n\nExpected Mean: 50.5\nSample Mean: 53.7\nSample Standard Deviation: 6.567\nStandard Error: 2.077\nSample Size: 10\nSignificance Level: .1\n\nSample 90% Confidence Interval: [49.89, 57.51]\np-Value: .079\nt-Statistic: 1.54\nCohen's D: .49\n\nCritical T: 1.383\n\n\nDo any of you know why this is happening?\n\nThanks,\n\nKevin\n\n\u2022 Your basic idea that a 90% confidence interval contains a hypothetical value not rejected at the 10% significance level is right. But I think you are mixing a one-sided test with a two-sided confidence interval. \u2013\u00a0BruceET Jan 8 at 10:29\n\nI think you may be using a two-sided 90% confidence interval along with a one-sided test at the 10% level. If so, your result is not surprising.\n\nYour data must have been something like my 10-vector y below. [Computations in R.]\n\ny\n[1] 51.5 48.3 49.7 54.8 60.3 53.7 51.3 48.8 49.3 69.3\nmean(y); sd(y)\n[1] 53.7 # sample mean\n[1] 6.559641 # sample standard deviation\n\n\nA (one-sided) t.test of $$H_0: \\mu \\le 50.5$$ vs. $$H_a: \\mu > 50.5$$ rejects at the 10% level because the P-value $$0.079 < 0.10 = 10\\%.$$\n\nt.test(y, mu=50.5, alt=\"greater\", conf.lev=.9)\n\nOne Sample t-test\n\ndata: y\nt = 1.5427, df = 9, p-value = 0.07865\nalternative hypothesis: true mean is greater than 50.5\n90 percent confidence interval:\n50.83113 Inf\nsample estimates:\nmean of x\n53.7\n\n\nThe one-sided 90% confidence interval amounts to a lower bound of 50.83, which does not include the hypothetical value 50.5. This is appropriate because $$H_0$$ is rejected.\n\nHowever, a two-sided test of $$H_0: \\mu = 50.5$$ against $$H_a: \\mu \\ne 50.5$$ is not rejected at the 10% level because the P-value $$0.1573 > .1 = 10\\%.$$ Accordingly, the corresponding two-sided 90% confidence interval, $$(49.8975, 57.5025),$$ does contain the hypothetical value $$\\mu = 50.5.$$\n\nt.test(y, mu=50.5, conf.lev=.9)\n\nOne Sample t-test\n\ndata: y\nt = 1.5427, df = 9, p-value = 0.1573\nalternative hypothesis: true mean is not equal to 50.5\n90 percent confidence interval:\n49.8975 57.5025\nsample estimates:\nmean of x\n53.7","date":"2021-04-18 12:16:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 9, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5812283754348755, \"perplexity\": 884.8935023669272}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038476606.60\/warc\/CC-MAIN-20210418103545-20210418133545-00365.warc.gz\"}"}
| null | null |
Q: Trying to install Rattle for R on my Mac I'm trying to install Rattle and have been running into some issues. Below is the initial code and results.
install.packages('rattle', repo='https://cran.cnr.Berkeley.edu/')
also installing the dependency 'RGtk2'
Package which is only available in source form, and may need compilation of
C/C++/Fortran: 'RGtk2'
Do you want to attempt to install these from sources?
I tried yes and the error it gave me is this
configure: error: GTK version 2.8.0 required
ERROR: configuration failed for package 'RGtk2'
* removing '/Library/Frameworks/R.framework/Versions/3.4/Resources/library/RGtk2'
Warning in install.packages :
installation of package 'RGtk2' had non-zero exit status
ERROR: dependency 'RGtk2' is not available for package 'rattle'
* removing '/Library/Frameworks/R.framework/Versions/3.4/Resources/library/rattle'
Warning in install.packages :
installation of package 'rattle' had non-zero exit status
The downloaded source packages are in
'/private/var/folders/47/39zt_y995czg2rgb3n485ty80000gn/T/Rtmpb8Eli0/downloaded_packages'
I also tried no and the error it gave me was this
ERROR: dependency 'RGtk2' is not available for package 'rattle'
* removing '/Library/Frameworks/R.framework/Versions/3.4/Resources/library/rattle'
Warning in install.packages :
installation of package 'rattle' had non-zero exit status
The downloaded source packages are in
'/private/var/folders/47/39zt_y995czg2rgb3n485ty80000gn/T/Rtmpb8Eli0/downloaded_packages'
Would anybody happen to know where I'm going wrong? Thank you in advance for any input.
A: My reputation is too low to comment, so I must post an answer. Zhiya's method worked for some of my students. It was not working for everyone, so we worked out a second method (thanks, Tugay!!)
The wrapper method of GW also was not working, perhaps because since it was compiled some of the versions have changed. (My theory is that all of the following must have versions that can interoperate: R, RGtk+, GTK, MacOS, Rattle. When any of them is updated on my system, Rattle may stop working. That's why you got this message
configure: error: GTK version 2.8.0 required
My blog post documenting what worked for me is here. It works with: MacOS 10.13.4,R 3.4.4, Rattle 5.1.3, RGtk2 2.24.32. YMMV.
I am sure that if I knew some UNIX, I could have gotten other methods to work as well. Again, the apparent problem is that as of today there are not up-to-date versions of RGtk2 and GTK that are compiled for the latest MacOS. A suitable compiled version does exist for Windows and is in the CRAN repository. Solving the error messages requires compiling the missing programs from their source code versions on CRAN.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,258
|
Q: ScintillaNet autocomplete weird behaviour with character '-' I'm trying to implement an autocomplete system for my code editor using ScintillaNet. Everything seems to works fine, but I realized that if the populated list contains strings with characters '_' i.e, the format X_XYZ, the focus is loosed and no string is selected (works for X, X_ and X_X but looses the focus for the following letters).
What I'm trying to do is to show the list after the text "this.".
Strings with this issue are for example:
p_M_Restart -> OK
p_MDWTest -> Fails at D and following letters
My code in the CharAdded event looks like this:
if (editor.GetWordFromPosition(editor.CurrentPos - 1) == TXT_THIS
&& e.Ch == '.')
{
editor.AutoComplete.List = Parameters.ToList();
editor.AutoComplete.SingleLineAccept = false;
editor.AutoComplete.FillUpCharacters = "([";
editor.AutoComplete.Show();
}
Is '_' a spcecial case? or is any consideration needed with this character?
Thanks,
A: Solved following the ScintillaNet issue:
https://scintillanet.codeplex.com/workitem/34165
Hope this will help other people.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,140
|
Q: How to run a sub-set of TestCases using --where for nunit For my project I want to run the exact same test cases twice, once locally and on a different VM in parallel in the cloud (Azure in my case).
I duplicated the TestCase and tagged one Category("Local") and the other Category("Cloud").
Running nunit3 from the console with --where="cat == Cloud" will thus run all TestCases of every test that has one or more TestCases tagged with Category("Cloud").
Is there a different way of only running selected TestCases by a commandline switch?
Simplified example:
[TestCase(TestName = "Canary, Run in cloud."), Category("Cloud")]
[TestCase(TestName = "Canary, Run locally."), Category("Local")]
public void Canary()
{
Assert.True(true);
}
A: Found a work-around.
Using --params:Cloud=true as command line argument and in the code
private bool ShallRunInCloud => TestContext.Parameters["Cloud"]?.ToLowerInvariant() == "true";
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,167
|
Home » Ads » BMW USED CARS FOR SALES » BMW 520i BLACK 2016 WITH MIND-BLOWING PRICE!!!!
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,651
|
{"url":"https:\/\/study.com\/academy\/answer\/an-energy-bar-contains-18-g-of-carbohydrates-if-the-energy-bar-was-his-only-fuel-how-far-could-a-68-kg-person-walk-at-5-km-h.html","text":"# An \"energy bar\" contains 18 g of carbohydrates. If the energy bar was his only fuel, how far...\n\n## Question:\n\nAn \"energy bar\" contains 18 g of carbohydrates. If the energy bar was his only fuel, how far could a 68 kg person walk at 5 km\/h?\n\n## Calories Inside the Energy Bar:\n\nCalories gained by eating the energy bar is different from the calories gained by the energy. The difference lies in the first alphabet. The lower case calorie denotes 1 calorie is gained and the upper case Calorie denotes 1000 calories.\n\nAccording to the data:\n\n{eq}1\\ \\textrm{carb} =4\\frac{\\textrm{Cal}}{g} {\/eq}\n\nGiven:\n\nAmount of carbohydrates in the energy bar is {eq}18\\ g {\/eq}\n\nMass of the person is {eq}m=68\\ kg {\/eq}\n\nSpeed of the person is {eq}v=5\\ km\/h {\/eq}\n\nFor the energy content in the energy bar:\n\n{eq}E_t=18\\times 4\\\\ E_t=72\\ \\textrm{Calories} {\/eq}\n\nFor the conversion of calories into joules:\n\n{eq}1\\ \\textrm{Cal}=4184\\ J\\\\ 72\\ \\textrm{Cal}=4184\\times 72\\\\ 72\\ \\textrm{Cal}=301248\\ J {\/eq}\n\nNow for the kinetic energy of the person:\n\n{eq}K.E=\\frac{1}{2}mv^2\\\\ K.E=\\frac{1}{2}68\\times (1.38)^2\\\\ K.E=64.7\\ J {\/eq}\n\nNow for the distance traveled by the person:\n\n{eq}S=\\frac{E_t}{K.E}\\\\ S=\\frac{301248}{64.7}\\\\ S=4656\\ m\\\\ S=4.656\\ km {\/eq}\n\nThus, the distance traveled by the person is 4.656 km","date":"2019-09-15 05:27:01","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8707025647163391, \"perplexity\": 10103.933799487388}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514570740.10\/warc\/CC-MAIN-20190915052433-20190915074433-00089.warc.gz\"}"}
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\section{Introduction and Preliminaries}
A real $C^*$-algebra is a Banach $*$-algebra $A$ over the real numbers such that $\| x^* x \| = \|x\|^2$ holds for all $x$ and such that every element of the form $1 + x^* x$ is invertible in the unitization of $A$ (see \cite{schroderbook}). In this paper, we will adopt the term {\it $R^*$-algebra} instead. Every $R^*$-algebra is isometrically isomorphic to a closed $*$-algebra of bounded operators on a Hilbert space over $\mathbb{R}$. In addition, every $R^*$-algebra is isomorphic to the $*$-algebra of fixed elements of a $C^*$-algebra with a conjugate linear involution.
In Kasparov's seminal paper \cite{kasparov81} introducing $\operatorname{KK}$-theory, he simultaneously considered both $R^*$-algebras and $C^*$-algebras. Since then, many alternate but equivalent or closely related pictures of $\operatorname{KK}$-theory have been introduced and developed by various authors
(\cite{ConnesHigson90}, \cite{cuntz87}, \cite{higson87}, \cite{higson90}, \cite{ManuilovThomsen04}, \cite{thomsen90}). The ability to move among the various pictures has contributed immensely to the utility of $\operatorname{KK}$-theory as a tool for solving problems. However, these authors have not following Kasparov's lead and the alternate pictures of $\operatorname{KK}$-theory have been developed primarily in the complex case.
In recent years, substantial progress has been made in developing the tools to study $R^{*}$-algebras including the development of united $K$-theory and the universal coefficient theorem in \cite{boersema02} and \cite{boersema04}. This has led to a classification of purely infinite simple $R^*$-algebras (in \cite{BRS}) and the classification of real forms of UHF-algebras that are stable over the CAR-algebra (in \cite{stacey}).
Given the centrality of $\operatorname{KK}$-theory for these projects, there has been a need to develop a systematic account of the various pictures of $\operatorname{KK}$-theory for $R^*$-algebras. In this paper, we will develop several of the alternate pictures of $\operatorname{KK}$-theory in the context of $R^*$-algebras and prove the appropriate equivalent theorems. In particular, in this paper we will consider the following pictures of $\operatorname{KK}$-theory and prove appropriate equivalence theorems for each: the standard Kasparov bimodule picture of $\operatorname{KK}$-theory, the Fredholm picture (both in Section~2), the universal property picture (Section~3), the extension picture (Section~4), and suspended $E$-theory using asymptotic morphisms (Section~5).
Furthermore, we prove a more general theorem which reduces the work required to replicate many of these equivalent theorems and
promises to ease the way for similar projects in the future. Suppose that
$\mu \colon F \rightarrow G$
is a natural transformation between homotopy invariant, stable, half exact functors. We prove that if $\mu$ is an isomorphism for all $C^*$-algebras, then it is an isomorphism for $R^*$-algebras. This is accomplished in Section~3 when we develop the universal properties of $\operatorname{KK}$-theory and $K$-theory for $R^*$-algebras.
In the last two sections, we will apply these ideas, using $\operatorname{KK}$-theory to prove the existence of certain asymptotic morphisms, which in turn is used to obtain new results for the problem of approximating a set of almost commuting matrices over the field of real numbers. In particular, let the Halmos number be the largest integer $d$ such that whenever $d$ real self-adjoint matrices almost commute (pairwise) they can be approximated by $d$ pairwise commuting matrices. More precisely, for all $\epsilon > 0$ there should be a $\delta > 0$ such that if $\{ H_i \}_{i = 1}^d$ is a collect of $d$ self-adjoint matrices such that
$$ \| H_r \| \leq 1 \text{ \qquad and \qquad} \| [ H_r, H_s ] \| \leq \delta \; ,$$
for all $r,s$, then there exists a collection $\{ K_i \}_{i = 1}^d$ of self-adjoint matrices such that
$$ \| K_r \| \leq 1 \text{ \qquad and \qquad} \| [ K_r, K_s ] \| = 0 \text{ \qquad and \qquad} \| H_r - K_r \| \leq \epsilon \; .$$
Furthermore, the dependence of $\delta$ on $\epsilon$ must be uniform, independent of the dimension of the matrices $H_r$. It is shown in
\cite{LorSorensenTorus} that in the context of real matrices, the statement is true for $d = 2$. We will show in Section 7 the statement is false for $d = 5$. Therefore, the Halmos number for real matrices is between 2 and 4, inclusive.
\section{The Standard and Fredholm Pictures of $\operatorname{KK}$-Theory}
We take the following definition from Section~2.3 of \cite{schroderbook} to be the standard definition of $\operatorname{KK}$-theory for $R^{*}$-algebras. It is essentially the same as that in \cite{kasparov81} where it was simultaneously developed for both $R^{*}$-algebras and $C^{*}$-algebras.
\begin{defin}\label{def:kbimod}
Let $A$ be a graded separable $R^{*}$-algebra and $B$ be an $R^{*}$-algebra with $\sigma$-unital.
\begin{itemize}
\item[(i)] A Kasparov ($A$-$B$)-bimodule is a triple $( E , \phi , T )$ where $E$ is a countably generated graded Hilbert $B$-module, $\ftn{ \phi }{ A }{ \mc{L} ( E ) }$ is a graded $*$-homomorphism, and $T$ is an element of $\mc{L} ( E )$ of degree 1 such that
\begin{equation*}
( T - T^{*} ) \phi ( a ), ( T^{2} - 1 ) \phi ( a ) , \ \mathrm{and} \ [ T , \phi ( a ) ]
\end{equation*}
lie in $\mathcal{K}(E)$ for all $a \in A$.
\item[(ii)] Two triples $( E_{i} , \phi_{i} , T_{i} )$ are \emph{unitarily equivalent} if there is unitary $U$ in $\mc{L} ( E_{0} , E_{1} )$, of degree zero, intertwining the $\phi_{i}$ and $T_{i}$.
\item[(iii)] If $(E, \phi, T)$ is a Kasparov ($A$-$B$)-bimodule and $\beta \colon B \rightarrow B'$ is a $*$-homomorphism of $R^{*}$-algebras, then the pushed-forward Kasparov ($A$-$B'$)-bimodule is defined by $$\beta_*(E,\phi,T) = (E \hat{\otimes}_\beta B', \phi \hat{\otimes} 1, T \hat{\otimes} 1) \; .$$
\item[(iv)] Two Kasparov ($A$-$B$)-bimodules $(E_{i} , \phi_{i} , T_{i} )$ for $i = 0,1$ are \emph{homotopic} if there is a Kasparov bimodule ($A$-$IB)$, say $(E, \phi , T )$, such that $(\varepsilon_i)_* (E, \phi, T)$ and $(E_i, \phi_i, T_i)$ are unitarily equivalent for $i = 0,1$, where $IB = C ([ 0 ,1] , B )$ and $\varepsilon_{i}$ denotes the evaluation map.
\item[(v)] A triple $(E, \phi , T )$ is \emph{degenerate} if the elements \begin{equation*}
( T - T^{*} ) \phi ( a ), ( T^{2} - 1 ) \phi ( a ) , \ \mathrm{and} \ [ T , \phi ( a ) ]
\end{equation*}
are zero for all $a \in A$. By Proposition~2.3.3 of \cite{schroderbook}, degenerate bimodules are homotopic to trivial bimodules.
\item[(vi)] $\operatorname{KK}( A , B )$ is defined to be the set of homotopy equivalence classes of Kasparov ($A$-$B$)-bimodules.
\end{itemize}
\end{defin}
The following theorem summarizes the principal properties of $\operatorname{KK}$-theory for $R^{*}$-algebras from Chapter~2 of \cite{schroderbook}.
\begin{propo} \label{kkproperties}
$\operatorname{KK}(A,B)$ is an abelian group for separable $A$ and $\sigma$-unital $B$.
As a functor on separable $R^{*}$-algebras
(contravariant in the first argument and covariant in the second argument), it is homotopy invariant, stable, and has split exact sequences in both arguments. Furthermore, there is a natural associate pairing (the intersection product)
$$\otimes_{KK} \colon \operatorname{KK}(A, C \otimes B) \otimes \operatorname{KK}(C \otimes A', B') \rightarrow \operatorname{KK}(A \otimes A', B \otimes B') \; .$$
\end{propo}
We now turn to the Fredholm picture of $\operatorname{KK}$-theory, which was developed in \cite{higson87} with only the situation of $C^{*}$-algebras in mind. However, the approach goes through the same for $R^{*}$-algebras, as follows.
\begin{defin}
Let $A$ and $B$ be separable $R^{*}$-algebras.
\begin{itemize}
\item[(i)] A triple $( \phi_{+} , \phi_{-} , U )$, where
$\ftn{ \phi_{ \pm } }{ A }{ \mc{M}({ \mathcal{K}_{\mathbb{R}} \otimes B }) }$ are $*$-homomorphisms, and $U$ is an element of $\mc{M}({ \mathcal{K}_{\mathbb{R}} \otimes B })$ such that
\begin{equation*}
U \phi_{+} ( a ) - \phi_{-} ( a ) U, \ \phi_{+} ( a ) ( U^{*} U - 1 ), \ \mathrm{and} \ \phi_{-} ( a ) ( U U^{*} - 1 )
\end{equation*}
lie in $\mathcal{K}_{\mathbb{R}} \otimes B$ for all $a \in A$ is called a $\operatorname{KK}(A,B)$-cycle.
\item[(ii)] Two $\operatorname{KK} ( A , B )$-cycles $( \phi_{+}^{1} , \phi_{-}^{1} , U^{1} )$ and $( \phi_{+}^{2} , \phi_{-}^{2} , U^{2} )$ are \emph{homotopic} if there is a $\operatorname{KK} ( A , IB )$-cycle $(\phi_{+} , \phi_{-} , U )$ such that $( \varepsilon_{i} \phi_{+}, \varepsilon_{i} \phi_{-} , \varepsilon_{i} ( U ) ) = ( \phi_{+}^{i} , \phi_{-}^{i} , U^{i} )$, where
$\ftn{ \varepsilon_{i} } { \mc{M}( \mathcal{K}_{\mathbb{R}} \otimes IB) }{ \mc{M}( \mathcal{K}_{\mathbb{R}} \otimes B ) }$ is induced by evaluation at $i$.
\item[(iii)] A $\operatorname{KK} ( A , B )$-cycle $(\psi_{+} , \psi_{-} , V )$ is \emph{degenerate} if the elements
\begin{equation*}
V \psi_{+} ( a ) - \psi_{-} ( a ) V, \ \psi_{+} ( a ) ( V^{*} V - 1 ), \ \mathrm{and} \ \psi_{-} ( a ) ( V V^{*} - 1 )
\end{equation*}
are zero for all $a \in A$.
\item[(iv)] The sum $( \phi_{+} , \phi_{-} , U ) \oplus ( \psi_{+} , \psi_{-} ,V )$ of two $\operatorname{KK} ( A , B )$-cycles is the $\operatorname{KK} ( A , B )$-cycle
\begin{equation*}
\left( \left(
\begin{matrix}
\phi_{+} & 0 \\
0 & \psi_{+}
\end{matrix}
\right),
\left(
\begin{matrix}
\phi_{-} & 0 \\
0 & \psi_{-}
\end{matrix}
\right),
\left(
\begin{matrix}
U & 0 \\
0 & V
\end{matrix}
\right)
\right)
\end{equation*}
where the algebra $M_2 ( \mc{M}( \mathcal{K}_{\mathbb{R}} \otimes B ) )$ is identified with $\mc{M}( \mathcal{K}_{\mathbb{R}} \otimes B )$ by means of some $*$-isomorphism $M_2 ( \mathcal{K}_{\mathbb{R}} ) \cong \mathcal{K}_{\mathbb{R}}$, which is unique up to homotopy by Section~1.17 of \cite{kasparov81}.
\item[(v)] Two cycles $( \phi_{+}^{0}, \phi_{-}^{0} , U^{0} )$ and $( \phi_{+}^{1}, \phi_{-}^{1} , U^{1} )$ are said to be \emph{equivalent} if there exist degenerate cycles $( \psi_{+}^{0} , \psi_{-}^{0} , V^{0} )$ and $( \psi_{+}^{1} , \psi_{-}^{1} , V^{1} )$ such that
\begin{equation*}
( \phi_{+}^{0} , \phi_{-}^{0} , U^{0} ) \oplus ( \psi_{+}^{0}, \psi_{-}^{0}, V^{0} ) \quad \mathrm{and} \quad ( \phi_{+}^{1} , \phi_{-}^{1} , U^{1} ) \oplus ( \psi_{+}^{1}, \psi_{-}^{1}, V^{1} )
\end{equation*}
are homotopic.
\item[(vi)] $\textbf{KK} ( A , B )$ is defined to be the set of equivalence classes of $\operatorname{KK} ( A , B )$-cycles.
\end{itemize}
\end{defin}
The following lemma is the real version of Lemma~2.3 of \cite{higson87}.
\begin{lemma} \label{lem:fhgrp}
$\textbf{KK}( A , B )$ is an abelian group, for separable $R^{*}$-algebras $A$ and $B$. As a functor it is contravariant in the first argument and covariant in the second argument.
\end{lemma}
\begin{proof}
For the first statement, we show that a cycle $(\phi_+, \phi_-, U)$ has inverse $(\phi_-, \phi_+, U^*)$. Indeed,
the sum
\begin{equation*}
\left(
\left(
\begin{matrix}
\phi_{+} & 0 \\
0 & \phi_{-}
\end{matrix}
\right),
\left(
\begin{matrix}
\phi_{-} & 0 \\
0 & \phi_{+}
\end{matrix}
\right),
\left(
\begin{matrix}
U & 0 \\
0 & U^*
\end{matrix}
\right)
\right)
\end{equation*}
is homotopic to a degenerate cycle via the operator homotopy
\begin{equation*}
W_{t} =
\left(
\begin{matrix}
\cos ( t ) U & - \sin(t) \\
\sin ( t ) & \cos ( t ) U^*
\end{matrix}
\right), \quad t \in \left[ 0 , \frac{ \pi }{ 2 } \right] \; .
\end{equation*}
The functoriality is established as in \cite{higson87} in Sections~2.4 through 2.7.
\end{proof}
The next proposition establishes the isomorphism between the two pictures of $\operatorname{KK}$-theory. First we review some preliminaries regarding graded $R^{*}$-algebras and Hilbert modules.
If $B$ is an $R^{*}$-algebra, then the \emph{standard even grading} on $M_2 ( B )$ is obtained by setting $M_2 ( B )^{(0)}$ to be the set of diagonal matrices and $M_2( B )^{(1)}$ the set of matrices with zero diagonal. The \emph{standard even grading} on $\mathcal{K}_{\mathbb{R}} \otimes B$ is obtained by choosing a $*$-isomorphism $\mathcal{K}_{\mathbb{R}} \otimes B \cong M_2 ( \mathcal{K}_{\mathbb{R}} \otimes B )$. This in turn induces a canonical (modulo a unitary automorphism) grading on $\multialg{ \mathcal{K}_{\mathbb{R}} \otimes B }$.
Let $\mathbb H_B$ be the Hilbert $B$-module consisting of all sequences $\{b_n\}_{n = 1}^\infty$ in $B$ such that $\sum_{n = 1}^\infty b_n^* b_n$ converges. Giving $B$ the trivial grading (that is $B^{(0)} = B$ and $B^{(1)} = \{0\}$), let $\hat{\mathbb H}_B = \mathbb H_B \oplus \mathbb H_B$ be the graded Hilbert $B$-module with $\hat{\mathbb H}_B^{(0)} = \mathbb H_B \oplus 0$ and $\hat{\mathbb H}_B^{(1)} = 0 \oplus \mathbb H_B$. Then the induced grading on $\mathcal{L}(\hat{ \mathbb H}_B)$ is identical with the standard even grading of $M_2 ( \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B } )$. Under the $*$-isomorphism $M_2 ( \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B } ) \cong \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B } $, this grading coincides with the one described in the previous paragraph.
\begin{theor}\label{prop:eqdef}
Let $A$ and $B$ be separable $R^{*}$-algebras. Then $\operatorname{KK} ( A , B )$ is isomorphic to $\textbf{KK} ( A , B )$.
\end{theor}
\begin{proof}
We give $A$ and $B$ the trivial grading and we give $\multialg{ \mathcal{K}_{\mathbb{R}} \otimes B }$ the standard even grading described above.
For a $\operatorname{KK} ( A , B )$-cycle $x = ( \phi_{+} , \phi_{-} , U )$ we define
\begin{equation*}
\alpha ( x ) = \left( \hat{ \mathbb H }_{ B } , \left( \begin{matrix} \phi_{+} & 0 \\ 0 & \phi_{-} \end{matrix} \right) , \left( \begin{matrix} 0 & U^{*} \\ U & 0 \end{matrix} \right) \right) \;.
\end{equation*}
It is readily verified that $\alpha(x)$ is a Kasparov ($A$-$B$) bimodule. Furthermore, for $x = ( \phi_{+} , \phi_{-} , U )$ and $y = ( \psi_{+} , \psi_{-} , V )$ it is easy to see that
$\alpha(x+y)$ and $\alpha(x) + \alpha(y)$ are unitarily equivalent via a degree 0 unitary.
We must show that $\alpha$ induces a well-defined homomoprhism
$$\overline{\alpha} \colon \textbf{KK} ( A , B ) \rightarrow \operatorname{KK}(A, B) \; .$$
Note first that $\alpha$ sends degenerate elements to degenerate elements. Next, suppose $( \phi_{+} , \phi_{-} , U )$ is a
$\operatorname{KK} ( A , IB )$-cycle implementing a homotopy between $x = ( \phi_{+}^{0} , \phi_{-}^{0} , U_{0} )$ and $y = ( \phi_{+}^{1} , \phi_{-}^{1} , U_{1} )$. That is, $\varepsilon_{i} ( \phi_{+} ) = \phi_{+}^{i}$, $\varepsilon_{i} ( \phi ) = \phi_{-}^{i}$, and $\varepsilon_{i} ( U ) = U_{i}$; where $\ftn{ \varepsilon_{t} }{ \multialg{ \mathcal{K}_{\mathbb{R}} \otimes IB } }{ \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B } }$ is the map induced by the evaluation map $\varepsilon_{t}$ at $t$.
Consider the Kasparov ($A$-$IB$)-bimodule
\begin{equation*}
z = \left( \hat{ \mathbb H }_{ IB } , \left( \begin{matrix} \phi_{+} & 0 \\ 0 & \phi_{-} \end{matrix} \right) , \left( \begin{matrix} 0 & U^{*} \\ U & 0 \end{matrix} \right) \right).
\end{equation*}
Since $
\hat{ \mathbb H }_{ B \otimes C[0,1] } \hat{ \otimes }_{ \epsilon_{i} } B \cong \hat{ \mathbb H }_{ B }$, it follows that
\begin{equation*}
(\varepsilon_0)_*(z) = \left( \hat{ \mathbb H }_{ B } , \left( \begin{matrix} \phi_{+}^{0} & 0 \\ 0 & \phi_{-}^{0} \end{matrix} \right) , \left( \begin{matrix} 0 & U_{0}^{*} \\ U_{0} & 0 \end{matrix} \right) \right)
\end{equation*}
and
\begin{equation*}
(\varepsilon_1)_*(z) =\left( \hat{ \mathbb H }_{ B } , \left( \begin{matrix} \phi_{+}^{1} & 0 \\ 0 & \phi_{-}^{1} \end{matrix} \right) , \left( \begin{matrix} 0 & U_{1}^{*} \\ U_{1} & 0 \end{matrix} \right) \right).
\end{equation*}
Therefore $\alpha(x)$ and $\alpha(y)$ are homotopic. Hence $\overline{\alpha}$ is well-defined.
To show that $\overline{ \alpha }$ is surjective, let $y = (E , \phi , T )$ be a Kasparov ($A$,$B$)-bimodule. By Proposition~2.3.5 of \cite{schroderbook}, we may assume that $E \cong \hat{ \mathbb H}_B$ and that $T = T^*$. Thus, with respect to the graded isomorphism $\mathcal{L}({\hat{ \mathbb H}_B}) \cong M_2(\multialg{ \mathcal{K}_{\mathbb{R}} \otimes B})$,
we can write
$$
\phi = \left( \begin{matrix} \phi_{+} & 0 \\ 0 & \phi_{-} \end{matrix} \right)
\qquad \text{and} \qquad
T = \left( \begin{matrix} 0 & U^{*} \\ U& 0 \end{matrix} \right)
$$
where $\phi_+$ and $\phi_-$ are $*$-homomorphisms from $A$ to $\multialg{ \mathcal{K}_{\mathbb{R}} \otimes B}$ and $U$ is an element of $\multialg{ \mathcal{K}_{\mathbb{R}} \otimes B}$. Then $y = \overline{\alpha}(x)$ where $x = (\phi_+, \phi_-, U)$.
Finally, we show that $\overline{\alpha}$ is injective. Suppose that $\overline{\alpha}(x) = \overline{\alpha}(y)$ where $x = (\phi_+, \phi_-, U)$ and $y = (\psi_+, \psi_-, V)$. Then there is a Kasparov $(A,B)$-bimodule $z = (E, \phi, T)$ such that
$$
(\varepsilon_0)_*(z) \cong \left( \hat{\mathbb H}_B, \sm{\phi_+ }{0}{0}{\phi_-}, \sm{0}{U^*}{U}{0} \right) $$
and
$$
(\varepsilon_1)_*(z) \cong \left( \hat{\mathbb H}_B, \sm{\psi_+ }{0}{0}{\psi_-}, \sm{0}{V^*}{V}{0} \right) \; .
$$
As above, we may assume that $E = \hat{\mathbb H}_{B \otimes C[0,1]}$ and that $T = T^*$. Then $z$ has the form
$$z = \left( \hat{\mathbb H}_{B \otimes C[0,1]}, \sm{\theta_+}{0}{0}{\theta_-}, \sm{0}{W^*}{W}{0} \right) $$
where $\theta_+$ and $\theta_-$ are $*$-homomorphisms to $\multialg{ \mathcal{K}_{\mathbb{R}} \otimes IB}$ and $W$ is an element of $\multialg{ \mathcal{K}_{\mathbb{R}} \otimes IB}$. Then $(\theta_+, \theta_-, W)$ is a Kasparov $(A, IB)$-bimodule implementing a homotopy between $x$ and $y$.
\end{proof}
\section{The Universal Property of $\operatorname{KK}$-Theory }
Let $F$ be a functor from the category {\textbf{C*$\mathbb{R}$-Alg}} of separable $R^{*}$-algebras to the cateogry {\textbf{Ab}} of abelian groups. We say that $F$ is
\begin{enumerate}
\item[(i)] {\it homotopy invariant} if $(\alpha_1)_* = (\alpha_2)_*$ whenever $\alpha_1$ and $\alpha_2$ are homotopic $*$-homomorphisms on the level of $R^{*}$-algebras.
\item[(ii)] {\it stable} if $(e_A)_* \colon F(A) \rightarrow F(\mathcal{K}_{\mathbb{R}} \otimes A)$ is an isomorphism for the inclusion $e_A \colon A \hookrightarrow \mathcal{K}_{\mathbb{R}} \otimes A$ defined via any rank one projection.
\item[(iii)] {\it split exact} if any split exact sequence of separable $R^{*}$-algebras
$$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$
induces a split exact sequence
$$0 \rightarrow F(A) \rightarrow F(B) \rightarrow F(C) \rightarrow 0 \; .$$
\item[(iv)] {\it half exact} if any short exact sequence of separable $R^{*}$-algebras
$$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$
induces an exact sequence
$$F(A) \rightarrow F(B) \rightarrow F(C) \; .$$
\end{enumerate}
In what follows we will see that if $F$ is homotopy invariant and half exact, then it is split exact.
\begin{propo} \label{stability}
If $F$ is a functor from {\textbf{C*$\mathbb{R}$-Alg}} to {\textbf{Ab}} that is homotopy invariant, then the functor $F_s$ defined by
$F_s(A) = F(\mathcal{K}_{\mathbb{R}} \otimes A)$ is homotopy invariant and stable.
\end{propo}
\begin{proof}
Just as in the complex case (Theorem~4.1.13 of \cite{jensenthomsenbook}), the map $e_{ \mathcal{K}_{\mathbb{R}}} \colon \mathcal{K}_{\mathbb{R}} \rightarrow \mathcal{K}_{\mathbb{R}} \otimes \mathcal{K}_{\mathbb{R}}$ is homotopic to an isomorphism.
\end{proof}
The following theorem is the version for $R^{*}$-algebras of Theorem~3.7 of \cite{higson87} and Theorem~22.3.1 of \cite{blackadarbook}.
\begin{theor} \label{thm:univprop1}
Let $F$ be a functor from {\textbf{C*$\mathbb{R}$-Alg}} to {\textbf{Ab}} that is homotopy invariant, stable, and split exact. Then there is a unique natural pairing $\alpha \colon F(A) \otimes \operatorname{KK}(A,B) \rightarrow F(B)$ such that $\alpha(x \otimes 1_A) = x$ for all $x \in F(A)$ and where $1_A \in \operatorname{KK}(A,A)$ is the class represented by the identity $*$-homomorphism.
Furthermore, the pairing respects the intersection product on $\operatorname{KK}$-theory in the sense that
$$\alpha(\alpha(x \otimes y) \otimes z) = \alpha(x \otimes ( y \otimes_B z)) \colon F(A) \otimes \operatorname{KK}(A,B) \otimes \operatorname{KK}(B,C) \rightarrow F(C) \; .$$
\end{theor}
\begin{proof}
Let $\Phi \in \operatorname{KK}(A,B)$. Using Theorem~\ref{prop:eqdef} we represent $\Phi$ with a $\operatorname{KK}(A,B)$ cycle and as in Lemma~3.6 of \cite{higson87}, we may assume that this cycle has the form $(\phi_+, \phi_-, 1)$. We use the same construction as in Definitions~3.3 and 3.4 in \cite{higson87}. In that setting $F$ is assumed to be a functor from separable $C^{*}$-algebras, but it goes through the same for functors from separable $R^{*}$-algebras to any abelian category. This construction produces a homomorphism $\Phi_* \colon F(A) \rightarrow F(B)$ and we then define $\alpha(x \otimes \Phi) = \Phi_*(x)$. The proof of Theorems~3.7 and 3.5 of \cite{higson87} carry over in the real case to show that $\alpha$ is natural, is well-defined, satisfies $\alpha(x \otimes 1_A) = x$, and is unique.
That $\alpha$ respects the Kasparov product follows from the uniqueness statement.
\end{proof}
We also note the contravariant version of the result above. If $F$ is a contravriant functor, otherwise satisfying the above hypotheses, then there is a pairing $\alpha \colon \operatorname{KK}(A, B) \otimes F(B) \rightarrow F(A)$ such that $\alpha(1_A \otimes x) = x$ for all $x \in F(A)$.
For any $R^{*}$-algebra $A$, we define $S A = \{ f \in C([0,1], A) \mid f(0) = f(1) = 0\}$, or equivalently up to $*$-isomorphism,
$SA = C_0(\mathbb{R}, A)$. We similarly define $S^{-1}A = \{ f \in C_0(\mathbb{R}, A_\C) \mid f(-x) = \overline{f(x)} \}$. By iteration, $S^n A$ is defined for all $n \in \mathbb{Z}$. Since $S S^{-1} \mathbb{R}$ is $\operatorname{KK}$-equivalent to $\mathbb{R}$, the formula $S^{n} S^m A \equiv S^{n+m} A$ holds up to $\operatorname{KK}$-equivalence for all $n,m \in \mathbb{Z}$. Then for any functor $F$ on {\textbf{C*$\mathbb{R}$-Alg}} and any integer $n$, we define $F_n(A) = F(S^n(A))$.
\begin{corol} \label{period}
Let $F$ be a functor from {\textbf{C*$\mathbb{R}$-Alg}} to {\textbf{Ab}} that is homotopy invariant, stable, and split exact. Then $F_*(A)$ has the structure of a graded module over the ring $K_*(\mathbb{R})$. In particular, $F(S^8 A) \cong F(A)$ and $F(S^{-1} S A) \cong F(A)$.
\end{corol}
\begin{proof}
For all separable $A$ and $\sigma$-unital $B$, the pairing of Proposition~\ref{kkproperties} gives $\operatorname{KK}_*(A,B)$ the structure of a module over $\operatorname{KK}_*(\mathbb{R}, \mathbb{R})$. Taking $A = B$, we define a graded ring homomorphism $\beta$ from $K_*(\mathbb{R}) \cong \operatorname{KK}_*(\mathbb{R}, \mathbb{R})$ to $\operatorname{KK}_*(A,A)$ by multiplication by $1_A \in \operatorname{KK}(A,A)$.
Then for any $x \in F_m(A)$ and $y \in K_n(\mathbb{R})$ we define
$x \cdot y = \alpha(x \otimes \beta(y)) \in F_{n+m}(A)$.
The second statement follows from the $\operatorname{KK}$-equivalence between $\mathbb{R}$ and $S^8 \mathbb{R}$, and that between $\mathbb{R}$ and $S^{-1} S \mathbb{R}$ from Section~1.4 of \cite{boersema02}.
\end{proof}
It also follows that the pairing Theorem~\ref{thm:univprop1} extends to a well-defined graded pairing
$$\alpha \colon F_*(A) \otimes \operatorname{KK}_*(A,B) \rightarrow F_*(B) \; .$$
Let {\textbf{KK}} be the category whose objects are separable $R^{*}$-algebras and the set of morphisms from $A$ to $B$ is $\operatorname{KK}(A , B )$. There is a canonical functor $\operatorname{KK}$ from {\textbf{C*$\mathbb{R}$-Alg}} to {\textbf{KK}} that takes an object $A$ to itself and which takes a $*$-homomorphism $f \colon A \rightarrow B$ to the corresponding element $[f] \in \operatorname{KK}(A,B)$.
\begin{corol}\label{thm:univprop2}
Let $F$ be a functor from {\textbf{C*$\mathbb{R}$-Alg}} to {\textbf{Ab}} that is homotopy invariant, stable, and split exact.
Then there exists a unique functor $\ftn{ \hat{F} }{ \textbf{KK} }{ \textbf{A} }$ such that $\hat{F} \circ \operatorname{KK} = F$.
\end{corol}
\begin{proof}
The functor $\hat{F}$ takes an object $A$ in {\textbf{KK}} to $F(A)$ in {\textbf{Ab}} and takes a morphism $y \in \operatorname{KK}(A,B)$ to the homomorphism $F(A) \rightarrow F(B)$ defined by $x \mapsto \alpha(x \otimes y)$. The composition $\hat{F} \circ \operatorname{KK} = F$ clearly holds on the level of objects. On the level of morphisms we must verify the formula $\alpha(x \otimes [f]) = f_*(x)$ for $f \colon A \rightarrow B$ and $x \in F(A)$. This formula follows by the naturality of the pairing $\alpha$, the formula $\alpha(x \otimes 1_A) = x$, and the formula $f_*(1_A) = [f] \in \operatorname{KK}(A,B)$ which is verified as in Section~2.8 of \cite{higson87}.
\end{proof}
\begin{propo} \label{les1}
Let $F$ be a functor from {\textbf{C*$\mathbb{R}$-Alg}} to {\textbf{Ab}} that is homotopy invariant and half exact. Then for any short exact sequence
$$0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \rightarrow 0$$
there is a natural boundary map $\partial \colon F(SC) \rightarrow F(A)$ that fits into a (half-infinite) long exact sequence
$$\dots \rightarrow F(SB) \xrightarrow{g_*} F(SC) \xrightarrow{\partial}
F(A) \xrightarrow{f_*} F(B) \xrightarrow{g_*} F(C) \; .$$
\end{propo}
\begin{proof}
Use the mapping cone construction as in Section~21.4 of \cite{blackadarbook}.
\end{proof}
\begin{corol} \label{splitexact}
A functor $F$ from {\textbf{C*$\mathbb{R}$-Alg}} to {\textbf{Ab}} that is homotopy invariant and half exact is also split exact.
\end{corol}
\begin{proof}
The splitting implies that $g_*$ is surjective. Thus in the sequence of Proposition~\ref{les1}, $\partial = 0$ and $f_*$ is injective.
\end{proof}
\begin{propo}
Let $F$ be a functor from {\textbf{C*$\mathbb{R}$-Alg}} to {\textbf{Ab}} that is homotopy invariant, stable, and half exact. Then for any short exact sequence
$$0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \rightarrow 0$$
there is a natural long exact sequence (with 24 distinct terms)
$$\dots \rightarrow F_{n+1}(C) \xrightarrow{\partial} F_n(A) \xrightarrow{f_*} F_n(B) \xrightarrow{g_*} F_n(C) \xrightarrow{\partial} F_{n-1}(A) \rightarrow \dots \; .$$
\end{propo}
\begin{proof}
From Corollary~\ref{splitexact} and Corollary~\ref{period}, $F$ is periodic; so Proposition~\ref{les1} gives the long exact sequence.
\end{proof}
We say that a homotopy invariant, stable, half-exact functor $F$ from {\textbf{C*$\mathbb{R}$-Alg}} to the category {\textbf{Ab}} of abelian groups
\begin{enumerate}
\item[(v)] satisfies the {\it dimension axiom} if there is an isomorphism $F_*(\mathbb{R}) \cong K_*(\mathbb{R})$ as graded modules over $K_*(\mathbb{R})$
\item[(vi)] is {\it continuous} if for any direct sequence of $R^{*}$-algebras $(A_n, \phi_n)$, the natural homomorphism
$$\lim_{n \to \infty} F_*(A_n) \rightarrow F_*(\lim_{n \to \infty} (A_n)) \; $$
is an isomorphism.
\end{enumerate}
\begin{theor}
Let $F$ be a functor from {\textbf{C*$\mathbb{R}$-Alg}} to {\textbf{Ab}} that is homotopy invariant, stable, half exact and satisfies the dimension axiom. Then there is a natural transformation $\beta \colon K_n(A) \rightarrow F_n(A)$. If $F$ is also continuous, then $\beta$ is an isomorphism for all $R^{*}$-algebras in the smallest class of separable $R^{*}$-algebras which contains $\mathbb{R}$ and is closed under $\operatorname{KK}$-equivalence, countable inductive limits, and the two-out-of-three rule for exact sequences.
\end{theor}
\begin{proof}
Let $z$ be a generator of $F(\mathbb{R}) \cong \mathbb{Z}$ and for $x \in K_n(A) \cong \operatorname{KK}(\mathbb{R}, S^nA)$ define
a $K_*(\mathbb{R})$-module homomorphism $\beta \colon K_*(A) \rightarrow F_*(A)$ by $\beta(x) = \alpha(z \otimes x)$. Taking $A = \mathbb{R}$, Theorem~\ref{thm:univprop1} yields that $\beta(1_0) = z$ where $1_0$ is the unit of the ring $K_*(\mathbb{R}) = \operatorname{KK}_*(\mathbb{R}, \mathbb{R})$. Therefore, $\beta$ is an isomorphism for $A = \mathbb{R}$. Then bootstrapping arguments show that $\beta$ is an isomorphism for all $R^{*}$-algebras in the class described.
\end{proof}
From Section~2.1 of \cite{boersema04} we have distinguished elements
\begin{align*}
c \in \operatorname{KK}_{0} ( \mathbb{R}, \mathbb{C} ), &\quad\quad r \in \operatorname{KK}_{0} ( \mathbb{C} , \mathbb{R} ) \\
\varepsilon \in \operatorname{KK}_{0} ( \mathbb{R} , T ), &\quad\quad \zeta \in \operatorname{KK}_{0} ( T, \mathbb{C} ) \\
\psi_{\scriptscriptstyle U} \in \operatorname{KK}_{0} ( \mathbb{C} , \mathbb{C} ), &\quad\quad \psi_{\scriptscriptstyle T} \in \operatorname{KK}_{0} ( T, T ) \\
\gamma \in \operatorname{KK}_{-1} ( \mathbb{C} , T ), &\quad\quad \tau \in \operatorname{KK}_{1} ( T, \mathbb{R}) \; .
\end{align*}
For any homotopy invariant, stable, split exact functor $F$ on {\textbf{C*$\mathbb{R}$-Alg}}, define the united $F$-theory of an $R^{*}$-algebra $A$ to be
$$F^{\scriptscriptstyle {\it CRT}}(A) = \{ F_*(A), F_*(\mathbb{C} \otimes A), F_*(T \otimes A) \} \; $$
together with the collection of natural homomorphisms \begin{align*}
c_n &\colon F_n(A) \rightarrow F_n(\mathbb{C} \otimes A) \\
r_n&\colon F_n(\mathbb{C} \otimes A) \rightarrow F_n(A) \\
\varepsilon_n &\colon F_n(A) \rightarrow F_n(T \otimes A) \\
\zeta_n &\colon F_n(T \otimes A) \rightarrow F_n(\mathbb{C} \otimes A) \\
(\psi_{\scriptscriptstyle U})_n &\colon F_n(\mathbb{C} \otimes A) \rightarrow F_n(\mathbb{C} \otimes A) \\
(\psi_{\scriptscriptstyle T})_n &\colon F_n(T \otimes A) \rightarrow F_n(T \otimes A) \\
\gamma_n &\colon F_n(\mathbb{C} \otimes A) \rightarrow F_{n-1}(\mathbb{C} \otimes A) \\
\tau_n &\colon F_n(T \otimes A) \rightarrow F_{n+1}(A) \\
\end{align*}
induced by the elements $c,r, \varepsilon, \zeta, \psi_{\scriptscriptstyle U}, \psi_{\scriptscriptstyle T}, \gamma, \tau$ via the
pairing of Theorem~\ref{thm:univprop1}.
\begin{propo}\label{thm:crtmod}
Let $F$ be a homotopy invariant, stable, split exact functor from {\textbf{C*$\mathbb{R}$-Alg}} to {\textbf{Ab}} and let $A$ be a separable $R^{*}$-algebra. Then $F^{\scriptscriptstyle {\it CRT}}(A)$
is a {\it CRT}-module. Moreover, if in addition $F$ is half exact, then $F^{\scriptscriptstyle {\it CRT}} ( A )$ is acyclic.
\end{propo}
\begin{proof}
To show that $F^{\scriptscriptstyle {\it CRT}}(A)$ is a {\it CRT}-module,
we must show that the {\it CRT} -module relations
\begin{align*}
rc &= 2 & \psi_{\scriptscriptstyle U} \beta_{\scriptscriptstyle U} &= -\beta_{\scriptscriptstyle U} \psi_{\scriptscriptstyle U} & \xi &= r \beta_{\scriptscriptstyle U}^2 c \\
cr &= 1 + \psi_{\scriptscriptstyle U} & \psi_{\scriptscriptstyle T} \beta_{\scriptscriptstyle T} &= \beta_{\scriptscriptstyle T} \psi_{\scriptscriptstyle T}
& \omega &= \beta_{\scriptscriptstyle T} \gamma \zeta \\
r &= \tau \gamma & \varepsilon \beta_{\scriptscriptstyle O} &= \beta_{\scriptscriptstyle T}^2 \varepsilon
& \beta_{\scriptscriptstyle T} \varepsilon \tau
&= \varepsilon \tau \beta_{\scriptscriptstyle T} + \eta_{\scriptscriptstyle T} \beta_{\scriptscriptstyle T} \\
c &= \zeta \varepsilon & \zeta \beta_{\scriptscriptstyle T} &= \beta_{\scriptscriptstyle U}^2 \zeta
& \varepsilon r \zeta &= 1 + \psi_{\scriptscriptstyle T} \\
(\psi_{\scriptscriptstyle U})^2 &= 1 & \gamma \beta_{\scriptscriptstyle U}^2 &= \beta_{\scriptscriptstyle T} \gamma
& \gamma c \tau &= 1 - \psi_{\scriptscriptstyle T} \\
(\psi_{\scriptscriptstyle T})^2 &= 1 & \tau \beta_{\scriptscriptstyle T}^2 &= \beta_{\scriptscriptstyle O} \tau & \tau &= -\tau \psi_{\scriptscriptstyle T} \\
\psi_{\scriptscriptstyle T} \varepsilon &= \varepsilon &
\gamma &= \gamma \psi_{\scriptscriptstyle U}
\qquad & \tau \beta_{\scriptscriptstyle T} \varepsilon &= 0 \\
\zeta \gamma &= 0 & \eta_{\scriptscriptstyle O} &= \tau \varepsilon
& \varepsilon \xi &= 2 \beta_{\scriptscriptstyle T} \varepsilon \\
\zeta &= \psi_{\scriptscriptstyle U} \zeta & \eta_{\scriptscriptstyle T} &= \gamma \beta_{\scriptscriptstyle U} \zeta
& \xi \tau &= 2 \tau \beta_{\scriptscriptstyle T} \;
\end{align*}
hold among the operations $\{c_n, r_n, \varepsilon_n, \zeta_n, (\psi_{\scriptscriptstyle U})_n, (\psi_{\scriptscriptstyle T})_n, \gamma_n, \tau_n\}$ on $F^{\scriptscriptstyle {\it CRT}}(A)$. But in the proof of Proposition~2.4 of \cite{boersema04}, it is shown that these relations hold at the level of $\operatorname{KK}$-elements. Therefore, using the associativity of the pairing of Theorem~\ref{thm:univprop1}, the same relations hold among the operations of $F^{\scriptscriptstyle {\it CRT}}(A)$.
Suppose now that $F$ is also half-exact. To show that $K^{\scriptscriptstyle {\it CRT}}(A)$ is acyclic,
we must show that the sequences
\begin{equation} \label{CRTequation1}
\dots \rightarrow F_n(A) \xrightarrow{\eta_{\scriptscriptstyle O}} F_{n+1}(A) \xrightarrow{c}
F_{n+1}(\mathbb{C} \otimes A) \xrightarrow{r \beta_{\scriptscriptstyle U}^{-1}} F_{n-1}(A) \rightarrow \dots
\end{equation}
\begin{equation} \label{CRTequation2}
\dots \rightarrow F_n(A) \xrightarrow{\eta_{\scriptscriptstyle O}^2} F_{n+2}(A) \xrightarrow{\varepsilon}
F_{n+2}(T \otimes A) \xrightarrow{\tau \beta_{\scriptscriptstyle T}^{-1}} F_{n-1}(A) \rightarrow \dots
\end{equation}
\begin{equation} \label{CRTequation3}
\dots \rightarrow F_{n+1}(\mathbb{C} \otimes A) \xrightarrow{\gamma} F_{n}(T \otimes A) \xrightarrow{\zeta}
F_{n}(\mathbb{C} \otimes A) \xrightarrow{1 - \psi_{\scriptscriptstyle U}} F_{n}(\mathbb{C} \otimes A) \rightarrow \dots \;
\end{equation}
are exact. These can be derived from the short exact sequences
$$0 \rightarrow S^{-1} \mathbb{R} \otimes A \rightarrow \mathbb{R} \otimes A \rightarrow \mathbb{C} \otimes A \rightarrow 0
$$
$$0 \rightarrow S^{-2} \mathbb{R} \otimes A \rightarrow \mathbb{R} \otimes A \rightarrow T \otimes A \rightarrow 0
$$
$$0 \rightarrow S \mathbb{C} \otimes A \rightarrow T \otimes A \rightarrow \mathbb{C} \otimes A \rightarrow 0
$$
from Sections~1.2 and 1.4 of \cite{boersema02}.
Indeed, focusing on the first one for our argument, the short exact sequence
$$0 \rightarrow S^{-1} \mathbb{R} \otimes A \xrightarrow{f} \mathbb{R} \otimes A \xrightarrow{g} \mathbb{C} \otimes A \rightarrow 0
$$
gives rise to the long exact sequence
$$\dots \rightarrow F_n(A) \xrightarrow{[f]} F_{n+2}(A) \xrightarrow{[g]}
F_{n+2}(\mathbb{C} \otimes A) \xrightarrow{\partial} F_{n-1}(A) \rightarrow \dots $$
where the homomorphisms are given by the multiplication of the elements $[f] \in \operatorname{KK}_2(\mathbb{R}, \mathbb{R})$, $[g] \in \operatorname{KK}_0(\mathbb{R}, \mathbb{C})$, and $\partial \in \operatorname{KK}_{-1}(\mathbb{C}, \mathbb{R})$. In the case of the functor $\operatorname{KK}(B,-)$, it was shown in the proof of Proposition~2.4 of \cite{boersema04} that the resulting sequence has the form
$$ \dots \rightarrow \operatorname{KK}_{n} ( B , A ) \xrightarrow{\eta_{\scriptscriptstyle O}}
\operatorname{KK}_{ n + 1} ( B , A ) \xrightarrow{c}
\operatorname{KK}_{n+1} ( B , \mathbb{C} \otimes A ) \xrightarrow{r \beta_{\scriptscriptstyle U}^{-1}}
\cdots $$
for all separable $R^{*}$-algebras $B$. It then follows easily that the $\operatorname{KK}$-element equalities $[f] = \eta_{\scriptscriptstyle O}$, $[g] = c$, $\partial = r \beta_{\scriptscriptstyle U}^{-1}$ hold. This proves that Sequence~\ref{CRTequation1} is exact. Sequences~\ref{CRTequation2} and \ref{CRTequation3} are shown to be exact the same way.
\end{proof}
\begin{theor} \label{main}
Let $F$ and $G$ be
homotopy invariant, stable, half exact functors from {\textbf{C*$\mathbb{R}$-Alg}} to {\textbf{Ab}} with a natural transformation $\mu_A \colon F(A) \rightarrow G(A)$. If $\mu_A$ is an isomorphism for all $C^{*}$-algebras $A$ in {\textbf{C*$\mathbb{R}$-Alg}}, then $\mu_A$ is an isomorphism for all $R^{*}$-algebras in {\textbf{C*$\mathbb{R}$-Alg}}.
\end{theor}
\begin{proof}
Let $A$ be a separable $R^{*}$-algebra. The natural transformation $\mu_A$ induces a homomorphism
$\mu_A^{\scriptscriptstyle {\it CRT}} \colon F^{\scriptscriptstyle {\it CRT}}(A) \rightarrow G^{\scriptscriptstyle {\it CRT}}(A)$
of acyclic {\it CRT}-modules which is, by hypothesis, an isomorphism on the complex part. Then the results in Section~2.3 of \cite{bousfield90} imply that $\mu_A^{\scriptscriptstyle {\it CRT}}$ is an isomorphism.
\end{proof}
\begin{corol}
Any homotopy invariant, stable, half exact functor from {\textbf{C*$\mathbb{R}$-Alg}} to {\textbf{Ab}} that vanishes on all $C^{*}$-algebras vanishes on all $R^{*}$-algebras.
\end{corol}
\section{Application: Isomorphism between $\operatorname{KK}_{-1}$ and $\mathrm{Ext}$}
In this section, we will use Theorem \ref{main} to show that $\operatorname{KK}_{-1}(A,B)$ is naturally isomorphic to $\mathrm{Ext}( A, B )^{-1}$ for separable $R^{*}$-algebras $A$ and $B$.
Let $\mathfrak{e} : 0 \to B \xrightarrow{\iota} E \overset{ \pi }{ \to } A \to 0$ be an exact sequence of $R^{*}$-algebras. As in the complex case, the associated mapping cone is defined to be
$$C_\pi = \{ (e, f) \in E \oplus C_{0}( [0,1] , B ) \mid f(1) = 0, f(0) = \pi(e) \} $$
and there are $*$-homomorphisms $\ftn{ \lambda_{\mathfrak{e}} }{ SA }{ C_{ \pi } }$, $\ftn{ \gamma_{ \mathfrak{e} } }{ B }{ C_{ \pi } }$, and $\kappa_{\mathfrak{e}} \colon C_\pi \rightarrow E$ defined by
\begin{align*}
\lambda_{\mathfrak{e}} ( f ) = ( 0, f ), \quad \quad \gamma_{ \mathfrak{e} } ( b ) = ( \iota b, 0 ), \quad \text{and} \quad \kappa_{\mathfrak{e}}(e,f) = e\; .
\end{align*}
As in the complex case, there is an exact sequence
$0 \rightarrow SA \xrightarrow{\lambda_{\mathfrak{e}}} C_\pi \xrightarrow{\kappa_{\mathfrak{e}}} E \rightarrow 0$.
Furthermore, $\kappa_{\mathfrak{e}} \circ \gamma_{\mathfrak{e}} = \iota$ and
$[\gamma_{\mathfrak{e}}]$ is an invertible element of $\operatorname{KK}_0(B, C_\pi)$.
\begin{lemma}\label{l:trivialext}
Let $\mathfrak{e} : 0 \to B \to E \overset{\pi}{\to} A \to 0$ be an exact sequence of $R^{*}$-algebras. If $\mathfrak{e}$ is trivial, i.e., there exists a $*$-homomorphism $\ftn{ \sigma }{ A }{ E }$ such that $\pi \circ \sigma = \mathrm{id}_{ A }$. Then $[ \lambda_{ \mathfrak{e} } ] = 0$ in $\operatorname{KK}_0(SA, C_{\pi})$.
\end{lemma}
\begin{proof}
Because of the section map $\sigma$, it follows that $\pi_*$ is surjective and that the boundary map vanishes in the sequence
$$\dots \rightarrow \operatorname{KK}_n(SA, B) \rightarrow \operatorname{KK}_n(SA, E) \xrightarrow{\pi_*} \operatorname{KK}_n(SA, A) \xrightarrow{\delta} \operatorname{KK}_{n-1}(SA, B) \rightarrow \dots \; .$$
By the comments above, the sequence
$0 \rightarrow SA \xrightarrow{\lambda_{\mathfrak{e}}} C_\pi \xrightarrow{\kappa_{\mathfrak{e}}} E \rightarrow 0 \; $
induces the same long exact sequence on $\operatorname{KK}$-theory. Thus the homomorphism
$\operatorname{KK}_n(SA, SA) \xrightarrow{ (\lambda_{ \mathfrak{e} })_* } \operatorname{KK}_n(SA, C_\pi) $ is zero.
Since $[ \lambda_{ \mathfrak{e} } ] = (\lambda_{ \mathfrak{e} })_* [1_{SA}]$ (where $[1_{SA}] \in \operatorname{KK}_0(SA, SA)$) it follows that $[ \lambda_{ \mathfrak{e} } ] = 0$.
\end{proof}
\begin{lemma} \label{l:uniteq}
Let $\mathfrak{e}_{i} : 0 \to B \to E_{i} \xrightarrow{\pi_{i}} A \to 0$ be an exact sequence of $R^{*}$-algebra with $B$ stable and $B$ an essential ideal of $E_{i}$. Suppose there exists a unitary $u \in \multialg{ B }$ such that
\begin{align*}
\mathrm{Ad} ( \pi_{B} ( u) ) \circ \tau_{ \mathfrak{e}_{1} } = \tau_{ \mathfrak{e}_{2} }
\end{align*}
where $\ftn{ \pi_{B} }{ \multialg{ B } }{ \corona{ B } = \multialg{ B } / B }$ is the canonical projection. Then
$
[ \lambda_{\mathfrak{e}_{1}} ] \times [ \gamma_{\mathfrak{e}_{1} } ]^{-1}
= [ \lambda_{\mathfrak{e}_{2}} ] \times [ \gamma_{\mathfrak{e}_{2} } ]^{-1}
\in \operatorname{KK}_0(SA, B)$.
\end{lemma}
\begin{proof}
As in the complex case, the conjugacy of the Busby invariants implies the corresponding extensions are equivalent. That is, we have the following commutative diagram:
\begin{align*}
\xymatrix{
0 \ar[r] & B \ar[r] \ar[d]^{ \mathrm{Ad} ( u ) } & E_{1} \ar[r]^{\pi_{1}} \ar[d]^{\mathrm{Ad} (u) } & A \ar[r] \ar@{=}[d] & 0 \\
0 \ar[r] & B \ar[r] & E_{2} \ar[r]_{\pi_{2}} & A \ar[r] & 0
}
\end{align*}
where we are identifying $E_{i}$ as a sub-algebra of $\multialg{B}$. Since the mapping cone construction is functorial, there exists a $*$-isomorphism $\ftn{ \alpha }{ C_{\pi_{1}} }{ C_{ \pi_{2} } }$ such that the diagram
\begin{align*}
\xymatrix{
S A \ar@{=}[d] \ar[r]^-{ \lambda_{\mathfrak{e}_{1} } } & C_{\pi_{1}} \ar[d]^{ \alpha } & B \ar[l]_-{ \gamma_{ \mathfrak{e}_{1} } } \ar[d]^{ \mathrm{Ad} (u) }\\
S A \ar[r]_{ \lambda_{ \mathfrak{e}_{2} } } & C_{\pi_{2}} & B \ar[l]^-{ \gamma_{ \mathfrak{e}_{2} } }.
}
\end{align*}
Since $[ \mathrm{Ad} (u) ] = [1_B] \in \operatorname{KK}_0(B, B)$, the result follows.
\end{proof}
As in the complex case, we define the sum of two extensions in terms of the Busby invariants. Let $B$ be stable and let $\ftn{ \pi_{B} }{ \multialg{ B } } { \corona{ B } }$ be the canonical projection. Then the sum of two extensions with Busby invariants $\tau_i \colon A \rightarrow \corona{B}$ is the unique extension (up to strong equivalence) with Busby invariant $\tau_1 \oplus \tau_2 \colon A \rightarrow \corona{B}$. The $*$-homomorphism $\tau_1 \oplus \tau_2$ is defined by fixing a $*$-isomorphism $M_2(B) \cong B$. More specifically,
$\tau_1 \oplus \tau_2 = \overline{\psi} \circ \sigma$
where $\sigma \colon A \rightarrow \corona{B} \oplus \corona{B} \subseteq \corona{B \oplus B}$ is defined by $\sigma = (\tau_1, \tau_2)$ and $\overline{\psi}$ is the composition
$$ \corona{B \oplus B} \rightarrow M_2( \corona{B}) \xrightarrow{\cong} \corona{B} \; .$$
\begin{lemma}
Let $\mathfrak{e}_{i} : 0 \to B \to E_{i} \xrightarrow{\pi_{i}} A \to 0$ be an exact sequence of $R^{*}$-algebras with $B$ stable.
Let $\mathfrak{e}_{1} \oplus \mathfrak{e}_{2} : 0 \to B \to E \to A \to 0$ be the sum as defined in the previous paragraph.
Then
\begin{align*}
[ \lambda_{ \mathfrak{e}_{1} \oplus \mathfrak{e}_{2} } ] \times [ \gamma_{ \mathfrak{e}_{1} \oplus \mathfrak{e}_{2} } ]^{-1} =[\lambda_{ \mathfrak{e}_{1} } ] \times [ \gamma_{\mathfrak{e}_{1}} ]^{-1} + [ \lambda_{ \mathfrak{e}_{2} } ] \times [ \gamma_{\mathfrak{e}_{2}} ]^{-1}
\in \operatorname{KK}_0(SA, B).
\end{align*}
\end{lemma}
\begin{proof}
Let $\tau_i$ be the Busby invariant corresponding to $\mathfrak{e}_{i}$ and $\tau_1 \oplus \tau_2$ the Busby invariant of the sum $\mathfrak{e}_1 \oplus \mathfrak{e}_2$, as above. Let $\sigma$ and $\overline{\psi}$ be as above also. Notice that $\sigma$ is the Busby invariant of an extension of the form
$$0 \rightarrow B \oplus B \rightarrow E_\sigma \rightarrow A \rightarrow 0 \; .$$
Since $\tau_1 \oplus \tau_2 = \psi \circ \sigma$, there exist morphisms $\psi$ and $\phi$ forming the following commutative diagram
\begin{align*}
\xymatrix{
0 \ar[r] & B \oplus B \ar[r] \ar[d]^{ \psi } & E_\sigma \ar[r] \ar[d]^{ \phi } & A \ar@{=}[d] \ar[r] & 0 \\
0 \ar[r] & B \ar[r] & E_{ \mathfrak{e}_{1} \oplus \mathfrak{e}_{2} } \ar[r] & A \ar[r] & 0.
}
\end{align*}
Let $\ftn{ \rho_{i} }{ B \oplus B }{ B }$ be the projection onto the $i$-coordinate and let
$\ftn{ \overline{ \rho }_{i} }{ \corona{ B \oplus B } }{ \corona{B} }$ be the induced map.
Then $\overline{ \rho }_{i} \circ \sigma = \tau_i$. Hence, there exists a $*$-homomorphisms $\ftn{ \phi_{i} }{ E_\sigma }{ E_{i} }$ such that
\begin{align*}
\xymatrix{
0 \ar[r] & B \oplus B \ar[r] \ar[d]^{ \rho_{i} } & E_{\sigma} \ar[r] \ar[d]^{ \phi_{i} } & A \ar[r] \ar@{=}[d] & 0 \\
0 \ar[r] & B \ar[r] & E_{i} \ar[r] & A \ar[r] & 0
}
\end{align*}
commutes for $i = 1, 2$.
By the above paragraphs, the following diagrams
\begin{align*}
\xymatrix{
S A \ar@{=}[d] \ar[r]^-{ \lambda_{\mathfrak{e} } } & C_{\rho} \ar[d]^{ \widetilde{\phi}_{i} } & B \oplus B \ar[l]_-{ \gamma_{ \mathfrak{e} } } \ar[d]^{ \rho_{i} }\\
S A \ar[r]_{ \lambda_{ \mathfrak{e}_{i} } } & C_{\pi_{i}} & B \ar[l]^-{ \gamma_{ \mathfrak{e}_{i} } }
}
\qquad \qquad
\xymatrix{
S A \ar@{=}[d] \ar[r]^-{ \lambda_{\mathfrak{e} } } & C_{\rho} \ar[d]^{ \widetilde{\phi} } & B \oplus B \ar[l]_-{ \gamma_{ \mathfrak{e} } } \ar[d]^{ \psi}\\
S A \ar[r]_{ \lambda_{ \mathfrak{e}_{1} \oplus \mathfrak{e}_{2} } } & C_{\pi_{3}} & B \ar[l]^-{ \gamma_{ \mathfrak{e}_{1} \oplus \mathfrak{e}_{2} } }
}
\end{align*}
are commutative. Thus,
\begin{align*}
[ \lambda_{ \mathfrak{e}_{1} \oplus \mathfrak{e}_{2} } ][ \gamma_{ \mathfrak{e}_{1} \oplus \mathfrak{e}_{2} } ]^{-1}
&= [ \lambda_{\mathfrak{e} } ][ \widetilde{\phi} ][ \gamma_{ \mathfrak{e}_{1} \oplus \mathfrak{e}_{2} } ]^{-1} \\
&=[ \lambda_{\mathfrak{e} } ][ \gamma_{ \mathfrak{e} } ]^{-1} [ \psi ] \\
&= [ \lambda_{\mathfrak{e} } ][ \gamma_{ \mathfrak{e} } ]^{-1} \left([ \rho_{1} ] + [ \rho_{2} ] \right) \\
&= [ \lambda_{ \mathfrak{e} } ]
\left( [ \widetilde{ \phi }_{1} ] [ \gamma_{ \mathfrak{e}_{1} } ]^{-1 }
+ [ \widetilde{ \phi }_{2} ][ \gamma_{ \mathfrak{e}_{2} } ]^{-1 } \right) \\
&= [ \lambda_{ \mathfrak{e}_{1} } ][ \gamma_{ \mathfrak{e}_{1} } ]^{-1 }
+ [ \lambda_{ \mathfrak{e}_{2} } ][ \gamma_{ \mathfrak{e}_{2} } ]^{-1 }.
\end{align*}
\end{proof}
The goal for the rest of this section is to produce an isomorphism between $\operatorname{KK}_{-1}(A,B)$ and the group $\operatorname{Ext}(A,B)^{-1}$ of invertible extensions.
For $r, s \in \mathbb{Z}_{ \geq 0 }$, let $\mathcal{C}\ell_{r,s}$ denote the Clifford algebra $\mathcal{C}\ell ( \mathbb{R}^{r+s} , Q_{r,s} )$ where $Q_{r,s}$ is the quadratic form
\begin{align*}
Q_{r,s} = - \sum_{ i = 1}^{r} x_{i}^{2} + \sum_{ j = r+1}^{ r+s} x_{j}^{2}.
\end{align*}
We will be using the fact that $\operatorname{KK}_*(A \hat{ \otimes } \mathcal{C}\ell_{p, q}, B \hat{ \otimes } \mathcal{C} \ell_{r, s})$ depends, up to natural isomorphism, only on $(p-q) - (r-s)$ (see Corollary~4.2.10 of \cite{schroderbook}). In particular, for any trivially graded $R^{*}$-algebras $A$ and $B$, we have $\operatorname{KK}_{-1}(A,B) \cong \operatorname{KK}_{0} (A , B \hat{ \otimes } \mathcal{C}\ell_{0,1} )$.
A Kasporov module $(A , B \hat{ \otimes } \mathcal{C}\ell_{0,1} )$-module is a triple from $( \phi,E, F )$ such that $\ftn{ \phi }{ A }{ \mathbb{B} ( E \hat{\otimes} \mathcal{C}\ell_{0,1} ) }$ is a graded $*$-homomorphism, $F$ is an operator in $\mathbb{B} ( E\hat{ \otimes } \mathcal{C}\ell_{0,1} )$ of degree $1$, and
\begin{align*}
F \phi ( a ) - \phi ( a ) F , ( F^{2} - 1 ) \phi ( a ) , ( F - F^{*} ) \phi ( a )
\end{align*}
are elements of $\mathcal{K}_{\mathbb{R}} \otimes \mathfrak{B}$ for all $a \in A$. Using an argument similar to that in the complex case, we may assume that $F^{*} = F = F^{-1}$ and $E = \mathbb{H}_{B}$ (see Section 17.4 and 17.6 of \cite{blackadarbook}). Since $\mathbb{B} ( \mathbb{H}_{B} \hat{ \otimes } \mathcal{C}\ell_{0,1} ) \cong \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B } \oplus \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B }$ with grading
\begin{align*}
( \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B } \oplus \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B } )^{(0)} &= \setof{ ( x, x ) }{ x \in \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B } } \\
( \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B} \oplus \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B } )^{(1)} &= \setof{ ( x , - x ) }{ x \in \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B } }
\end{align*}
we have that $\phi = \psi \oplus \psi$ for some $*$-homomorphism $\ftn{ \psi }{ A }{ \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B } }$ and $F = T \oplus (-T)$ for some self-adjoint unitary $T$ in $\multialg{ \mathcal{K}_{\mathbb{R}} \otimes B }$.
Set $\ftn{ \pi }{ \multialg{ \mathcal{K}_{\mathbb{R}} \otimes B } }{ \corona{ \mathcal{K}_{\mathbb{R}} \otimes B } }$ be the canonical projection. Set $p = \frac{ T + 1 }{ 2 }$. Then $p$ is a projection and $\pi ( p ) ( \pi \circ \psi ( a ) ) = ( \pi \circ \psi ( a ) ) \pi ( p )$ for all $a \in A$. Define $\ftn{ \tau_{ ( E , \phi, F ) } }{ A }{ \corona{ \mathcal{K}_{\mathbb{R}} \otimes B } }$ by
$$
\tau_{ ( E , \phi, F ) } ( a ) = \pi ( p \psi( a ) p ) \; .
$$
Then $\tau_{ ( E, \phi , F ) }$ is a $*$-homomorphism from $A$ to $\corona{ \mathcal{K}_{\mathbb{R}} \otimes B }$. Set $\mathfrak{e}_{ (E, \psi , F ) }$ be the exact sequence
\begin{align*}
\mathfrak{e}_{ ( E, \psi , F ) } : 0 \to \mathcal{K}_{\mathbb{R}} \otimes B \to X( E, \psi, F) \to A \to 0
\end{align*}
induced by $\tau_{ ( E, \phi, F ) }$. This construction is well-defined and gives a natural transformation
$$\Theta_{A,B} \colon \operatorname{KK}_{-1}(A,B) \rightarrow \operatorname{Ext}(A,B)^{-1} \; .$$
By Lemma~\ref{l:uniteq}, we also have a well-defined natural transformation
$$\ftn{ \Lambda_{A, B } }{ \operatorname{Ext}( A , B)^{-1} }{ \operatorname{KK}( S A , B ) }$$
given by
$\Lambda_{ A , B } ( [ \mathfrak{e} ] ) = [\lambda_{ \mathfrak{e}_{ ( E, \psi , F ) } } ][ \gamma_{ \mathfrak{e}_{ ( E, \psi , F ) } } ]^{-1}.$
\begin{theor}\label{t:extkk}
If $A$ and $B$ are separable $R^{*}$-algebras with $B$ stable, then $\Theta_{A,B}$ and $\Lambda_{A,B}$ are isomorphisms.
\end{theor}
\begin{proof}
Let $\mu_{A,B} = \Lambda_{A,B} \circ \Theta_{A,B}$. For separable $C^*$-algebras $A$ and $B$ there is a similarly defined homomorphism $\mu_{A, B} ^\C \colon \operatorname{KK}^\C_{-1}(A, B) \rightarrow \operatorname{KK}^\C(SA, B)$. According to Proposition~19.5.7 of \cite{blackadarbook}, this composition $\mu_{A,B}$ is exactly the usual isomorphism from $\operatorname{KK}^\C_{-1}(A, B)$ to $\operatorname{KK}^\C(SA, B)$.
Suppose now that $B$ is a separable $C^*$-algebra. Then as in Lemma~4.3 of \cite{boersema04}, there is a natural isomorphism $\iota$ from $\operatorname{KK}_*^\C(A_\C, B)$ to $\operatorname{KK}_*(A,B)$. Furthermore, the diagram below commutes,
\begin{align*}
\xymatrix{
\operatorname{KK}_{-1}^\C(A _\C, B) \ar[rr]^{\mu^\C_{A_\C,B}} \ar[d]^{\iota} &&
\operatorname{KK}^\C(SA _\C, B) \ar[d]^{\iota} \\
\operatorname{KK}_{-1}(A, B) \ar[rr]^{\mu_{A,B}} &&
\operatorname{KK}(SA, B)
}
\end{align*}
showing that $\mu_{A,B}$ is an isomorphism when $B$ is a $C^*$-algebra. It then follows by Theorem~\ref{main} that $\mu_{A,B}$ is an isomorphism for all separable $R^*$-algebras $A$ and $B$.
It remains now to prove that $\Theta_{A,B}$ is surjective. Let $\ftn{ \tau }{ A }{ \corona{B} }$ be a $*$-homomorphism such that there exists a $*$-homomorphism $\ftn{ \tau^{-1} }{ A }{ \corona{B} }$ with $\tau \oplus \tau^{-1}$ is a trivial extension. Then there exists a $*$-homomorphism
\begin{align*}
\ftn{ \phi =
\begin{bmatrix}
\phi_{11} & \phi_{12} \\
\phi_{21} & \phi_{22}
\end{bmatrix} }{ A }{ M_{2} ( \multialg{B} ) \; ,}
\end{align*}
which is a lifting of $\tau \oplus \tau^{-1}$. Set $q = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \in M_{2} ( \multialg{B} )$. Then $q$ is a projection with $q \phi (a) - \phi ( a ) q \in M_{2} ( B )$. Moreover, if $\ftn{ \pi_{2} }{ M_{2} ( \multialg{B} ) }{M_{2} ( \corona{B} ) }$ is the natural projection, then $\pi_{2} ( q \phi(a) q ) = \tau(a) \oplus 0$.
Let $\ftn{ \pi }{ \multialg{B} }{ \corona{B} }$ be the natural projection. Using the $*$-isomorphism $M_{2} ( \multialg{ B } ) \cong \multialg{B}$, we have a $*$-homomorphism $\ftn{ \psi }{ A } { \multialg{ B } }$ and a projection $p \in \multialg{B}$ such that $p \psi ( a ) - \psi ( a ) p \in B$ and $\pi ( p \psi(a) p ) = \tau ( a )$. Set $T = 2p - 1$. Then
\begin{align*}
\Theta_{A,B} \left( \left( \mathbb{H}_{B} \hat{ \otimes } \mathcal{C}\ell_{0,1}, \psi \oplus \psi , T \oplus -T \right) \right) = [ \tau ].
\end{align*}
Hence, $\Theta_{A,B}$ is surjective, which completes the proof.
\end{proof}
\section{Asymptotic Morphisms and $E$-theory}
In this section, we will use Theorem \ref{main} to show that $\operatorname{KK} (A, B )$ is naturally isomorphic to $E(A,B)$ for separable $R^{*}$-algebras $A$ and $B$, when $A$ is nuclear.
The following definition of an asymptotic morphism for $R^{*}$-algebras is exactly analogous to that for $C^{*}$-algebras. This definition also appears in Section~8 of \cite{BRS}.
\begin{defin}\label{d:asymorp}
Let $A$ and $B$ be $R^{*}$-algebras. An \emph{asymptotic morphism} from $A$ to $B$ is a family $\asy{ \phi_{t} }, \ ( t \in [1,\infty) )$ of maps from $A$ to $B$ with the following properties:
\begin{enumerate}
\item for all $a \in A$, $t \mapsto \phi_{t} ( a )$ is bounded and continuous; and
\item The set $\asy{ \phi_{t} }$ is asymptotically $*$-linear and multiplicative, i.e.\
\begin{enumerate}
\item $\displaystyle \lim_{ t \to \infty } \norm{ \phi_{t} ( \lambda a + b ) - ( \lambda \phi_{t} ( a ) + \phi_{t} ( b ) ) } = 0$;
\item $\displaystyle \lim_{ t \to \infty } \norm{ \phi_{t} ( a^{*} ) -\phi_{t} ( a )^{*} } = 0$; and
\item $\displaystyle \lim_{ t \to \infty } \norm{ \phi_{t} ( a b ) - \phi_{t} ( a ) \phi_{t} ( b ) } = 0$
\end{enumerate}
for all $a, b \in A$ and $\lambda \in \mathbb{R}$.
\end{enumerate}
Two asymptotic morphisms $\ftn{ \asy{ \phi_{t} }, \asy{ \psi_{t} } }{ A }{ B }$ are \emph{equivalent} if
\begin{align*}
\lim_{ t \to \infty } \left[ \phi_{t} ( a ) - \psi_{t} ( a ) \right] = 0
\end{align*}
for all $a \in A$.
\end{defin}
\begin{defin}
Let $A$ be an $R^{*}$-algebra. Set
\begin{align*}
\beta B &= \setof{ \ftn{f}{ [ 1, \infty ) }{ B } }{ \text{$f$ is bounded and continuous} } \\
\beta_{0} B &= \setof { f \in \beta B }{ \lim_{t \to \infty } f(t) = 0} \\
\alpha B &= \beta B / \beta_{0} B
\end{align*}
\end{defin}
Let $\ftn{ \asy{\phi_{t}} }{ A }{ B }$ be an asymptotic morphism. By Property (1), for each $a \in A$, the function $\hat{a} : t \mapsto \phi_{t} (a)$ is an element of $\beta B$. Let $\ftn{ \rho_{B} }{ \beta B }{ \alpha B }$ be the natural projection. By Property (2), the map
\begin{align*}
a \mapsto \rho_{B} ( \hat{a} )
\end{align*}
is a $*$-homomorphism from $A$ to $\alpha B$, which we denote by $\overline{ \asy{\phi_{t}} }$. If $\asy{ \phi_{t} }$ is equivalent to $\asy{ \psi_{t} }$, then $\overline{ \asy{ \phi_{t} } } = \overline{ \asy{ \psi_{t} } }$. Let $\ftn{ \phi }{ A }{ \alpha B }$ be a $*$-homomorphism and let $\psi \colon A \rightarrow \beta B$ be a lifting of $\phi$. Set $\phi_{t} = \mathrm{ev}_{t} \circ \psi$. Then $\asy{ \phi_{t} }$ is an asymptotic morphism from $A$ to $B$ such that $\overline{ \asy{ \phi_{t} } } = \phi$.
From the above paragraph, we have the following proposition (compare to Remark~25.1.4 (a) of \cite{blackadarbook}).
\begin{propo}\label{p:bijection}
There is a bijection from the set of $*$-homomorphism from $A$ to $\alpha B$ to the set of equivalence classes of asymptotic morphisms from $A$ to $B$.
\end{propo}
\begin{remar}
Every $*$-homomorphism from $A$ to $B$ defines an asymptotic morphism as follows: Let $\ftn{ \phi }{ A }{ B }$ be a $*$-homomorphism. The $\phi$ induces an asymptotic morphism $\asy{ \phi_{t} }$ by $\phi_{t} (a) = \phi ( a )$ for all $t \in [1, \infty )$ and $a \in A$. We will denote this asymptotic morphism by $\asy{ \phi }$.
\end{remar}
\begin{defin}
Let $\ftn{ \asy{ \phi_{t} } , \asy{ \psi_{t} } }{ A }{B}$ be asymptotic morphisms. $\asy{ \phi_{t} }$ is \emph{homotopic} to $\asy{ \psi_{t} }$ if there exists an asymptotic morphism $\ftn{ \asy{ \Phi_{t} } }{ A }{ IB }$ such that for each $t \in [1, \infty )$ and $a \in A$,
\begin{align*}
\mathrm{ev}_{0}^{[0,1]} ( \Phi_{t} (a) ) = \phi_{t} ( a ) \quad \text{and} \quad
\mathrm{ev}_{1}^{[0,1]} ( \Phi_{t} (a) ) = \psi_{t} ( a ).
\end{align*}
\end{defin}
Set $[[A, B]]$ to be the set of homotopy classes of asymptotic morphisms from $A$ to $B$. If $\asy{ \phi_{t} }$ is an asymptotic morphism, then $[ \asy{ \phi_{t} }]$ will denote the class in $[[A,B]]$ represented by $\asy{ \phi_{t} }$. The following proposition is analogous to Remark~2.5.1.2(g) of \cite{blackadarbook}.
\begin{propo}\label{p:eqhomotopy}
Let $\asy{ \phi_{t} }$ and $\asy{ \psi_{t} }$ be equivalent asymptotic morphisms from $A$ to $B$. Then $\asy{ \phi_{t} }$ and $\asy{ \psi_{t} }$ are homotopic.
\end{propo}
\begin{proof}
Let $a \in A$ and let $t \in [1, \infty )$. Set $\Phi_{t} ( a ) (s) = s \psi_{t} ( a ) + ( 1 - s ) \phi_{t} ( a )$ for all $s \in [0,1]$. Since $\Phi_{t} ( a )$ is the straight line homotopy from $\phi_{t}(a)$ and $\psi_{t} ( a )$, we have that $\Phi_{t} ( a ) \in I B$. Since
\begin{align}\label{eqhomotopy}
\norm{ \Phi_{t} ( a ) - \Phi_{t'} ( a ) } \leq \norm{ \phi_{t} ( a ) - \phi_{t'} ( a ) } + \norm{ \psi_{t} ( a ) - \psi_{t'} ( a ) }
\end{align}
for all $a \in A$ and for all $t, t' \in [1, \infty )$, we have that
\begin{align*}
t \mapsto \Phi_{t} (a)
\end{align*}
is a continuous function for all $a \in A$.
Define $\Psi_{t} ( a )(s) = \psi_{t} ( a )$ for all $a \in A$, $t \in [1, \infty )$, and $s \in [0,1]$. It is clear that $\asy{ \Psi_{t} }$ is an asymptotic morphism from $A$ to $IB$. Note that
\begin{align*}
\norm{ \Phi_{t} ( a ) - \Psi_{t} ( a ) } \leq \norm{ \phi_{t} ( a ) - \psi_{t} ( a ) }
\end{align*}
for all $a \in A$ and $t \in [1, \infty )$. Thus, $\lim_{ t \to [1, \infty ) } ( \Phi_{t} ( a ) - \Psi_{t} ( a ) ) = 0$ which implies that $\asy{ \Phi_{t} }$ is an asymptotic morphism from $A$ and $IB$. By construction, $\mathrm{ev}_{0}^{[0,1]} \circ \Phi_{t} = \phi_{t}$ and $\mathrm{ev}_{1}^{[0,1]} \circ \Phi_{t} = \psi_{t}$ for all $t \in [1, \infty )$.
\end{proof}
Let $s \in [0,1]$. Then $\mathrm{ev}_{s}^{[0,1]} : f \mapsto f(s)$ is a $*$-homomorphism from $IB$ to $B$. Define
$\widetilde{ \mathrm{ev} }_{s}^{[0,1]} \colon \beta IB \rightarrow \beta B$ by $\widetilde{ \mathrm{ev} }_{s}^{[0,1] } ( f ) (t) = f( t) (s)$ for all $f \in \beta IB$. Let $f \in \beta IB$. Then the function
\begin{align*}
t \mapsto f(t)(s)
\end{align*}
is continuous and bounded by $\norm{ f }$. Hence, $\widetilde{ \mathrm{ev} }_{s}^{[0,1] } ( f ) \in \beta B$. Thus, $\ftn{ \widetilde{ \mathrm{ev} }_{s}^{[0,1]} }{ \beta IB }{ \beta B }$ is a $*$-homomorphism for all $s \in [0, 1]$. Note that $\widetilde{ \mathrm{ev} }_{s}^{[0,1]} ( \beta_{0} IB ) = \beta_{0} B$. Thus, there exists a $*$-homomorphism $\ftn{ \overline{ \mathrm{ev} }_{s}^{[0,1]} }{ \alpha IB }{ \alpha B }$ such that the diagram
\begin{align*}
\xymatrix{
0 \ar[r] & \beta_{0} IB \ar[r] \ar[d]^{ \widetilde{ \mathrm{ev} }_{s}^{[0,1]} }& \beta IB \ar[r]^{ \rho_{IB} } \ar[d]^{ \widetilde{ \mathrm{ev} }_{s}^{[0,1]} } & \alpha IB \ar[r] \ar[d]^{ \overline{ \mathrm{ev} }_{s}^{[0,1]} }& 0 \\
0 \ar[r] & \beta_{0} B \ar[r] & \beta B \ar[r]_{ \rho_{B} } & \alpha B \ar[r] & 0
}
\end{align*}
is commutative.
\begin{propo}\label{p:homotopequal}
Let $A$ and $B$ be $R^{*}$-algebras and let $\asy{ \phi_{t} }$ and $\asy{ \psi_{t} }$ be asymptotic morphisms. Suppose there exists a $*$-homomorphism $\ftn{ \Phi }{ A }{ I \alpha B}$ such that $\overline{ \mathrm{ev} }_{0}^{[0,1]} \circ \Phi = \overline{ \asy{ \phi_{t} } }$ and $\overline{ \mathrm{ev} }_{1}^{[0,1]} \circ \Phi = \overline{ \asy{ \psi_{t} } }$. Then $\asy{ \phi_{t} }$ and $\asy{ \psi_{t} }$ are homotopic.
\end{propo}
\begin{proof}
Note that there exists an asymptotic morphism $\ftn{ \asy{ \Phi_{t} } }{ A }{ I B }$ such that $\overline{ \asy{ \Phi_{t} } } = \Phi$. Then $\asy{ \mathrm{ev}_{0}^{[0,1]} \circ \Phi_{t} }$ and $\asy{ \mathrm{ev}_{1}^{[0,1]} \circ \Phi_{t} }$ are asymptotic morphism from $A$ to $B$ that are homotopic. Since $\overline{ \mathrm{ev} }_{0}^{[0,1]} \circ \Phi = \overline{ \asy{ \phi_{t} } }$, we have that $\asy{ \phi_{t} }$ and $\asy{ \mathrm{ev}_{0}^{[0,1]} \circ \Phi_{t} }$ are equivalent. Similarly, $\asy{ \psi_{t} }$ and $\asy{ \mathrm{ev}_{1}^{[0,1]} \circ \Phi_{t} }$ are equivalent. By Proposition \ref{p:eqhomotopy}, $\asy{ \phi_{t} }$ and $\asy{ \mathrm{ev}_{0}^{[0,1]} \circ \Phi_{t} }$ are homotopic and $\asy{ \psi_{t} }$ and $\asy{ \mathrm{ev}_{1}^{[0,1]} \circ \Phi_{t} }$ are homotopic. Hence, $\asy{ \phi_{t} }$ and $\asy{ \psi_{t} }$ are homotopic.
\end{proof}
\begin{lemma}\label{l:quasicentral}
Let $I$ be an ideal of a separable $R^{*}$-algebra $A$. Then there exists a continuous approximate identity $\{ u_{t} \}_{t \in [1, \infty)}$ of $I$ which is quasi-central in $A$. In other words, for all $t$, $u_t$ is positive and norm-bounded by $1$; and for all $a \in A$,
$\lim_{t \to \infty } \norm{ u_{t} a - a u_{t} } = 0$.
\end{lemma}
\begin{proof}
Note that the complexification $I_{ \mathbb{C} }$ of $I$ is an ideal of the complexification $A_{\mathbb{C}}$ of $A$ and $A_{\mathbb{C}}$ is separable since $A$ is separable. Let $\tau$ be a conjugate-linear involution of $A_{\mathbb{C}}$ such that $A$ can be identified with $\setof{ x \in A_{\mathbb{C}} }{ \tau ( x ) = x }$. Then $I$ can be identified as $\setof{ x \in I_{ \mathbb{C} } }{ \tau ( x ) = x }$. Since $I_{\mathbb{C}}$ is an ideal of a separable $C^{*}$-algebra $A_{\mathbb{C}}$, there exists a continuous quasi-central approximate identity $\{ w_{t} \}_{ t \in [1, \infty ) } $ of $I_{\mathbb{C}}$.
Set $u_{t} = \frac{1}{2} ( w_{t} + \tau ( w_{t} ) ) \in A$. Then it is easy to verify that $\{ u_{t} \}_{t \in [1, \infty)}$ is again a continuous quasi-central approximate identity of $I_{\mathbb{C}}$ (hence of $I$) and is again quasi-central in $A_\mathbb{C}$ (hence in $A$).
\end{proof}
\begin{lemma}\label{l:approxunit}(See Lemma 25.5.2 of \cite{blackadarbook}).
Let $A$ be an $R^{*}$-algebra. Let $\{ u_{t} \}_{ t \in [1, \infty ) }$ be given such that $u_t$ is positive and $\norm{ u_{t} } \leq 1$ for all $t$. Let $x \in A$ and let $f \in S\mathbb{R}$.
\begin{itemize}
\item[(a)] If $\displaystyle \lim_{ t\to \infty } ( x u_{t} - u_{t} x ) = 0$, then $\displaystyle \lim_{ t\to \infty } ( x f(u_{t}) - f(u_{t}) x ) = 0$.
\item[(b)] If $\displaystyle \lim_{ t\to \infty } u_{t} x = x$, then $\displaystyle \lim_{ t\to \infty } f(u_{t}) x = 0$.
\end{itemize}
\end{lemma}
\begin{proof}
Note that,
\begin{align*}
u_{t}^{k} x - x u_{t}^{k} = \sum_{ n= 0}^{k-1} ( u_{t}^{k-n} x u_{t}^{n} - u_{t}^{k - (n +1) } x u_{t}^{n+1} ).
\end{align*}
Therefore,
\begin{align*}
\norm{ u_{t}^{k} x - x u_{t}^{k} } \leq k \norm{ u_{t} x - x u_{t} }
\end{align*}
which implies that $\displaystyle \lim_{ t \to \infty } ( u_{t}^{k} x - x u_{t}^{k} ) = 0$, whenever $\displaystyle \lim_{ t\to \infty } ( x u_{t} - u_{t} x ) = 0$.
Also,
\begin{align*}
u_{t}^{k} x - x = \sum_{ n = 0}^{ n-1 } ( u_{t}^{k-n} x - u_{t}^{k-n-1} x ).
\end{align*}
Therefore,
\begin{align*}
\norm{ u_{t}^{k} x - x } \leq k \norm{ u_{t} x - x }
\end{align*}
which implies that $\displaystyle \lim_{ t \to [1, \infty ) } ( u_{t}^{k} x - x ) = 0$ whenever $\displaystyle \lim_{ t\to \infty } u_{t} x = x$.
The lemma now follows from the Stone-Weierstrass Theorem and the above observations.
\end{proof}
\begin{propo}\label{p:extasy}(See Proposition 25.5.1 in \cite{blackadarbook}.)
Let $\mathfrak{e} : 0 \to B \overset{ \iota }{ \to } E \overset{ \pi }{ \to } A \to 0$ be an exact sequence of $R^{*}$-algebras. Suppose $B$ has a continuous approximate identity $\{ u_{t} \}_{ t \in [1, \infty ) }$ which is quasi-central in $E$.
Let $\sigma$ be a bounded continuous cross-section of $\pi$. Then there exists an asymptotic morphism $\ftn{ \asy{ \phi_{t}^{ \mathfrak{e} } } }{ S A }{ B }$ such that
\begin{align*}
\lim_{ t \to \infty } ( \phi_{t}^{ \mathfrak{e} } ( f \otimes a ) - f( u_{t} ) \sigma (a) ) = 0,
\end{align*}
for all $f \in S \mathbb{R}$ and $a \in A$, where we are identifying $S A$ with $S \mathbb{R} \otimes A$. Moreover, if $\tau$ is another cross section of $\pi$, then the associated asymptotic morphisms are equivalent.
\end{propo}
We note that the existence of the cross-section $\sigma$ is given by the Bartle-Graves selection theorem.
\begin{proof}
Define $\ftn{ \theta_{t} }{ S \mathbb{R} \times A }{ B}$ by $\theta _{t} ( f, x ) = f( u_{t} ) \sigma ( x )$. Since $\{ u_{t} \}_{ t \in [1, \infty ) }$ is continuous, for each $( f, x ) \in S \mathbb{R} \times A$, the map $t \mapsto \theta_{t} ( f, x )$ is continuous and bounded. Note that if $\tau$ is another cross section of $\pi$, then $\sigma ( x ) - \tau ( x ) \in B$ for all $x \in A$. Then for each $( f, x ) \in S \mathbb{R} \times A$, by Lemma \ref{l:approxunit},
\begin{align*}
\lim_{ t \to \infty } \norm{ f( u_{t} ) \sigma (x) - f( u_{t} ) \tau(x) } = 0.
\end{align*}
Therefore, $\theta_{t}$ does not depend on the choice of cross section (up to equivalence of asymptotic morphisms). Let $x \in A$, then the function $f \mapsto \theta_{t} ( f, x )$ is linear for each $t \in [1, \infty )$. Let $f \in S \mathbb{R}$. Since
\begin{align*}
\sigma ( x + y ) - \sigma ( x ) - \sigma (y), \ \sigma ( x^{*} ) - \sigma ( x )^{*} , \ \text{and} \ \sigma (\lambda x ) - \lambda \sigma ( x )
\end{align*}
are elements of $B$ for all $x, y \in A$ and $\lambda \in \mathbb{R}$, by Lemma \ref{l:approxunit}
\begin{align*}
\lim_{ t \to \infty } \norm{ \theta_{t} ( f, x + y ) - \theta_{t} ( f, x ) - \theta_{t} ( f, y ) } = 0 \\
\lim_{ t \to \infty } \norm{ \theta_{t} ( f, x^{*} ) - \theta_{t} ( f,x )^{*} } = 0\\
\lim_{ t \to \infty } \norm{ \theta_{t} ( f, \lambda x ) - \lambda \theta_{t} ( f, x ) } = 0
\end{align*}
for all $x, y \in A$ and $\lambda \in \mathbb{R}$. Since $\sigma ( x y) - \sigma ( x ) \sigma (y) \in B$ for all $x, y \in A$ and $( fg)( u_{t}) = f( u_{t}) g( u_{t} )$ for all $f, g \in S \mathbb{R}$ and $t \in [1, \infty )$ and since
\begin{align*}
&\norm{ ( f g ) ( u_{t} ) \sigma (x y) - f(u_{t}) \sigma (x) g( u_{t}) \sigma (y ) } \\
&\qquad \leq \norm{ ( fg )( u_{t} ) \sigma ( xy ) - ( fg)( u_{t}) \sigma(x) \sigma (y) } + \norm{ f } \norm{ \sigma(y) } \norm{ g( u_{t} ) \sigma (x) - \sigma(x) g( u_{t} ) }
\end{align*}
by Lemma \ref{l:approxunit},
\begin{align*}
\lim_{ t \to \infty } \norm{ \theta_{t} ( fg, x y ) - \theta_{t} ( f, x ) \theta_{t} ( g, y ) } = 0
\end{align*}
for all $f, g \in S \mathbb{R}$ and $x, y \in A$.
Since $\norm{ f( u_{t} ) \sigma ( x ) } \leq \norm{ f } \norm{ \sigma(x) }$ for all $f \in S \mathbb{R}$ and $x \in A$ and since $\theta_{t}$ is independent of the choice of $\sigma$, we may choose a cross section of $\pi$ such that $\norm{ \sigma(x) } = \norm{ x }$ for all $x \in A$. Therefore, $\theta_{t}$ defines a $*$-homomorphism from $\ftn{ \psi }{ S \mathbb{R} \otimes A }{ \alpha B }$. Let $\ftn{ \phi }{ S \mathbb{R} \otimes A }{ \beta B }$ be a lifting of $\psi$. Set $\phi_{t}^{ \mathfrak{e} } = \mathrm{ev}_{t} \circ \phi \colon SA \rightarrow B$.
By construction,
\begin{align*}
\lim_{ t \to \infty } ( \phi_{t}^{ \mathfrak{e} } ( f \otimes a ) - f( u_{t} ) \sigma (a) ) = 0,
\end{align*}
for all $f \in S \mathbb{R}$ and $a \in A$. Also, by the above observation, $\tau$ is another cross section of $\pi$, then the induce $*$-homomorphism from $S \mathbb{R} \otimes A$ to $\alpha B$ is $\psi$. Thus, the induced asymptotic morphism is equivalent to $\asy{ \phi_{t}^{ \mathfrak{e} } }$.
\end{proof}
\begin{propo} \label{p:extasy2}
Let $\mathfrak{e} : 0 \to B \overset{ \iota }{ \to } E \overset{ \pi }{ \to } A \to 0$ be an exact sequence of separable $R^{*}$-algebras. Then the class in $[[S A, B ]]$ given by the asymptotic morphism $\asy{ \phi_{t}^{\mathfrak{e}} }$ from Proposition \ref{p:extasy} is independent of the choice of approximate identity that is quasi-central in $E$.
\end{propo}
\begin{proof}
Let $\{ u_{t} \}_{ t\in [1, \infty ) }$ and $\{ v_{t} \}_{ t \in [1, \infty ) }$ be continuous approximate identities for $B$ that are quasi-central in $E$. Then $w_{t} ( s ) = s u_{t} + ( 1 - s ) v_{t}$ for all $s \in [0,1]$ is an approximate identity for $IB$ and is quasi-central in $IE$. Set $D = \setof{ f \in IE }{ \text{$\pi ( f(t) ) = \pi ( f(s) )$ for all $t, s \in [0,1]$} }$. Then $D$ is a sub-$R^{*}$-algebra of $IE$, $IB$ is an ideal of $D$, and $D / IB \cong A$. By Proposition \ref{p:extasy}, there exists an asymptotic morphism $\ftn{ \asy{ \Phi_{t} } }{ S A }{ IB }$, built using the approximate identity $\{w_t\}$ and the cross-section $\sigma \colon A \rightarrow SB$. By construction, $\overline{ \mathrm{ev} }_{0}^{ [0,1] } \circ \overline{ \asy{ \Phi_{t} } }$ is equal to the $*$-homomorphism from $S A$ to $B$ constructed from $\{ u_{t} \}_{ t \in [1, \infty ) }$ and $\overline{ \mathrm{ev} }_{0}^{ [0,1] } \circ \overline{ \asy{ \Phi_{t} } }$ is equal to the $*$-homomorphism from $S A$ to $B$ constructed from $\{ v_{t} \}_{ t \in [1, \infty ) }$. By Proposition \ref{p:homotopequal}, we have that the asymptotic morphism constructed from $\{ u_{t} \}_{ t \in [1, \infty ) }$ and $\{ v_{t} \}_{ t \in [1, \infty ) }$ are homotopic.
\end{proof}
As in the complex case there is a natural isomorphism between $[[A, \mathcal{K}_\R \otimes B]]$ and $[[\mathcal{K}_\R\otimes A, \mathcal{K}_\R \otimes B]]$ (see Section~25.4 of \cite{blackadarbook}). Also as in the complex case, the set $[[A,\mathcal{K}_\R \otimes S B]]$ has the structure of an abelian group. The group operation is defined equivalently either in terms of a chosen $*$-isomorphism $M_2(\mathcal{K}_\R) \cong \mathcal{K}_\R$ or in terms of concatenation of loops in $\mathcal{K}_\R \otimes B$ based at $0$.
\begin{defin} \label{def:e}
For $R^{*}$-algebras $A$ and $B$, we define $E(A,B) = [[SA, \mathcal{K}_\R \otimes SB]]$.
\end{defin}
Given an asymptotic morphism $\asy{ \phi_{t} } \colon A \rightarrow B$ and an element $\gamma \in SA$, the composition $\phi_t \circ \gamma$ is an element of $SB$ for all $t \in [1, \infty)$. We let $S \phi_t$ denote the resulting map $SA \rightarrow SB$ and using the compactness of the circle, is it easy to show that $\asy{S \phi_{t} }$ is an asymptotic morphism. Furthermore it is easy to show if two asymptotic morphisms
$\asy{ \phi_{t} }$ and $\asy{ \psi_{t} }$ are homotopic, then $\asy{S \phi_{t} }$ and $\asy{S \psi_{t} }$ are also homotopic. Thus, every asymptotic morphism from $A$ to $B$ induces a unique element of $E(A,B)$. It also follows that there is a well-defined map $\Sigma \colon E(A, B) \rightarrow E(SA, SB)$, which can easily be shown to be a group homomorphism.
In fact, the map $\Sigma$ is a special case of the tensor product construction for asymptotic morphisms described in Lemma II.B.$\beta$.5 of \cite{connesbook}, which carries over to the case of $R^*$-algebras. Given two asymptotic morphisms $\asy{\phi_t} \colon A \rightarrow C$ and $\asy{\psi_t} \colon B \rightarrow D$, there is a tensor product asymptotic morphism
$\asy{(\phi \otimes \psi)_t} \colon A \otimes_{\rm{max}} B \rightarrow C \otimes_{\rm{max}} D$
which satisfies
$\lim_{t \to \infty} \left((\phi \otimes \psi)_t(a \otimes b) - \phi_t(a) \otimes \psi_t(b) \right) = 0$
for all $a \in A$ and $b \in B$.
We also will need to make use of the associative product structure on $E$-theory described in Proposition II.B.$\beta$.4 of \cite{connesbook}, which also carries over to the case of $R^*$-algebras. Given two asymptotic morphisms $\asy{\phi_t} \colon A \rightarrow B$ and $\asy{\psi_t} \colon B \rightarrow C$, there is a composition asymptotic morphism $\asy{\psi \circ \phi}_t \colon A \rightarrow C$, defined uniquely up to homotopy. In the special case that $\psi$ or $\phi$ is an actual $*$-homomorphism, then this product is a literal composition. In the general case, a reparametrization is necessary to construct an asymptotic morphism from the composition (see Lemma II.B.$\beta$.3). The resulting product induces a natural homomorphism $E(A, B) \otimes E(B, C) \rightarrow E(A, C)$.
\begin{theor} \label{Eproperties}
$E(A,B)$ is a bivariant functor from separable $R^{*}$-algebras to abelian groups. In both arguments, it is homotopy invariant, stable, half exact, and has a degree 8 periodicity isomorphism.
\end{theor}
\begin{proof}
The homotopy invariance is immediate and the stability follows from Proposition~\ref{stability}. By Propositions~\ref{p:extasy} and \ref{p:extasy2}, any extension
$$\mathfrak{e} : 0 \rightarrow J \rightarrow A \xrightarrow{q} B \rightarrow 0$$
gives rise to a well-defined asymptotic morphism $\asy{ \phi_{t}^{ \mathfrak{e} } }$ from $S B$ to $J$. Then the proofs leading up to and including Corollary~25.5.7 of \cite{blackadarbook} carry over to the real case to show that the functor $E(A,\cdot)$ is a split exact functor for fixed separable $A$. Then by Theorem~\ref{thm:univprop1} (of the present paper) there is a bilinear pairing $E(A,B) \otimes \operatorname{KK}(B,C) \rightarrow E(A,C)$. Since this map is associative, multiplication by the Bott element in $\operatorname{KK}(\mathbb{R}, S^8 \mathbb{R})$ induces a periodicity isomorphism in the second argument of $E(\cdot, \cdot)$. Similarly, $E( \cdot, B)$ is also split exact, so by the comments following Theorem~\ref{thm:univprop1} there is a natural pairing, $\operatorname{KK}(A, B) \otimes E(B, C) \rightarrow E(A,C)$ proving periodicity in the first argument.
We postpone the proof of half-exactness until after a series of lemmas.
\end{proof}
For any elements $x \in E(A, B)$ and $z \in \operatorname{KK}(B, C)$, the pairing described in the proof above gives an element in $E(A, C)$ which we will denote by $x \otimes_\alpha z$. Taking $1_A \in E(A, A)$ we obtain a homomorphism $\epsilon \colon \operatorname{KK}(A, B) \rightarrow E(A, B)$. For $x \in E(A, B)$ and $y \in E(B, C)$, we let $x \otimes_E y$ denote the product in $E(A, C)$. Similarly for $z \in \operatorname{KK}(A, B)$ and $w \in \operatorname{KK}(B, C)$, we let $z \otimes_{KK} w$ denote the product in $\operatorname{KK}(A, C)$.
\begin{lemma} \label{epsilonringhom}
For $x \in \operatorname{KK}(A, B)$ and $y \in \operatorname{KK}(B, C)$ we have $\epsilon(x \otimes_{KK} y) = \epsilon(x) \otimes_E \epsilon (y)$.
\end{lemma}
\begin{proof}
First we show that $\epsilon(x \otimes_{KK} y) = \epsilon(x) \otimes_\alpha y$. Indeed, using Theorem~\ref{thm:univprop1} we have
$$\epsilon ( x \otimes_{KK} y) = 1_A \otimes_\alpha (x \otimes_{KK} y)
= (1_A \otimes_\alpha x ) \otimes_\alpha y
= \epsilon(x) \otimes_\alpha y \; .$$
Secondly, we prove that $z \otimes_\alpha y = z \otimes_E \epsilon(y)$ for $z \in E(A, B)$. Indeed, this clearly holds for $y = 1_B \in \operatorname{KK}(B,B)$ so it holds in general by the uniqueness statement of Theorem~\ref{thm:univprop1}.
From these facts, we complete the proof with the calculation
$$\epsilon(x \otimes_{KK} y) = \epsilon(x) \otimes_\alpha y = \epsilon(x) \otimes_E \epsilon(y) \; .$$
\end{proof}
\begin{lemma} \label{suspension}
For any $R^*$-algebras, $\Sigma \colon E(A, B) \rightarrow E(SA, SB)$ is an isomorphism.
\end{lemma}
\begin{proof}
Let $\alpha \in E(\mathbb{R}, S^8 \mathbb{R})$ and $\beta \in E(S^8 \mathbb{R}, \mathbb{R})$ be Bott elements arising from the corresponding Bott elements in $\operatorname{KK}$-theory via $\epsilon$. Then it follows from Lemma~\ref{epsilonringhom} that $\alpha \otimes_E \beta = 1_\mathbb{R}$ and $\beta \otimes \alpha = 1_{S^8 \mathbb{R}}$. It follows that the map $z \mapsto (\alpha \otimes 1_A) \otimes_E z \otimes (\beta \otimes 1_B)$ is an isomorphism from $E(S^8 A, S^8 B)$ to $E(A, B)$. We call this isomorphism $\Theta$.
Now consider the homomorphism $\Sigma^8 \colon E(A, B) \rightarrow E(S^8 A, S^8 B)$. From the comments following Definition~\ref{def:e}, this homomorphism can be expressed as $z \mapsto 1_{S^8 \mathbb{R}} \otimes z$. Then for $z \in E(A, B)$ the composition is given by
\begin{align*}
(\Theta \circ \Sigma^8)(z)
&= (\alpha \otimes 1_A) \otimes_E (1_{S^8 \mathbb{R}} \otimes z) \otimes_E (\beta \otimes 1_B) \\
&= (\alpha \otimes_E \beta) \otimes z \\
&= 1_\mathbb{R} \otimes z \\
&= z \; .
\end{align*}
It follows that $\Sigma \colon E(A, B) \rightarrow E(SA, SB)$ is injective.
Since $\Theta \circ \Sigma^8 = \ensuremath{\operatorname{id}}_{E(A,B)}$ and since $\Theta$ is already known to be an isomorphism, it also follows that $\Sigma^8 \circ \Theta = \ensuremath{\operatorname{id}}_{E(S^8A, S^8B)}$. Hence $\Sigma \colon E(S^7 A, S^7 B) \rightarrow E(S^8A, S^8 B)$ is surjective.
Replacing $A$ and $B$ with suspensions of appropriate degrees, we obtain that $\Sigma \colon E(S^8 A, S^8 B) \rightarrow E(S^9 A, S^9 B)$ is an isomorphism.
Finally, consider the diagram
\begin{equation*}
\xymatrix{
E(A, B) \ar[rr]^\Sigma
&& E(SA, SB) \\
E(S^8 A, S^8 B) \ar[rr]^\Sigma \ar[u]^\Theta
&& E(S^9 A, S^9 B) \ar[u]^\Theta
}
\end{equation*}
where the vertical maps and the lower map are all known to be isomorphisms.
This diagram commutes modulo a homomorphism induced by the rearrangement of the order of the suspension factors of $S^9 A$ and $S^9 B$. Since the rearrangement of the factors corresponds to an even permutation, it is homotopic to the identity and induces an identity homomorphism on $E$-theory. It follows that the $\Sigma \colon E(A, B) \rightarrow E(S A, S B)$ is an isomorphism.
\end{proof}
\begin{proof}[Completion of proof of Theorem~\ref{Eproperties}.]
Finally, to prove half-exactness let $0 \to J \overset{ \iota }{ \to } A \overset{ q }{ \to } B \to 0$ be an extension of separable $R^{*}$-algebras and $D$ be an $R^{*}$-algebra. Suppose $h$ is an asymptotic morphism from $SD \otimes \mathcal{K}_{ \mathbb{R} }$ to $SA \otimes \mathcal{K}_{ \mathbb{R} }$ such that $[ q \circ h ] = [ 0 ]$. Lemma~25.5.12 of \cite{blackadarbook} and its proof carry over to the real case, so there exists an asymptotic morphism $k$ from $S^2 D \otimes \mathcal{K}_{ \mathbb{R} }$ to $S^2 J \otimes \mathcal{K}_{ \mathbb{R} }$ such that $[ S \iota \circ k ] = [ S h ]$.
In the commutative diagram
\begin{equation*}
\xymatrix{
E( D , J ) \ar[r]^{ \iota_{*} } \ar[d]^{\Sigma} & E( D , A ) \ar[r]^{ q_{*} } \ar[d]^{\Sigma} & E( D , B ) \ar[d]^{\Sigma} \\
E( S D , S J ) \ar[r]^{ \iota_{*} } & E( S D , S A ) \ar[r]^{ q_{*} } & E ( S D , S B ) \\
}
\end{equation*}
the vertical maps are isomorphisms, so there exists an asymptotic morphism $g$ from $SD \otimes \mathcal{K}_{ \mathbb{R} }$ to $SJ \otimes \mathcal{K}_{ \mathbb{R} }$ such that $[ \iota \circ g ] = [ h ]$ in $E( D , A )$.
Half-exactness in the first argument is proved in a similar way.
\end{proof}
\begin{theor}\label{t:kkthyethyiso}
Let $A$ be a separable, nuclear $R^{*}$-algebra and let $B$ be a separable $R^{*}$-algebra. Then the homomorphism
$\epsilon \colon \operatorname{KK}( A , B ) \rightarrow E ( A , B )$ is an isomorphism.
\end{theor}
\begin{proof}
By Theorem~\ref{main}, it suffices to show that $\operatorname{KK}(A,B) \rightarrow E(A,B)$ is an isomorphism when $B$ is a $C^{*}$-algebra. In the diagram
\begin{equation*}
\xymatrix{
\operatorname{KK}^{\mathbb{C}}(A_\C, B) \ar[r]^{~} \ar[d] &
E^{\mathbb{C}}(A_\C, B) \ar[d] \\
\operatorname{KK}(A, B) \ar[r] &
E(A, B) }
\end{equation*}
we use $\operatorname{KK}^{\mathbb{C}}(-,-)$ and $E^{\mathbb{C}}(-,-)$ to denote the versions of these functors on $C^{*}$-algebras (for example $E^{\mathbb{C}}(-,-)$ consists of homotopy classes of asymptotic morphisms that are asymptotically linear over $\mathbb{C}$). Since $A_\C$ is nuclear, the top horizontal homomorphism is an isomorphism by Theorem~25.6.3 of \cite{blackadarbook}. The left vertical homomorphism is an isomorphism by Lemma~4.3 of \cite{boersema04}. The right vertical homomorphism is defined by restriction---a complex asymptotic morphism defined on $S A_\C \otimes \mathcal{K}$ restricts to a real asymptotic morphism defined on $S A \otimes \mathcal{K}_\R$---and it is an isomorphism since every real asymptotic morphism on $S A \otimes \mathcal{K}_\R$ can be extended uniquely to a complex asymptotic morphism on
$S A_\C \otimes \mathcal{K}$. The square commutes by Theorem~3.7 of \cite{higson87}, since the two directions around the square give natural transformations $\operatorname{KK}^{\mathbb{C}}(A_\C, -) \rightarrow E(A, -)$ of functors defined on separable $C^{*}$-algebras, each of which sends $1_A \in \operatorname{KK}^{\mathbb{C}}(A_\C, A_\C)$ to the asymptotic morphism represented by the inclusion of $A$ into $A_\C$. Therefore, the bottom row is an isomorphism as desired.
\end{proof}
Finally, we present below the real analog of Theorem 5.8 of \cite{ManuilovThomsen04}, showing that $E$-theory is a special case of $\operatorname{KK}$-theory. That we can obtain this result without reproducing the extensive technicalities of the proof in \cite{ManuilovThomsen04} shows the utility of Theorem~\ref{main}.
Since $\mathcal{M}(B \otimes \mathcal{K}_\R)$ is $\operatorname{KK}$-trivial, it follows that $E(A, \mathcal{M}(B \otimes \mathcal{K}_\R)) = 0$.
Then we use the long exact sequence arising from
$$0 \rightarrow B \otimes \mathcal{K}_\R \rightarrow \mathcal{M}(B \otimes \mathcal{K}_\R) \rightarrow \mathcal{Q}(B \otimes \mathcal{K}_\R) \rightarrow 0$$
to get an isomorphism
$E_0(A, \mathcal{Q}(B \otimes \mathcal{K}_\R)) \cong E_{-1}(A, B \otimes \mathcal{K}_\R)$. Combining this with stability and with Lemma~\ref{suspension}, there is an isomorphism
$$\gamma \colon E( S^{-1}A , \mathcal{Q}(B \otimes \mathcal{K}_\R) ) \rightarrow E ( A , B ) $$
for $R^*$-algebras $A$ and $B$.
\begin{theor}
Let $A$ and $B$ be a separable $R^{*}$-algebras. Then there is an isomorphism \\
$$\varepsilon' \colon \operatorname{KK}( S^{-1}A , \mathcal{Q}(B \otimes \mathcal{K}_\R) ) \rightarrow E ( A , B ) \; .$$
\end{theor}
\begin{proof}
Let $\varepsilon' = \gamma \circ \varepsilon$ where $\varepsilon$ is the isomorphism of Theorem~\ref{t:kkthyethyiso}. We know that $\varepsilon'$ is an isomorphism in the complex case by Theorem 5.8 of \cite{ManuilovThomsen04}. So it suffices by Theorem~\ref{main} to show that $\varepsilon'$ fits in a commutative square in the same way that $\varepsilon$ does in the proof of Theorem~\ref{t:kkthyethyiso}. The square in question can be factored into two squares as follows:
\begin{equation*}
\xymatrix{
\operatorname{KK}^{\mathbb{C}}(S^{-1} A_\C, \mathcal{Q}(B \otimes \mathcal{K}_\R)) \ar[r]^{~} \ar[d] &
E^{\mathbb{C}}(S^{-1} A_\C, \mathcal{Q}(B \otimes \mathcal{K}_\R)) \ar[r] \ar[d] &
E^{\mathbb{C}}(A_\C, B) \ar[d] \\
\operatorname{KK}(S^{-1} A, \mathcal{Q}(B \otimes \mathcal{K}_\R)) \ar[r] &
E(S^{-1} A, \mathcal{Q}(B \otimes \mathcal{K}_\R)) \ar[r] &
E(A, B) }
\end{equation*}
The first square is just a specialization of the square in Theorem~\ref{t:kkthyethyiso}. The second square commutes since the horizontal maps are homomorphisms that arise from stabilization, from long exact sequences, and from suspensions; all of which commute with the vertical restriction map.
\end{proof}
\section{Application: asymptotic morphisms on spheres}
The goal of this section is to prove the following theorem, using the results of the previous sections, the universal coefficient theorem for real $C^{*}$-algebras, and a united $K$-theory analysis.
\begin{theor}\label{t:asynonvanishing}
There exists asymptotic morphisms $ \asy{ \phi_{t} }$ that induce non-trivial homomorphisms on $K$-theory of the following forms:
\begin{enumerate}
\item ${ S^d \mathbb{R} } \rightarrow { \mathcal{K}_{\mathbb{R}} }$ where $d > 0$ and $d \equiv 0, 4, 6, 7 \pmod 8$,
\item ${ S^d \mathbb{R} } \rightarrow { \mathcal{K}_{\mathbb{R}} \otimes \mathbb{H} }$ where $d > 0$ and $d \equiv 0, 2, 3, 4\pmod 8$
\end{enumerate}
Furthermore, if $n \in \mathbb{N}$ does not satisfy the specified condition in each case, then there does not exist a corresponding asymptotic morphism that is non-trivial on $K$-theory.
\end{theor}
Before we prove Theorem \ref{t:asynonvanishing} we need to establish some relationships between asymptotic morphisms and mapping cones which are well-known for $C^{*}$-algebras. Note that if $\ftn{ \phi }{ B } {C }$ is a $*$-homomorphism and $\ftn{ \asy{ \psi_{t} } }{ A }{ B }$ is an asymptotic morphism, then $\ftn{ \asy{ \phi \circ \psi_{t} } }{ A }{ C }$ is an asymptotic morphism.
\begin{propo}\label{p:amorpmapc}
Let $\mathfrak{e} : 0 \to B \overset{\iota}{\to} E \overset{ \pi }{ \to } A \to 0$ be a short exact sequence of $R^{*}$-algebras. Then $[ \asy{ \gamma_{ \mathfrak{e} } \circ \phi_{t}^{ \mathfrak{e} } } ] = [ \asy{ \lambda_{ \mathfrak{e} } } ]$ in $[[ S A , C_{ \pi } ]]$, where $\asy{ \phi_{t}^{ \mathfrak{e} }}$ is the asymptotic morphism associated to $\mathfrak{e}$.
\end{propo}
\begin{proof}
Let $\ftn{ \sigma }{ A }{ E }$ be a cross section of $\pi$ such that $\norm{ \sigma ( x ) } = \norm{ x }$. Let $\{ u_{t} \}_{ t \in [1, \infty ) }$ be an approximate identity of $B$ that is quasi-central in $E$. For $f \in S \mathbb{R}$ and $x \in A$ and $s \in [0,1]$,
\begin{align*}
\phi_{t} ( f , x ) ( s ) &= f( s u_{t} + (1-s) 1_{ \widetilde{E} } ) \sigma (x) \\
\psi_{t} ( f , x )(s)(r) &= f( r + 1 - s ) x, \ r \in [0,1]
\end{align*}
where $\widetilde{E} = E$ if $E$ unital and $\widetilde{E}$ is the unitization of $E$ if $E$ is not unital, and $f( r + 1 - s ) = 0$ if $r + 1 - s \geq 1$. By the continuity of continuous functional calculus, we have that $\psi_{t} ( f,x)$ is continous and $\phi_{t} ( f, x )$ is continuous. Since
\begin{align*}
\pi ( \phi_{t} (f,x)(s) ) &= f( 1 - s ) \pi ( \sigma ( x ) ) = f( 1 - s ) x
= \psi_{t} ( f, x )(s)( 0 ) \\
\text{and} \\
\psi_t(f,x)(s)(1) & = 0,
\end{align*}
we have that $( \phi_{t} ( f,x ) , \psi_{t} ( f, x ) ) \in IC_{\pi}$.
For all $f \in S \mathbb{R}$ and $x \in A$, it is clear that $t \mapsto \psi_{t} ( f, x )$ is continuous and bounded in $I CA$ and that $t \mapsto \phi_{t}(f,x)$ is the same in $IE$. Hence, the map $t \mapsto ( \phi_{t} ( f, x ) , \psi_{t} ( f, x ) ) \in I C_{\pi}$, is continuous and bounded on $[1, \infty )$.
Define $\ftn{ \Psi }{ S \mathbb{R} \otimes A }{ \beta I C_{\pi} }$ by $\Psi ( f, x ) ( t ) = ( \phi_{t} (f, x ), \psi_{t} ( f, x ) )$. Let $b \in B$, $x \in E$, and $t \in [1, \infty )$. Then
\begin{align*}
\sup\setof{ \norm{ \left( ( 1 - s ) u_{t} + s 1_{ \widetilde{ E } } \right) b - b} }{ s \in [0,1] } &= \sup \setof{ \norm{ ( 1 - s ) u_{t} b - ( 1 - s ) b } }{ s \in [0,1] } \\
&= \norm{ u_{t} b - b }
\end{align*}
and
\begin{align*}
&\sup \setof{ \norm{ \left( ( 1 -s ) u_{t} + s 1_{ \widetilde{ E } } \right) x - x \left( ( 1 -s ) u_{t} + s 1_{ \widetilde{ E } } \right) } }{ s \in [0,1] } \\
&\qquad = \sup \setof{ \norm{ ( 1 - s ) ( u_{t} x - x u_{t} ) } }{ s \in [0,1] } \\
&\qquad = \norm{ u_{t} x - x u_{t} }.
\end{align*}
Set $f_{t} \in I \widetilde{E}$ by $f_{t} ( s ) = ( 1 -s ) u_{t} + s 1_{ \widetilde{ E } }$. Then by the above paragraph, $\{ f_{t} \}_{ t \in [1, \infty ) }$ is a family of positive elements in $I \widetilde{E}$ with $\norm{ f_{t} } \leq 1$ such that
\begin{align*}
\lim_{ t \to \infty } f_{t} b = b
\end{align*}
for all $b \in B$ and
\begin{align*}
\lim_{ t \to \infty } ( f_{t} x - x f_{t} ) = 0
\end{align*}
for all $x \in E$, where we identify $b$ and $x$ in $I \widetilde{E}$ as constant functions. Therefore, for all $f , g \in S \mathbb{R}$, for all $x, y \in A$, and for all $\lambda \in \mathbb{R}$,
\begin{align*}
\lim_{ t \to \infty } ( \phi_{t} ( f, x + y ) - \phi_{t} ( f, x ) - \phi_{t} ( f, y ) ) = 0 \\
\lim_{ t \to \infty } ( \phi_{t} ( f, x^{*} ) - \phi_{t} ( f, x )^{*} ) = 0 \\
\lim_{ t \to \infty } ( \phi_{t} ( f, \lambda x ) - \lambda \phi_{t} ( f, x ) ) = 0 \\
\lim_{ t \to \infty } ( \phi_{t} ( fg, x y ) - \phi_{t} ( f, x ) \phi_{t} ( g, y ) ) = 0,
\end{align*}
using Lemma~\ref{l:approxunit} as in the proof of Proposition~\ref{p:extasy}.
By the definition of $\psi_{t}$, for all $f , g \in S \mathbb{R}$, for all $x, y \in A$, and for all $\lambda \in \mathbb{R}$,
\begin{align*}
\lim_{ t \to \infty } ( \psi_{t} ( f, x + y ) - \psi_{t} ( f, x ) - \phi_{t} ( f, y ) ) = 0 \\
\lim_{ t \to \infty } ( \psi_{t} ( f, x^{*} ) - \phi_{t} ( f, x )^{*} ) = 0 \\
\lim_{ t \to \infty } ( \psi_{t} ( f, \lambda x ) - \lambda \psi_{t} ( f, x ) ) = 0 \\
\lim_{ t \to \infty } ( \psi_{t} ( fg, x y ) - \phi_{t} ( f, x ) \psi_{t} ( g, y ) ) = 0.
\end{align*}
It is clear from the definition of both $\phi_{t}$ and $\psi_{t}$ that $\phi_{t} ( \cdot , x )$ and $\psi_{t} ( \cdot , x )$ are linear functions.
From the above observations, there exists a $*$-homomorphism $\ftn{ \Phi }{ S \mathbb{R} \otimes A }{ \alpha IC_{\pi} }$ such that if $\widetilde{ \Phi }$ is a set-theoretical lifting of $\Phi$ and $ \Phi_{t} = \mathrm{ev}_{t} \circ \widetilde{ \Phi } $ is the associated asymptotic morphism, then for each $f \in S \mathbb{R}$ and $x \in A$,
\begin{align*}
\lim_{ t \to \infty } ( \Phi_{t} ( f \otimes x ) - \Psi ( f, x ) ( t ) ).
\end{align*}
Note that
\begin{align*}
\mathrm{ev}_{0}^{[0,1]} ( \Psi ( f, x ) ( t ) ) &= ( \phi_{t} ( f, x ) ( 0 ) , \psi_{t} ( f, x ) ( 0 ) ) \\
&= ( f( u_{t}) \sigma ( x ), 0 ) \\
&= \gamma_{ \mathfrak{e} } ( f( u_{t} ) \sigma ( x ) )
\end{align*}
and
\begin{align*}
\mathrm{ev}_{1}^{[0,1]} ( \Psi ( f, x ) ( t ) ) &= ( \phi_{t} ( f, x ) ( 1) , \psi_{t} ( f, x ) (1) ) \\
&= ( 0 , f \otimes x ) \\
&= \lambda_{ \mathfrak{e} } ( f \otimes x ).
\end{align*}
Hence, for all $f \in S \mathbb{R}$ and $x \in A$,
\begin{align*}
\lim_{ t \to \infty } \norm{ \mathrm{ev}_{0}^{[0,1]} \circ \Phi_{t} ( f \otimes x ) - \gamma_{ \mathfrak{e} } \circ \phi_{t}^{ \mathfrak{e} } ( f \otimes x ) }
= \lim_{ t \to \infty } \norm{ \mathrm{ev}_{0}^{[0,1]} \circ ( \Phi_{t} ( f \otimes x ) - \Psi ( f, x ) ( t ) ) } = 0
\end{align*}
and
\begin{align*}
\lim_{ t\to \infty } \norm{ \mathrm{ev}_{1}^{[0,1]} \circ \Phi_{t} ( f \otimes x ) - \lambda_{\mathfrak{e}}( f \otimes x ) } = \lim_{ t \to \infty } \norm{ \mathrm{ev}_{1}^{[0,1]} \circ ( \Phi_{t} ( f \otimes x ) - \Psi ( f, x ) ( t ) ) } = 0.
\end{align*}
Thus, $\asy{ \mathrm{ev}_{0}^{[0,1]} \circ \Phi_{t} }$ is homotopic to $\asy{ \gamma_{ \mathfrak{e} } \circ \phi_{t}^{ \mathfrak{e} } }$ and
$\asy{ \mathrm{ev}_{1}^{[0,1]} \circ \Phi_{t} }$ is homotopic to $\asy{ \lambda_{ \mathfrak{e} } }$.
Since the asymptotic morphism $\asy{ \Phi_{t} }$ gives a homotopy between $\asy{ \mathrm{ev}_{0}^{[0,1]} \circ \Phi_{t} }$ and $\asy{ \mathrm{ev}_{1}^{[0,1]} \circ \Phi_{t} }$, we have that
$\asy{ \gamma_{ \mathfrak{e} } \circ \phi_{t}^{ \mathfrak{e} } }$ and $\asy{ \lambda_{ \mathfrak{e} } }$ are homotopic. Thus, $[ \asy{ \gamma_{ \mathfrak{e} } \circ \phi_{t}^{ \mathfrak{e} } } ] = [ \asy{ \lambda_{ \mathfrak{e} } } ]$ in $[[ S A , C_{ \pi } ]]$.
\end{proof}
Let $B$ be an $R^{*}$-algebra. Then we have the following exact sequence of $R^{*}$-algebras
\begin{align*}
0 \to \beta_{0} B \to \beta B \xrightarrow{\rho_B} \alpha B \to 0.
\end{align*}
The above exact sequence induces a long exact sequence in $K$-theory
\begin{align*}
\dots \to K_{*} ( \beta_{0} B ) \to K_{*} ( \beta B ) \xrightarrow{ ( \rho_{B} )_* } K_{*} ( \alpha B ) \to K_{*-1} ( \beta_{0} B ) \to \dots
\end{align*}
for each $n \in \mathbb{Z}$. Since $ \beta_{0} B$ is contractible, $K_{*} ( \beta_{0} B ) = 0$. Hence, $( \rho_{B} )_*$ is an isomorphism.
\begin{defin}
Let $A$ and $B$ be $R^{*}$-algebras and let $\ftn{ \asy{ \phi_{t} } }{ A }{ B }$ be an asymptotic morphism.
Let $\asy{ \phi_t}_*$ denote the composition
\begin{align*}
( \mathrm{ev}^{[1, \infty)}_{1} )_* \circ ( \rho_{B} )^{-1}_* \circ \overline{ \asy{\phi_{t}} } _*
\end{align*}
from $K_* ( A )$ to $K_* ( B )$.
\end{defin}
The next three lemmas are well-known in the context of $C^{*}$-algebras. We are not able to find a reference for $R^{*}$-algebras so for the convenience of the reader we provided the proof here.
\begin{lemma}\label{l:homotopkthy}
Let $A$ and $B$ be $R^{*}$-algebras and let $\ftn{ \asy{ \phi_{t} } , \asy{ \psi_{t} } }{ A }{ B }$ be asymptotic morphisms.
If $\asy{ \phi_{t} }$ is homotopic to $\asy{ \psi_{t} }$,
then $\asy{ \phi_{t} }_* = \asy{ \psi_{t} }_*$.
\end{lemma}
\begin{proof}
Let $\ftn{ \asy{ \Phi_{t} } }{ A } { C( [ 0,1 ] , B ) }$ be an asymptotic morphism such that $\mathrm{ev}_{0}^{[0,1]} \circ \Phi_{t} = \phi_{t}$ and $\mathrm{ev}_{1}^{[0,1]} \circ \Phi_{t} = \psi_{t}$ for all $t \in [1, \infty )$. Since
$$
( \mathrm{ev}_{s}^{[0,1] } )_* \asy{ \Phi_{t} }_* =
( \mathrm{ev}_{s}^{[0,1] } )_* ( \mathrm{ev}_{1}^{[1,\infty)} )_* ( \rho_{ IB} )^{-1}_* \overline{ \asy{ \Phi_{t} } } _* $$
for all $s \in [0,1]$ and since for each $f \in \beta IB$, the function
\begin{align*}
s \mapsto \mathrm{ev}_{s}^{[0,1] } \circ \mathrm{ev}_{1}^{ [1,\infty) } (f)
\end{align*}
is continuous,
\begin{align*}
( \mathrm{ev}_{0}^{[0,1]} )_* ( \mathrm{ev}_{1}^{[1, \infty) } )_*
= ( \mathrm{ev}_{1}^{[0,1]} )_* ( \mathrm{ev}_{1}^{[1, \infty) } )_* .
\end{align*}
Therefore,
\begin{align*}
( \mathrm{ev}_{0}^{[0,1] } )_* \asy{ \Phi_{t} } _*
&= ( \mathrm{ev}_{0}^{[0,1] } )_* ( \mathrm{ev}_{1}^{[1,\infty)} )_* ( \rho_{ IB} )_*^{-1} \overline{ \asy{ \Phi_{t} } } _* \\
&= ( \mathrm{ev}_{1}^{[0,1] } )_* ( \mathrm{ev}_{1}^{[1,\infty)} )_* ( \rho_{ IB} )_*^{-1} \overline{ \asy{ \Phi_{t} } } _* \\
&= ( \mathrm{ev}_{1}^{[0,1] } )_* \asy{ \Phi_{t} } _*.
\end{align*}
Note that $\overline{ \mathrm{ev} }_{0}^{ [0,1] } \circ \overline{ \asy{ \Phi_{t} } }= \overline{ \asy{ \phi_{t} } }$, $\overline{ \mathrm{ev} }_{1}^{ [0,1] } \circ \overline{ \asy{ \Phi_{t} } } = \overline{ \asy{ \psi_{t} } }$, $\overline{\mathrm{ev} }_{s}^{ [0,1] } \circ \rho_{ IB } = \rho_{B} \circ \widetilde{ \mathrm{ev} }_{s}^{[0,1]}$, and $\mathrm{ev}_{1}^{[1,\infty) } \circ \widetilde{\mathrm{ev}}_{s}^{[0,1]} = \mathrm{ev}_{s}^{[0,1]} \circ \mathrm{ev}_{1}^{[1,\infty)}$. Therefore,
\begin{align*}
( \mathrm{ev}_{0}^{[0,1] } )_* \asy{ \Phi_{t} }_*
&= ( \mathrm{ev}_{0}^{[0,1] } )_* ( \mathrm{ev}_{1}^{[1,\infty)} )_* ( \rho_{ IB} )^{-1}_* \overline{ \asy{ \Phi_{t} } } _* \\
&= ( \mathrm{ev}_{1}^{[1, \infty) } )_* ( \widetilde{ \mathrm{ev} }_{0}^{[0,1]} )_* (\rho_{ IB} )^{-1}_*
\overline{ \asy{ \Phi_{t} } }_* \\
&= ( \mathrm{ev}_{1}^{[1, \infty) } )_* ( \rho_{ B} )^{-1}_* ( \overline{\mathrm{ev}}_{0}^{ [0,1] } )_*
\overline{ \asy{ \Phi_{t} } } _* \\
&= ( \mathrm{ev}_{1}^{ [1, \infty ) } )_* ( \rho_{B} )^{-1}_* \overline{ \asy{ \phi_{t} } } _* \\
&= \asy{ \phi_{t} }_*.
\end{align*}
A similar computation shows that
\begin{align*}
( \mathrm{ev}_{1}^{[0,1] } )_* \asy{ \Phi_{t} } _* = \asy{ \psi_{t} } _*,
\end{align*}
which implies that
\begin{align*}
\asy{ \phi_{t} } _* = ( \mathrm{ev}_{0}^{[0,1] } )_* \asy{ \Phi_{t} } _* = ( \mathrm{ev}_{1}^{[0,1] } )_* \asy{ \Phi_{t} }_* = \asy{ \psi_{t} } _*.
\end{align*}
\end{proof}
\begin{lemma}\label{l:inducekthyhom}
Let $A$ and $B$ be $R^{*}$-algebras and let $\ftn{ \phi }{ A } {B }$ be a $*$-homomorphism.
Then $\phi_* = \asy{\phi}_* \colon K_*(A) \rightarrow K_*(B)$.
\end{lemma}
\begin{proof}
Note that $\ftn{ \widetilde{ \asy{ \phi } } }{ A }{ \beta B }$ which maps $a$ to the constant function $\phi(a)$ is a $*$-homomorphism and $\overline{ \asy{ \phi } } = \rho_{B} \circ \widetilde{ \asy{ \phi_{t} } }$. Therefore,
\begin{align*}
\asy{ \phi }_* = ( \mathrm{ev}_{1}^{[0,1]} )_* ( \rho_{B} )^{-1}_* \overline{ \asy{ \phi } } _*
= ( \mathrm{ev}_{1}^{[0,1]} )_* \widetilde{ \asy{ \phi } } _* = \phi_*.
\end{align*}
\end{proof}
\begin{lemma}\label{l:inducekthycomp}
Suppose $\ftn{\phi}{B}{C}$ is a $*$-homomorphism and $\ftn{ \asy{ \psi_{t} } }{ A }{ B }$ is an asymptotic morphism.
Then $ \asy{ \phi \circ \psi_{t} } _* = \phi_* \circ \asy{ \psi_{t} } _*$.
\end{lemma}
\begin{proof}
Define $\ftn{ \eta_{ \phi } }{ \beta B }{ \beta C }$ by $\eta_\phi( f )( t ) = \phi ( f(t) )$. Then $\eta_{\phi}$ is a $*$-homomorphism such that $\eta_{ \phi } ( \beta_{0} B ) \subseteq \beta_{0} C$. Thus, there exists a $*$-homomorphism $\ftn{ \overline{\eta}_{\phi} }{ \alpha B }{ \alpha C }$ such that
\begin{align*}
\overline{\eta}_{\phi} \circ \rho_{B} = \rho_{C} \circ \eta_{ \phi } \quad \text{and} \quad \overline{\eta}_{\phi} \circ \overline{ \asy{ \psi_{t} } } = \overline{ \asy{ \phi \circ \psi_{t} } }.
\end{align*}
Hence,
\begin{align*}
\asy{ \phi \circ \psi_{t} } _*
&= (\mathrm{ev}^{[1, \infty)}_{1}) _* (\rho_{C} )^{-1}_* \overline{ \asy{ \phi \circ \psi_{t} } }_* \\
&= ( \mathrm{ev}^{[1, \infty)}_{1} )_* ( \rho_{C} )^{-1}_* ( \overline{\eta}_{\phi} )_* \overline{ \asy{ \psi_{t} } } _* \\
&= ( \mathrm{ev}^{[1, \infty)}_{1} )_* ( \eta_{\phi} )_* ( \rho_{B} )^{-1}_* \overline{ \asy{ \psi_{t} } }_* \\
&= \phi _* ( \mathrm{ev}_{1}^{ [1, \infty ) }) _* ( \rho_{B} )^{-1}_* \overline{ \asy{ \psi_{t} } } _* \\
&= \phi_* \circ \asy{ \psi_{t} } _*.
\end{align*}
\end{proof}
As an aside, we note that it is possible to prove a more general statement: the formula $ \asy{ \phi_t \circ \psi_{t} } _* = \asy{ \phi_{t} } _* \circ \asy{ \psi_{t} } _*$ holds for asymptotic morphisms $\ftn{ \asy{ \phi_{t} } }{B}{C}$ and $\ftn{ \asy{ \psi_{t} } }{ A }{ B }$.
\begin{theor}\label{t:CRTnonvanishing}
There exists non-trivial {\it CRT}-homomorphisms of degree 0 of the following forms:
\begin{enumerate}
\item $K^{\scriptscriptstyle {\it CRT}}(S^d \mathbb{R}) \rightarrow K^{\scriptscriptstyle {\it CRT}}(\mathbb{R})$ where $d > 0$ and $d \equiv 0, 4, 6, 7 \pmod 8$
\item $K^{\scriptscriptstyle {\it CRT}}(S^d \mathbb{R}) \rightarrow K^{\scriptscriptstyle {\it CRT}}(\mathbb{H})$ where $d > 0$ and $d \equiv 0, 2, 3, 4\pmod 8$.
\end{enumerate}
Furthermore, if $n \in \mathbb{N}$ does not satisfy the specified condition in each case, then there does not exist a corresponding non-trivial {\it CRT}-homomorphism.
\end{theor}
\begin{proof}
The ${\it CRT}$-module $K^{\scriptscriptstyle {\it CRT}}(S^d \mathbb{R})$ is a free {\it CRT}-module with a single generator in the real part in degree $-d$. Hence there exists a non-trivial {\it CRT}-module homomorphism $K^{\scriptscriptstyle {\it CRT}}(S^d \mathbb{R}) \rightarrow M$ if and only if $M_{\scriptscriptstyle O}^{-d} \neq 0$. Now $KO_*(\mathbb{R})$ is non-zero in and only in degrees $0, 1, 2, 4 \pmod 8$; and $KO_*(\mathbb{H})$ is non-zero in and only in degrees $0, 4, 5, 6 \pmod 8$. Thus parts (1) and (2) follow as well as the converse statements.
\end{proof}
We are now ready to prove Theorem~\ref{t:asynonvanishing}.
\begin{proof}[Proof of Theorem \ref{t:asynonvanishing}]
For $d \geq 1$, where $d$ is not in one of the required congruence classes, Theorem~\ref{t:CRTnonvanishing} implies that no asymptotic morphism can exist that induces a non-trivial homomorphism on $K$-theory.
Now suppose that $d \geq 1$ and suppose that $A = S^d \mathbb{R}$ and $B = \mathbb{R}$ or $B = \mathbb{H}$ are algebras such that the form $A \rightarrow B$ matches the conditions of the statement of Theorem~\ref{t:asynonvanishing}. By Theorem~\ref{t:CRTnonvanishing}, there is a non-zero homomorphism $K^{\scriptscriptstyle {\it CRT}}(A) \rightarrow K^{\scriptscriptstyle {\it CRT}}(B)$ and by the Universal Coefficient Theorem for real $C^{*}$-algebras (Theorem~1.1 of \cite{boersema04} ), this {\it CRT}-module homomorphism is induced by a nonzero element $\xi \in \operatorname{KK}(A, B)$.
By Theorem \ref{t:extkk}, there exists an extension
$\mathfrak{e} \in \operatorname{Ext} (S^{d-1} \mathbb{R}, B)^{-1}$
such that
$\Lambda_{S^{d-1} \mathbb{R}, B }[ { \mathfrak{e} } ] = \xi$.
Since $[\lambda_{\mathfrak{e}} ][ \gamma_{ \mathfrak{e} } ]^{-1} = \xi$ this implies that $[\lambda_{\mathfrak{e}} ] \neq 0$ where $[\lambda_{\mathfrak{e}} ] \in \operatorname{KK}(A, C_\pi)$.
Furthermore, the homomorphism of $\mathcal{CRT}$-modules
$(\lambda_{\mathfrak{e}})_* \colon K^{\scriptscriptstyle {\it CRT}}(A) \rightarrow K^{\scriptscriptstyle {\it CRT}}(C_\pi)$ is nonzero. This is because $K^{\scriptscriptstyle {\it CRT}}(A)$ is a free $\mathcal{CRT}$-module so the Universal Coefficient Theorem implies that
$$\operatorname{KK}(A, C_\pi) \rightarrow \operatorname{Hom}_{\scriptscriptstyle {\it CRT}}(K^{\scriptscriptstyle {\it CRT}}(A), K^{\scriptscriptstyle {\it CRT}}(C_\pi))$$
is an isomorphism. Even more, this implies that the homomorphism on $K$-theory $(\lambda_{\mathfrak{e}})_* \colon K_*(A) \rightarrow K_*(C_\pi)$ is nonzero, by Theorem~4 of \cite{realuct2}.
Let $\ftn{ \asy{ \phi_{t}^{ \mathfrak{e} } } }{ S^d \mathbb{R} }{ B }$ be the asymptotic morphism associated to $\mathfrak{e}$.
Then by Proposition \ref{p:amorpmapc}, $\asy{ \gamma_{ \mathfrak{e} } \circ \phi_{t}^{ \mathfrak{e} } }$ is homotopic to $\asy{ \lambda_{\mathfrak{e}} }$. Therefore, by Lemmas \ref{l:homotopkthy}, \ref{l:inducekthyhom}, and \ref{l:inducekthycomp}; we have
\begin{align*}
( \gamma_{\mathfrak{e} } )_* \circ \asy{ \phi_{t}^{\mathfrak{e} } } _* = ( \lambda_{\mathfrak{e}} )_*.
\end{align*}
Since $( \lambda_{\mathfrak{e}} )_* \neq 0$, it follows that $ \asy{ \phi_{t}^{\mathfrak{e} } } _* \neq 0$.
\end{proof}
\section{Almost commuting matrices}
We can use all this machinery to find novel examples of almost
commuting real symmetric matrices. Our approach is to use commutative
$R^{*}$-algebras and create asymptotic morphisms out of these. On
the one hand these carry $K$-theory data that can distinguish them
from actual $*$-homomorphisms. On the other, in the image the relations
that make the $R^{*}$-algebra commutative get turned into the property
of being almost commuting.
Let $\mathbb{F}$ denote either $\mathbb{R}$ or $\mathbb{C}$ or
$\mathbb{H}$. We generalize a question of Halmos \cite{DavidsonEssentially2010}
to ask about $d$ almost commuting self-adjoint matrices over $\mathbb{F}$.
\begin{probl}
\label{HalmosProblem} For all $\epsilon>0$, does there exist $\delta>0$
so that, for all $n$, given $d$ self-adjoint contractions $H_{r}$
in $\mathbf{M}_{n}(\mathbb{F})$ such that
\[
\left\Vert \left[H_{r},H_{s}\right]\right\Vert \leq\delta,
\]
there exist $d$ self-adjoint contractions $K_{r}$ with
$\left\Vert K_{r}-H_{r}\right\Vert \leq\epsilon$
and
\[
\left[K_{r},K_{s}\right]=0?
\]
\end{probl}
Lin \cite{LinAlmostCommutingHermitian} showed that in the complex
case the anwer is ``yes'' for $d=2$ while it was known much earlier
\cite{VoiculescuHeisenberg} that the answer is ``no'' in the
complex case for $d=3$.
For the quaternionic case, the result is the same: yes for $d=2$
\cite{LorSorensenTorus} and no for $d=3$ \cite{HastLorTheoryPractice}.
These leaves the real case, arguably the most important case. We know the
answer is ``yes'' for $d=2$ (\cite{LorSorensenTorus}). We will show that the answer is ``no'' for $d = 5$, leaving open the cases $d = 3,4$.
The proof techniques used for a negative result for $d=3$ in the complex and quaternionic cases rely on the fact
that $K_{-2}(\mathbb{H}) \neq 0$ and
$K_{-2}(\mathbb{C}) \neq 0$ and so will not
work for $\mathbb{F}=\mathbb{R}$ since $K_{-2}(\mathbb{R}) = 0$.
However, since $K_{-4}(\mathbb{R})$ is nontrivial, we will see that these methods will apply for $d = 5$.
We start by connecting this problem to a problem couched in the theory of $R^*$-algebras. For any sequence $B_n$ of $R^*$-algebras, let $\pi$ be the quotient map from the product
$\prod_{n=1}^\infty B_n$ to its quotient by the sum $\left.\prod_{n=1}^{\infty} B_n \right/\bigoplus_{n=1}^{\infty} B_n $.
\begin{probl}
\label{HalmosProblem2} Does every $*$-homomorphism of the form
$$\psi \colon S^{d-1} \mathbb{R} \rightarrow \left.\prod_{n=1}^{\infty} \mathbf{M}_{m(n)}(\mathbb F) \right/\bigoplus_{n=1}^{\infty} \mathbf{M}_{m(n)}(\mathbb F) $$
where $\{m(n)\}_{n=1}^\infty$ is a sequence of integers,
lift to a $*$-homomorphism
$$\tilde{\psi} \colon S^{d-1} \mathbb{R} \rightarrow \prod_{n=1}^\infty \mathbf{M}_{m(n)}(\mathbb F)$$
such that $\psi = \pi \circ \widetilde{\psi}$?
\end{probl}
\begin{thm} \label{2Problems}
For a fixed positive integer $d$ and division algebra $\mathbb F$, if the answer to Problem~\ref{HalmosProblem} is ``yes'' then the answer to Problem~\ref{HalmosProblem2} is also ``yes''.
\end{thm}
We prove Theorem~\ref{2Problems} below, following two lemmas. Then, the rest of the section is devoted to showing that the answer to Problem~\ref{HalmosProblem2} is ``no'' when $d = 5$, using a $K$-theoretic obstruction. We start with a lemma stating the universal properties of $C(S^{d-1}, \mathbb{R})$. This is a standard result that can be proven using the techniques of Chapter~3 of \cite{loringbook}.
\begin{lemma} \label{classifyingspaces}
Let $A$ be an $R^*$-algebra.
If $h_1, \dots, h_d$ are commuting self-adjoint contractions in $A$ that satisfy $\sum_{j = 1}^d h_i = 1$, then there is a unique $*$-homomorphism
$\psi \colon C(S^{d-1}, \mathbb{R}) \rightarrow A$ sending the $j$th coordinate function $f_j$ to $h_j$.
\end{lemma}
\begin{lemma} \label{approxcircle}
For each $\varepsilon > 0$ there exists a $\delta > 0$ such that the following holds.
Let $k_1, \dots, k_d$ be commuting self-adjoint elements in a unital $R^*$-algebra $A$ that satisfy
$\|\sum_{i = 1}^d k_i^2 - 1 \| < \delta$. Then there exist commuting self-adjoint elements $k'_1, \dots, k'_d$ in $A$ such that $\|k'_i - k_i \| < \varepsilon$ and $\sum_{i = 1}^d (k'_i)^2 = 1$.
\end{lemma}
\begin{proof}
Assume that $0 < \delta < .5$ and that the elements $k_i$ satisfy $\|\sum_{i = 1}^d k_i^2 - 1 \| < \delta$.
Since the elements $k_i$ commute, we can treat them as real-valued functions in $C(X, \mathbb{R})$ for some compact space $X$. Let $r(x) = \sqrt{1/ \sum_{i = 1}^d k_i^2(x) }$. Then $k'_i(x) = r(x) k_i(x)$ satisfies $ \sum_{i = 1}^d (k'_i)^2 = 1$. A short calculation shows that
$\|k_i - k'_i \| = \left\|1 - \sqrt{\frac{1}{\sum_{i = 1}^d k_i^2}} \right\|$. Thus for any $\varepsilon > 0$, we can find a $\delta$ small enough to ensure that $\|k_i - k'_i \| < \varepsilon$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{2Problems}]
Suppose that the answer to Problem~\ref{HalmosProblem} is ``yes'' for some $d$ and $\mathbb F$, and let
$$\psi \colon C(S^{d-1}, \mathbb F) \rightarrow \left.\prod_{n=1}^{\infty} \mathbf{M}_{m(n)}(\mathbb F) \right/\bigoplus_{n=1}^{\infty} \mathbf{M}_{m(n)}(\mathbb F) $$
be a $*$-homomorphism. Then taking the images of the coordinate functions in $C(S^{d-1}, \mathbb F)$ we find that there exist sequences of matrices $H_{in} \in \mathbf{M}_{m(n)} (\mathbb F)$ ($i \in \{1, \dots, d\}$, $n \in \mathbb{N}$) that are asymptotically (as $n \to \infty$) self-adjoint contractions, that satisfy $\sum_{i = 1}^d H_{in}^2 = 1$ asymptotically, and such that $H_{in}$ and $H_{jn}$ asymptotically commute for each $i,j \in \{1, \dots, d\}$. In fact, we may assume that each $H_{in}$ is exactly a self-adjoint contraction. This is achieved by replacing $H_{in}$ by $\tfrac{1}{2} \left( H_{in} + H_{in}^* \right)$ and then by $f(H_{in})$ where $f \in C(\mathbb{R}, \mathbb{R})$ is defined by
$$f(x) = \begin{cases}
-1 & x \leq -1 \\
x & -1 < x < 1 \\
1 & x \geq 1. \end{cases} $$
We may also assume without loss of generality that $ \| \sum_{i = 1}^d h_{in}^2 - 1 \| \leq .5$ for all $n$.
By our hypothesis, there exists self-conjugate contractions $K_{in} \in \mathbf{M}_{m(n)} ( \mathbb F)$ that exactly commute and satisfy $\lim_{n \to \infty} (H_{in} - K_{in}) = 0$ for each $i$. By Lemma~\ref{approxcircle}, there exists matrices $L_{in} \in \mathbf{M}_{m(n)} (\mathbb F)$ such that $\lim_{n \to \infty} (K_{in} - L_{in}) = 0$ for all $i$ and $\sum_{i = 1}^d L_{in}^2 = 1$ for each $n$.
Then by Lemma~\ref{classifyingspaces}, there exist $*$-homomorphisms $\psi_n \colon C(S^{d-1}, \mathbb{R}) \rightarrow \mathbf{M}_{m(n)} (\mathbb F)$ that map the $d$ coordinate functions to $L_{in}$. These maps form a $*$-homomorphism
$$\tilde{\psi} \colon C(S^{d-1}, \mathbb{R}) \rightarrow \prod_{n =1 }^\infty \mathbf{M}_{m(n)} (\mathbb F) \; .$$
Since $\lim_{n \to \infty} \| L_{in} - H_{in} \| = 0$ it follows by the uniqueness statement of Lemma~\ref{classifyingspaces} that $\pi \circ \tilde{\psi} = \psi$.
\end{proof}
\begin{lemma}
\label{lem:isZero}
Suppose $\mathbb{F}$ is a division algebra and $d\geq1$. Any
$*$-homorphism
\[
S^k \mathbb{R}
\rightarrow\prod_{n=1}^{\infty}\mathbf{M}_{m(n)}(\mathbb{F})
\]
is homotopic to zero.
\end{lemma}
\begin{proof}
It suffices to show that any $*$-homomorphism
$\varphi: S^{d}\mathbb{R}
\rightarrow
\mathbf{M}_{m(n)}(\mathbb{F})$
is homotopic to the zero map. As the codomain is finite dimensional,
this $\varphi$ must factor through $C(X,\mathbb{R})$ where $X$
is finite. Thus $\varphi$ is a sum of point evaluations, each of which is homotopic to $0$.
\end{proof}
Let
$\Theta$ be the map from
$K_* \left(\prod_{n=1}^{\infty}B_i \right)$ to $\prod_{n=1}^{\infty}K_* (B_i)$ induced by the collection of projection maps
$$\pi_i \colon \prod_{n=1}^{\infty}B_i \rightarrow B_i \; .$$
In \cite{DadarlatEilers}, there is a thorough analysis of the circumstances under which $\Theta$
is surjective and injective on $K_0$ and $K_1$ for $C^*$-algebras $B_i$. The following lemma states that the situation for $R^*$-algebras partially reduces to that for $C^*$-algebras.
\begin{lemma} Let $\{B_i\}_{i \in \mathbb{N}}$ be a sequence of $R^*$-algebras and let $(B_i)_\C$ be their respective complexifications. Then
the map
$$\Theta^{\scriptscriptstyle {\it CRT}} \colon K^{\scriptscriptstyle {\it CRT}} \left(\prod_{n=1}^{\infty}B_i \right) \rightarrow \prod_{n=1}^{\infty}K^{\scriptscriptstyle {\it CRT}}(B_i) \; $$
is an isomorphism if and only if
$$\Theta \colon K_* \left(\prod_{n=1}^{\infty} (B_i)_\C \right) \rightarrow \prod_{n=1}^{\infty}K_*((B_i)_\C) \; $$
is an isomorphism.
\end{lemma}
\begin{proof}
Since $K^{\scriptscriptstyle {\it CRT}}$ is a {\it CRT}-module homomorphism, this follows immediately from the results of Section~2.3 of \cite{bousfield90}.
\end{proof}
We shall say that a family of $R^*$-algebras $\{B_i\}_{i \in \mathbb{N}}$ has {\it property $\Theta$} if $\Theta^{\scriptscriptstyle {\it CRT}}$ is an isomorphism. We shall say further that an $R^*$-algebra $B$ has {\it property $\Theta$} if $\{B_i\}_{i \in \mathbb{N}}$ has property $\Theta$ where $B_i = B$ for all $i \in \mathbb{N}$. In particular $\mathbb F \otimes \mathcal{K}_\mathbb{R} \cong \mathcal{K}_\mathbb F$ has property $\Theta$.
We note that if $B$ has property $\Theta$, then the $*$-homomorphism
\[
\Delta: B \rightarrow \prod_{n=1}^{\infty} B
\]
and the resulting map
\[
\hat{\Delta}:
B \rightarrow
\left.\prod_{n=1}^{\infty} B \right/\bigoplus_{n=1}^{\infty} B
\]
into the quotient are both injective on $K$-theory and united $K$-theory.
Recall that we have $\beta B = C_b([1, \infty), B)$ and $\alpha B = C_b([1, \infty), B)/C_0([1, \infty), B)$.
We can evaluate an element of $\beta B$ at the natural numbers, and
get a surjection
\[
\delta_{\mathbb{N}}:\beta B\rightarrow\prod_{n=1}^{\infty}B \; ;
\]
which passes to a $*$-homomorphism
$$\hat{\delta}_{\mathbb{N}} \colon \alpha B \rightarrow \left.\prod_{n=1}^{\infty}B\right/\bigoplus_{n=1}^{\infty}B \; .$$
\begin{lemma}
\label{asymorphismtoproduct}
Suppose $A$ is a separable $R^{*}$-algebra and $B$ is a separable $R^*$-algebra with property $\Theta$. If
\[
\left\langle \phi_{t}\right\rangle :A\rightarrow B
\]
is an asympotic morphism, then there is a $*$-homomorphism
\[
\psi: A \rightarrow
\left.\prod_{n=1}^{\infty}B\right/\bigoplus_{n=1}^{\infty}B
\]
so that on $K$-theory
\[
\psi_{*}=\hat{\Delta}_{*}\circ \left\langle \phi_{t}\right\rangle _*.
\]
In particular, if $\left\langle \phi_{t}\right\rangle _*$ is non-zero, then so is $\psi_*$.
\end{lemma}
\begin{proof}
Define $\psi =
\hat{\delta}_{\mathbb{N}} \circ \overline{\left\langle \phi_{t}\right\rangle } $. Consider the following diagram. The square commutes exactly. The triangle on the top commutes at the level of $K$-theory, as can be seen using the fact that $\delta_1$ is homotopic to $\delta_k$ for any $k \in \mathbb{N}$ and identifying the $K$-theory of the product with the product of the $K$-theory.
\[
\xymatrix{
&& && B \ar[d]^{\Delta} \\
&& \beta B \ar[rr]^{\delta_\mathbb{N}} \ar[rru]^{\delta_1} \ar[d]^{\rho_B}
&& { \displaystyle \prod_{n=1}^{\infty}B } \ar[d]^{\pi} \\
A \ar[rr]^{ \overline{\left\langle \phi_{t}\right\rangle } }
&& \alpha B \ar[rr]^{\hat{\delta}_{\mathbb{N}} ~~~~~}
&& {\left.\displaystyle \prod_{n=1}^{\infty}B\right/\displaystyle \bigoplus_{n=1}^{\infty}B }
}
\]
Using the relation
$
\left\langle \phi_{t}\right\rangle _*
=
\delta_{1}\circ\left(\rho_{B}\right)_{*}^{-1}\circ\overline{\left\langle \phi_{t}\right\rangle }_{*}
$,
an easy calculation shows that
$\hat{\Delta}_{*}\circ\left\langle \phi_{t}\right\rangle _*
= \psi_*
$
\end{proof}
\begin{lemma} \label{fgsubalgebra}
Suppose $B$ is a separable $R^{*}$-algebra, and $A$ is a finitely generated $R^{*}$-subalgebra of
\[
A \subseteq
\left.\prod_{n=1}^{\infty} (B\otimes \mathcal{K}_\mathbb{R} )\right/\bigoplus_{n=1}^{\infty} (B\otimes \mathcal{K}_\mathbb{R}) \; .
\]
Then there is
a sequence $m(1)<m(2)<$ on natural numbers so that
\[
A \subseteq
\left.\prod_{n=1}^{\infty}\mathbf{M}_{m(n)}(B)\right/\bigoplus_{n=1}^{\infty}\mathbf{M}_{m(n)}(B).
\]
\end{lemma}
\begin{proof}
Given a single element $a \in A$, write
\[
a = (a_{1},a_{2},\dots)+\bigoplus (B\otimes \mathcal{K}_\mathbb{R})
\]
where $a_i \in B \otimes \mathcal{K}_\R$.
We can choose an increasing sequence $p_{1},p_{2}, \dots$ of standard projections
in $1\otimes\mathbb{K}_{\mathbb{R}}$ (with $1$ in $\tilde{B}$ if
needed) so that
\[
\left\Vert p_{n}b_{n}p_{n}-b_{n}\right\Vert \leq\frac{1}{n}
\]
and so
\[
(b_{1},b_{2},\dots)+\bigoplus (B\otimes \mathcal{K}_\mathbb{R})
=
(p_{1}b_{1}p_{1},p_{2}b_{2}p_{2},\dots)+ \bigoplus (B\otimes\mathcal{K}_\mathbb{R}).
\]
More generally, for a finite set of elements in $A$, we can use a single sequence of
projections as above to show that
$$A \subseteq \left.\ \prod_{n=1}^\infty p_n (B \otimes \mathcal{K}_\R) p_n \right/ \bigoplus_{n=1}^\infty p_n(B \otimes \mathcal{K}_\R)p_n \; .$$
\end{proof}
We now put these results together for our main theorem.
\begin{thm}
Suppose that $d \in \mathbb{N}$ and $\mathbb F \in \{\mathbb{R}, \mathbb{H}\}$ satisfies the hypotheses of Theorem~\ref{t:asynonvanishing}. Then there is a sequence of integers $m(1),m(2),\dots$
and a unital $*$-homomorphism
\[
\varphi:C(S^{d},\mathbb{R})\rightarrow\left. \prod_{n=1}^{\infty}\mathbf{M}_{m(n)}(\mathbb F)\right/\bigoplus_{n=1}^{\infty}\mathbf{M}_{m(n)}(\mathbb F)
\]
that cannot be lifted to a unital $*$-homomorphism
\[
\psi:C(S^{d},\mathbb{R})\rightarrow \prod_{n=1}^{\infty} \mathbf{M}_{m(n)}(\mathbb F).
\]
\end{thm}
\begin{proof}
Let $d$ be as above. By Theorem~\ref{t:asynonvanishing}, there exists an asymptotic morphism
\[
\left\langle \phi_{t}\right\rangle :S^d \mathbb{R} \rightarrow \mathcal{K}_\mathbb F
\]
that induces a non-zero map on $K$-theory.
Then by Lemma~\ref{asymorphismtoproduct},
we obtain a $*$-homomorphism of the form
$$
\phi' \colon S^d \mathbb{R} \rightarrow
\left.\prod_{n=1}^{\infty} \mathcal{K}_\mathbb F \right/\bigoplus_{n=1}^{\infty} \mathcal{K}_\mathbb F \;
$$
that is non-zero on $K$-theory. By Lemma~\ref{fgsubalgebra}, we have that this $*$-homomorphism factors through a $*$-homomorphism of the form
$$
\phi: S^d \mathbb{R} \rightarrow
\left.\prod_{n=1}^{\infty}\mathbf{M}_{m(n)}(\mathbb F)\right/\bigoplus_{n=1}^{\infty}\mathbf{M}_{m(n)}(\mathbb F) \; .
$$
Since $\phi' = \iota \circ \phi$, where $\iota$ is the inclusion
\[
\iota :
\left.\prod_{n=1}^{\infty}\mathbf{M}_{m(n)}(B)\right/\bigoplus_{n=1}^{\infty}\mathbf{M}_{m(n)}(B)
\rightarrow
\left.\prod_{n=1}^{\infty}B\otimes \mathcal{K}_\mathbb F \right/\bigoplus_{n=1}^{\infty}B\otimes \mathcal{K}_\mathbb F \; ,
\]
it follows that $\phi_*$
is also nonzero.
Now $\phi$ cannot be lifted to a $*$-homomorphism
$\psi$ with values in $\prod_{n=1}^{\infty} \mathbf{M}_{m(n)}(\mathbb F)$
since such a lift would have to be non-zero on $K$-theory contradicting Lemma~\ref{lem:isZero}.
Finally, extend unitally to form a $*$-homomorphism
$$
\phi: C(S^d, \mathbb{R}) \rightarrow
\left.\prod_{n=1}^{\infty}\mathbf{M}_{m(n)}(\mathbb F)\right/\bigoplus_{n=1}^{\infty}\mathbf{M}_{m(n)}(\mathbb F) \;
$$
that similarly cannot be lifted.
\end{proof}
\begin{corol}
For $d = 5$, $\mathbb F = \mathbb{R}$ and for $d = 3$, $\mathbb F = \mathbb{H}$; the answer to Problem~\ref{HalmosProblem2}, and hence also to Problem~\ref{HalmosProblem}, is ``no''.
\end{corol}
The result above for $d = 3$, $\mathbb F = \mathbb{H}$, replicates results from \cite{ LorSorensenTorus}. Considering this case more closely, we can clarify the $K$-theory behind the invariant for 3D topological insulators considered in
\cite{HastLorTheoryPractice}. We consider $\mathbb{H}$ as sitting inside its complexification,
so is to be regarded as the set of
\[
\left(\begin{array}{cc}
a & -\overline{b}\\
b & \overline{a}
\end{array}\right)
\]
where $a$ and $b$ are complex numbers. More generally, we identify
$\mathbf{M}_N(\mathbb{H})$ with $2N$-by-$2N$ complex matrices of the form
\[
\left(\begin{array}{cc}
A & -\overline{B}\\
B & \overline{A}
\end{array}\right).
\]
See \cite{LoringQuaternions}.
The first several homotopy groups for $\mathrm{Sp}(n)$ stablize early.
The connection with $K$-theory is that
$K_{1}(C(S^{3},\mathbb{H}))$
is given by homotopy classes of maps from the three sphere into the
symplectic unitary matrices of various sizes. Let $x_{1},\dots,x_{4}$
be the coordinate functions in $\mathbb{R}^{4}$ restricted to the
unit sphere $S^{3}$. From
\cite[\S10.6]{ViroHomotopyYheory} we learn that
\[
\left(\begin{array}{cc}
x_{1}+ix_{2} & -x_{3}+ix_{4}\\
x_{3}+ix_{4} & x_{1}-ix_{2}
\end{array}\right)\in C(S^{3},\mathbb{H})
\]
is a generator of
\[
\pi_3(\mathrm{Sp}(1))
\]
and so gives us a generator for
\[
K_{-3}(C(S^{3},\mathbb{R}))\cong K_{1}(C(S^{3},\mathbb{H})) \cong \mathbb{Z}/2.
\]
The $K_1$ group of
\[
\left.\prod_{n=1}^{\infty}\mathbf{M}_{m(n)}(\mathbb{R})\right/\bigoplus_{n=1}^{\infty}\mathbf{M}_{m(n)}(\mathbb{R})
\]
is easy enough to work out, since it is entirely governed by the
determinants of real othogonal matrices. It is a subgroup of
\[
\left. \prod_{n=1}^{\infty} \mathbb{Z}/2 \right/ \bigoplus_{n=1}^{\infty} \mathbb{Z}/2.
\]
When
$d=3$, the induced map $\varphi_{*}$ on $K$-theory is such that
the generator of
\[
K_{-2}\left(C(S^{3},\mathbb{R})\right)
\]
is sent to the class of
\[
\left(\overline{1},\overline{1},\dots\right)
\]
in
\[
K_{-2}\left(\left.\prod_{n=1}^{\infty}\mathbf{M}_{m(n)}(\mathbb{H})\right/\bigoplus_{n=1}^{\infty}\mathbf{M}_{m(n)}(\mathbb{H})\right)
=
\left.\prod\mathbb{Z}_{2}\right/\bigoplus\mathbb{Z}_{2}.
\]
This means that the $\mathbb{Z}_{2}$-index \cite{HastLorTheoryPractice}
can be non-trivial for arbitrarily small commutators. Now we switch
to considering $\mathbf{M}_{N}(\mathbb{H})$ as a real part of
$\mathbf{M}_{2N}(\mathbb{C})$
with respect to the dual operation
\[
\left[\begin{array}{cc}
A & B\\
C & D
\end{array}\right]^{\sharp}=\left[\begin{array}{cc}
D^{\mathrm{T}} & -B^{\mathrm{T}}\\
-C^{\mathrm{T}} & A^{\mathrm{T}}
\end{array}\right].
\]
That is, there are self-dual self-adjoint matrices $H_{1},\dots,H_{4}$
with
\[
\left\Vert \left[H_{r},H_{s}\right]\right\Vert \leq\delta,\ \left\Vert \sum_{r=1}^{4}H_{r}^{2}-I\right\Vert \leq\delta
\]
for any small $\delta$, so that
\[
\det\left[\begin{array}{cc}
H_{1}+iH_{2} & -H_{3}+iH_{4}\\
H_{3}+iH_{4} & H_{1}-iH_{2}
\end{array}\right]<0.
\]
This is evidence that the phenomena of 3D systems with topologically
protected states as witnessed in \cite{HastLorTheoryPractice} would
persist in larger system size.
\section{The ten-fold way}
Some of the motivation of this work is to explain index studies of
topological insulators. The symmetries present in topological insulators
are essential in their behavior. Physicists think of time-reversal
(TR) symmetry
in terms of commutating relations with anti-unitary operators, but it
is equivalent to think in terms of anti-multiplicative involutions.
For single particle systems, this picture suffices. There can be
no TR symmetry, where the relevant algebra for a finite model
is the $C^*$-algebra $\mathbf{M}_N(\mathbb{C})$. When
TR symmetry is present, the TR symmetry comes in two forms, and
leading to either the transpose or the dual operation. We are looking
at Dyson's ``three-fold'' way, or real, complex or quaternionic matrices.
In more complex systems, one needs to track additional symmetries,
in particular particle-hole symmetry. The picture here, in terms
of algebras, is that there are two anti-multiplicative involutions
$\tau$ and $\gamma$ and the Hamiltonian $H \in \mathbf{M}_N(\mathbb{C})$
now satisfies
\[
H^* = H,\quad H^{\tau} = H,\quad H^{\gamma} = -H.
\]
There are more general symmetry groups to consider, acting by unitary and
anti-unitary operators. See \cite{FidkowskiKitaev}. For now,
we consider
just the symmetries beyond being self-adjoint.
In physics, it is expected that these two
``$\tau$-operations'' will commute. As one can have either symmetry
non-existant, or equivalent to the transpose, or equivalent to
the dual operation, it seems that we will get nine symmetry
classes. Instead, we find ten as the combined symmetry of
$\tau$ followed by $\gamma$ can be preserved, or broken, when
each is individually broken. So condensed matter physics is now
investigated in terms of the ten-fold way. See \cite{Hofmann-Credner}
for a nice introduction to the matrix classes that arise.
It is easy to check that when $\tau$ and $\gamma$ commute,
\[
\alpha(a) = (a^\tau)^\gamma
\]
defines an order-two automorphism. If we are to follow the
ten-fold way, we need to be working with graded real
$C^*$-algebras.
It was in the context of graded real $C^*$-algebras $\operatorname{KK}$-theory
was introduced. In work on the standard model \cite{ConnesStandardModel},
Connes considered real structures on spectral triples. It
seems likely that progress in graded real noncommutative
geometry will have application in the study of topological
insulators.
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\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
|
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| 733
|
Thank you for your consideration of a gift to the UIC Center of Excellence in Maternal and Child Health. We are committed to developing the next generation of leaders in maternal and child health who are skilled at community collaboration, use evidence-based information and engage in practice steeped in social justice. Your gift will directly support our efforts to strengthen maternal and child health education for MPH and PhD students.
For more information about making a gift, please contact the Center at coemch@uic.edu.
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This scholarship is awarded to an African American student at the master or doctoral level whose studies focus on women's health. This fund was established in memory of Dr. Lacey, a nationally renowned alumna and faculty member of the School of Public Health.
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Learn about how financial support of the Center strengthens scholarship and financial assistance offerings for MPH and PhD students in maternal and child health.
|
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"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,492
|
"Culpa al Corazón" (English: "Blame the Heart") is a 2015 song by American singer Prince Royce. The song was released on November 13, 2015 as the lead single taken from Royce's fifth studio album, Five (2017). It received a Lo Nuestro nomination for Tropical Song of the Year.
The music video premiered on January 22, 2016 with Shadowhunters' co-star Emeraude Toubia (Royce's then-girlfriend, now ex-wife) playing the female lead role.
Charts
Weekly charts
Year-end charts
Certifications
See also
List of Billboard number-one Latin songs of 2016
References
2015 singles
2015 songs
Prince Royce songs
Sony Music Latin singles
Songs written by Prince Royce
Spanish-language songs
Songs written by Daniel Santacruz
Music videos directed by Jessy Terrero
|
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| 3,038
|
Shipping: L:14.567"x W:2.559"x H:18.307" 3.280 lbs.
Jack London Kitchen and Bath is proud to present this bone finished toilet seat, by American Standard. The 5259B65C.021 is made from premium materials, this Toilet Seat offers great function and value for your home. This fixture is part of American Standard's decorative SEATS Collection, so make sure to check out other styles of fixtures to accessorize your room.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,384
|
Q: How to set uiculture for AjaxControl I have an asp.net application that is entirely in the Dutch language (Culture and UICulture) This is set in the header of the aspx:
<%@ Page Language="vb" AutoEventWireup="false" UICulture="nl" Culture="nl-NL" CodeBehind="CSSSR_BlokjesSchema.aspx.vb" EnableEventValidation="false" Inherits="CS_SSR_RAIO.CSSSR_BlokjesSchema" %>
This workes perfectly, but not for the Ajax calendar extender. In that component, the dates and month names are still displayed in English.
What am i forgetting?? What setting in Ajax makes my calendar use the Culture i set?
A: I found it myself
The toolkit scriptmanager needs to be explicitly told that is has to use localisation:
<ajaxControlToolkit:ToolkitScriptManager ID="ToolkitScriptManager1" EnableScriptGlobalization="true" EnableScriptLocalization="true" runat="server" CombineScripts="true"></ajaxControlToolkit:ToolkitScriptManager>
After i changed this, the calendar was showing the date descriptions in Dutch
|
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"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,923
|
Q: How to publish multiple runtimes with dotnet cli to custom output folders I have a csproj file which contains a few target runtimes, and I was hoping to have a singular command I could run which would build them all and put them in a designated directory. Something like this:
dotnet publish -c release -o ../../some-dist-folder
If I run this command I get a some-dist-folder full of good stuff, but I am unsure if it is just outputting a single runtime or all runtimes, as the folder contains:
|- *.dll
|- Refs
|- *.dll
|- Runtimes
|- Lots of folders
So there is no exe (not that surprised as thats only a windows thing) and the Runtimes seems to contain loads of folders relating to various linux/mac/windows runtimes etc.
However if I was to run:
dotnet publish -c release -r win-x64 -o ../../some-dist-folder
I get some apphost.exe and some other files and the runtimes folder is gone, so I am left wondering if the former approach where I am trying to just build everything at once is not actually working and I need to explicitly publish for each platform or if I need to manually pick out the bits I care about from the output.
Ideally I was hoping to run a command like the ones above and get an output folder like:
|- some-dist-folder
|- win-x64
|- osx-some-version
|- some-linux-distro-version
|- other-runtimes-version
So can I get close to this in a single command or is it just safer to manually run each publish explicitly?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,878
|
Q: mvc jquery paging search results with only 1 query I have a search page that I'm implementing as part of an ASP.NET + MVC + jquery site. The search query scores each result that is returned as part of the results set. i.e. Higher score for closer match.
Because the results set can change, and because the algorithm that scores the matches is somewhat intensive, I only want to load the results once and to find a way to click through the pages of results.
I figured that I would make a page that contained all of the search results (each result has its own ) and just show/hide a subset by clicking on buttons associated with jquery code. I think I can see how to do this before I start trying to make it work (I'm still fairly new to jquery), but thought I'd first see if anyone has any better ideas.
Thanks in advance for any suggestions.
Edit: Francisco's suggestion was just what I needed. I implemented it with some minor changes and created some jquery buttons to navigate to first/prev/next/last page along with a 'Viewing results n-m out of N' label. Thanks for all the suggestions.
A: Cache the results and page through the cached resultset. Maybe using a combination of jqGrid and json.
However you do not really include enough information for us to give the best approach.
You need to do some analysis first.
Factors to consider :
*
*How many results is a user likely to
look through?
*Can a user deal with so much information on one page?
*Is the page going to be slow if you render the whole result set?
UPDATE
Another option could be to use Infinite Scroll. Basically as the user scrolls the results and is towards the end of the screen you auto load in x more results. More here and here
A: Have you considered using something like Lucene.NET for your indexing? Lucene.NET does a very good job in dealing with offsets for paging. The Document instances that you return from your search could contain the key values for the individual records or you could store enough information to represent the visual aspects for each search result for each item, as well.
When the user pages, you just adjust your offset. Things like query caching and optimization are built-in. I pulled down and compiled a few weeks ago and had a POC to build the index, add and remove items from the index, and return results in just a few minutes by cobbling the samples together. I much prefer using a dedicate index for indexing, leaving the database to work more as a dedicated storage medium - but that is my personal preference.
If you decide to check it out, pull and compile the source from their SVN. The "stable" version is a bit outdated.
Edit:
Here is a brief example on paging with Lucene.NET: Paging using Lucene.net
A: I think your idea is good enough.
Just put all the elements you want from each page into differents divs. Then give the div a class or id attribute that is easily identifiable, such as:
<div id="page1" class="divpage">
page 1 html here
</div>
<div id="page2" class="divpage">
page 2 html here
</div>
etc
The jQuery code would look like this:
$("#page1").show();
$(".divpage").hide(); //this hides all other divs
Edit:
To do this dynamically, creat a strongly-typed view associated to your list of results. The header of the view should be something like:
<%@ Page Title="" Language="C#" MasterPageFile="~/Views/Shared/Site.Master" Inherits="System.Web.Mvc.ViewPage<IEnumerable<Result>>" %>
With "result" being your class:
public class Result
{
int score {get;set;}
string data {get;set;}
}
Then in your view, you put this:
<%int resultsPerPage = 10;
int currentResults = 0;
int numPage=0;
foreach(var item in Model)
{
if (currentResults==0) Response.Write("<div id=\"page"+numPage+"\" class=\"divpage\">");
%>
<%=item.score.ToString()%>
<%=item.data%>
<%currentResults++;
if (currentResults>resultsPerPage)
{
numPage++;
currentResults=0;
Response.Write("</div>");
}
}%>
<%if (currentResults!=0) Response.Write("</div>");%>
Then for your buttons:
<%for (int i=0;i<numPage;i++)
{
%>
<a href="" onclick="showPage('<%=i%>');return false;">Page <%=i%></a>
<%}%>
<script type="text/javascript">
function showPage(num)
{
$("#page"+num).show();
$(".divpage").hide();
}
</script>
Finally, in your controller:
public ActionResult ShowResults()
{
List<Result> results = getResultsOrdererdByScoreFunction();
return View(results);
}
|
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"redpajama_set_name": "RedPajamaStackExchange"
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| 1,453
|
Best of South America
Top Sights, Authentic Experiences
# Contents
## Plan Your Trip
### Welcome to South America
### South America's Top 20
### Need to Know
### Hot Spots for...
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### Month by Month
### Get Inspired
### Itineraries
### Family Travel
## On The Road
### Rio de Janeiro, Brazil
#### Copacabana & Ipanema Beaches
#### Pão de Açúcar
#### Cristo Redentor
#### Walking Tour: Historic Centro
#### Sights
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#### Tours
#### Shopping
#### Eating
#### Drinking & Nightlife
#### Entertainment
### Salvador, Brazil
#### Afro-Brazilian Rhythms
#### Pelourinho
#### Sights
#### Courses
#### Tours
#### Shopping
#### Eating
#### Drinking & Nightlife
### Brazilian Amazon
#### Riverboat Travel
#### Jungle Expeditions & Wildlife-Watching
#### Belém
#### Santarém
#### Alter do Chão
#### Manaus
### The Pantanal, Brazil
#### Exploring the Pantanal
#### Cuiabá
#### Parque Nacional da Chapada dos Guimarães
#### Campo Grande
### Iguazú Falls, Brazil & Argentina
#### Visiting the Falls
#### Foz do Iguaçu, Brazil
#### Puerto Iguazú, Argentina
### Buenos Aires, Argentina
#### Tango in Buenos Aires
#### Plaza de Mayo
#### Cementerio de la Recoleta
#### Parrillas
#### Walking Tour: Historical Saunter
#### Sights
#### Shopping
#### Eating
#### Drinking & Nightlife
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### Southern Patagonia, Argentina
#### Glaciar Perito Moreno
#### Hiking the Fitz Roy Range
#### El Calafate
#### El Chaltén
### Santiago, Chile
#### Cerro San Cristóbal
#### Maipo Valley Wineries
#### Cajón del Maipo Outdoor Activities
#### Sights
#### Shopping
#### Eating
#### Drinking & Nightlife
### The Lakes District, Chile
#### Adrenalin Activities
#### Visiting Huilliche Communities
#### Osorno
#### Puerto Varas
#### Cochamó
#### Frutillar
#### Puerto Octay
### Torres Del Paine, Chile
#### Hiking the Parque Nacional Torres del Paine
#### Puerto Natales
#### Sights
#### Activities
#### Shopping
#### Eating
#### Drinking
#### Entertainment
### The Atacama Desert, Chile
#### Visiting the Desert
#### San Pedro de Atacama
#### Sights
#### Activities
#### Shopping
#### Eating
### La Paz, Bolivia
#### Markets
#### Tiwanaku
#### Sights
#### Tours
#### Shopping
#### Eating
#### Drinking & Nightlife
### Salar De Uyuni, Bolivia
#### Visiting the Salt Flats
#### Uyuni
#### Sights
#### Tours
#### Shopping
#### Eating
#### Drinking & Nightlife
### Lake Titicaca, Peru
#### Islas Uros
#### Chullpas
#### Sillustani
#### Cutimbo
#### Isla Taquile
#### Puno
### Cuzco & The Sacred Valley, Peru
#### Festivals & Events
#### Pisac Ruins
#### Ollantaytambo Ruins
#### Walking Tour: Central Cuzco
#### Cuzco
#### Pisac
#### Ollantaytambo
### Machu Picchu, Peru
#### Visiting Machu Picchu
#### The Inca Trail
#### Aguas Calientes
#### Sights
#### Activities
#### Eating
### The Cordilleras, Peru
#### Cordillera Blanca
#### Chavín de Huántar
#### Huaraz
#### Sights
#### Activities & Tours
#### Shopping
#### Eating
#### Drinking & Nightlife
### Quito & the Central Highlands, Ecuador
#### Quito's Old Town
#### Adventures in Baños
#### Parque Nacional Cotopaxi
#### Quito
#### Baños
### Galápagos Islands, Ecuador
#### Visiting the Galápagos
#### Puerto Ayora
#### Sights
#### Eating
#### Drinking
### Cartagena, Colombia
#### Cartagena's Old Town
#### Islas del Rosario
#### Sights
#### Activities & Tours
#### Shopping
#### Eating
#### Drinking & Nightlife
#### Entertainment
## In Focus
### South America Today
### History
### People & Culture
### Outdoor Activities
## Survival Guide
### Directory A–Z
#### Directory A–Z
#### Accessible Travel
#### Accommodations
#### Customs Regulations
#### Electricity
#### Discount Cards
#### Emergency & Important Numbers
#### Food
#### Health
#### Insurance
#### Internet Access
#### Legal Matters
#### LGBT Travelers
#### Maps
#### Money
#### Opening Hours
#### Photography
#### Post
#### Safe Travel
#### Telephone
#### Toilets
#### Tourist Information
#### Visas
#### Women Travelers
### Transport
#### Getting There & Away
#### Getting Around
### Language
### Behind the Scenes
### Our Writers
## Welcome to South America
Andean peaks, Amazonian rainforest, Patagonian glaciers, Inca ruins, colonial towns, white-sand beaches and vertiginous nightlife: the wonders of South America set the stage for incredible adventures
Tourists at Iguazú Falls, Brazil | R.M. NUNES / GETTY IMAGES ©
From the snow-capped peaks of the Andes to the undulating waterways of the Amazon, South America spreads a dazzling array of natural wonders. This is a continent of lush rainforests, towering volcanoes, misty cloud forests, bone-dry deserts, redrock canyons and ice-blue glaciers. Take in some of the incredible natural wonders found in every country in South America, then head to the coast for an idyllic retreat among palm-fringed, whitesand beaches and photogenic tropical islands.
Such settings offer tantalizing adventures. You can hike past ancient temples first laid down by the Incas, contemplate the awe-inspiring power of Iguazú Falls, or spend the day watching wildlife from a dugout canoe on one of the Amazon's countless igarapés (narrow waterways).
South America's diversity doesn't end with landscapes. You'll find colonial towns whose cobblestone streets lead past gilded churches and stately plazas little changed since the 18th century. You can haggle over colorful textiles at indigenous markets, share meals with traditional dwellers of the rainforest and follow the pounding rhythms of Afro-Brazilian drum corps. South America is home to an astounding variety of modern and ancient cultures, and experiencing it first hand is as easy as showing up.
# Plan Your Trip
South America's Top 20
## 1Machu Picchu, Peru
**_South America's fabled Inca ruins_**
This fantastic Inca citadel, lost to the world until its discovery in the early 20th century, stands as a ruin among ruins. With its emerald terraces backed by steep peaks and Andean ridges echoing on the horizon, the sight simply surpasses the imagination. This marvel of engineering has withstood six centuries of earthquakes, foreign invasion and howling weather. Discover it for yourself: wander through its stone temples and scale its dizzying heights.
GO GA / 500PX ©
Girls dressed in traditional Peruvian clothing, Machu Picchu | JOEL SHAWN / SHUTTERSTOCK ©
South America's Top 20 PLAN YOUR TRIP
## 2Brazilian Amazon
**_The world's greatest rainforest_**
Home to the greatest collection of plant and animal life on earth, the awe-inspiring Amazon has outdoor excursions of all sorts: from easy nature hikes to scaling 50m trees, from luxury lodges to makeshift camps in the forest. Spend your day plying the winding waterways in a canoe, slow-boating between towns or scanning the canopy for monkeys. Whatever your interest, there's a jungle trip in the Amazon waiting to blow your mind.
Toucan | AGUSTAVOP / GETTY IMAGES ©
STREETFLASH / GETTY IMAGES ©
South America's Top 20 PLAN YOUR TRIP
## 3Galápagos Islands, Ecuador
**_Enchanting islands packed with wildlife_**
The famous Galápagos Islands, with their volcanic, other-worldly landscapes, are a magnet for wildlife lovers. Here, you can get up close and personal with massive lumbering tortoises, scurrying marine iguanas (the world's only seagoing lizard), prancing blue-footed boobies and a host of other unusual species. On snorkeling trips, you'll see playful doe-eyed sea lions, lightning-fast penguins and graceful sea turtles winging through the equatorial waters. Sea lions
MAKINGSAUCE / GETTY IMAGES ©
South America's Top 20 PLAN YOUR TRIP
## 4Iguazú Falls, Brazil & Argentina
**_A jaw-dropping array of 275 waterfalls_**
The thunderous roar, the dramatic cascades, the refreshing sprays, the absolute miraculous work of Mother Nature – nothing prepares you for that first moment you set eyes upon Iguazú Falls. On the Brazilian side, enjoy the wide-eyed view of the whole astounding scene stretching out before you in all its panoramic wonder. In Argentina, get up close and personal with the deafening Devil's Throat for a mind-blowing experience.
JPSANCHEZ / GETTY IMAGES ©
South America's Top 20 PLAN YOUR TRIP
## 5Rio de Janeiro, Brazil
**_Samba-charged city of beaches and mountains_**
On privileged real estate flanked by striking Atlantic-blue waters, sugary-white sands and a mountainous backdrop of Crayola-green rainforest, Rio's cinematic cityscape has few rivals. And once its soundtrack kicks in – a high-on-life siren song of bossa nova and samba – Rio's raw energy seizes you with the come-hither allure of a tropical fantasy. You'll have no choice but to follow.
View of Cristo Redentor (Christ the Redeemer) and Pão de Açúcar (Sugarloaf Mountain) | R.M. NUNES / SHUTTERSTOCK ©
South America's Top 20 PLAN YOUR TRIP
## 6Buenos Aires, Argentina
**_A grand dame with a romantic soul_**
Whip together a beautiful Argentine metropolis with gourmet cuisine, awesome shopping, frenzied nightlife and gorgeous locals, and you'll get Buenos Aires. It's a European-like, cosmopolitan city encompassing both slick and unsafe neighborhoods, but that's the appeal. You can experience classic cafes, amazing steaks, surprising architecture, energizing _fútbol_ games and, of course, that sultry tango. Buenos Aires is elegant, seductive, emotional, confounding, frustrating and chock-full of attitude.
A colorful facade in La Boca | STUART DEE / GETTY IMAGES ©
South America's Top 20 PLAN YOUR TRIP
## 7Torres del Paine, Chile
**_Hike amid dramatic Patagonian landscapes_**
The wind whips dark clouds overhead as the trail suddenly opens to reveal a stunning vista of granite spires soaring high above the Patagonian steppe. These are the Torres del Paine, the proud centerpiece of Chile's famous national park. Trekking through this Unesco Biosphere Reserve isn't for the faint of heart – guides say the park sees all four seasons in a single day – but hiking the 'W' remains a rite of passage for adventurous travelers.
Hiker enjoying the view of Cuernos del Paine and Lake Pehoé | MICHELE FALZONE / GETTY IMAGES ©
South America's Top 20 PLAN YOUR TRIP
## 8Cuzco & the Sacred Valley, Peru
**_Former capital of the Inca empire_**
With ancient cobblestone streets, grandiose baroque churches and the remnants of Inca temples with centuries-old carvings, no place looms larger in Andean history than Cuzco, a city that has been inhabited continuously since pre-Hispanic times. Mystic, commercial and chaotic, this unique city is still a stunner. It's also the gateway to the picturesque Sacred Valley, with its famed temples and colorful markets, as well as Machu Picchu just beyond.
Quechua woman in Cuzco | KEREN SU / GETTY IMAGES ©
South America's Top 20 PLAN YOUR TRIP
## 9Southern Patagonia, Argentina
**_Walk beside glaciers and shark-tooth summits_**
The rugged, mountainous wilderness of Southern Patagonia is a world-class trekking destination. In the Fitz Roy Range, you can plan challenging multiday treks, or plan shorter outings to take in stunning views just a day's hike from town. South of Fitz Roy, Perito Moreno is one of the most accessible glaciers on the planet. Watch from steel catwalks as building-sized icebergs calve from the face and crash into Lago Argentino.
PAOLOMARCHETTI / GETTY IMAGES ©
Magellanic penguin | BUENAVENTURAMARIANO / GETTY IMAGES ©
South America's Top 20 PLAN YOUR TRIP
## 10Quito & the Central Highlands, Ecuador
**_Architectural jewel and gateway to Andean adventure_**
History lurks around every corner of Quito's vibrant Centro Histórico. Delve into the past by stepping off the cobblestones into beautifully maintained museums, historic mansions and jaw-dropping sanctuaries. Afterwards, join the festivities on lively La Ronda. Head further afield to outdoor adventures (and thermal springs) in Baños or the snowcapped scenery of Cotopaxi.
Plaza Grande | JESS KRAFT / SHUTTERSTOCK ©
PXHIDALGO / GETTY IMAGES ©
Quito's colonial-style facades | ALESSANDRO_PINTO / GETTY IMAGES ©
South America's Top 20 PLAN YOUR TRIP
## 11Lakes District, Chile
**_Adventure playground of lakes and mountains_**
While turquoise glacial lakes dominate the landscape, they're hardly the only attraction in this photogenic southern Chile region. Play on towering, perfectly conal, snowcapped volcanoes. Visit charming lakeside hamlets such as Frutillar. Admire the green umbrella of national parks like Parque Nacional Huerquehue. A long list of outdoor adventures and a unique, German-influenced Latin culture make for a cinematic region that appeals to all.
Pier at Frutillar | PAULO COSTA / 500PX ©
South America's Top 20 PLAN YOUR TRIP
## 12Salvador, Brazil
**_World capital of Afro-Brazilian culture_**
Salvador is famous for _capoeira_ (martial-art dance), Candomblé, Olodum, colonial Portuguese architecture, African street food and one of the oldest lighthouses in the Americas. The city's past, marked by gritty stories of Portuguese seafaring and the heartbreaking history of the African slave trade, is one of hardship, but today's lively Bahian capital offers a unique fusion of two vibrant cultures. The music and nightlife scene culminates when Salvador hosts one of Brazil's best Carnavals.
LAZYLLAMA / SHUTTERSTOCK ©
South America's Top 20 PLAN YOUR TRIP
## 13The Atacama Desert, Chile
**_Otherworldly landscapes in northern Chile_**
See the desert don its surrealist cloak as you stand atop a giant sand dune, with the sun slipping below the horizon and multicolored hues bathing the sands, all with a backdrop of distant volcanoes and the rippling Cordillera de la Sal. This is just one small part of the Atacama Desert, a mesmerizing landscape that encompasses red rock formations, jagged mountains, glittering salt lakes and sputtering geysers, plus dazzling star-filled skies.
Salar de Atacama | HELDER GERALDO RIBEIRO / SHUTTERSTOCK ©
Mountain biking in the desert | JOHN WARBURTON-LEE / GETTY IMAGES ©
Tourists at El Tatio Geysers | PAKAWAT THONGCHAROEN / GETTY IMAGES ©
South America's Top 20 PLAN YOUR TRIP
## 14The Cordilleras, Peru
**_Ground zero for outdoor adventure in Peru_**
The Cordilleras are one of the preeminent hiking, trekking and backpacking spots in South America. Every which way you throw your gaze, perennially glaciated white peaks razor their way through expansive mantles of lime-green valleys. The Cordillera Blanca is one of the highest mountain ranges in the world and boasts the enigmatic 3000-year-old ruins of Chavín de Huántar.
GALYNA ANDRUSHKO / SHUTTERSTOCK ©
South America's Top 20 PLAN YOUR TRIP
## 15The Pantanal, Brazil
**_Wildlife-watching in the wetlands_**
Few places on earth can match the wildlife-watching of the Pantanal, a wondrously remote wetland in the heart of Brazil. From cute capybaras to stately storks, the animal life simply abounds and is remarkably easy to see in the open marshy surroundings, whether you are traveling on foot, on horseback or by boat. Among the big draws is the elusive jaguar – this is one of the best places in South America to spot one.
WALTER MARIO STEIN / SHUTTERSTOCK ©
South America's Top 20 PLAN YOUR TRIP
## 16Santiago, Chile
**_Chile's art- and food-loving capital_**
Santiago is the center of the nation's cultural and intellectual universe. See the places that inspired the great poet Pablo Neruda, followed by visits to the grand collections in the Museo Nacional de Bellas Artes. By evening delve into Santiago's avant-garde restaurant scene, followed by a late night exploring raucous beer halls, candlelit poetry houses and just about anything else your inner Bacchus desires.
Plaza de Armas and the Catedral Metropolitana de Santiago | F11PHOTO / SHUTTERSTOCK ©
TIFONIMAGES / GETTY IMAGES ©
'Yellow Spiral' sculpture by Osvaldo Pena | KRZYSZTOF DYDYNSKI / GETTY IMAGES ©
South America's Top 20 PLAN YOUR TRIP
## 17Salar de Uyuni, Bolivia
**_The surreal salt flats of Bolivia_**
While a three-day jeep tour through the world's largest salt flat will leave your bones chattering, it could quite possibly be the defining experience of your Bolivian adventure. The vastness, austerity and crystalline perfection of the salt flat will inspire you. An early-morning exploration of rock gardens, geyser fields and piping-hot springs along with the camaraderie of three days on the road with your fellow 'Salterians' will create a lasting memory.
ANTONIO SALINAS L / GETTY IMAGES ©
South America's Top 20 PLAN YOUR TRIP
## 18Lake Titicaca, Peru
**_Floating reed islands and traditional living_**
Less a lake than a highland ocean, Titicaca is home to fantastical sights – none more surreal than the floating islands crafted entirely of tightly woven totora reeds. Requiring near constant renovation, the reeds are also used to build thatched homes and elegant boats. There are plenty of other islands to choose from, such as Isla Taquile, where rural Andean life from centuries long gone lives on, and where the quinoa soup recipe has been perfected. Puno
OLGA GAVRILOVA / SHUTTERSTOCK ©
South America's Top 20 PLAN YOUR TRIP
## 19Cartagena, Colombia
**_Colonial grandeur on the Caribbean_**
The hands of the clock on the Puerta del Reloj wind back 400 years in an instant as visitors enter the walled old town of Cartagena. Strolling the cobblestone streets is like stepping into the pages of a Gabriel García Márquez novel. The pastel-toned balconies overflow with bougainvillea and the streets are abuzz with food stalls around magnificent Spanish-built churches, squares and historic mansions.
Colonial building facade | RAFAL CICHAWA / SHUTTERSTOCK ©
South America's Top 20 PLAN YOUR TRIP
## 20La Paz, Bolivia
**_High-energy city famed for its markets_**
The world's highest capital city, La Paz is a dynamic whirl of tootling taxis, bustling indigenous markets and fast-moving pedestrians all mixing it up beneath the blinding _altiplano_ sun. Welcome to one of South America's most fascinating cities: a strangely intoxicating metropolis of rich ethnographic museums and stunning hilltop lookouts with a packed festival calendar. La Paz is also the gateway to intriguing pre-Columbian ruins outside of town.
Women in traditional Bolivian clothing | PEEK CREATIVE COLLECTIVE / SHUTTERSTOCK ©
# Plan Your Trip
Need to Know
Currency
Argentine peso (AR$)
Bolivian boliviano (B$)
Brazilian real (R$)
Chilean peso (CH$), Colombian peso (COP$) Ecuador: US dollar (US$)
Peru: nuevo sol (S)
Languages
Portuguese, Spanish, and over 500 indigenous languages
Visas
Visas are sometimes required, or in some cases other charges must be paid, for example for reciprocity fees or tourist cards.
Money
ATMs are available in major towns and cities; stock up on funds before visiting remote areas. Credit cards are widely accepted.
Cell Phones
Local SIM cards can be used in unlocked European and Australian (GSM) phones. Or you can purchase a phone when you arrive.
Time
Ranges from GMT minus three hours (Brazil's east coast) to GMT minus six hours (the Galapagos Islands).
### When to Go
### High Season _(Dec–Mar)_
o It's high season in Brazil and the Atlantic coast; beaches and festivals (such as Carnaval) are big draws.
o The best time to visit Patagonia, though expect higher prices.
### Shoulder _(Oct–Nov)_
o It's dry season in the Amazon, making for fine wildlife-watching.
o Fewer crowds and lower prices make this a good time to visit Buenos Aires, Rio and other coastal destinations.
### Low Season _(Jul–Aug)_
o In Chile and Argentina, many services close at beach resorts, and mountain passes can be blocked by snow.
### Daily Costs
#### Budget: Less than US$50
o Dorm beds: from US$12
o Double rooms: from US$30
o Shopping at markets, eating inexpensive set meals: from US$5
#### Midrange: US$50–120
o Budget jungle lodge in the Amazon per day: US$50–80
o Hiking and cycling tours per day: from US$50
o 3½-day Manaus–Belem boat trip (hammock fare): from US$100
#### Top End: More than US$120
o Hiking the Inca Trail (four-day trek) per person: US$600
o Multiday Galápagos cruise per day: around US$200
### Useful Websites
**Lonely Planet** (www.lonelyplanet.com/south-america) Destination information, hotel bookings, traveler forum and more.
**Latin American Network Information Center** (www.lanic.utexas.edu) Links to all things Latin American.
**UK Foreign Travel Advice** (www.gov.uk/foreign-travel-advice) Travel advisories.
**US State Department** (www.state.gov) Travel advice and warnings.
**Thorn Tree** (lonelyplanet.com/thorntree) Get trip recommendations and destination tips from other travelers.
### Opening Hours
On Sunday, nearly everything is closed. In the Andean countries, businesses tend to close earlier.
**Banks** Monday to Friday (for money changing).
**Businesses** 8am or 9am–noon and 2pm–8pm or 9pm Monday to Friday. Shorter hours on Saturday.
### Arriving in South America
#### Aeropuerto Internacional Ministro Pistarini, Buenos Aires
Frequent shuttle buses head downtown (AR$240); taxis cost around AR$1200.
#### Aeropuerto Internacional Arturo Merino Benítez, Santiago
Frequent shuttle connections go to downtown Santiago hotels (40 minutes, CH$7000), plus local buses (one hour to downtown, then transfer to the metro or Transantiago bus; CH$1700) and taxis (CH$18,000).
#### Galeão International Airport, Rio de Janeiro
Bus 2018 (R$16, one to two hours, every 30 minutes) goes to Copacabana and Ipanema. Radio taxis cost R$130 (45 to 90 minutes). Metered yellow-and-blue _comum_ (common) taxis cost between R$82 and R$100. Shuttle service costs R$25.
### Getting Around
**Bus** Extensive services throughout the continent, except for the Amazon. You'll find reclinable seats (and super-powered air-conditioning on long hauls).
**Plane** Useful for crossing immense distances; can save days of travel; prices are generally high, but airfare promotions are frequent.
**Car** Useful for traveling at your own pace, though cities can be difficult to navigate and secure parking is a must.
**Boat** Slow, uncomfortable, but brag-worthy transport between towns in the Amazon, with trips measured in days rather than hours. You'll need a hammock, snacks, drinking water and a high tolerance for boredom.
**Train** Limited networks, generally geared toward tourists.
For more on, Getting Around A
# Plan Your Trip
Hot Spots for...
## Dramatic Scenery
Thundering waterfalls, volcanoes and otherworldly desert landscapes. When gazing upon these natural wonders, you might feel like you've stepped back a few million years.
MATT MUNRO / LONELY PLANET ©
Iguazú Falls
Spread between Argentina and Brazil, these are some of the most spectacular waterfalls on earth.
Garganta del Diablo
A close-up view of the powerful torrent.
Torres del Paine
In southern Patagonia, sparkling glaciers, topaz lakes and sheer granite cliff-faces dominate the landscape.
The 'W'
Photographing the dramatic scenery on this multiday trek.
Salar de Uyuni
The world's largest salt flats are a dazzling remnant of a vast prehistoric lake.
Isla Incahuasi
Gazing over the cacti and endless salt horizon.
## Urban Allure
Life spills into the streets with pop-up graffiti murals, sprawling food markets, and buzzing music-filled cafes in South America's colorful city neighborhoods.
TETIANA TUCHYK / SHUTTERSTOCK ©
Rio de Janeiro
The Cidade Maravilhosa is full of lovely beaches, samba-fueled nightlife and jaw-dropping scenery.
Pão de Açúcar
Admire the view over one of the world's most captivating cities.
Buenos Aires
A place that's hard to leave, with old-world cafes, sultry tango clubs, and French and Italianate architecture.
Esquina Carlos Gardel
Nothing captures the city's essence like the tango.
Santiago
Grand architecture, vast green spaces and a thriving arts scene: Chile's capital has it all.
Centro Gabriela Mistral
Grab the cultural pulse at this performing-arts center.
## Wildlife
South America is home to more plant and animal species than any other place on earth; it has countless settings to watch wildlife.
GUENTERGUNI / GETTY IMAGES ©
The Amazon
An unforgettable experience whether you slow-boat along the river or base at an upscale jungle lodge.
Mamirauá Reserve
Marveling at the abundant wildlife in this remote reserve.
The Pantanal
At these wildlife-rich wetlands, you can spot countless birds and mammals.
Pantanal Jaguar Camp
Spotting felines at this aptly named wilderness lodge.
The Galápagos
These volcanic islands are home to creatures so tame, you'll practically be tripping over all the sea lions.
Rancho Primicias
Admiring tortoises as they lumber through the highlands.
## Outdoor Adventures
Get your adrenaline fix clambering past snow-capped peaks, paddling along rushing rivers or mountain biking down slopes. There are many ways to get your heart racing.
CHRISTIAN KOBER / ROBERTHARDING / GETTY IMAGES ©
Volcan Cotopaxi
If you're fit you can try for the ascent. Or go horseback riding from a hacienda.
Condor Trekk
Take a guided trek to the summit for a magnificent sunrise.
Lakes District
Against a mountainous backdrop, you can take an adventure-filled trip on class IV rapids.
Al Sur
Splashing through the white water on a guided rafting trip.
The Atacama Desert
The desert oasis town of San Pedro de Atacama offers daytime adventures followed by nights of stargazing.
Valle de la Muerte
Sandboarding down dunes against a red-rock backdrop.
# Plan Your Trip
Essential South America
### Activities
There's a whole range of adventures awaiting in South America. You can go hiking amid the soaring peaks of the Andes, go rafting along rushing jungle-lined rivers and overnight in a rainforest lodge with the sounds of the Amazon all around you. And you'll find many more astounding options in every country on the continent. There's memorable diving off of the Galápagos, trekking and mountaineering in the Andes, wildlife-watching in the Pantanal and legendary mountain biking outside of La Paz.
### Shopping
South America has a treasure trove of arts and crafts. All across the continent, you'll find lively markets packed with crafts and clothing. In the Andes, you'll find alpaca wool sweaters and scarves, woven textiles, ceramics, masks and gold and silver jewelry. Brazil has its own enormous crafts scene, with some of the finest works emerging from the Northeast. Neighboring Argentina has myriad temptations for shoppers, from street fairs and antique shops to indie clothing boutiques. Wherever you roam, when you need essentials, large North American–style malls and sizable supermarkets can be found in the big cities.
Traditional Peruvian handmade textiles | SL-PHOTOGRAPHY / SHUTTERSTOCK ©
### Eating
South America has staggering variety when it comes to cuisine. You'll find sizzling steaks in Buenos Aires, creative twists on age-old dishes in the Andes, and tender freshwater fish in the Amazon. Every country has its own specialties, and within countries you'll find a great range (with coastal recipes quite different from cooking traditions in the interior). The capitals and big cities of South America are all fertile grounds for foodies, though increasingly, you can also find fantastic meals in smaller towns. Memorable meals can sometimes be had at markets, as well.
Peruvian ceviche | Larisa Blinova / Shutterstock ©
### Drinking & Nightlife
You'll find dynamic nightlife in all of South America's big cities, particularly in Rio, Buenos Aires and Santiago. You'll find easygoing beach bars, DJ-fueled clubs, stylishly decked speakeasies and cocktail bars, as well as craft-beer bars tapping into the recent craze for local microbrews. Wine bars are all the rage in Argentina and Chile, and nearly every region has its specialty drinks – from _canelazo_ (a hot rum, cinnamon and orange juice concoction) in the Andean highlands to _mate_ (hot _yerba mate_ drunk from a special gourd) in Argentina, Chile and southern Brazil.
o Live Music Venues
Rio Scenarium
Tango Porteño
Espaço D'Venetta
Jallalla
Café Havana
### Entertainment
Live music is the lifeblood of South America. Brazil's music scene is unrivaled, with buzzing samba clubs in Rio and heart-pounding drum corps filling the streets of Salvador. Buenos Aires has its magnificent tango scene, which can mean watching or dancing depending on the place (plenty of places offer lessons), while Peru and Bolivia have their _peñas_ , venues for traditional folkloric music. Aside from live rhythms, there are countless other draws in South America, including theater, dance and indie cinema, as well as live football matches, which bring die-hard fans to local (sometimes massive!) sporting arenas in every corner of the continent.
# Plan Your Trip
Month by Month
## January
It's peak season in Brazil and Argentina. Expect higher prices, bigger crowds and sweltering temperatures as city dwellers head to the coast. This is also the most popular time to travel to Patagonia.
### 3 Santiago a Mil
This 17-day theater and dance fest (www.fundacionteatroamil.cl/en/santiagoamil)features dozens of shows and events around the Chilean capital, staged by international and local companies.
## February
The sizzling summer is still in full swing in the southern half of the continent, with exorbitant prices and sparse accommodations during Brazilian Carnaval. Elsewhere, it's fairly wet in the northern Andes and the Amazon region.
### z Carnaval
The famous bacchanalian event happens across South America, though the pre-Lenten revelry is most famous in Brazil. Rio and Salvador throw the liveliest bashes, with street parades, costume parties and round-the-clock merriment.
Carnaval in Rio de Janeiro | EFE NEWS AGENCY / ALAMY STOCK PHOTO ©
### z Fiesta de la Virgen de Candelaria
Celebrated across the highlands in Bolivia and Peru, this festival features music, drinking, eating, dancing, processions and fireworks. Some of the liveliest celebrations take place in Puno (Peru). The big day is February 2.
## March
While the weather is still warm in the south, the crowds thin and prices fall a bit at beach destinations. It's still rainy in the northern Andes.
### 3 Lollapalooza Chile
Chile's rock fest (www.lollapaloozacl.com) kicks off in Santiago in late March or early April, and features an impressive line-up of homegrown and international groups on par with the North American version of Lollapalooza.
Lenny Kravitz performing at Lollapalooza Chile | LUOMAN / GETTY IMAGES ©
### z Concurso Nacional de Belleza
Also known as the Carnaval de Cartagena, this beauty pageant and festival (www.srtacolombia.org), Cartagena's most important annual bash, celebrates the city's independence day and the crowning of Miss Colombia. Festivities include street dancing, music and fancy-dress parades.
## April
In the Andes, crowds and high-season prices mark Holy Week, a boon of national tourism in March or April. Elsewhere on the continent, you'll find generally fewer crowds and good post-summer prices.
### 1 Semana Santa
Throughout Latin America, Holy Week is celebrated with fervor. One of the most colorful processions happens in Quito (Ecuador) when purple-robed penitents parade through the streets on Good Friday.
o Best Festivals
**Carnaval,** February
**Fiesta de la Virgen de Candelaria,** February
**Inti Raymi,** June
**Tango BA Festival y Mundial,** August
**Reveillon,** December
## May
Buenos Aires and Rio head into low season, with cooler weather and lower prices; the rain begins to taper off in the Andes, making it a fine time to go trekking.
### z La Festividad de Nuestro Señor Jesús del Gran Poder
Held in late May or early June, La Festividad de Nuestro Señor Jesús del Gran Poder in La Paz involves candle processions, elaborate costumes and dancing.
## June
High season in the Andean nations corresponds with the North American summer (June to August), when the weather is also sunniest and driest. Book major tours (such as hiking the Inca Trail) well in advance.
### z Inti Raymi
This millennia-old indigenous celebration of the solstice and harvest is celebrated in many Andean towns. In Cuzco it's the event of the year, attracting thousands of visitors for street fairs, open-air concerts and historical reenactments.
## August
It's dry in many parts of the continent, making August a fine time to visit the Amazon, the Pantanal or the Andes. It's chilly to freezing south of the Tropic of Capricorn.
### 3 Tango BA Festival y Mundial
World-class tango dancers perform throughout Buenos Aires during this two-week festival (<http://festivales.buenosaires.gob.ar>). Competition is fierce for the title of 'world's best tango dancer.' You can also hone your own moves at classes and workshops.
## September
The weather remains dry and sunny (but chilly) in the Andes, though you'll find fewer crowds. September is also a good (less rainy) time to visit the Amazon.
### z Fiestas Patrias
Chilean Independence is feted during Fiestas Patrias (week of September 18), with a week of big barbecues, _terremotos_ (potent wine punch) and merrymaking all over Chile.
## October
Heavy rains make for tough traveling in Colombia, while the Andes generally have milder weather. In Bolivia, Brazil, Chile and Argentina, temperatures are mild, making it a pleasant time to visit.
### z Oktoberfest
Join the swillers and oompah bands in Puerto Varas, Chile, for live music in lederhosen and beer festivals.
## November
Rainier days are on the horizon in the Amazon. Generally November nets better prices, good weather and fewer crowds than December in most parts of South America.
### z Puno Week
Starting November 5, this week-long festival involves several days of spectacular costumes and street dancing to celebrate the legendary emergence of the first Inca, Manco Cápac.
### 3 Buenos Aires Jazz Festival Internacional
BA's big jazz festival (www.buenosairesjazz.gob.ar) showcases the talents of more than 200 musicians who play in 70 different concerts around town.
## December
December marks the beginning of summer, with beach days (and higher prices) on both the Atlantic and Pacific coasts. It's fairly rainy in the Andes.
### z Founding of Quito Festival
Quito's biggest bash is a much anticipated event, with parades, street dances and open-air concerts throughout town.
### z Reveillon
There are many great spots in South America to celebrate New Year's Eve, but Rio is a perennial favorite. Some two million revelers, dressed in white to bring good luck, pack the sands of Copacabana Beach to watch fireworks light up the night sky.
# Plan Your Trip
Get Inspired
### Read
**Gabriel García Márquez & Mario Vargas Llosa** García Márquez _(Love in the Time of Cholera)_ and Vargas Llosa _(War of the End of the World)_ , both Nobel Prize winners, are among the continent's greatest writers.
**Jorge Luis Borges** This legend of modern literature is best known for his labyrinthine tales and playful melding of myth and truth, such as in _Ficciones_.
**Jorge Amado** Colorful, ribald stories set in Bahia, such as the classic _Dona Flor and Her Two Husbands_.
**Bruce Chatwin** Poignant and beautifully written travel narrative, _In Patagonia,_ blends fact and fable.
### Watch
**The Motorcycle Diaries** (2004) The road trip that made a revolutionary.
**Central do Brasil** (1998) Moving tale of a homeless boy and an older woman on a journey across Brazil.
**Neruda** (2016) The communist Chilean poet goes on the run.
**Aguirre, the Wrath of God** (1972) Werner Herzog's historical masterpiece about a fruitless search for El Dorado.
**Amargo Mar** (1984) Highly regarded look at the loss of Bolivia's coastline to Chile.
**Qué Tan Lejos** (2006) Road movie about two women in the Ecuadorian highlands.
### Listen
**África Brasil** (Jorge Ben Jor) Celebratory album from the 1970s that blends funk, samba and blues.
**Amanecer** (Bomba Estéreo) A blend of African, indigenous and vibrant dance beats by an inventive Colombian band.
**Lunático** (Gotan Project) Brilliant fusing of tango with electronic grooves.
**Roots of Chicha** (various artists) Wild Peruvian cumbias that channel psychedelic, rock and melodic sounds.
**Tropicália ou Panis et Circencis** A famed Brazilian collaboration between Gilberto Gil, Caetano Veloso, Gal Costa and Tom Zé.
Glaciar Grey, Torres del Paine, Chile | CIRCUMNAVIGATION / SHUTTERSTOCK ©
# Plan Your Trip
Seven-Day Itineraries
## Forests, Falls & Urban Allure
Careful planning will allow you to take in some of the continent's most majestic sights on a one-week itinerary. You'll get a taste of the Amazon, see awe-inspiring waterfalls and explore two of South America's most dynamic cities.
VALMOL48 / GETTY IMAGES ©
ALEX BERGER / VIRTUALWAYFARER.COM / GETTY IMAGES ©
# Seven-Day Itineraries
## Colonial Towns & Lava Landscapes
On a seven-day trip in the northwest, you can visit two colonial cities followed by a memorable cruise around the Galápagos Islands. The varied terrain encompasses the Caribbean Sea, jagged Andean peaks and volcanoes in the Pacific.
ROCHARIBEIRO / SHUTTERSTOCK ©
WESTEND61 / GETTY IMAGES ©
# 10-Day Itinerary
## Grand Adventures in the South
A paradise for lovers of the great outdoors, the south beckons with glacial lakes and dramatic peaks. You'll get a dose of city life before heading down to Chile's photogenic Lakes District, then on to the big open skies of Patagonia.
JOSE LUIS STEPHENS / EYEEM / GETTY IMAGES ©
ICYS / GETTY IMAGES ©
MICHELE FALZONE / GETTY IMAGES ©
# Two-Week Itinerary
## From the Andes to the Atacama
Over two weeks, you'll join an unforgettable hike along the Inca Trail and experience Andean wonders: ancient indigenous temples, floating islands, and one (literally) breathtaking metropolis. The trip ends amid the otherworldly landscapes of the Atacama Desert.
SUNSINGER / SHUTTERSTOCK ©
KAZUKI KIMURA / EYEEM / GETTY IMAGES ©
DANTE BUSQUETS / SHUTTERSTOCK ©
# Plan Your Trip
Family Travel
In general, wherever you roam in South America, you'll find a warm welcome when traveling with children. Family culture is strong in Latin America and locals generally do their best to accommodate young travelers. That said, infrastructure can be lacking. Baby-change facilities are rare outside of big cities and pushing strollers (prams) around can be a challenge amid broken pavements and missing sidewalks (a wearable baby carrier is a better idea). Public transport can often be quite crowded. For insider tips check out Lonely Planet's _Travel with Children_.
Tourist photographing marine iguanas on Isla Fernandina, Galápagos Islands | MARIDAV / SHUTTERSTOCK ©
### Lodging
The great majority of hotels accept children without any problems; the most upscale may even offer babysitting services. The only places with possible minimum age restrictions are small boutique hotels or guesthouses. During summer, reserving a hotel with a pool can be a good idea. Also look for places with kitchenettes.
Apartments are available in nearly every city; in less-urban holiday destinations you can look for _cabañas_ (cabins) with full kitchens. Larger campgrounds often have _cabañas,_ common cooking facilities and sometimes play structures.
Hostels are usually not the best environment for kids, but a few welcome them.
### Dining
While restaurants don't offer special kids' meals, most offer a variety of dishes suitable for children; none are spicy. It is perfectly acceptable to order a meal to split between two children or an adult and a child; most portions are abundant.
The only challenge to dining families is the late hours in Argentina and Chile, where restaurants open for dinner no earlier than 7pm, sometimes 8pm, and service can be quite slow. Bring a journal or scribble book and crayons for the kids to pass the time.
Local snacks are handy for hungry little ones. Empanadas make good, healthy snacks that are fun to eat. You'll find tropical juices, coconut water and snacks such as _pão de queijo_ (cheese bread), served at ubiquitous juice bars in Brazil. And ice cream is found everywhere on the continent – an essential treat on hot days!
Parque Bicentenario, Santiago, Chile | DFLC PRINTS / SHUTTERSTOCK ©
### Before You Go
Keep the kids in mind as you plan your itinerary or include them in the trip planning from the get-go. If renting a car, ask ahead if you can book a child's seat, as they are not always available. It's not necessary to be tied down to a schedule while traveling in the continent; plenty of activities can be booked just a few days in advance.
o Best Destinations for Kids
Galápagos Islands
Iguazú Falls
Buenos Aires
Santiago
Rio de Janeiro
### Need to Know
o Disposable diapers (nappies) are found in big-city supermarkets, but variety may be limited, so come prepared.
o Cribs (cots) aren't widely available in hotel rooms.
o Children get in free or half-price to most major sights around South America.
o Baby formula isn't always available outside major cities; bring your own supply.
o In Argentina and southern Chile, outdoor activities are best experienced outside the winter months of June through August (with the exception of skiing, of course).
# Rio de Janeiro, Brazil
#### Copacabana & Ipanema Beaches
#### Pão de Açúcar
#### Cristo Redentor
#### Walking Tour: Historic Centro
#### Sights
#### Activities
#### Tours
#### Shopping
#### Eating
#### Drinking & Nightlife
#### Entertainment
#
Rio de Janeiro, Brazil
_Flanked by gorgeous mountains, golden beaches and verdant rainforests fronting deep-blue sea, Rio de Janeiro occupies one of the most spectacular settings of any metropolis in the world. Tack on one of the sexiest populations on the planet and you have an intoxicating tropical cocktail._
_Rio's residents, known as _cariocas_ , have perfected the art of living well. Prepare to be seduced as you stroll world-famous beaches like Copacabana and Ipanema, enjoy the views from atop Corcovado or Pão de Açúcar, and join the party at dance halls, bars and open-air cafes, which proliferate around the city._
Aerial view of Rio de Janeiro | MICROGEN / GETTY IMAGES ©
Two Days in Rio de Janeiro
Spend the first day enjoying the sand and sun off Ipanema Beach. Have dinner at Zazá Bistrô Tropical and drinks at Nosso. On day two, get active along Copacabana Beach, followed by a trip to Pão de Açúcar for fabled views over the city. Cap the day with live samba at Rio Scenarium in Lapa.
Four Days in Rio de Janeiro
On your third day, visit the lush Jardim Botânico, then head skyward for fabulous views beneath open-armed Cristo Redentor. End the day with drinks at lakeside Palaphita Kitch. On the last day, check out the museums and baroque churches of Centro, art-gaze along the Boulevard Olímpico and have a sunset dinner and drinks at atmospheric Xian.
Arriving in Rio de Janeiro
Rio's main airport, Galeão international airport (GIG), is 15km north of the city center. Bus 2018 links the airport with Copacabana, Ipanema, Leblon and other neighborhoods (75 minutes to two hours, R$16). Radio taxis charge a set fare of R$130 to Copacabana and Ipanema (45 to 90 minutes). Less-expensive metered yellow-and-blue taxis cost around R$100. Ride-sharing services cost around R$60. There's also shuttle service (R$25).
Where to Stay
Rio has a wide range of lodgings, including B&Bs, hostels and guesthouses; there are dozens of high-end high-rises with ocean views, particularly in Copacabana. An abundance of options keeps rates from going sky high, except during Carnaval and other major events.
For information on what each Rio neighborhood has to offer, see the table on.
TOP EXPERIENCE
# Copacabana & Ipanema Beaches
Rio's loveliest beaches are a magnificent confluence of land and sea, with golden sands anchored by green peaks. The city's sun-kissed backyard has amusements of all kinds, from surfing and jogging to sitting back, agua de côco (coconut water) in hand, and watching Rio at play.
Great For...
pgl
yDon't Miss
Enjoying the sunset from Ponta do Arpoador, a rocky outcropping at Ipanema's eastern end.
8Need to Know
Reach Copacabana Beach via the metro stations of Cardeal Arcoverde, Siqueira Campos and Cantagalo. For Ipanema Beach, use NS da Paz and General Osório metro stations.
5Take a Break
Pick up sandwiches with fresh grilled chicken plus cold drinks from Barraca do Uruguai (map Google map; Posto 9; sandwiches R$14-20; hnoon-5pm) on Ipanema Beach.
oTop Tip
Go early to stake out a spot, and bring just enough cash for the day.
### Beachside Activities
Aside from frolicking in the waves and people-watching, the main attraction for visitors to the beach is running or cycling the beach path; the beachside lane closes to traffic on Sundays. Early risers can greet the sunrise while exercising. Another way to get physical is to have a go on a stand-up paddle board, which you can hire from the south end of Copacabana Beach. Hire outfits also offer lessons if you need a bit of a primer.
Surfing at Ipanema Beach | GIOVANI CORDIOLI / GETTY IMAGES ©
### The Scene
The beaches are demarcated by _postos_ (posts), which mark off subcultures as diverse as the city itself. Ipanema's Posto 9, near Rua Vinícius de Moraes, is Garota de Ipanema, where Rio's most lithe and tanned bodies tend to migrate. In front of Rua Farme de Amoedo the beach is known as Praia Farme, and is the stomping ground for gay society. Arpoador, at Ipanema's eastern end, is Rio's most popular surf spot.
In Copacabana, young football and _futevôlei_ (soccer volleyball) players hold court near Rua Santa Clara. Postos 5 and 6 are a mix of favela kids and _carioca_ retirees, while the spot next to Forte de Copacabana is the _colônia dos pescadores_ (fisherfolk's colony). As Copacabana Beach curves north you get into the quieter sands of Leme (Av Princesa Isabel forms the boundary between Copacabana and Leme).
### Beach Kiosks
By afternoon and nightfall, the beach kiosks – particularly in Copacabana – are great places to relax, rehydrate over _agua de côco_ , and refuel before the night begins. On weekends there's also live music. Things get decidedly livelier when football games are screened, with kiosks setting up TVs, and passers-by and beach-goers joining the celebratory crowds over ever-flowing drinks.
Aerial view of Copacabana Beach | STUART DEE / GETTY IMAGES ©
### Relaxing & Refreshment
Once on the beach, head to your favorite _barraca_ (beach stall), where you can hire chairs and a sun umbrella. There are also food and drink stalls, though roving vendors will come to you, proffering cold drinks (try the sweet tea-like _maté_ ) and snacks (such as crunchy _globos_ – a crispy biscuit made from manioc flour).
If you're in the mood for something more substantial, head up to one of the kiosks lining the beachfront. Some Copacabana kiosks have full menus (the kitchens are cleverly concealed underground).
TOP EXPERIENCE
# Pão de Açúcar
From the summit of Pão de Açúcar (Sugarloaf Mountain), it's clear why Rio is called the Cidade Maravilhosa (Marvelous City). Following the ascent by aerial cable car up the mountain, you'll be rewarded with magnificent views over Rio.
Great For...
gAb
yDon't Miss
The view of planes landing beneath you at Santos Dumont Airport in Guanabara Bay.
8Need to Know
(map Google map) Sugarloaf Mountain; %21 2546-8400; www.bondinho.com.br; Av Pasteur 520, Urca; adult/child R$85/42; h8am-8pm
5Take a Break
Linger over cocktails and excellent snacks from the mountaintop Clássico Beach Club (map Google map; %21 2542-8966; www.classicobeachclub.com/urca; Av Pasteur 520; h9am-9pm).
oTop Tip
To avoid queues, come first thing in the morning or late in the day. Save 10% by purchasing a ticket online.
### The Ascent by Cable Car
The most traditional way to reach the top is to board the two-stage cable car that departs from Urca every 20 minutes or so. The glass-and-steel cars ascend 215m to Morro da Urca and are good fun in themselves. From here, you can see Baía de Guanabara and along the winding coastline; on the ocean side of the mountain is Praia Vermelha, in a small, calm bay. Aside from the traditional visit, you can also arrange for a guided tour that delves into the history of the sight; there's also a backstage tour that allows a behind-the-scenes look at the cable-car engine rooms.
Morro da Urca has a restaurant, souvenir shops, outdoor theater and a helipad. In the summer, concerts are sometimes staged in the amphitheater. The second cable car continues up to Pão de Açúcar.
Cable car to Pão de Açúcar | RAPHAEL KOERICH / GETTY IMAGES ©
### The View
At the top, the city unfolds beneath you, with Corcovado mountain and Cristo Redentor off to the west, the twinkling lights of Niteroi across the bay to the east, and the long curve of Copacabana Beach to the south. If the breathtaking heights unsteady you, there's a stylish restaurant and bar on hand serving caipirinhas and snacks.
For prime views of the Cidade Maravilhosa, go around sunset on a clear day.
### Climbing to the Top
Those who'd rather take the long way to the top should sign up with one of the granite-hugging climbing tours offered by various outfits in Rio, including Crux Eco ( %21 3322-8765, 21 99392-9203; www.cruxeco.com.br; c). Morro da Urca is much easier to climb and you can do it on your own. The short but steep path takes about 30 minutes to climb. You'll find the unmarked trail along the Pista Cláudio Coutinho.
### What's Nearby
#### Praia Vermelha
Beneath Morro da Urca, narrow Praia Vermelha has superb views of the rocky coastline from the shore. Its coarse sand gives the beach the name _vermelha_ (red). Because the beach is protected by the headland, the water is usually calm.
#### Pista Cláudio Coutinho
This paved 1.2km trail winds along the southern contour of Morro da Urca. It's a lush area, with the craggy mountain rising on one side, and waves crashing on rocks below. To get here, walk 100m north along the edge of Praia Vermelha (with your back to the cable-car station) and you'll see the entrance to the path straight ahead, just past the beach.
TOP EXPERIENCE
# Cristo Redentor
One of Rio's most identifiable landmarks, the magnificent 38m-high Cristo Redentor (Christ the Redeemer) looms large atop the forest-covered mountain of Corcovado. From the statue's base, you'll have mesmerizing views over all of Rio.
Great For...
Agb
yDon't Miss
The splendid views from the Cog Train (sit on the right side of the train for the best scenery).
8Need to Know
(map Google map) Christ the Redeemer; %21 2558-1329; www.tremdocorcovado.rio; cog train station, Cosme Velho 513; adult/child R$75/49; h8am-7pm
5Take a Break
Although touristy, the restaurant just below the statue serves decent Brazilian food and the terrace views are fabulous.
oTop Tip
Avoid going on weekends when the crowds are biggest, and don't go on overcast days.
### The Views
Corcovado, which means 'hunchback,' rises straight up from the city to a height of 710m, and at night the brightly lit statue is visible from nearly every part of the city. When you reach the top, you'll notice the Redeemer's gaze directed at Pão de Açúcar (Sugarloaf Mountain), with his left arm pointing toward the Zona Norte; Maracanã football stadium crowds the foreground. You can also see the international airport on Ilha do Governador just beyond and the Serra dos Órgãos mountain range in the far distance. Beneath Christ's right arm you can see Lagoa Rodrigo de Freitas, Hipódromo de Gávea, Jardim Botânico, and over to Ipanema and Leblon.
SAN HOYANO / SHUTTERSTOCK ©
### Cog Train
The most popular way up to the statue is to take the red narrow-gauge train that departs every 30 minutes from the cog station. Note that you must buy your cog train tickets in advance (either online or at various sale points in town). It takes approximately 20 minutes to get to the top. To reach the cog station, take any 'Cosme Velho' bus; from Copacabana, Ipanema or Leblon bus 583 will get you here.
S J FRANCIS /SHUTTERSTOCK ©
### Historical Background
The Redeemer, which opened in 1931, is considered to be the world's largest art deco statue. It's not a gift from the French, as is popularly believed. However, chief sculptor Paul Landowski was of French-Polish origin and carried out much of the construction in France. He collaborated with the Rio architect-engineer Heitor Silva Costa (1873–1947) on the project. Many organizations helped make the statue a reality, including several individuals who went door-to-door asking for contributions.
The idea of the statue originated in 1921 when a group called Círculo Carioca held a competition for a religious monument to commemorate Brazil's upcoming 100 years of independence. Heitor's winning project, which took 10 years to build, was considered particularly ambitious at the time – naysayers doubted whether it could be accomplished at all. Heitor's original idea depicted Christ as a vertical form with a long cross held against his side but the committee wanted something recognizable from a great distance, so the crosslike outstretched arms were chosen instead. Today it's one of Brazil's most frequented attractions, welcoming more than one million visitors a year.
Flamengo, Botafogo & Urca
1Sights
1Cristo RedentorA3
2Pão de AçúcarF2
3Parque do FlamengoD1
2Activities, Courses & Tours
4Sail in RioD3
5Eating
5Adega PérolaC4
6Bar UrcaE2
7BrouD1
8Churrascaria PalaceD4
9MarchezinhoD3
10Meza BarB3
11OtequeC3
12SaloméD4
13Sírio LibanezaD1
14South FerroD3
15Tacacá do NorteD1
6Drinking & Nightlife
Bar Urca(see 6)
Clássico Beach Club(see 2)
16CoLABD3
17ComunaC3
18Pavão AzulC4
1Sights
### 1Gávea, Jardim Botânico & Lagoa
Lagoa Rodrigo de FreitasLake
(map Google map)
One of the city's most picturesque spots, Lagoa Rodrigo de Freitas is encircled by a 7.2km walking and cycling path. Bikes are available for hire from stands along the east side of the lake, as are paddleboats. For those who prefer _caipirinhas_ (cocktails made from limes, sugar, ice and high-proof sugarcane alcohol) to plastic swan boats, the lakeside kiosks on either side of the lake offer alfresco food and drinks, sometimes accompanied by live music on warm nights.
Rio's Iconic Streetcar
The bonde (map Google map; return ticket R$20; h9am-5pm Mon-Fri, 10am-5:30pm Sat, 11am-4:30pm Sun; mCarioca) is the last of the historic streetcars that once crisscrossed the city. Its romantic clatter through the cobbled streets is the archetypal sound of bohemian Santa Teresa. Currently the bonde travels every 15 to 20 minutes from the cable-car station in Centro over the scenic Arcos da Lapa and as far as Largo do Guimarães in the heart of Santa Teresa.
### 1Flamengo
Museu da RepúblicaMuseum
(map; %21 2127-0324; <http://museudarepublica.museus.gov.br>; Rua do Catete 153; R$6, Wed & Sun free; h10am-5pm Tue-Fri, 11am-6pm Sat & Sun)
The Museu da República, located in the Palácio do Catete, has been wonderfully restored. Built between 1858 and 1866, and easily distinguished by the bronze condors on its eaves, the palace was home to the president of Brazil from 1896 until 1954, when President Getúlio Vargas committed suicide here. The museum has a good collection of art and artifacts from the Republican period. It also houses a small cafe, an art-house cinema and a bookstore.
Parque do FlamengoPark
(map Google map; Av Infante Dom Henrique)
Officially called Parque Brigadeiro Eduardo Gomes, Parque do Flamengo was the result of a landfill project that leveled the São Antônio hill in 1965. It now spreads all the way from downtown Rio through Glória, Catete and Flamengo, and on around to Botafogo. The 1.2 sq km of land reclaimed from the sea now sees every manner of outdoor activity by _cariocas_.
Centro, Santa Teresa & Lapa
1Sights
1Arcos da LapaC6
2BondeB8
3Boulevard OlímpicoB1
4Centro Cultural Banco do BrasilD3
5Escadaria SelarónC7
6Igreja São Francisco da Penitência & Convento de Santo AntônioC5
7Museu Chácara do CéuB7
8Museu de Arte do RioC2
9Museu do AmanhãC2
10Museu Histórico NacionalE4
11Paço ImperialD4
12Praça FlorianoD5
13Real Gabinete Português de LeituraC4
14Theatro MunicipalD5
15Travessa do ComércioD3
7Shopping
16CRABB5
17Favela HypeB8
18La VeredaB8
5Eating
19Bar do MineiroB8
20Cais do OrienteD3
21Casa MomusB5
22Confeitaria ColomboD4
23Espírito SantaB8
24Govardhana HaríD4
25SobrenaturalB8
26XianF7
6Drinking & Nightlife
27Explorer BarB8
3Entertainment
28Carioca da GemaC6
Favela Hype(see 17)
29Lapa 40 GrausB6
30Pedra do SalB2
31Rio ScenariumB5
32SacrilégioC6
33TribOzD7
### 1Centro & Praça Mauá
Museu Histórico NacionalMuseum
(map Google map; %21 3299-0324; <http://mhn.museus.gov.br>; off General Justo, near Praça Marechal Âncora; adult/child R$10/5, Sun free; h10am-5:30pm Tue-Fri, 1-5pm Sat & Sun)
Housed in the colonial arsenal, which dates from 1764, the impressive Museu Histórico Nacional contains relics relating to the history of Brazil from its founding to its early days as a republic. Highlights include gilded imperial coaches, the throne of Dom Pedro II, massive oil paintings depicting the horrific war with Paraguay and a full-sized model of a colonial pharmacy.
Museu de Arte do RioMuseum
(map Google map; MAR; %21 3031-2741; www.museudeartedorio.org.br; Praça Mauá 5; adult/child R$20/10, Tue free; h10am-5pm Tue-Sun)
Looming large over Praça Mauá, the MAR is an icon for the rebirth of Rio's once derelict port. The huge museum hosts wide-ranging exhibitions that focus on Rio in all its complexity – its people, landscapes, beauty, challenges and conflicts. Start by taking the elevator to the top (6th) floor and absorbing the view over the bay. There's also an excellent restaurant here. Then work your way down through the galleries, taking in a mix of international and only-in-Rio exhibitions.
Museu do AmanhãMuseum
(map Google map; www.museudoamanha.org.br; Rodrigues Alves 1; adult/child R$20/free, Tue free; h10am-6pm Tue-Sun)
Designed by famed Spanish architect Santiago Calatrava, this thoughtfully conceived science museum has interactive exhibitions on outer space, the earth and its biodiversity, humans (and our impact on the world), and the major global trends and challenges that lie in the future. In all, the museum takes a somewhat pessimistic view of the human species and its long-ranging impact on the world.
Boulevard OlímpicoArea
(map Google map)
Rio's formerly derelict port district has been reborn as a wide promenade lined with massive street art. A handful of renowned artists have painted spectacular murals on the old warehouses, though Brazilian artist Eduardo Kobra (www.eduardokobra.com) deserves special mention for his jaw-dropping work entitled _Etnias_ (Ethnicities). The massive mural stretches for 190m and features photogenic portraits of indigenous people from around the globe.
Centro Cultural Banco do BrasilCultural Center
(map Google map; CCBB; %21 3808-2020; www.culturabancodobrasil.com.br; Primeiro de Março 66, Centro; h9am-9pm Wed-Mon) F
Housed in a beautifully restored 1906 building, the Centro Cultural Banco do Brasil hosts some of Rio's top exhibitions. Facilities include a cinema, two theaters and a permanent display of the evolution of currency in Brazil. There is always something going on, from exhibitions and lunchtime and evening concerts to film screenings, so look up details in _O Globo's_ entertainment listings or the 'Veja Rio' insert in _Veja_ magazine before you go.
Jardim Botânico
The exotic 137-hectare botanical gardens (map Google map; %21 3874-1808; www.jbrj.gov.br; Jardim Botânico 920; R$15; hnoon-6pm Mon, 8am-6pm Tue-Sun), with more than 8000 plant species, was designed by order of the Prince Regent Dom João (later to become Dom João VI) in 1808. The garden is quiet and serene on weekdays and blossoms with families on weekends. Highlights of a visit here include the row of palms (planted when the garden first opened), the Amazonas section, the lake containing the huge Vitória Régia water lilies and the enclosed _orquidário_ , home to 600 species of orchids.
Before or after a visit to the gardens, you can grab a bite or a drink at Jarbô (map Google map; %21 2259-2924; www.jarbocafe.com; mains R$38-67; hnoon-6:30pm Mon, 8am-6:30pm Tue-Sun; W v). It's set in a lovely colonial building and has outdoor seating on a peaceful veranda.
Avenue of Royal Palms | DIEGO GRANDI / SHUTTERSTOCK ©
### 1Santa Teresa & Lapa
Escadaria SelarónLandmark
(map Google map; btwn Joaquim Silva & Pinto Martins, Lapa)
One of Rio's best-loved attractions, the steps leading up from Joaquim Silva became a work of art when Chilean-born artist Jorge Selarón decided to cover them with colorful mosaics. A dedication to the Brazilian people, the 215 steps are a vivid riot of color.
Arcos da LapaLandmark
(map Google map; Lapa)
A much-photographed symbol of Lapa, the arches date back to the mid-18th century, when the structure served as an aqueduct to carry water from the Carioca River to downtown Rio. In a style reminiscent of ancient Rome, the 42 arches stand 64m high. Today the arches carry the bonde cable car on its way between Centro and Santa Teresa. Located near Av Mem de Sá.
Museu Chácara do CéuMuseum
(map Google map; %21 3970-1093; www.museuscastromaya.com.br; Murtinho Nobre 93; adult/child R$6/free, Wed free; hnoon-5pm Wed-Mon)
The former mansion of art patron and industrialist Raymundo Ottoni de Castro Maya contains a small but diverse array of modern art, formerly Ottoni's private collection, which he bequeathed to the nation. In addition to works by Portinari, Di Cavalcanti and Lygia Clark, the museum displays furniture and Brazilian maps dating from the 17th and 18th centuries, and hosts temporary exhibitions.
2Activities
If you weigh less than 100kg (about 220lb) and have a spare R$500, you can do the fantastic hang glide off 510m Pedra Bonita – one of the giant granite slabs that tower above Rio – onto Pepino Beach in São Conrado. Flights last about seven to 10 minutes, and no experience is necessary. Guest riders are secured in a kind of pouch that is attached to the hang glider.
A reliable operator is Just Fly ( %21 3593-4362; www.justflyinrio.blogspot.com; R$520).
TTours
Jungle MeHiking
( %21 4105-7533; www.jungleme.com.br; tours from R$195; c)
This top-notch outfit offers excellent hiking tours through Parque Nacional da Tijuca led by knowledgeable guides. The eight-hour Tijuca Circuit (Peaks & Waterfalls tour) offers challenging walks up several escarpments with stunning views of Rio, followed by a refreshing dip in a waterfall. The Prainha & Grumari tour consists of a hike between scenic beaches in Rio's little-visited western reaches.
Rio by BikeCycling
(map Google map; %21 96871-8933; www.riobybike.com; Av NS de Copacabana 1085, Copacabana; tours R$100-135)
Two Dutch journalists operate this biking outfit, and their excellent pedaling tours combine scenery with cultural insight. It's a great way to get an overview of the city, with guides pointing out landmarks and describing key events that have shaped Rio. Tours generally last three to four hours and travel mostly along bike lanes separated from traffic.
Eat Rio Food ToursWalking
(www.eatrio.net; tours from US$75)
On these highly recommended small-group tours, you'll visit markets, snack bars and other foodie hot spots to taste a wide range of fruits, juices and street food little known outside Brazil. The English-speaking guides are excellent and provide culinary as well as cultural insights. The cost of the tour covers all food, including snacks and a big meal at the end.
Sail in RioBoating
(map Google map; %21 99998-3709; www.sailinrio.com; Av Pasteur 333, Iate Clube do Rio de Janeiro; sailing tours from R$250)
This recommended outfit offers memorable sailing tours around Rio. When the weather cooperates, there are daily departures (at 11am) from the yacht club in Urca. You can also arrange your own private sailings with up to 10 people; you choose the date and help design the itinerary (private sailings cost R$1500 for three hours, R$1800 for five hours).
Maracanã Football Stadium
Rio's Maracanã stadium (map; %21 2334-1705; www.suderj.rj.gov.br/maracana.asp; Av Maracanã, São Cristóvão) is hallowed ground among football lovers. The massive arena has been the site of legendary victories and crushing defeats. Maracanã played a starring role both in the 2014 World Cup and the 2016 Summer Olympics. No matter who takes the field, the 78,800-seat arena comes to life in spectacular fashion on game days. Seeing a game is the best way to experience the excitement; you can also get a behind-the-scenes look during a stadium tour with Maracanã Tour ( %21 98341-1949; www.tourmaracana.com.br; tour guided/unguided R$60/50; h9am-4:30pm), available daily.
DAVIDE GRECO / 500PX ©
7Shopping
CRABArts & Crafts
(map Google map; %21 3380-1850; www.facebook.com/crabsebrae; Praça Tiradentes 69; h10am-5pm Tue-Sat) S
One of Rio's best new handicraft stores stocks an exquisite collection of pieces from every corner of Brazil. You'll find ceramics with pre-Columbian imagery found in the Serra da Capivara, lovely eco-jewelry made of seeds and nuts from indigenous groups and brightly painted folk art from the Northeast, as well as sustainably made animal-free bags, vegetable-fiber woven textiles and all-rubber footwear.
There's much to discover here, and all of the products are fair trade.
Pé de BoiArts & Crafts
( %21 2285-4395; www.pedeboi.com.br; Ipiranga 55, Laranjeiras; h10am-7pm Mon-Fri, to 1pm Sat)
Although everything is for sale here, Pé de Boi feels more like an art gallery than a handicrafts shop, owing to the high quality of the wood and ceramic works, and the tapestries, sculptures and weavings. This is perhaps Rio's best place to see one-of-a-kind pieces by artists from Bahia, Amazonia, Minas Gerais and other parts of Brazil.
Don't miss the upstairs area, which has photos of some of the artists whose work is on sale.
Gilson MartinsFashion & Accessories
(map Google map; %21 2227-6178; shop.gilsonmartins.com.br; Visconde de Pirajá 462; h9am-8pm Mon-Fri, to 7pm Sat) S
Designer Gilson Martins transforms the Brazilian flag and silhouettes of Pão de Açúcar (Sugarloaf Mountain) and Corcovado into eye-catching accessories in his flagship store in Ipanema. This is the place for unique glossy handbags, wallets, passport covers, key chains and iPad covers. Products are durable and use recycled and sustainable materials – and are not available outside Rio.
OsklenClothing
(map Google map; %21 2227-2930; www.osklen.com; Maria Quitéria 85; h9am-8pm Mon-Fri, 10am-7pm Sat, 11am-5pm Sun)
One of Brazil's best-known fashion labels outside the country, Osklen is notable for its stylish and well-made beachwear (particularly men's swim shorts and graphic T-shirts), sneakers and outerwear. The company was started in 1988 by outdoor enthusiast Oskar Metsavaht, the first Brazilian to scale Mont Blanc.
Lenny NiemeyerFashion & Accessories
(map Google map; %21 2227-5537; www.lennyniemeyer.com; Garcia d'Ávila 149; h10am-8pm Mon-Fri, to 6pm Sat)
When it comes to swimwear, few designers have earned a following like Lenny Niemeyer. She creates beautiful, form-fitting bikinis and one-pieces in both classic and avant-garde styles. You'll also find plenty of other essentials for street and beach, including dresses, tops, caftans and sarongs.
Maria OiticicaJewelry
(map Google map; %21 3875-8025; www.mariaoiticica.com; Av Afrânio de Melo Franco 290, Shopping Leblon; h10am-10pm Mon-Sat, 1-9pm Sun) S
Using native materials found in the Amazon, Maria Oiticica has created some lovely handcrafted jewelry inspired by indigenous art. Seeds, plant fibers and tree bark are just some of the ingredients of her bracelets, necklaces, earrings and sandals. There are even sandals and handbags made from fish 'leather.'
Chocolate QChocolate
(map Google map; %21 2274-1001; www.chocolateq.com; Shop B, Garcia d'Ávila 149; h10am-8pm Mon-Sat)
This is serious chocolate. Cacao beans, harvested from a single, environmentally conscious grower in Bahia, are processed into a range of exquisite dark-chocolate products (milk chocolate fans be advised, there is nothing for you here). All the bars and pastilles are beautifully packaged, with colorful illustrations of Brazilian wildlife, making them eminently gift-worthy.
La VeredaArts & Crafts
(map Google map; %21 2507-0317; www.lavereda.art.br; Almirante Alexandrino 428, Santa Teresa; h10am-8pm)
La Vereda stocks a colorful selection of handicrafts from local artists and artisans in a spot near Largo do Guimarães. Handpainted clay figurines by Pernambuco artists, heavy Minas ceramics, delicate sterling-silver jewelry and loosely woven tapestries cover the interior of the old store.
Favela HypeFashion & Accessories
(map Google map; www.facebook.com/favelahype; Paschoal Carlos Magno 103, Santa Teresa; h10am-11pm Tue-Sun)
This ultrahip but very friendly gathering spot is part fashion boutique, part restaurant and (at certain times) all party space. The art-filled shop sells swimwear, dresses, jewelry, sunglasses, ecofriendly flip-flops, floral-print button-downs (for the guys) and original artworks. Most products are made by local designers.
Ipanema, Leblon & Lagoa
1Sights
1Jardim BotânicoA1
2Lagoa Rodrigo de FreitasC1
2Activities, Courses & Tours
3Rio by BikeF3
7Shopping
4Chocolate QC3
5Gilson MartinsC3
6Hippie FairE4
Lenny Niemeyer(see 4)
7Maria OiticicaB3
8OsklenD3
5Eating
9Barraca do UruguaiD4
10Casa CamoleseA1
11Giuseppe GrillA3
12JarbôA1
13QuitériaD4
14VeroD4
15Zazá Bistrô TropicalD4
16ZukaA3
6Drinking & Nightlife
17Baretto-LondraE4
18CanastraE3
Nosso(see 8)
19Palaphita KitchE2
3Entertainment
20Bip BipF3
21Clube ManoucheB1
5Eating
### 5Ipanema & Leblon
VeroIce Cream $
(map Google map; %21 3497-8754; Visconde de Pirajá 229; ice creams R$11-16, pizza slices R$13-16; h10am-12:30am; v)
This artisanal Italian-run gelateria whips up Rio's best ice cream. You'll find more than two dozen rich and creamy temptations, including _gianduia_ (chocolate with hazelnut), _caramelo com flor de sal_ (caramel with sea salt), _figo com amêndoas_ (fig with almond) and classic flavors such as _morango_ (strawberry).
QuitériaBrazilian $$
(map Google map; %21 2267-4603; www.ipanemainn.com.br/en/quiteria; Maria Quitéria 27; mains around R$53, lunch buffet weekday/weekends R$39/53; h7am-11pm)
Inside the Ipanema Inn, this humble-looking place serves delectable contemporary Brazilian cooking. Seasonal, high-quality ingredients feature in the extensive lunch buffet, which draws mostly locals. By night the à la carte menu offers hits such as nicely turned out pork ribs with mashed baroa potatoes and cashew sauce, or grilled catch of the day with creamy rice and black-eyed beans.
Zazá Bistrô TropicalFusion $$$
(map Google map; %21 2247-9101; www.zazabistro.com.br; Joana Angélica 40; mains R$65-87; h7pm-midnight Mon-Thu, from noon Fri-Sun)
Inside an art-filled and whimsically decorated converted house, Zazá serves beautifully prepared dishes with Asian accents, and uses organic ingredients when possible. Favorites include chicken curry with jasmine rice, flambéed prawns with risotto, and grilled fish served with caramelized plantain. Don't miss brilliantly creative cocktails like the Caramba Carambola, with Amazonian gin (or vodka), _carambola_ (star fruit), ginger, mint and lemon.
ZukaInternational $$$
(map Google map; %21 3205-7154; www.zuka.com.br; Dias Ferreira 233; mains R$76-120; h7pm-1am Mon, noon-4pm & 7pm-1am Tue-Fri, 1pm-midnight Sat & Sun)
One of Rio's best restaurants, Zuka prepares mouthwatering cuisine. Try zingy ceviche with Brazilian _piquinho_ pepper or the confection-like delicacy of Zuka's original foie gras to start, followed by tender octopus over a roast potato crisp, honey-glazed duck breast with Moroccan couscous, grilled fish of the day with truffle sauce, or one of many other outstanding dishes.
Giuseppe GrillBrazilian $$$
(map Google map; %21 2249-3055; www.bestfork.com.br; Bartolomeu Mitre 370; mains R$88-150; hnoon-4pm & 7pm-midnight Mon-Thu, noon-1am Fri & Sat, noon-11pm Sun)
One of Rio's best restaurants, Giuseppe Grill aims for perfection and it rarely disappoints. Chargrilled meat and fish plates are outstanding, as are Brazilian classics like _bobó de camarão_ (manioc paste with dried shrimp and coconut milk). The setting is all class, with black-garbed waitstaff, dark-wood tables and artful touches (an open kitchen, a soaring bar that reaches to the ceiling and a Zen-like wall of water near the entrance).
Hippie Fair
The Zona Sul's most famous market, the Hippie Fair (map Google map; Praça General Osório; h9am-6pm Sun) has artwork, jewelry, handicrafts, clothing and souvenirs for sale. Stalls in the four corners of the plaza sell tasty plates of _acarajé_ (croquettes made from mashed black-eyed peas, with a sauce of _vatapá_ – manioc paste, coconut and _dendê_ oil – and shrimp), plus excellent desserts. Don't miss it.
Performer at Hippie Fair | JOHN MAIER JR / LONELY PLANET ©
### 5Copacabana & Leme
SaloméBistro $$
(map Google map; %21 2541-2416; www.facebook.com/salomebistro; Av Atlântica 994; mains R$48-65; h6:30pm-midnight Tue-Thu, noon-midnight Fri-Sun; v)
Run by the same people behind Ipanema's enormously popular Canastra, Salomé brings a dash of style to the Leme beachfront. The easygoing open-sided bistro has vintage mirrors and curious wall hangings, potted palms and atmospheric lighting – though on fine days, the outdoor tables are the best place to enjoy high-quality small plates, grilled meats and seafood, and wines by the glass.
Adega PérolaTapas $$
(map Google map; %21 2255-9425; www.facebook.com/adegaperolariodejaneiro; Siqueira Campos 138; sharing plates R$24-46; h11am-1am Mon-Thu, to 2am Fri & Sat)
Going strong since 1957, this atmospheric bottle-lined eating and drinking den serves outstanding Spanish and Portuguese small plates, including marinated octopus, whole sardines, stuffed olives, codfish balls, fried squid and marinated shrimp. Just step inside and check out the culinary bounty on the long front counter.
Churrascaria PalaceChurrascaria $$$
(map Google map; %21 2541-5898; www.churrascariapalace.com.br; Rodolfo Dantas 16; all you can eat R$140; hnoon-midnight; v)
This elegantly set dining room is one of the best _churrascarias_ (traditional barbecue restaurants) in town. Aside from juicy, high-quality cuts of meat, the included buffet nearly steals the show with fresh oysters, grilled heart of palm, sushi and sashimi, rice with wild mushrooms and goji berries, codfish casseroles, smoked fish, grilled prawns and octopus and amazing salads.
Snacking on the Seawall
Bar Urca (map Google map; %21 2295-8744; Cândido Gaffrée 205, Urca; h6:30am-11pm Mon-Sat, 8am-8pm Sun) is a much-loved neighborhood snack spot with a marvelous setting near Urca's bayside waterfront. At night, young and old crowd along the seaside wall to enjoy cold drinks, appetizers and fine views. For something more upscale, Bar Urca's elegant upstairs dining room (mains R$57-170; hnoon-3pm & 7-11pm Mon-Sat, noon-8pm Sun) serves outstanding seafood.
JOHN MAIER JR / LONELY PLANET ©
### 5Botafogo & Urca
MarchezinhoCafe $$
(map Google map; www.facebook.com/marchezinho; Voluntários da Pátria 46, Botafogo; sandwiches R$28-38; h9am-1am Mon-Wed, to 2am Thu & Fri, 10am-2am Sat) S
This multipurpose space operates as a cafe, bar and minimarket. The owner's French roots show through on the menu and store shelves, with the excellent cheese, bread and coffee. The carefully selected ingredients and produce are exclusively Brazilian, however, with most coming from small local producers. Come for creative tapas, delectable sandwiches, craft beers and first-rate cocktails.
Meza BarTapas $$
(map Google map; %21 3239-1951; www.mezabar.com.br; Capitão Salomão 69, Humaitá; tapas R$16-42; h6pm-1am)
Humaitá's see-and-be-seen hot spot serves delectable, Brazilian-slanted tapas to a sophisticated and trendy crowd. Creative cocktails and delightful staff round out the fun.
South FerroAsian, Pizza $$
(map Google map; %21 3986-4323; www.facebook.com/pg/SouthFerro; Arnaldo Quintela 23, Botafogo; mains R$41-62; h11:30am-3pm Mon, 11:30am-3pm & 6:30-11pm Tue-Sat; v)
Chef Sei Shiroma made his name with Ferro e Farinha, widely regarded as Rio's best pizza place. South Ferro, Shiroma's second restaurant, adds a healthy pinch of Asian spice in vegetable dumplings, tempura, fish curry and outstanding bowls of ramen. You'll also find a few pizza selections and well-made cocktails (R$32).
OtequeGastronomy $$$
(map Google map; %21 3486-5758; www.oteque.com; Conde de Irajá 581; 8-course tasting menu R$250; h7:30-11:30pm Tue-Sat)
Much to the delight of _cariocas,_ celebrated São Paulo chef Alberto Landgraf recently moved to Rio and opened this small, beautifully designed restaurant, complete with a theatrically lit open kitchen. The menu changes daily but features exquisite seafood (grilled octopus, scallops, monkfish, oysters) artfully prepared with Brazilian accents like hearts of palm and _tucupi_ (a sauce made from wild manioc).
### 5Flamengo
Sírio LibanezaMiddle Eastern $
(map Google map; %21 2146-4915; Largo do Machado 29, Loja 16-19; snacks R$6-28; h8am-11pm Mon-Sat; W)
Always packed, this bustling place serves tasty and cheap Syrian-Lebanese cuisine and great juices. Try the hearty _kibe de forno_ (an oven-baked ground-beef dish with spices), a hummus platter or _kafta_ (spiced meat patty), followed by baklava and other sweets. It's inside the Galleria Condor on Largo do Machado.
Tacacá do NorteAmazonian $
(map Google map; %21 2205-7545; Barão do Flamengo 35, Flamengo; tacacá R$27; h9am-11pm Mon-Sat)
In the Amazonian state of Pará, people order their _tacacá_ late in the afternoon from their favorite street vendor. In Rio, you don't have to wait until the sun is setting. The fragrant soup of manioc paste, lip-numbing leaves of _jambú_ (a Brazilian vegetable), and fresh and dried shrimp isn't for everyone. But then again, neither is the Amazon.
Ferro e FarinhaPizza $$
(Andrade Pertence 42; pizzas R$40-50; h7-11:30pm Tue-Sat, from 6:30pm Sun)
Sei Shiroma, an expat from NYC, and a dexterous team of dough handlers serve Rio's best pizza at this atmospheric and delightfully ramshackle spot in Catete. Seats are few, with just a handful of bar stools crowding around the pizza makers and oven at center stage, plus a few outdoor tables, so go early to try to beat the crowds.
BrouBrazilian $$
(map Google map; %21 2556-0618; www.facebook.com/casa.brou; Senador Vergueiro 2, Flamengo; mains lunch R$18-30, dinner R$32-56; h11:30am-3pm & 6pm-midnight)
A brilliant new addition to Flamengo, this bright and attractive eating and drinking spot features a daytime menu of creative salads topped with grilled veggies, salmon or roast meat, as well as quiches and soups. By night, crowds come for nicely prepared prime rib, grilled octopus and thin-crust pizzas fired up in a wood-burning oven.
Ipanema's Best Street Party
Run by a trio of Frenchmen, Canastra (map Google map; Jangadeiros 42; h6:30pm-1am Tue-Sat) has become one of the top nightlife destinations in Ipanema. Most nights, the crowd spills out onto the streets, and plenty of socializing _cariocas_ never even make it inside. This, however, would be a mistake, as the food (cheese and charcuterie plates, grilled squid and octopus) and drinks (Brazilian wines, sangria, caipirinhas) is outstanding. There's also a hidden bar downstairs – an artfully decorated speakeasy where you can escape the chattering crowds. The big night is Tuesday, when fresh oysters from Santa Catarina arrive (a fine deal at R$40 a dozen).
Caipirinha | MARKO POPLASEN / SHUTTERSTOCK ©
### 5Centro & Praça Mauá
XianAsian $$
(map Google map; %21 2303-7080; www.xianrio.com.br; top fl, Almirante Silvio de Noronha 365, Bossa Nova Mall; mains R$65-90; hnoon-11pm Mon-Thu, to midnight Fri & Sat, noon-10pm Sun; W)
One of Rio's loveliest new additions is this spacious, theatrically decorated dining room, open-air cocktail bar and nightclub, set rather dramatically on the edge of the bay. Come for mouthwatering seafood and tender meats with Asian accents, such as fresh oysters with ponzu sauce, salmon truffle tartar or marinated pork ribs with miso.
Cais do OrienteBrazilian $$
(map Google map; %21 2203-0178; www.caisdooriente-rj.com.br; Visconde de Itaboraí 8, Centro; mains R$55-75; h11am-10pm Mon-Sat, to 5pm Sun)
Brick walls lined with tapestries stretch high to the ceiling in this almost-cinematic 1870s mansion. Set on a brick-lined street, hidden from the masses, it blends elements of Brazilian and Mediterranean cooking in dishes such as duck breast with Brazil-nut _farofa_ (garnish of manioc flour sautéed with butter) and _açaí_ sauce.
Govardhana HaríVegetarian $$
(map Google map; %21 2544-2636; 2nd fl, Rodrigo Silva 6; lunch R$35; h11:30am-3:30pm Mon-Fri; v)
Amid artwork and decorations from India, the always-welcoming Hare Krishnas whip up delicious vegetarian dishes made with care. The restaurant is tucked down a narrow lane just off Rua São José and is always packed, so arrive early to get a seat.
### 5Santa Teresa & Lapa
Casa MomusMediterranean $$
(map Google map; %21 3852-8250; www.casamomus.com.br; Rua do Lavradio 11, Lapa; mains R$42-58; h11:30am-5pm Mon, to midnight Tue & Wed, to 2am Thu-Sat)
One of the best and loveliest restaurants in Lapa, Casa Momus has a small but well-executed menu of Mediterranean-influenced dishes. Start with prawn croquettes, fried polenta with spicy Gorgonzola or a Moroccan lamb _kafta_ (spiced meat patty) with tabbouleh and yogurt sauce, then feast on oxtail risotto with watercress, seared tuna with couscous, grilled beef tenderloin, and other rich main courses.
Bar do MineiroBrazilian $$
(map Google map; %21 2221-9227; Paschoal Carlos Magno 99, Santa Teresa; mains R$65-125, weekday lunch special R$22-35; hnoon-2am Tue-Sat, to midnight Sun)
Black-and-white photographs of legendary singers cover the walls of this old-school _boteco_ (small open-air bar) in the heart of Santa Teresa. Lively crowds have been filling this spot for years to enjoy traditional Minas Gerais dishes. The _feijoada_ (bean-and-meat stew served with rice) is tops and served every day, along with appetizers, including _pasteis_ (savory pastries) _._
Bar do Mineiro | CANNON PHOTOGRAPHY LLC / ALAMY STOCK PHOTO ©
Espírito SantaAmazonian $$$
(map Google map; %21 2507-4840; www.espiritosanta.com.br; Almirante Alexandrino 264, Santa Teresa; mains R$55-98; hnoon-11pm)
At this beautifully restored mansion, you can sit on the back terrace with its sweeping views or in the charming, airy dining room, and feast on rich, expertly prepared meat and seafood dishes from the Amazon and Northeast. Top picks include the _moqueca_ (fish stew), made with pintado (a delicious river fish), and the slow-roasted pork ribs served with sweet potato.
SobrenaturalSeafood $$$
(map Google map; %21 2224-1003; Almirante Alexandrino 432, Santa Teresa; mains for 2 R$80-155; hnoon-11pm Thu-Tue)
The old hardwood ceiling and exposed brick set the stage for feasting on _frutas do mar_ (seafood). Lines gather at weekends for crabmeat appetizers, fresh grilled fish and flavorful platters of _moqueca_ (fish stew). During the week, stop by for tasty lunchtime specials. There's live music most nights.
Samba Clubs in Lapa
Samba is the great soundtrack of Rio. It plays all across town, but if you're looking for its heart, you'll probably find it in the bohemian neighborhood of Lapa. In this neighborhood, addictive rhythms spill out of old-fashioned dance halls, drawing music lovers from far and wide.
Rio Scenarium (map Google map; %21 3147-9000; www.rioscenarium.art.br; Rua do Lavradio 20, Lapa; R$35-60; h7pm-4am Tue-Sat) One of the city's most photogenic nightspots, Rio Scenarium has three floors, each lavishly decorated with antiques. Balconies overlook the stage on the 1st floor, where dancers keep time to the jazz-infused samba filling the air.
Carioca da Gema (map Google map; %21 2221-0043; www.barcariocadagema.com.br; Av Mem de Sá 79, Lapa; R$20-40; h7:30pm-1:30am Sun-Tue & Thu, to 3am Fri, from 9pm Sat) One of Lapa's pioneers, this small, warmly lit club still attracts some of the city's best samba bands, and you'll find a festive mixed crowd filling the dance floor most nights.
Sacrilégio (map Google map; %21 3970-1461; www.sacrilegio.com.br; Av Mem de Sá 81, Lapa; R$20-40; h7pm-2am Tue-Thu, to 6am Fri, 8pm-6am Sat) A short stroll past the Arcos da Lapa you'll find a key spot for live samba in an intimate setting. The garden makes a fine setting for a few cold chopes while the music filters through the windows.
Rio Scenarium | JOHN MAIER JR/LONELY PLANET / GETTY IMAGES ©
6Drinking & Nightlife
Palaphita KitchLounge
(map Google map; %21 2227-0837; www.palaphitakitch.com.br; Av Epitácio Pessoa s/n; h6pm-midnight Mon-Fri, from 10am Sat & Sun)
A great spot for a sundowner, Palaphita Kitch is an open-air, thatched-roof wonderland with rustic bamboo furniture, flickering tiki torches and a peaceful setting on the edge of Lagoa Rodrigo de Freitas. It's a popular spot with couples, who come for the view and the creative (but pricey) cocktails: the caipirinhas, made from unusual fruits from the Northeast and Amazonia, are a hit.
NossoCocktail Bar
(map Google map; %21 99619-0099; www.facebook.com/nossoipanema; Maria Quitéria 91, Ipanema; h7pm-1am Tue-Thu, to 2am Fri & Sat, 6-11pm Sun)
Located in the heart of Ipanema, this supremely stylish cocktail bar and restaurant, complete with swanky rooftop terrace, feels like it has been plucked directly from downtown Manhattan. The meticulously crafted cocktails are made with high-quality spirits and often arrive adorned with foams and dehydrated fruit slices. The menu offers equally intricate yet delicious dishes.
Pavão AzulBar
(map Google map; %21 2236-2381; Hilário de Gouveia 71, Copacabana; hnoon-midnight Mon-Sat, to 8pm Sun)
A Copacabana classic, Pavão Azul is a simple open-sided bar where huge crowds gather on the sidewalk out front to drink ice-cold _chope_ (draft beer) and chat late into the night. It's been so successful in fact that the owners have opened a similar _boteco,_ Pavãzinho, across the street. Don't miss the fantastically good, inexpensive _pataniscas_ (codfish fritters).
CoLABBar
(map Google map; %21 3592-0470; www.colab-rio.com; Fernandes Guimarães 66; h10am-1am Tue-Sat)
CoLAB is one of Botafogo's increasing number of lively bar/restaurants attracting a predominantly young crowd with a combination of international cuisine, cocktails and craft beers. The menu features curries (both Indian and Thai) as well as samosas, falafel sandwiches, burgers and other temptations. Calm by day, CoLAB picks up in the evenings – with DJs and the occasional band keeping things lively.
ComunaBar
(map Google map; %21 3579-6175; www.facebook.com/comunacc; Sorocaba 585, Botafogo; hnoon-1am Tue-Fri, 6:30pm-1:30am Sat, 6:30pm-midnight Sun)
This creative space is equal parts bar, art gallery and independent bookseller (and indie publishing house). There's always something afoot in the delightfully off-the-beaten-path locale, with workshops, music sessions, readings, exhibitions and fashion shows. It's also just a great spot for a local microbrew and a bite (try one of the award-winning burgers).
Explorer BarCocktail Bar
(map Google map; %21 3264-9665; www.explorerbar.com; Almirante Alexandrino 399, Santa Teresa; h5pm-midnight Tue-Fri, from 2pm Sat & Sun)
A gorgeous multilevel garden with fairy lights is the centerpiece of this charming cocktail bar. Beautifully conceived cocktails (around R$30) showcase exotic flavors in elixirs like TLV, with orange, tamarind, ginger foam and grated nutmeg. The first-rate food menu features an eclectic mix of Sicilian seafood pasta, _sabich_ (a roasted eggplant and hummus sandwich) and other globally inspired dishes (mains R$32 to R$56).
Baretto-LondraLounge
(map Google map; %21 3202-4000; www.fasano.com.br/gastronomia/baretto-londra; Av Vieira Souto 80, Hotel Fasano; h8pm-2am Thu-Sat)
You'll find one of Rio's most glam bars here inside the Hotel Fasano, offering a vision of decadence matched by few of the city's night spots. The intimate space, designed by Philippe Starck, has an enchantingly illuminated bar, leather armchairs and divans, and a DJ spinning world electronica.
Baretto-Londra | JOHN MAIER JR/LONELY PLANET / GETTY IMAGES ©
3Entertainment
Bip BipLive Music
(map Google map; %21 2267-9696; www.facebook.com/barbipbip; Almirante Gonçalves 50, Copacabana; h7pm-midnight Sun-Fri)
For years Bip Bip has been one of the city's favorite spots to catch a live _roda de samba_ (informal samba played around a table), despite it being just a storefront with a few battered tables. As the evening progresses the tree-lined neighborhood becomes the backdrop to serious jam sessions, with music and revelers spilling into the street.
Pedra do SalSamba
(map Google map; Largo João da Baiana, Gamboa; h8pm-midnight Mon & Fri)
The Monday- and Friday-night street parties here are major draws for lovers of samba, whether they're Brazilian or foreign, rich or poor. The lively _samba da mesa_ features a handful of changing players who belt out well-known songs to swaying, joyful crowds surrounding the tiny plaza.
TribOzJazz
(map Google map; %21 2210-0366; www.triboz-rio.com; Conde de Lages 19, Lapa; R$15-30; h6-8pm & 9pm-1am Thu-Sat)
Not for lazy ears, this avant-garde jazz house known to the sonic cognoscenti is unique in Rio for its serious approach to performances. Run by an Australian ethnomusicologist, it sits in a shadier part of old Lapa in a signless mansion, which transforms into a beautiful showcase space.
Clube ManoucheLive Music
(map Google map; www.casacamolese.com.br; Jardim Botânico 983; R$60-90)
Downstairs from the main restaurant in Casa Camolese (map Google map; %21 3514-8200; mains R$42-70; hnoon-midnight Mon-Sat, to 11pm Sun; v), Clube Manouche is an intimate music bar designed in the tradition of West Village clubs in New York City. Sight lines are excellent no matter where you sit, and you can order drinks throughout the show. The club lines up some top-notch jazz artists, as well as eclectic sounds from Brazil and beyond.
Lapa 40 GrausLive Music
(map Google map; %21 3970-1338; www.lapa40graus.com.br; Rua do Riachuelo 97, Lapa; R$20-50; h8pm-4am Thu-Sat)
This impressive multistory music venue and pool hall has tables for lounging on the 1st floor, more than a dozen pool tables on the 2nd floor, and a small stage and dancing couples on the top floor. There are usually two shows nightly. Pop, rock, samba and _choro_ (romantic, improvised, samba-related music) typically kick off around 9pm.
8INFORMATION
Riotur's multilingual website (www.visit.rio) is a good source of information. Its offices distribute maps and guides about major events; the main branch is in **Copacabana** (map; %21 2541-7522; Av Princesa Isabel 183; h9am-6pm Mon-Fri).
8GETTING THERE & AWAY
Rio's main airport is Galeão international airport, located 15km north of the city center on Ilha do Governador.
**Aeroporto Santos Dumont** (map; %21 3814-7222; www4.infraero.gov.br; Praça Senador Salgado Filho), used by some domestic flights, is by the bay, in the city center, 1km east of Cinelândia metro station.
8GETTING AROUND
**Metro** The most convenient way to get around. Trains run from 5am to midnight Monday through Saturday, and from 7am to 11pm on Sunday and holidays. Single rides cost R$4.30.
**Bus** Buses are frequent and cheap; destinations are listed above the windscreen. Fares are around R$3.50.
**Taxi** Useful at night. Fares start at R$5.50, plus R$2.50 per kilometer. Rates rise to R$3 per kilometer at night and on Sunday.
**Bike** Rio's bike-share scheme has numerous stations around town. You'll need a local cell phone (mobile) number to release the bikes at each station.
**Ride-sharing apps** Uber, but not Lyft, is widely available in Rio de Janeiro.
# iWhere To Stay
_Rio has a wide range of lodging, including B &Bs, hostels and guesthouses, plus scores of luxury hotels, particularly in Copacabana. Abundant options keeps rates from soaring, except during Carnaval and other major events. Book well ahead during high season._
Neighbourhood | Atmosphere
---|---
**Ipanema & Leblon** | Great location near the beach and the lake, with Rio's best restaurants and bars nearby. Pricier than other locations, and fewer options than Copacabana.
**Copacabana & Leme** | Numerous hotels packed into a long, narrow high-rise-lined neighborhood facing the beach. Good restaurants, shops and bars, plus decent transport links to other parts of Rio.
**Botafogo & Urca** | Rio's hippest neighborhood, Botafogo has creative restaurants and drinking spots that attract a local crowd. It's also a short metro ride to the beaches. On the downside, it has noisy, traffic-clogged avenues.
**Flamengo & Around** | Better prices than the beachside districts, but very few restaurants and limited nightlife.
**Santa Teresa & Lapa** | Santa Teresa is a charming district with bohemian shops and eateries, but is far from the coastline. Lapa is the epicenter of Rio's samba-fueled nightlife, but gritty by day.
**Barra da Tijuca & Western Rio** | Great beaches with a laid-back vibe, though it's very spread out and lacks the energy of other parts of Rio.
# Salvador, Brazil
#### Afro-Brazilian Rhythms
#### Pelourinho
#### Sights
#### Courses
#### Tours
#### Shopping
#### Eating
#### Drinking & Nightlife
#
Salvador, Brazil
_Salvador da Bahia has an energy and unadorned beauty that few cities can match. Once the magnificent capital of Portugal's New World colony, today Salvador is the pulsating heart of the country's Afro-Brazilian community. Its brilliantly hued center is a living museum of 17th- and 18th-century architecture and gold-laden churches. Wild festivals happen frequently, with drum corps pounding out rhythms against the backdrop of colonial buildings almost daily. Aside from the many attractions within Salvador, a gorgeous coastline lies right outside the city – a suitable introduction to the tropical splendor of Bahia._
Colorful buildings, Pelourinho | UEBERKUNST / GETTY IMAGES ©
Two Days in Salvador
Spend two full days exploring the Pelourinho. Visit baroque churches like Igreja e Convento São Francisco and learn about Salvador's Carnaval traditions at the Casa do Carnaval da Bahia. Take a cooking class and indulge in Bahian delicacies at Restaurante do Senac and catch a performance at the famed Balé Folclórico da Bahia.
Four Days in Salvador
On your third day, go souvenir shopping and snacking at the Mercado Modelo, browse modern art and catch a concert at the Solar do Unhão.
On the fourth day visit the Igreja NS do Bonfim, then head down to Barra for fun on the beach, a lighthouse visit and seafood at Du Chef Arte e Gastronomia.
Arriving in Salvador
The airport is located about 30km east of the center. For a taxi going to Barra or the Pelourinho (40 to 60 minutes), you'll pay around R$140 in advance at the airport via the official Taxi COMTAS and Taxi COOMETAS stands.
An executive bus called First Class (www.firstclassbus.com.br) travels between the airport and key tourist points (R$25 per person).
Where to Stay
Staying in the Pelourinho means being close to the action, but the beach suburbs are mellower (and just a short bus or taxi ride away). Santo Antônio is a peaceful neighborhood with classy _pousadas_ (guesthouses) in renovated old buildings just a short walk from the Pelourinho. Reservations during Carnaval are essential.
TOP EXPERIENCE
# Afro-Brazilian Rhythms
Salvador is the pulsing center of an incredible music and dance scene, which blends African and Brazilian traditions in a wide range of musical styles including live percussion and samba.
Great For...
hrd
yDon't Miss
Live outdoor concerts at Largo de Tereza Batista, Largo do Pedro Arcanjo and Praça Quincas Berro d'Água.
8Need to Know
During the high season, there are almost nightly concerts in the inner courtyards of the Pelourinho.
5Take a Break
A good place for creative Bahian and international fare is at Bar Zulu (map Google map; Rua das Laranjeiras 15; mains R$22-56; h11am-10pm; W v) in the Pelourinho.
oTop Tip
You can frequently hear drum corps, which rehearse by walking through the Pelourinho and gathering a following as they go.
### Balé Folclórico da Bahia
The most astounding professional show is put on by this world-renowned folkloric ballet company ( %71 3322-1962; www.balefolcloricodabahia.com.br; Gregório de Mattos 49, Teatro Miguel Santana; R$60; hshows 8pm). Incredibly talented performers showcase dances of the _orixás_ (the gods in the Afro-Brazilian religion of Candomblé), _capoeira_ (a dance-like martial art) and _maculêlê_ (stick dance) among many other highlights.
Balé Folclórico da Bahia | KIKE CALVO / ALAMY STOCK PHOTO ©
### Espaço D'Venetta
This buzzed-about cultural center, bar and art space (www.dvenetta.com.br; Rua dos Abodes 12, Santo Antônio; cover varies; h6pm-midnight Wed-Fri, noon-midnight Sat, noon-6pm Sun) in Santo Antônio features live samba, samba de roda and jazz. Some events are free.
### Teatro Castro Alves
For the biggest acts, keep your eye on Salvador's finest venue, the Teatro Castro Alves (www.tca.ba.gov.br; Praça 2 de Julho, Campo Grande). Its Concha Acústica (amphitheater) has weekly concerts throughout summer.
### Jam no MAM
Saturday-evening jazz and bossa nova at MAM (www.jamnomam.com.br; Av Contorno s/n, Museu de Arte Moderna; adult/child R$8/4; h6-9pm Sat) is a must for music lovers. Go early to see the museum first and catch the views at sunset. Though the venue is located within walking distance of the Pelourinho, muggings are common along this quiet stretch; taking a taxi is recommended.
### Topázio
The popular show that used to be staged at the Solar do Unhão, featuring 18 dancers, musicians and _capoeiristas_ , now happens here after dinner at O Coliseu (map Google map; %71 3321-5585; www.ocoliseu.com.br; 2nd fl, Cruzeiro de São Francisco 9; lunch buffet per person R$65, dinner & cultural show R$200; h11:30am-4pm & 7-10pm Mon-Sat; v). Dinner starts at 7pm and the show begins around 8:30pm. Reserve ahead.
### Olodum
Founded in 1979, Olodum is famous for its _afro bloco_ , which takes inspiration from the musical styles of reggae, salsa and Brazilian samba. Catch them playing around town, or take a peek into the group's headquarters, the Escola Olodum (map Google map; %71 3321-4154; www.olodum.com.br; Gregório de Mattos 22; tours by donation, workshops free; h9am-6pm Mon-Fri, to 1pm Sat).
TOP EXPERIENCE
# Pelourinho
Packed with colorful buildings and magnificent churches, the cobblestone-lined Pelourinho is not just for tourists. Cultural centers and schools of music, dance and capoeira pack these colonial buildings.
Great For...
hvr
yDon't Miss
The lavish interior of the Igreja e Convento São Francisco, one of Brazil's most magnificent churches.
8Need to Know
To see what's on while you're in town, log onto Festa da Semana (www.festadasemana.com.br/salvador).
5Take a Break
A Cubana (map Google map; www.acubana.com.br; Portas do Carmo 12; cones R$7-13; h9:30am-10:30pm) is one of Salvador's oldest and best ice-cream shops.
oTop Tip
The Pelourinho shifts quickly into sketchy areas, so avoid wandering off the beaten path.
### Largo do Pelourinho
Picture-perfect Largo do Pelourinho map Google map is a sloping, triangle-shaped square, once the site of the _pelourinho_ (whipping post) – one of several nearby locations where slaves were exposed and punished. After slavery was outlawed in 1835, the neighborhood fell into disrepair; in the 1990s major restoration efforts were initiated to preserve the cobblestone square's colonial houses and churches. Today, the square is the heart of Salvador's historic center.
GONZALO AZUMENDI / GETTY IMAGES ©
### Igreja e Convento São Francisco
The baroque Igreja e Convento São Francisco (map Google map; Cruzeiro de São Francisco; R$5; h10am-5pm) is filled with displays of wealth and splendor. An 80kg silver chandelier dangles over ornate wood carvings covered in gold leaf, and the convent courtyard is paneled with hand-painted _azulejos_ (Portuguese tiles). The complex was finished in 1723.
Interior detail of Igreja e Convento São Francisco | LEOKLEEMANN / SHUTTERSTOCK ©
### Museu Afro-Brasileiro
Holding one of Bahia's most important collections, the Museu Afro-Brasileiro (map Google map; %71 3283-5540; www.facebook.com/museuafro2; Terreiro de Jesus; adult/child R$6/3; h9am-5pm Mon-Fri, 10am-5pm Sat) exhibits wood carvings, baskets, pottery and other artwork and crafts linking Brazilian and African artistic traditions. Don't miss the breathtaking carved wooden panels depicting _orixás_ (spirits common in Afro-Brazilian spirituality) by Argentine-born Carybé.
Museu Afro-Brasileiro | DIDI / ALAMY STOCK PHOTO ©
### Igreja NS do Rosário dos Pretos
The king of Portugal gave the Irmanidade dos Homens Pretos (Brotherhood of Black Men) the land for the periwinkle-blue Igreja NS do Rosário dos Pretos (map Google map; Largo do Pelourinho; admission by donation; h8am-5pm Mon-Fri, to 7pm Sat) in 1704. Building in their free time, it took these slaves and freed slaves almost 100 years to complete it.
### Catedral Basílica
The Catedral Basílica (map Google map; Terreiro de Jesus; admission by donation; h9am-5pm Mon-Sat, from 1pm Sun) dates from 1672 and is a marvelous example of Jesuit architecture. The interior is elegant and simple, with marble-covered walls and towering pillars. The sacristy has a beautiful carved jacaranda archway and a painted dome and floor.
### Fundação Casa de Jorge Amado
Literary types shouldn't miss a quick visit to the **Fundação Casa de Jorge Amado** ( %71 3321-0070; www.jorgeamado.org.br; Largo do Pelourinho 51; R$5, Wed free; h10am-6pm
Mon-Fri, to 4pm Sat), offering an overview of the life of one of Brazil's best-known writers. The museum cafe is a good place to grab a coffee and pastry.
Salvador
1Sights
1Casa do Carnaval da BahiaB4
2Catedral BasílicaC4
3Elevador LacerdaB5
4Escola OlodumD4
5Forte de Santo Antônio da BarraA2
6Igreja e Convento São FranciscoD4
7Igreja NS do Rosário dos PretosC2
8Largo do PelourinhoC3
9Mercado ModeloA6
10Museu Afro-BrasileiroC4
Museu de Arqueologia e Etnologia(see 10)
Museu Náutico da Bahia(see 5)
11Plano Inclinado GonçalvesB4
12Terreiro de JesusC4
2Activities, Courses & Tours
13Associação Artística e Cultural DiasporaD3
14Associação Brasileira de Capoeira AngolaC3
15Salvador BusA2
16SENACC2
7Shopping
Mercado Modelo(see 9)
5Eating
17A CubanaC3
18Bar ZuluD4
19Dona Chika-kaC3
20Du Chef Arte e GastronomiaA2
21Maria Mata MouroD4
22O ColiseuC4
Pelô Bistrô(see 24)
23Restaurante do SenacC2
6Drinking & Nightlife
24Casa do Amarelindo BarC3
25PereiraA1
3Entertainment
26Balé Folclórico da BahiaC3
1Sights
Casa do Carnaval da BahiaMuseum
(map Google map; %71 3324-6760; Praça Ramos de Queirós; R$30; h11am-7pm Tue-Sun)
The story of Salvador's world-famous Carnaval is told through wonderfully evocative archival video and photographs at this museum which opened in 2018. Even just walking through the museum with its explosions of color and displays of musical instruments and folk-art statues is worth the trip. Downstairs you'll find an excellent gift shop.
Museu de Arqueologia e EtnologiaMuseum
(map Google map; Archaeology & Ethnology Museum; %71 3283-5533; www.mae.ufba.br; Faculdade de Medicina, Terreiro de Jesus; adult/child R$6/3; h9am-5pm Mon-Fri)
Below the Museu Afro-Brasileiro (one admission ticket gets you into both), the Museu de Arqueologia e Etnologia exhibits indigenous Brazilian pottery, bows and arrows, masks and feather headpieces. Also tucked between the building's arching stone foundations is 19th-century glass and porcelain found during the excavations for the metro.
Mercado ModeloMarket
(map Google map; www.mercadomodelobahia.com.br; Praça Visconde de Cayru; h9am-6pm Mon-Sat, to 2pm Sun)
The original 1861 Customs House, where slaves were housed when they arrived in Salvador, was partly destroyed in a fire in 1986. After reconstruction, it was transformed into the Mercado Modelo. The market is a great spot to pick up tourist trinkets, and a couple of cafes out back provide seating with great views of the bay.
Terreiro de JesusSquare
(map Google map; Praça 15 de Novembro)
A colorful intersection of vendors, tourists, capoeiristas and colorful locals, the Terreiro de Jesus is a historic site of religious celebrations, and is ringed by four churches, as well as the 19th-century **Faculdade de Medicina Building**. The plaza feeds into the **Cruzeiro de São Francisco** , named for the cross in the square's center.
Solar do UnhãoHistoric Building
(map; %71 3117-6139; Av Contorno s/n; h1-6pm Tue-Sun) F
This well-preserved, 18th-century complex served as a transfer point for sugar shipments: legend says it's haunted by the ghosts of murdered slaves. Today, the building houses the Museu de Arte Moderna (MAM; www.jamnomam.com.br/mam; h1-6pm Tue-Sun) F, with a changing display of avant-garde exhibits, a hillside **sculpture garden** , and popular Saturday-evening jazz and bossa nova concerts (JAM no MAM with stunning views over the bay. Take a taxi – the area is known for tourist muggings.
Forte de Santo Antônio da BarraHistoric Building
(map Google map; Largo do Farol da Barra s/n)
Built in 1698, Bahia's oldest fort is more commonly called the Farol da Barra for the lighthouse (South America's oldest) within its walls. In addition to having superb views, the fort houses an excellent nautical museum (Nautical Museum of Bahia; %71 3264-3296; www.museunauticodabahia.org.br; adult/student & senior R$15/7.50; h9am-6pm Tue-Sun, daily Jan & Jul), with relics and displays from the days of Portuguese seafaring, plus exhibits on the slave trade.
Beach-going in Barra
**Praia Porto da Barra** , on a horseshoe-shaped stretch of coast, is a small, picturesque beach, that's a great place to be at sunset. The bay's waters are clear and calm, and the people-watching is fantastic. You won't lack for sustenance, with vendors selling everything imaginable.
To the left of the lighthouse, **Praia do Farol da Barra** has a beach break popular with surfers at high tide, and tidal pools popular with children and families at low tide.
Praia Porto da Barra | RUY BARBOSA PINTO / GETTY IMAGES ©
CCourses
Classes in _capoeira_ , African dance and percussion are easily arranged through guesthouses in the Pelourinho. Generally you can expect to pay anywhere from R$55 to R$120 per hour for a class.
SENACCooking
(map Google map; %71 3186-4000; www.ba.senac.br; Largo do Pelourinho 13-19; from R$180; h9-11am & noon-4:30 Mon-Fri)
A must-see for foodies, SENAC is a Bahian culinary school where both locals and tourists take cooking classes – one course focuses solely on the preparation of popular street foods like _acarajé_ (Bahian fritters made of brown beans and dried shrimp fried in palm oil).
Associação Brasileira de Capoeira AngolaCapoeira
(map Google map; %71 8824-7869; www.facebook.com/abcangola; Maciel de Baixo 38; prices vary; hclasses 6pm Mon & Wed; c)
This friendly _capoeira_ association (known as ABCA) runs classes open to all ages, including children five and older. Be sure to wear comfortable, loose-fitting pants you can move around in.
Associação Artística e Cultural DiásporaDancing
(map Google map; Diáspora Art Center; %71 3323-0016; Rua das Laranjeiras 44; prices vary; h10am-6pm Mon-Fri)
Stop in to inquire about the latest schedule of classes in traditional and contemporary Afro-Brazilian dance, _capoeira_ and percussion.
TTours
Salvador BusTours
(map Google map; %71 3356-6425; www.salvadorbus.com.br; Av Sete de Setembro s/n; adult/child R$65/50; h9:30am-6:45pm Mon-Sat)
If you're short on time, climb aboard the open-air Salvador Bus at Farol da Barra, the Mercado Modelo or a number of other downtown destinations; the hop-on, hop-off sightseeing bus offers multilingual tours.
7Shopping
Mercado ModeloArts & Crafts
(map Google map; Praça Cayru, Cidade Baixa; h9am-6pm Mon-Sat, to 2pm Sun)
In Salvador's unpleasant past the Mercado Modelo was the site where slaves coming into the city were detained. Now the building houses dozens of tourist-oriented stalls selling local handicrafts, as well as food stalls and restaurants.
5Eating
Restaurante do SenacBuffet $$
(map Google map; %71 3324-8101; www.ba.senac.br; Largo do Pelourinho 13; buffet per kg R$50, típico per kg R$51.90; hbuffet a quilo 11:30am-3:30pm Mon-Fri, buffet típico 11:30am-3:30pm Mon-Sat; v)
The best Bahian buffet in town. The cooking school SENAC spreads a tempting array of regional dishes, including several varieties of seafood, _moqueca_ (fish stew) _,_ and traditional desserts. The impressive _buffet tipico_ (traditional Bahian food) is on the top floor, not to be confused with the more general street-level _buffet a quilo_ , which is also good for a quick lunch.
_Moqueca_ (fish stew) | ROCHARIBEIRO / SHUTTERSTOCK ©
Du Chef Arte e GastronomiaSeafood $$
(map Google map; %71 3042-4433; www.facebook.com/duchefartegastronomia; Afonso Celso 70, Barra; mains R$45-90; hnoon-11pm Tue-Thu, to 1am Fri & Sat, to 5pm Sun; v)
Fresh seafood and upscale plating make the dishes at this swish Barra restaurant pop. You're likely to make the acquaintance of the chef, Lucius Gaudenzi, while you dine on tender shrimp or one of the vegan dishes on the menu.
Dona Chika-kaAfro-Brazilian $$
(map Google map; %71 3321-1712; Rua João Castro Rabelo 10; mains R$30-80; h6-10pm Mon-Sat)
Flickering candles lead the way up to a charmingly decorated 2nd-floor dining room, where you can feast on Dona Chika-ka's famous _bobó de camarão_ (shrimp chowder) and other Bahian delights. During quieter months you may find reduced business hours and only the small downstairs dining area open.
Paraíso TropicalBahian $$$
( %71 3384-7464; www.restauranteparaisotropical.com.br; Edgar Loureiro 98B, Cabula; mains R$53-98; hnoon-10pm Tue-Sat, to 5pm Sun)
Far off the beaten path in the residential neighborhood of Cabula, but foodies don't mind the detour to Paraíso Tropical: the classic Brazilian restaurant has long been considered one of Salvador's top choices for beautifully prepared Bahian cuisine with a gourmet twist. Come for a leisurely lunch; expect long waits on weekends.
Maria Mata MouroBrazilian $$$
(map Google map; %71 3321-3929; www.mariamatamouro.com.br; Rua da Ordem Terceira 8; mains R$40-80; hnoon-11pm)
The picture-perfect garden patio and elegant dining room are fine settings to enjoy one of Pelô's top menus. You'll find Bahian classics, fresh seafood and Portuguese dishes prepared with a gourmet twist, plus a varied wine list and a talented bartender shaking up unique cocktails.
Igreja NS do Bonfim
This famous 18th-century church ( %71 3316-2196; www.santuariosenhordobonfim.com; Praça Senhor do Bonfim; h6:30am-6.30pm Mon-Thu & Sat, from 5:30am Fri & Sun), located a few kilometers north of Comércio on the Itapagipe Peninsula, is the source of the _fitas_ (colored ribbons) you see everywhere in Salvador, a souvenir of the church and a symbol of Bahia itself. Bonfim's fame derives from its power to effect miraculous cures, making it a popular shrine.
In the Sala dos Milagres (Room of Miracles) devotees leave photos, letters and ex-votos – wax replicas of body parts representing those that were cured or need curing.
Due to Candomblistas' syncretization of Jesus Christ (Nosso Senhor do Bonfim) with Oxalá, their highest deity, Bonfim is their most important church. Huge services are held here on Friday, Oxalá's favorite day of the week.
If you tie a _fita_ around your wrist, you are making a commitment that lasts for months. With each of the three knots a wish is made, which will come true by the time the _fita_ falls off. Cutting it off is inviting doom.
Check their website for a full schedule of various services and events held daily.
Y.LEVY / ALAMY STOCK PHOTO ©
6Drinking & Nightlife
Casa do Amarelindo BarCocktail Bar
(map Google map; %71 3266-8550; www.casadoamarelindo.com; Rua das Portas do Carmo 6, Hotel Casa do Amarelindo; hnoon-late)
The chic tropical-style bar at the lovely Pelô Bistrô map Google map at Casa do Amarelindo is the ideal spot for a nightcap; better still is the panoramic terrace where a skilled bartender shows up after dark to mix classic cocktails.
PereiraBar
(map Google map; %71 3264-6464; www.pereirarestaurante.com.br; Sete de Setembro 3959, Barra; hnoon-4pm & 5pm-midnight Tue-Fri & Sun, noon-4pm & 5:30pm-1am Sat)
Up a staircase from the seaside road that curves around the tip of Barra, Pereira is a stylish restaurant and wine bar. Excellent _chope_ (draft beer) is on tap and the sunset views over the ocean are beautiful.
Beco dos ArtistasBar
(Artist's Alley; Av Cerqueira Lima 4, Garcia; hhours vary)
Art mavens and a young, queer-friendly crowd flock to Beco dos Artistas, a small alley with several bars and a nightclub. Take a taxi and enter from Rua Leovigildo Filgueira.
8INFORMATION
**Bahiatursa** The tourism authority is friendly but not very organized. The **Pelourinho office** ( %71 3321-2463; www.bahiatursa.ba.gov.br; Rua Francisco Muniz Barreto 12, Pelourinho; h8:30am-9pm Mon-Thu, to 10pm Fri-Sun), which has maps and listings of what's happening around town, is your best bet.
8GETTING THERE & AWAY
**Aeroporto Deputado Luis Eduardo do Magalhães** (Praça Gago Coutinho s/n, São Cristóvão; %71 3204-1010) is served by domestic airlines like **GOL** ( %0300 115-2121; www.voegol.com.br; airport), **LATAM** ( %1-866-435-9526; www.latam.com), and **Azul** ( %1-888-587-2985; www.voeazul.com.br). There are several direct flights to/from foreign cities such as Miami or Lisbon via **TAP** (Air Portugal; %1-800-221-7370; www.flytap.com), and Buenos Aires via **Aerolíneas Argentinas** ( %1-800-333-0276; www.aerolineas.com.ar).
Salvador's **bus station** (Bus Station; %71 3616-8300; www.rodoviariadesalvador.com.br) is 8km east of the city center.
8GETTING AROUND
Linking Cidade Alta (the Pelourinho) and Cidade Baixa (Comércio and the ferry terminals) are the **Elevador Lacerda** ( %71 3322-7049; fare R$0.15; h6am-11pm) and the **Plano Inclinado Gonçalves** (Praça da Sé & Guindaste dos Padres, Comércio; R$0.15; h7am-7pm Mon-Fri, to 1pm Sat).
Public buses crisscross the city; particularly useful to tourists are those that run between Barra and Praça da Sé (R$3.70).The destinations are clearly labeled on the front of the bus.
Taxis can be taken at meter price (legal) or negotiated, but you might not get to choose which. Uber is also available.
# Brazilian Amazon
#### Riverboat Travel
#### Jungle Expeditions & Wildlife-Watching
#### Belém
#### Santarém
#### Alter do Chão
#### Manaus
#
Brazilian Amazon
_What traveler drawn to the wild places of the planet hasn't imagined a trip to the Amazon, not only to admire the towering trees, secretive wildlife and awesome river, but to enter, in a real sense, the very life spring of the planet?_
_In fact, the Amazon's quintessential experiences are more sublime than superlative: canoeing through a flooded forest, dozing in a hammock on a boat chugging upriver, waking to the otherworldly cry of howler monkeys. On a river whose size is legendary, it's actually the little things that make it special._
Rio Amazonas | MANTAPHOTO / GETTY IMAGES ©
Two Days in the Brazilian Amazon
With limited time, fly into Manaus and head upriver for a couple of nights along the Rio Urubu, the closest you'll get to deep Amazon immersion in this short time frame. There you'll get a chance to see plentiful wildlife on jungle hikes and canoe trips, and to visit local communities.
Four Days in the Brazilian Amazon
Spend day three in Manaus. Browse Amazonian handicrafts at Galeria Amazônica, tour the jaw-dropping Teatro Amazonas and stroll the lush paths of the Jardim Botânico Adolpho Ducke.
On the fourth day, make a half-day trip into the forest with Amazon Tree Climbing, where you can also see the unusual Encontro das Águas.
Arriving in the Brazilian Amazon
The main airport of Manaus, Aeroporto Internacional Eduardo Gomes, is located 13km north of the city center, and it has a few international flights, as well as excellent domestic connections.
Belém's Aeroporto Internacional Val-de-Cans, located 8km north of the center, is a hub for international, domestic and regional flights.
Where to Stay
The Amazon has everything from five-star city hotels to basic lodges in the remote reaches of the network of rivers. Away from the cities, many places offer packages that include food, accommodations and all activities, as well as transfers from the nearest town. There are many jungle lodges, including the celebrated Pousada Uacari inside the Mamirauá Reserve.
TOP EXPERIENCE
# Riverboat Travel
Rivers are roads in Amazonia, and the slow pace of a boat trip is a uniquely Amazonian experience: you'll be sleeping in a hammock, watching the river, forest and local life glide by.
Great For...
Agr
yDon't Miss
The view of the bluffs around Monte Alegre on the two-day boat trip between Belém and Santarém.
8Need to Know
You'll need to bring your own hammock and some rope to attach it.
5Take a Break
Be sure to bring your own snacks; food on the boat (rice and beans!) doesn't offer much variety.
oTop Tip
Get to the boat six to eight hours early to secure a good hammock spot–preferably on the middle deck.
### Life on the Boat
Trips are long and languid, measured in days instead of hours. Much of the time is spent on the boat's upper deck, watching the scenery glide by, knocking back beers, talking and laughing over the ever-blasting music. The middle deck is a place to read, nap in a hammock, or practice Portuguese with your neighbor. Night falls quickly and decisively, and night skies on the river can be spectacular.
### Riverboat Realities
For most people a two- to three-day trip is plenty. For all its romantic appeal, riverboat travel can get rather tedious, especially with the constant pounding of the music and engine. The boats typically travel far from shore so there's virtually no chance of seeing wildlife. Consider taking a boat for one leg, and flying the others; air and boat fares can be surprisingly comparable, and you'll have more time for tours and other activities.
Riverboat at sunset | LUC KOHNEN / SHUTTERSTOCK ©
### Suggested Itineraries
The two-day boat trip between Belém and Santarém is an interesting one, passing through the tidal zone around Belém, the narrows of Breves, the high bluffs around Monte Alegre and finally the main channel near Santarém. (The Santarém–Manaus leg, by contrast, follows the main channel only.) The Rio Negro, especially the upper regions, has little boat traffic and winds through massive archipelagos. Porto Velho to Manaus, along the Rio Madeira, is another good choice, a scenic backdoor route to the Amazon.
### Trip Preparations
Most boats have a few _camarotes_ (cabins with two to four bunks and a fan) and suites, with air-con and bathrooms. The advantage is you sleep in a bed instead of a hammock, can lock up your gear, and usually get better food. The disadvantage is you miss out on the camaraderie (and bragging rights) that come from sleeping in a hammock alongside everyone else. If you book a cabin or suite, avoid those on the top deck, where there's blasting music.
Buffet-style meals may or may not be included in the fare. Either way, consider skipping them – you really don't want the runs during a boat trip. Instead, get made-to-order burgers and sandwiches from the kitchen on the upper deck, and pack plenty of food that travels well, such as apples, nuts and energy bars. Don't forget to bring drinking water.
TOP EXPERIENCE
# Jungle Expeditions & Wildlife-Watching
Whether based at a jungle lodge or on a touring riverboat, you can have the experience of a lifetime spying wildlife in the mother of all rainforests. While anything's possible, the most common trip is three to five days with hiking, canoeing, spotting caiman and visiting local villages.
Great For...
fgA
yDon't Miss
Not-to-be-missed wildlife spotting.
8Need to Know
Manaus (and nearby destinations) serves as the gateway to jungle trips.
5Take a Break
Be sure to bring along your favorite snacks, for which you'll be grateful on long hikes.
oTop Tip
Bring binoculars. Even a small pair makes a big difference in observing wildlife.
### Mamirauá Reserve
The remarkable Mamirauá Sustainable Development Reserve has pristine rainforest, abundant animal life and fairly easy access, located just 1½ hours by boat from Tefé, with its reliable air and boat service. The primary lodge, the Pousada Uacari ( %97 3343-4160; www.uakarilodge.com.br; all-inclusive 3/4/7 nights s R$2740/3210/4410, d R$4980/5840/8020; W) in the reserve, runs outstanding tours and has comfortable accommodations. It's one of the Amazon's best ecotourism operations.
### Parque Nacional de Anavilhanas
Stretching along the Rio Negro for almost 130km and with 400 islands, this national park, centered on the Anavilhanas Archipelago, is one of the most rewarding excursions in the Rio Negro Basin. When water levels are low, island beaches appear and camping overnight becomes possible.
The best tours here are run by Visit Amazônia ( %92 99114-6038, 92 99215-1648; www.visitamazonia.org; Av Presidente Getulio Vargas) and Expedição Katerre ( %92 3365-1644; www.katerre.com; Francisco Cardoso).
Aerial view of Parque Nacional de Anavilhanas | MARCOS AMEND / SHUTTERSTOCK ©
### Reserva Extrativista Baixo Rio Branco-Jauaperi
This newly minted extractive **reserve** (Xixuaú-Xipariná) has excellent wildlife-watching – if you're here to see the Amazon in its pristine state, this place should be high on your list. The reserve extends north from the Rio Negro and is most easily reached from Novo Airão.
Most tours here are handled by top-notch Visit Amazonia, based in Novo Airão. Other groups offering tours here include Expedição Katerre and Amazonia Expeditions.
### Tropical Tree Climbing
Tours with this warm French–Venezuelan outfit ( %92 99245-3669; www.tropicaltreeclimbing.com; Hwy BR-178, Km 144) include garden-fresh meals, hiking through verdant forest and, of course, climbing a tree (or two) – usually a huge _angelim_ or _samaúma_. Overnight in comfy guest rooms or even up in a tree!
Tours of one to seven days are available, including dedicated photo safaris. Longer trips include visits to waterfalls and other area sights.
Squirrel monkey | DANITA DELIMONT / ALAMY STOCK PHOTO ©
### Amazon Eco Adventures
Amazon Eco Adventures (map Google map; %92 98831-1011; www.amazonecoadventures.com; 10 de Julho, at Tapajós) is an excellent choice for jungle trips. It offers tours with small groups, comfortable speedboats, its own overnight boat and top guides. You'll go swimming at the meeting of the waters, and activities like visiting indigenous villages don't have a voyeuristic feel.
They also have a lodge up on the Rio Urubu. Although this outfit is pricier than others, it's well worth it.
### Swallows & Amazons
This long-established South African–Brazilian agency (www.swallowsandamazonstours.com; Ramos Ferreira 922) specializes in riverboat tours, with different boats available for varying levels of comfort. Tours mostly go up the Rio Negro, including to the Anavilhanas Archipelago and Jaú National Park, exploring smaller tributaries along the way, with plenty of hiking, canoeing and fishing.
### Amazônia Expeditions
This experienced operator ( %92 3671-2731; www.amazoniaexpeditions.com.br; Rua Miguel Ribas 1339) has spent decades leading tour groups as well as prestigious scientific expeditions. It's one of few groups that goes up the Rio Negro to the Reserva Extrativista Baixo Rio Branco-Jauaperi, and guides are professional and knowledgeable about wildlife and botany.
### Lo Peix
Tours by Lo Peix ( %92 98182-4793; www.lopeix.com; Av Coronel Teixera Puerto, Ponta Negra; per person R$435-1680, 3-12 days) go further up the Rio Negro than other Manaus-based groups and hit places like the Anavilhanas Archipelago, Jaú National Park, prehistoric sites near Airão Velho and more, with frequent stops for canoeing, hiking, snorkeling and visiting local communities.
The custom-built boat has small comfortable berths, solar power and up-to-date safety equipment.
Hiking in the Brazilian Amazon | GUENTER FISCHER / GETTY IMAGES ©
### Juma Lodge
The deluxe lakefront cabins at Juma Lodge ( %92 3232-2707; www.jumalodge.com; 1 night all-inclusive s R$2245-3265, d R$3326-4838) stand dramatically on 15m stilts, connected by wood walkways, with huge screened windows and private patios. West-facing units can get hot in the late afternoon, but that's when you might be sipping a caipirinha on the lodge's shady deck or in the spacious communal dining area. Tours make use of comfortable motorized canoes, though groups can be large.
### Amazon Antonio's Lodge
Antonio Gomes has built a splendid lodge ( %92 99961-8314; www.antonio-jungletours.com; Rio Urubu; per person in dm/room with private bathroom incl meals from R$330/375; a) on the mosquito-free banks of the Rio Urubu, surrounded by a pristine tract of riverine forest rich in wildlife and tall trees. The rooms are outstanding, especially the chalets with expansive terraces and hammocks. The three-storey observation tower has glorious views, especially at sunset. Activities include fishing, jungle hikes, canoe trips and overnight camping. Combine it with a trip downriver to Antonio's other lodge, Pousada Cumaru (per person in dm/d with shared bathroom incl meals R$360/400; p).
oDid You Know?
The Amazon Basin spans nine countries and three standard time zones. It has an area of more than 7 million sq km, accounting for 40% of the entire South American continent.
### Amazon Gero Tours
The effusive Gero Mesquita ( %92 99983-6273, 92 99198-0111; www.amazongerotours.com; Tapajós 27), an all-round good guy, runs a popular lodge in the Juma-Mamori area with comfortable dorms and private rooms, and a cadre of skilled guides. Besides standard tours, Gero arranges multiday treks into untouched forest and organizes 'social sustainability' programs where travelers work on needed community projects.
yWhen to Go
The best time to visit is during the dry season (June to December), when river levels drop and the wildlife-spotting is excellent.
## Belém
The eastern gateway to the Amazon region and a destination in its own right, Belém is worth at least a couple of days of your life. It's a rewarding city, with streets and parks shaded by mango trees, the pastel facades of once-decadent mansions now fading in the tropical sun, as well as a number of fascinating museums, pungent markets and decent restaurant-bars.
Belém
1Sights
1Estação das DocasB1
2Forte do PresépioA3
2Activities, Courses & Tours
3ValeverdeB2
7Shopping
4Feira de ArtesanatoD2
5Mercado Ver-o-PesoA2
5Eating
Estação das Docas(see 1)
La em Casa(see 1)
6Rei Do BacalhauB2
6Drinking & Nightlife
Amazon Beer(see 1)
1Sights
Estação das DocasMarket
(map Google map; %91 3212-5525; www.estacaodasdocas.com.br; Blvd Castilho França; h9am-1am)
An ambitious renovation project converted three down-at-heel riverfront warehouses into a popular gathering spot, with restaurants, bars, shops and even an art-house theater. There are river views and displays about Belém's history, plus a post office and numerous ATMs. Enjoy live music most nights, performed from a moving platform in the rafters, rolling slowly the length of the dining area.
Basílica Santuario de NazaréChurch
(www.basilicadenazare.com.br/site; Praça Justo Chermont; h5:30am-8pm) F
Rather humdrum from the outside, the Basílica Santuario de Nazaré has a truly spectacular interior. Sink into a cushioned pew and admire the soaring marble columns, brilliant stained-glass windows and ornate wood and tile work in every direction, even the ceiling. The basilica is the focal point of Brazil's largest religious festival, Círio de Nazaré, which draws more than a million worshippers to Belém on the second Sunday in October.
Forte do PresépioFortress
(map Google map; Praça Fr Brandão; R$4, free Tue; h10am-6pm Tue-Fri, to 2pm Sat & Sun)
The city of Belém was founded in 1616 with the construction of this imposing fort, which was intended to protect Portuguese interests upriver from incursions by the French and Dutch. Today it houses a small but excellent museum, primarily about Pará state's indigenous communities (displays in Portuguese only), and has great river and city views from atop its thick stone walls.
TTours
Amazon Star Turismo ( %91 3212-6244; www.amazonstar.com.br; Henrique Gurjão 56; h8am-6pm Mon-Fri, to noon Sat) Organizes day trips around Belém, including birdwatching, nature walks and city tours (per person R$150 to R$300), plus overnight tours combining these options, and multiday packages to Ilha de Marajó.
ValeverdeBoating
(map Google map; %91 3218-7333; www.valeverdeturismo.com.br; Estação das Docas)
Runs a variety of short tours on the river (per person R$50 to R$175), including sunrise trips and pleasant evening cruises. Valeverde has an office and daily schedule at the pier at Estação das Docas; in most cases you can simply show up.
7Shopping
Mercado Ver-o-PesoMarket
(map Google map; Av Castilho França; h5am-6pm)
This is Belém's best place to browse and shop, whether for long pants, lacquered piranha or anything in between. Pará has gained national attention with _technobrega_ music, a defiantly from-the-streets genre whose best collections aren't sold in stores but on amateur CDs in Ver-o-Peso (and on YouTube). Be wary of pickpockets in the early and late hours.
Feira de ArtesanatoMarket
(map Google map; Praça da República; h7am-3pm Fri-Sun)
A large crafts fair that has the city's biggest range of attractive artwork, a lot of which is homemade. It's especially busy on Sundays.
5Eating
Rei Do BacalhauSeafood $
(map Google map; %91 3241-5824; Travessa Campos Sales 216; mains from R$15; h8am-6pm)
The 'King of Cod' is a Belém classic, with cheap and cheerful service and fresh street-food staples straight from the nearby fish market. It's no-frills, which makes us love it all the more.
La em CasaBrazilian $$
(map Google map; %91 3212-5588; www.laemcasa.com; Av Marechal Hermes, Estação das Docas; mains R$28-70, set menu for 1/2 people R$72/120)
Around since 1972, La em Casa is an excellent choice for Pará specialties centered around Amazon fish and duck dishes, as well as Brazilian barbecued meats. Try the Pará tasting menu to sample the local flavors; if you snag a riverside table on a warm Amazonian night, you'll love being here even more.
MangoCafe $$
( %91 3199-2731; www.mangoalimentacaosaudavel.com.br; Av Brás de Aguiar 593; mains R$14-45; h11:30am-4pm Tue-Sun & 6-10pm Tue-Sat; v)
Wildly popular with Belém's hip young crowd, Mango is devoted to the art of healthy eating, with fab dishes such as duck burger, and grilled mango for dessert. Terrific juices, salads, light quiches and sandwiches round off a fine choice, with plenty of options for vegetarians.
3Entertainment
Casa do GilsonLive Music
( %91 3272-7306; www.facebook.com/casadogilson; Travessa Padre Eutíquio 3172; h8pm-3am Fri, from noon Sat & Sun)
Come here for Belém's best live music. Opened in 1987, Gilson's attracts crowds with first-rate samba, _choro_ (improvised samba-like genre) and other music, and terrific food and atmosphere. It's between ruas Nova and Tambés.
8GETTING THERE & AWAY
AIR
Belém's Aeroporto Internacional Val-de-Cans ( %91 3210-6000; www.infraero.gov.br), 8km north of the center, is a hub for international, domestic and regional flights. Taxis to the center are R$50.
BOAT
All long-distance boats leave Belém from the centrally located Terminal Hidroviária (Av Marechal Hermes). You can purchase tickets from the booths inside the terminal, but an easier and more reliable option is to contact Amazon Star Turismo, whose multilingual staff can book boat tickets over the phone or internet.
AR Transporte ( %91 3224-1225; www.artransporte.com.br) and San Marino ( %in Manaus 93 99179-1222; www.facebook.com/Ferryboat.sanmarinoII) offer boat service to Manaus and other points along the way. At last check, boats to Manaus left Wednesday and Saturday, plus every other Tuesday, at 6pm with stops at Monte Alegre (hammock/hammock with air-con/cabin with air-con/cabin with air-con and bathroom R$200/230/750/950, three days), Santarém (R$230/270/800/1000, four days), Parintins (R$330/370/950/1100, five days), and Manaus (R$370/400/1000/1200, six days); cabins rates are for two people. Meals are not included, but you can purchase food on board.
Brazilian dishes, Belém | CANNON PHOTOGRAPHY LLC / ALAMY STOCK PHOTO ©
## Santarém
Santarém is a terrific destination, a languid riverside town with a breezy waterfront promenade, parks and several good restaurants. More than that, it's a fine gateway to several quintessential Amazon experiences – getting to primary rainforest, in the Floresta Nacional (FLONA) do Tapajós, is much easier here than from Manaus, and Santarém has its very own 'meeting of the waters,' where the Rio Tapajós flows into the Amazon. Not far to the east, Lago Maicá is a gorgeous floodplain rich in birds and other wildlife, including pink dolphins and sloths.
1Sights
Lago MaicáLake
The floodplains east and southeast of Santarém are among the Amazon's most rewarding excursions. Flooded for much of the year, the plains are home to fabulous birdlife (including toucans and macaws), pink dolphins, howler monkeys, sloths and anacondas. The sunrise and sunset views are pure magic and there's a real sense of tranquility out here. Take an overnight boat trip, go canoeing through the flooded forest and relax far from the tourist hordes. Gil Serique organizes especially enjoyable excursions here.
Praça Mirante do TapajósViewpoint
This pleasant oval-shaped plaza and viewpoint has two open-air food stalls and good river views. An observation tower affords an even better view, including of Santarém's own 'meeting of the waters.' Look for a set of stairs just east of Brisa Hotel.
Estação das Docas
An ambitious renovation project converted three down-at-heel riverfront warehouses into a popular gathering spot (www.estacaodasdocas.com.br; Blvd Castilho França; hnoon-midnight Sun-Wed, noon-3am Thu-Sat), with restaurants, bars, shops and even an art-house theater. There are river views and displays about Belém's history, plus a post office and ATMs. Enjoy live music most nights, performed from a moving platform in the rafters, rolling slowly the length of the dining area.
Favorite places include Lá em Casa, serving pricey but outstanding regional food, and Amazon Beer (map Google map; %91 3212-5400; www.amazonbeer.com.br; Av Marechal Hermes, Estação das Docas; h5pm-1am), with delectable pub grub to accompany its artisanal beer.
PULSAR IMAGENS / ALAMY STOCK PHOTO ©
5Eating
Quiosque da PraçaStreet Food $
(Praça Barão de Santarém; from R$8; h7am-1pm & 4-11pm)
This small kiosk just west of the center serves the best _tacacá_ (local soup with shrimp) we've tasted in Brazil. A late-afternoon bowl of this local favorite will set you up nicely for the night ahead.
Restaurante PiracemaBrazilian $$
( %93 3522-7461; www.restaurantepiracema.com.br; Av Mendonça Furtado 73; R$25-80; h11am-11:30pm Tue-Sat, to 3pm Sun)
Considered by many to be the best restaurant in town, Piracema uses regional ingredients and flavors to create dishes you'll find nowhere else. The specialty is the _peixe á Piracema,_ a spherical construction of layered smoked _pirarucú_ (a freshwater fish), banana and cheese. It's strange but delicious, and large enough for two.
8GETTING THERE & AWAY
Aeroporto de Santarém – Maestro Wilson Fonseca is 14km west of town. It handles flights to/from Manaus and Belém. Buses (R$3.50, 30 minutes) run to the city every half-hour; taxis cost R$60.
The main port, Docas do Pará, 2.5km west of the center, has boats to Belém (hammock R$200, double cabin R$600 to R$800, 48 hours) at 10am Friday and every second Sunday, and 11am Monday, which stops at Monte Alegre (hammock R$50, five to seven hours) along the way; to Manaus (hammock R$180, double cabin R$600, 40 to 48 hours) at noon Monday to Wednesday, Friday and Saturday, and at 11am on Thursday.
## Alter do Chão
Just a short bus ride from Santarém, Alter do Chão is a cool little town with white-sand river beaches and a laid-back vibe. It's best known for its Ilha do Amor (Island of Love), a picturesque island ringed by a white-sand beach directly in front of town.
But Alter do Chão is much more than a beach town. The lagoon it fronts (Lago Verde) can be explored by canoe or stand-up paddle board. It's also the departure point for boat tours to nearby forest reserves and isolated communities.
2Activities
Adventure sports are a natural fit for many visitors here. Rewarding options include stand-up paddling (SUP), windsurfing, kayaking, mountain biking and tree climbing ( _arbolismo_ in Portuguese). Ask at Mãe Natureza ( %93 3527-1264, 93 99131-9870; www.maenaturezaecoturismo.com.br; Praça 7 de Setembro; h8:30am-1pm & 4-11pm), Areia Branca Ecotour ( %93 3527-1386, 93 99121-5646; www.areiabrancaecotour.com.br; Lago Verde; h8am-noon & 2-7pm Mon-Fri, to 1pm Sat & Sun) or Gil Serique for prices and availability.
TTours
Gil SeriqueEcotour
( %93 99130-5298; www.gilserique.com; Av Copacabana 45, at PA-457; per person per day US$150)
Gil is a true character, a lithe teller of tales and one of the area's top naturalists. Born and raised nearby, Gil's tours are part history, part ecology and part family lore, related with infectious enthusiasm and excellent English. Visits to Lago Maicá, a gorgeous floodplain teeming with birds and other wildlife, are especially memorable. Pricier than others, but worth it.
5Eating
Espaço GastronômicoBrazilian $$
( %93 98401 6144; www.espacoalter.com.br; Lauro Sodré 74; mains R$25-60; h7-11pm; W) S
What a fabulous place! It's made almost entirely from recycled wood and other materials, with a commitment to using local ingredients and indigenous sauces, which might include Brazil nuts, honey and manioc, or ceviche with mango. Upstairs is the place to be, with diaphanous curtains blowing in the sea breeze. Joana is a marvelous host.
8GETTING THERE & AWAY
Hourly buses to Alter do Chão depart from Santarém's bus station ( %93 3523-4940; Hwy BR-163) (R$3.50, one hour), located 2.5km west of the center.
## Manaus
Manaus is the Amazon's largest city, an incongruous urban metropolis in the middle of the jungle and a major port for seafaring vessels that's 1500km from the ocean. The Amazonian rainforest has a population density half that of Mongolia's, but the journey there invariably begins in (or passes through) this gritty, bustling city.
Manaus is no architectural gem, but does have some genuinely rewarding sights. It's also a place to stock up on anything you forgot to pack, make reservations and begin your journey out into the jungle.
Manaus
1Sights
1Centro Cultural Usina ChaminéC3
2Mercado Municipal Adolfo LisboaB3
3Teatro AmazonasB1
2Activities, Courses & Tours
4Amazon Eco AdventuresB1
5Amazon Gero ToursB1
6Swallows & AmazonsB1
5Eating
7Tacacaria AmazôniaB1
6Drinking & Nightlife
8Bar do ArmandoB1
1Sights
Jardim Botânico Adolpho DuckePark
( %92 3582-3188; www.jardimbotanicodemanaus.org; Av Margarita s/n; R$10, incl tower R$30; h8am-noon & 1-4:30pm Tue-Sun, last entry 4pm) F
Spanning over 100 sq km, this 'garden' is actually the world's largest urban forest. There's a network of five short trails (guides and closed shoes required, two to three hours, free with admission) and an open-air museum that includes rotating exhibits on Amazonian flora and fauna and a spectacular 42-meter-high observation tower. It's busier on weekends and free on Tuesday.
Teatro AmazonasTheater
(map Google map; %92 3232-1768; Praça São Sebastião; guided tour R$20; h9:15am-5pm, tours every 30 mins until 4pm)
This gorgeous theater was built at the height of the rubber boom, using European designers, decorators and even raw materials. The original driveway was Brazilian, though, made of Amazonian rubber to soften the clatter of late-arriving carriages. The theater's performance schedule includes an excellent opera festival (Festival Amazonas de Ópera; hlate Apr–early May) in April and May. Hour-long guided tours offer an up-close look at the theater's opulent construction.
Centro Cultural Usina ChaminéMuseum
(map Google map; %92 3633-3026; Av Lourenço da Silva Braga; h8am-2pm Tue-Fri, 9am-1pm Sat) F
Also known as the Museu dos Cinco Sentidos (Museum of the Five Senses), this innovative museum uses the five senses to evoke and illustrate indigenous and Caboclos life and culture. You can hear recordings of native languages, smell Amazonian spices, admire indigenous folk art and more.
Mercado Municipal Adolfo LisboaMarket
(map Google map; %92 3232-9210; Rua dos Barés 46; h6am-5pm Mon-Sat, to 7pm Fri, to noon Sun)
Manaus' historic city market was inaugurated in 1882, a downscaled replica of Paris' famed Les Halles market. Safe and bustling, the central building has mostly handicraft shops, with wares ranging from predictable kitsch to high quality. A side building houses a working fish market – visit early in morning to see the vast array of Amazonian fish on display. Great little places to eat throughout.
Floresta Nacional do Tapajós
One of the last and most accessible stretches of primary rainforest in the region, this protected area is the Amazon you always imagined. Behemoth samaúma and other trees, some with trunks too big for even 20 people to stretch their arms around, are a highlight of this 5440-sq-km reserve on the east side of the Rio Tapajós.
Wildlife includes numerous species of birds (king vultures are a highlight), while the forest is also home to elusive cats (including jaguar and ocelot, although they're rarely seen), as well as howler monkeys, squirrel monkeys and capuchins.
The villages of Maguarí, Jamaraquá and São Domingo near the reserve have rustic lodging and forested trails (though it's a 2½-hour hike to reach primary rainforest from the villages). There are some lovely _igarapés_ (inlets), too, with good canoeing and possibly animal-spotting when the water is high.
You can get to FLONA by bus from Santarém or boat from Alter do Chão. The recommended guide Gil Serique, whose family once lived in the forest, leads tours (from US$150 per person per day).
Man resting on a samaúma tree | MARIANO VILLAFANE / SHUTTERSTOCK ©
TTours
Amazon Tree ClimbingEcotour
( %92 8195-8585; www.amazontreeclimbing.com; Caravelle 22a, Tarumã)
Yellow-shirted guides lend a youthful vibe to this outfit, whose tours range from half-day trips near Manaus with views of the Meeting of the Waters to all-day excursions that may also include visiting an indigenous village. Getting to the top of the massive trees can be quite challenging physically, but the experience is unforgettable.
5Eating
Amazônico Peixaria RegionalSeafood, Brazilian $$
( %92 3236-0546; www.amazonico.com.br; Av Darcy Vargas 222; mains R$36-98; hnoon-3pm & 6:30-11pm Mon-Sat, 11:30am-4pm Sun)
However you get your _tambaqui_ (perhaps the tastiest of all Amazonian fish) prepared here – stewed, grilled, ribs – the execution is perfect. They serve all manner of other fish and steak dishes, and fab desserts too. The atmosphere is casual, the service attentive and the cooking top-notch. It will be one of the best meals you'll have in the Amazon.
Tacacaria AmazôniaBrazilian $$
(map Google map; %92 99112-3730; www.facebook.com/TacacariaAmazonia; 10 de Julho 503; mains R$20-45; h10:30am-10:30pm Mon-Sat, from 3pm Sun)
This bright, appealing place serves a range of local dishes, specializing in _tacacá_ (local
soup with shrimp) and well-prepared fish from the Amazon, including _tambaqui_ and _pirarucú_. English is spoken and they're good at putting an order together without the hard sell.
BanzeiroBrazilian $$$
( %92 3234-1621; www.restaurantebanzeiro.com.br; Libertador 102, NS das Gracas; mains R$75-140; h11:30am-3pm & 7-11pm, to 10pm Sun)
One of Manaus' top gourmet restaurants, especially for fish. Varieties such as _pirarucú,_ _tambaqui_ and other Amazonian specialties are served in various preparations, from cheese and banana to parsley and _formigas_ (ants). Pricey but truly one of a kind. Online reservations recommended.
6Drinking
Bar do ArmandoBar
(map Google map; %92 3232-1195; www.facebook.com/bardoarmando; 10 de Julho 593; h5pm-2am)
Near the opera house, this is a traditional rendezvous for Manaus' intellectual and bohemian types, but all sorts of people crowd around the outdoor tables for beers and conversation.
Encontro das Águas
Just beyond Manaus, the warm dark Rio Negro pours into the cool creamy Rio Solimões, but because of differences in temperature, speed and density, their waters don't mix, instead flowing side by side for several kilometers. The bi-color phenomenon occurs throughout the Amazon, but nowhere as dramatically as here. Day trips always include a stop here, and many tour operators at least pass by en route to their lodges. Never disappoints.
Waters mixing from Rios Negro and Solimões | MARIANO VILLAFANE / SHUTTERSTOCK ©
8GETTING THERE & AROUND
AIR
**Aeroporto Internacional Eduardo Gomes** ( %92 3652-1210; www4.infraero.gov.br/aeroportos/aeroporto-internacional-de-manaus-eduardo-gomes/; Av Santos Dumont 1350) is located 13km north of the city center. Bus 813 'Aeroporto-Ejecutivo' (R$8) runs every half-hour between the airport and Praça da Matriz in the center of town; taxis charge R$75.
BOAT
There are several terminals. Passenger boats going downstream to Belém depart from the central **Estação Hidroviária de Manaus** ( %92 3233 7061; www.portodemanaus.com.br; Porto Flutuante) and usually make stops in Santarém and Monte Alegre. To Santarém (36 hours), these depart at 11am Monday to Saturday (hammock/cabin from R$180/8000); boats to Belém (4 days) depart at 11am Wednesday and Friday (hammock/cabin from R$350/R$1100).
# The Pantanal, Brazil
#### Exploring the Pantanal
#### Cuiabá
#### Parque Nacional da Chapada dos Guimarães
#### Campo Grande
#
The Pantanal, Brazil
_The Pantanal, one of the most important and fragile ecosystems on the planet, truly shines as a top destination for wildlife-watching. The world's largest wetland, in the heart of the continent, covers some 210,000 sq km. The majority is in Brazil, split between the states of Mato Grosso and Mato Grosso do Sul._
_The Pantanal has few people and no towns. Car travel is restricted by the seasons. You can either penetrate the Pantanal from the north, where the Transpantaneira runs deep into the region, or from the south, where Estrada Parque cuts across the wetlands._
_Victoria amazonica_ (giant water lilies) | CORBIS DOCUMENTARY / GETTY IMAGES ©
Two Days in the Pantanal
Fly into Cuiabá, and head out on a two-day excursion in the northern Pantanal. Look for macaws from lookout towers, spy capybara along the Transpantaneira and go for a jaguar-spotting boat trip near Porto Jofre. Overnight in an eco-minded _fazenda_ , like Pantanal Jaguar Camp.
Four Days in the Pantanal
On your third day in the region, visit the spectacular red rock formations and shimmering waterfalls of Parque Nacional Chapada dos Guimarães. End the trip with a day taking in the sights of Cuiabá, including the indigenous-focused Museu Rondon de Etnologia e Arqueologia. Treat yourself to a memorable meal at Mahalo.
Arriving in the Pantanal
Cuiabá is the main gateway to the northern Pantanal. Campo Grande is the principal southern launch point into the Pantanal.
There are direct flights to Cuiabá, Campo Grande and Corumbá from other Brazilian destinations.
Cuiabá and Campo Grande are easily reached by direct buses from numerous Brazilian cities.
Where to Stay
Pantanal accommodations are divided into roughly three types: _pousadas_ , which include all meals and range from simple to top-end; _fazendas,_ which are ranch-style hotels that usually have horses and often boats for use; and _pesqueiros,_ which cater for anglers and usually have boats and fishing gear for rent.
If you travel independently, you can rent a car and book a stay at various Pantanal lodges.
TOP EXPERIENCE
# Exploring the Pantanal
The Amazon gets the press coverage, but the Pantanal is a better place to see wildlife. The dense foliage of the Amazon makes it difficult to observe the animals, but in the open marshes of the Pantanal, wildlife is much easier to spot. If you like to see animals in their natural environment, don't miss this place.
Great For...
fgA
yDon't Miss
Spotting jaguars–the largest big cat in the Americas–from Porto Jofre.
8Need to Know
Quality tours start from around R$600 per day, including food and lodging.
5Take a Break
Before or after a Pantanal trip, indulge in delectable river fish at Lélis Peixaria ( %65 3322-9195; www.lelispeixaria.com.br; Marechal Mascarenhas de Morães 36; rodízio lunch/dinner R$60/90; h11:30am-3pm & 6:30-11pm Mon-Sat, 11:30am-3pm Sun) in Cuiabá.
oTop Tip
Be prepared: pack sunglasses, binoculars, insect repellent, a hat, sunscreen and a jacket for the early morning chill.
### Northern Pantanal
From Cuiabá, the capital of Mato Grosso, small tour operators arrange safaris along the Transpantaneira that include transportation, ranch accommodations on farms, and guides. Tours from Cuiabá tend to be slightly more expensive, but more professional, with smaller groups and better-trained guides than those from Campo Grande.
Jabiru storks | ZWILLING330 / GETTY IMAGES ©
### Transpantaneira
This raised dirt road sectioned by small wooden bridges begins at Poconé and ends 145km south at Porto Jofre. It's a spectacular drive, with plentiful wildlife the whole way. At Porto Jofre, jaguar-spotting boat trips are a major drawcard in the dry season, while great fishing is also a possibility.
### Pantanal Nature
Superb agency ( %65 99925-2265, 65 3322-0203; www.pantanalnature.com.br; Av Historiador Rubens de Mendonça 1856, Cuiabá; 4-day nature tour per person R$4150, jaguar tour R$5800; hoffice hours 7-11:30am & 1:30-5pm Mon-Thu, 8am-noon Fri & Sat) run by Ailton Lara that has a sterling reputation for its professional tours and expert guides. It runs Pantanal nature tours and, in season, jaguar tours from the Pantanal Jaguar Camp in Porto Jofre.
Wild jaguar | JAMI TARRIS / GETTY IMAGES ©
### Ecoverde Tours
This relaxed budget agency ( %65 99638-1614; www.ecoverdetours.com.br; Pedro Celestino 391, Cuiabá; 4-day nature tour s/d R$2400/4000, jaguar tour R$3000/5000) has decades of service and experienced guides. Working with local pousadas toward an ecofriendly approach, Joel Souza can guide you in several languages. He runs nature tours year-round and jaguar-spotting tours from June to October.
### Focus Tours
In operation for nearly four decades, this trailblazing ecotourism operator ( %Brazil 31 9134-3833, USA +1 505-216-7780; www.focustours.com) S was founded by ecologist Douglas Trent. As well as customizable bird-watching and nature-watching trips, it runs jaguar-watching excursions on the Rio Paraguai and operates a major conservation project.
### Southwild
This innovative high-end operator (www.southwild.com; 7-day jaguar-spotting packages per person US$5000-7000) has been a pioneer in Pantanal wildlife tourism. It has upmarket packages involving stays at Southwild Pantanal Lodge (Fazenda Santa Tereza; www.southwild.com; Transpantaneira, Km 68; d incl full board R$820; a W s) S and Flotel and Jaguar Suites (www.southwild.com; Rio Piquirí; 6-night packages per person US$5000-6700; hJun-Nov; a i W). Tours cater for serious nature-watchers with on-site biologists.
### Araras Eco Lodge
This pioneering ranch ( %65 3682-2800; www.araraslodge.com.br; Transpantaneira, Km 32; s/d incl full board & excursions R$1591/2482; a W s) S offers great comfort and luxury in a peaceful setting. There is a treetop tower for bird-watching (hyacinth macaws), while welcoming owners André and Akhila make an immense effort to be sustainable.
Araras Eco Lodge | DAVIDCALLAN / GETTY IMAGES ©
### Pantanal Jaguar Camp
The Pantanal Jaguar Camp ( %65 99925-2265; www.pantanaljaguarcamp.com.br; Transpantaneira, Km 145, Porto Jofre; d with meals & excursion R$1035, camping per person R$60, with own tent R$45; a W) S is an intimate, solar-powered wilderness lodge with seven comfortable rooms, camping and an on-site restaurant. This is a great Porto Jofre base for jaguar-seeking boat trips affiliated with Pantanal Nature. Other activities include night safaris and bird-watching.
### Southern Pantanal
The southern Pantanal has a fine choice of lodges from backpacker favorites to exclusive eco-havens. Campo Grande is the major launch pad for the southern Pantanal. This is also the best place for budget-friendly trips, with prices starting at around R$500 per day. All accommodations arrange nature-watching trips on both land and water, as well as horseback riding and other Brazilian countryside experiences.
### Pousada & Camping Santa Clara
This working ranch ( %67 99939-3570; www.pantanalsantaclara.com.br; Estrada Parque, Km 22; 3-day, 2-night package incl full board & excursions camping/dm/s/d R$580/860/1220/1980; a W s) is enthusiastically run and has a host of activities (hikes, piranha fishing, night safaris, horseback riding), accommodations to suit all budgets and hearty _pantaneiro_ cooking. It is deservedly popular, with compact rooms, a plunge pool and games, plus macaws and peccaries about.
### Pantanal Discovery
Owner Gil of Pantanal Discovery ( %67 99163-3518; www.gilspantanaldiscovery.com.br; Hotel Mohave, Afonso Pena 602, Campo Grande; 3-day, 2-night package hammock/dm/s/d R$1000/1200/1550/2800) is the pick of budget operators in town. He's used to dealing with short notice or tight schedules and has his own transportation.
### Pantanal Viagens
A reputable agency ( %67 3321-3143; www.pantanalviagens.com.br; Room 9, Old Bus Terminal, Joaquim Nabuco 200, Campo Grande) working with Pousada Passo do Lontra and other Pantanal lodgings. Caters to mid- to high-end budgets and has professional and reliable packages.
y When to Go
It's possible to visit the Pantanal year-round, but it's best to go during the dry season (May to September) as the wildlife converges on the reduced water.
### Pantanal Jungle Lodge
This boardwalk lodge ( %67 3242-1488; www.pantanal-jungle-lodge.com; Estrada Parque, Km 8; 3-day, 2-night package dm/s/d R$910/1380/2310; a W s) is a backpacker favorite, thanks to its enviable riverside location and well-organized activities – from canoeing and piranha fishing to night safaris on the river and wildlife-spotting treks. Lodge either in one of the breezy dorms or in a private room with air-conditioning. All have an insect-screened porch to relax in.
oBird-Watching
If you are interested in birds, it's worth bringing the _Birds of the Pantanal_ field guide, which is on sale at most lodges (R$150).
## Cuiabá
A lively place with a vibrant dining scene and some beautiful colonial architecture around its main square, Cuiabá is an excellent starting point for excursions to the Pantanal.
1Sights
Museu Rondon de Etnologia e ArqueologiaMuseum
(MUSEAR; %65 3313-7391; www.museurondonufmt.blogspot.com; Av Fernando Corrêa da Costa 2367; h7:30-11am & 1-5pm Mon-Fri) F
The small Museu Rondon has exhibits on indigenous culture and is well worth a visit to check out the ornate headdresses and weaponry. The museum is located on the grounds of the Federal University of Mato Grosso (UFMT), behind the swimming pool. To get here, catch a C01 Universitária bus (R$3.85) from Praça Alencastro. Ring ahead to check it's open.
5Eating
MahaloFusion $$
( %65 3028-7700; www.mahalocozinhacriativa.com.br; Presidente Castelo Branco 359; mains from R$45; h11am-2:30pm & 7:30pm-midnight Mon-Sat)
Inside a converted mansion, the city's big splurge is one of Brazil's top restaurants, thanks to the efforts of Parisian-trained chef Ariani Malouf. Go for the set weekday lunch (R$55 for two courses) or else choose from the likes of pintado (giant catfish) encrusted with Brazil nuts or a perfectly-seared rack of lamb with sweet potato puree. Dress nicely and book ahead.
8GETTING THERE & AWAY
Cuiabá's **international airport** (CGB; %65 3614-2511; Av João Ponce de Arruda, Várzea Grande) is in Várzea Grande, 7km from central Cuiabá.
Cuiabá's **bus station** (Jules Rimet) is 3km north of the center.
## Parque Nacional da Chapada dos Guimarães
Picture the scene: red rock buttresses soaring up from a green valley; lines of palm trees marking the location of clear rivers and pools for snorkeling; waterfalls ranging from immense to petite and swimmable; dusty hikes through parched land, observed only by yellow-eyed burrowing desert owls; mysterious caves to explore...
The top-notch Chapada Explorer ( %65 3301-1290; www.chapadaexplorer.com.br; Praça Dom Wunibaldo 57; h8-11am Mon-Sat) S runs excursions to all of the area's attractions in small groups.
1Sights
Véu de NoivaWaterfall
(Bridal Veil; h9am-5pm, last entry 4pm) F
The impressive Véu de Noiva, an 86m free-falling waterfall, provides the park's characteristic postcard moment. A small trail leads to the lookout, perched on top of rocks with the canyon below. This is one of Chapada's most dazzling spots; no guide necessary.
Cidade de PedraMountain
(Stone City; www.icmbio.gov.br/parnaguimaraes/guia-do-visitante.html; entry only with authorized guide; h9am-5pm Sun-Fri, from 3pm Sat) F
Cidade de Pedra provides the region's most transcendent moment. Jagged red sandstone rock formations jut up into the sky from the tops of enormous cliffs that drop down into the vast green valley beneath. You follow the short footpath that skirts the edge of the cliff through scrubland, peering at the abyss below. Morning is the best time to visit, when the sunlight illuminates most of the cliffs. It's 25km northwest of Chapada.
Vale do Rio ClaroSnorkeling
(Rio Claro Valley; www.icmbio.gov.br; h8am-5pm, last entry 1pm)
A steep scramble takes you to a viewpoint overlooking the lush valley and the razor-thin rock formations of the Crista do Galo (rooster's crest). Then you swim and snorkel in clear, deep, rapid-fed pools in the forest before embarking on a 500m float, following your guide along the twists and turns of the narrow river, the somber underwater world revealing itself beneath you.
8GETTING THERE & AWAY
**CMT buses** ( %65 3301-2679) leave Cuiabá's bus station for Chapada town (R$17, 1¼ hours, nine daily). Buses will also drop off and pick up at the entrance to Véu de Noiva.
## Campo Grande
Campo Grande is a vast, modern metropolis, where high-rises tower above the shopping malls and streets are lined with restaurants.
1Sights
Museu das Culturas Dom BoscoMuseum
( %67 3326-9788; www.mcdb.org.br; Parque das Nações Indígenas, Av Afonso Pena; R$10; h8am-4:30pm Tue-Sat)
Built on the site of a Bororo burial ground, this superb museum is divided into two parts. One is a collection of over 10,000 insects and stuffed flora and fauna, while the other is a visually striking, unmissable introduction to the indigenous people of the Mato Grosso region, with subtly lit underfloor and suspended-glass displays showcasing shaman paraphernalia, weaponry, everyday tools, splendid adornments made of feathers and funerary objects. The enlarged black-and-white photos are almost equally striking.
5Eating
Casa do PeixeSeafood $$$
( %67 3382-7121; www.casadopeixe.com.br; Av João Rosa Pires 1030; mains for 2 people R$95, rodízio R$86; h11am-2pm daily, 6-11pm Mon-Sat)
This place rightfully deserves its reputation as the best _peixaria_ (fish restaurant) in town, with a superior buffet accompanying the hearty mains. The _rodízio_ is an awesome way to acquaint yourself with _pintado,_ _pacu_ and other fish of the Pantanal, prepared in many sublime ways, while the inclusion of sushi and sashimi pays homage to the city's large Japanese population.
Circuito das Cachoeiras
The **Circuito das Cachoeiras** (Waterfall Circuit; www.icmbio.gov.br; entry only with authorized guide; h8:30am-5pm, last entry noon) F involves a gentle 6.5km hike through a parched red landscape, covered with scrubland and low trees, with six waterfall stops en route; the whole thing takes four to six hours, depending on how easy you want to take it. The first waterfall is the highest, while the others are better for swimming, with deep pools and cascades forming natural Jacuzzis. Entry is free, but you must go with a registered guide.
Andorinhas Waterfall | SERGIOROCHA / SHUTTERSTOCK ©
8GETTING THERE & AWAY
**Aeroporto Internacional de Campo Grande** (CGR; %67 3368-6000; Av Duque de Caxias) is 7km west of town (a R$30 to R$40 taxi ride).
Campo Grande's **bus station** ( %67 3026-6789; Av Gury Marques 1215) is located 6km south of the center on the road to São Paulo.
# Iguazú Falls, Brazil & Argentina
#### Visiting the Falls
#### Foz do Iguaçu, Brazil
#### Puerto Iguazú, Argentina
#
Iguazú Falls, Brazil & Argentina
_One of the planet's most awe-inspiring sights, the Iguazú Falls are simply astounding. A visit is a jaw-dropping, visceral experience, and the power and noise of the cascades – a chain of hundreds of waterfalls nearly 3km in length – live forever in the memory. An added benefit is the setting: the falls lie split between Brazil and Argentina in a large expanse of national park, much of it rainforest teeming with unique flora and fauna. The falls are easily reached from either side of the Argentine-Brazilian border._
View of Río Iguazú and the Garganta del Diablo on the left | GCOLES / GETTY IMAGES ©
Two Days in Iguazú Falls
Spend the first day taking in the sweeping views of the waterfalls from the Brazilian side. On day two, cross over to Argentina for a close-up look at the Garganta del Diablo, take a boat trip along the river and then visit Isla San Martín.
Four Days in Iguazú Falls
Spend day three on the Argentine side of the falls, exploring the rainforest-lined Sendero Macuco, where you can spy birds, monkeys and other wildlife, then take a swim at the base of the small waterfall. On day four, head out to Itaipú Binacional for a look at one of the largest dams on the planet.
Arriving in Iguazú Falls
Located 9km south of the Argentine side of the falls, **Cataratas del Iguazú International Airport** (www.aa2000.com.ar/iguazu) has flights to Buenos Aires and Cordoba.
On the Brazilian side, Foz do Iguaçu International Airport has flights to major Brazilian cities as well as Lima.
Where to Stay
The towns of Foz do Iguaçu, Brazil, and Puerto Iguazú, Argentina have abundant accommodation options. There are also two high-end hotels near the falls: **Meliá Iguazú** (www.melia.com) in Argentina and **Belmond Hotel das Cataratas** (www.belmond.com) in Brazil.
Visitors at the falls, Argentina | SHARPTOYOU / SHUTTERSTOCK ©
TOP EXPERIENCE
# Visiting the Falls
While the Argentine side, with its variety of trails and boat rides, offers many more opportunities to see individual falls close up, the Brazilian side yields the more panoramic views. You can easily make day trips to both sides of the falls, no matter which side of the border you're on.
Great For...
gAf
yDon't Miss
The platform overlooking the deafening cascade of the Garganta del Diablo.
8Need to Know
**Parque Nacional Iguazú** ( %03757-491469; www.iguazuargentina.com; adult foreigners/Mercosur/Argentines AR$600/480/310, child AR$150/120/100; h8am-6pm) , Argentina; Parque Nacional do Iguaçu ( %45 3521-4400; www.cataratasdoiguacu.com.br; Hwy BR-469, Km 18; adult foreigners/Mercosur/Brazilians R$62/49/36, child R$10; h9am-5pm), Brazil.
5Take a Break
On the Brazilian side, have a buffet lunch on the terrace at Restaurante Porto Canoas (www.cataratasdoiguacu.com.br; Parque Nacional do Iguaçu; buffet R$59.90; hnoon-4pm).
oTop Tip
To beat the crowds (and congestion along the narrow gangways), get to the falls by 9am.
### Parque Nacional Iguazú, Argentina
On the Argentine side of the falls, this park has loads to offer and involves a fair amount of walking. The spread-out entrance complex ends at a train station, with departures every half-hour to the Cataratas train station, where the waterfall walks begin, and to the Garganta del Diablo. You may prefer to walk: it's only 650m along the Sendero Verde path to the Cataratas station, and a further 2.3km to the Garganta.
UNDIVIDED / SHUTTERSTOCK ©
#### Garganta del Diablo
A 1.1km walkway across the placid Río Iguazú leads to one of the planet's most spectacular sights, the 'Devil's Throat.' The lookout platform is perched right over this amazingly powerful, concentrated torrent of water.
From Cataratas train station, train it or walk the 2.3km to the Garganta del Diablo stop. The last train to the Garganta leaves at 4pm and we recommend taking it, as it'll be a somewhat less crowded experience. If you walk, you'll see quite a lot of wildlife around this time of day, too.
#### Circuito Inferior
This circuit (1400m) descends to the river, passing delightfully close to falls on the way. At the end of the path prepare for a drenching at the hands of Salto Bossetti if you're game. Just short of here, a free launch makes the short crossing to Isla San Martín. At the same junction you can buy tickets for the popular boat rides under the falls.
#### Circuito Superior
The Circuito Superior (1750m) is entirely level and gives good views of the tops of several cascades and across to more. A recently constructed final section crosses a large swath of the Iguazú river, ending above the powerful Salto San Martín before wending its way back across river islands.
#### Isla San Martín
From the end of the Circuito Inferior, a free launch takes you across to this island with a trail of its own that gives the closest look at several falls, including Salto San Martín, a huge, furious cauldron of water. When the water is high – and this is the case more often than not – island access is shut off.
#### Iguazú Jungle Explorer
Offers three adventure tours ( %03757-421696; www.iguazujungle.com). Most popular is the short boat trip leaving from the CircuitoInferior that takes you right under one of the waterfalls for a high-adrenalin soaking (AR$1000), while the Paseo Ecológico (AR$400) is a wildlife-oriented tour in inflatable boats upstream from the falls.
#### Sendero Macuco
This 3.5km jungle trail leads through dense forest to a nearly hidden waterfall. It's a rare opportunity to explore the park independently. Six interpretive stations explain the flora, including bamboo, palmitos and pioneer plants. The white-bearded manakin and toco toucan live in these parts, as does a troupe of brown capuchin monkeys.
You can swim at the base of the waterfall but take care and don't head out into the river.
#### Safaris Rainforest
Using knowledgeable guides, this outfit ( %03757-491074; www.rainforest.iguazuargentina.com) is the best option for appreciating Parque Nacional Iguazú's flora and fauna. It offers combined driving-walking excursions: the Safari a la Cascada takes you to the Arrechea waterfall (AR$560, 90 minutes); the Safari en la Selva (AR$660, two hours) is better, a trip in a less-touristed part of the park that includes explanations of Guaraní culture.
### Parque Nacional do Iguaçu, Brazil
Brazil's section of the falls offers fabulous panoramic views. You can't miss the shiny entrance to Parque Nacional do Iguaçu, which houses bathrooms, ATMs, lockers, souvenir shops, left-luggage facilities and a vast parking lot. You can purchase your ticket in advance on the website and pick it up in the preferential line at the ticket windows, or buy from on-site machines (if you have a chip-in-pin card). Once ticketed, you will be directed to board a free double-decker bus, which departs every 10 minutes. In addition to the main event, the park and its environs offers trails, boat trips, whitewater rafting and other nature-focused adventures.
#### Waterfall Trail & Views
Once you're in the park and ready to visit the falls, take the Parque Nacional do Iguaçu bus to the third stop at Belmond Hotel das Cataratas. Here you can pick up the main waterfall observation trail, Trilha das Cataratas ('Waterfall Trail'), a 1200m trail following the shore of the Iguaçu River, terminating at the stunning Garganta do Diabo.
An elevator heads up to a viewing platform at the top of the falls at Porto Canoas, the last stop of the double-decker buses. Porto Canoas has a gift shop, a couple of snack bars and an excellent buffet restaurant.
Boat trip to the waterfalls | JAKUB BARZYCK / SHUTTERSTOCK ©
#### Macuco Safari
Under concession from the national park, Macuco ( %45 3529-6262; www.macucosafari.com.br; Av das Cataratas, Km 25, Parque Nacional do Iguaçu; boat trip R$215, excursions R$61-515) is the designated adventure operator on the Brazilian side of the falls. The main event is the namesake safari, which involves a wet-and-wild boat ride on rapids and waterfalls, but they also offer trekking, bird-watching, whitewater-rafting and other nature-centric activities. To reach Macuco, get off the double-decker bus at the second stop.
oDid You Know?
Thousands of years before they were 'discovered' by Europeans, the falls were a holy burial place for the Tupi-Guaraní and Paraguas tribes.
#### Trilha das Bananeiras
The Bananeiras Trail is a 1.5km walk passing lagoons and observing aquatic wildlife, ending at a jetty where you can take boat rides or silent floating excursions in kayaks down to Porto Canoas. If you plan to do any of these, chat with one of the agents touting them around the park visitor center; they can get you a discount.
#### Trilha do Poço Preto
A 9km guided hike (get off the bus at the first stop) through the jungle on foot, by bike or on an electric cart. The trail ends at Taquara Island, where you can kayak or take a boat cruise to Porto Canoas. You can also return via the Bananeiras Trail.
oWildlife
On the falls trails you'll see large lizards, coatis, monkeys and birds, but you'll see much more on one of the few trails through the dense forest.
## Foz do Iguaçu, Brazil
The Brazilian city of Foz do Iguaçu went through a period of frenzied growth during the 18 years that Itaipú Dam was under construction (completed in 1982), when the population increased more than fivefold. These days, it's a fairly average town, but it makes a fine base for exploring the famed waterfalls nearby as well other area attractions such as Itaipú Binacional and Parque das Aves.
1Sights
Parque das AvesBird Sanctuary
(Bird Park; %45 3529-8282; www.parquedasaves.com.br; Av das Cataratas, Km 17.1; R$45; h8:30am-5pm)
This 5-hectare bird park, located 300m from the entrance to Parque Nacional do Iguaçu, is home to 800-plus species of birds, including red ibis, bare-throated bellbird and flamingos galore. They live in 8m-high aviaries that are constructed right in the forest, some of which you can walk through. Kids and adults freak out alike. Well worth it.
Enthusiasts can go deeper here with Guaraní forest (R$250) and behind-the-scenes (R$200) experiences.
Marco das Três FronteirasViewpoint
( %45 3132-4100; www.marcodastresfronteiras.com.br; Av Gen Meira s/n; 2-11pm; hadult/child R$22/12)
Once little more than an uneventful obelisk, the Parque Nacional do Iguaçu concession folks have now built a small tourist complex around the viewpoint of all three tri-border countries that includes a children's park, a light and water show, cultural shows and a memorial to Álvar Núñez Cabeza de Vaca – the first European to have recorded the existence of the falls around 1540.
2Activities
In addition to the nature and water-related activities inside Parque Nacional do Iguaçu, Foz offers skydiving (over Itaipú Binacional; R$590) as well as other excursions.
Aguaray Eco EsportesOutdoors
( %45 99158-8826; www.aguaray.com.br; Alameda Caete s/n; R$150)
Arranges a recommended three-hour excursion on the Rio Iguaçu that involves a nature hike, kayak or stand-up paddle board, and visits to waterfalls.
5Eating
Castelo LibanêsMiddle Eastern $
( %45 3526-1218; Vinícius de Moraes 520; mains R$16-60; hnoon-10pm)
Foz's Lebanese community is second in size only to São Paulo's; come here to sample the Middle Eastern cuisine. The bakery does great _esfihas_ (open-faced or triangular meat pies); other well-done staples include roast chicken, Lebanese _kafta_ (beef kebabs), shawarma and hummus.
Empório com ArteCafe $$
(Av das Cataratas 569; mains $30-62; h3-11pm Tue-Fri, 2-11pm Sat, 2-10pm Sun; W)
This _mineiro_ -run highlight of the Foz scene ('scene' used loosely) is full of country charm and swarming with town sophisticates gossiping among a potpourri of mostly Minas Gerais art and handicrafts (all for sale). Decent espresso, elaborate mixed-fruit _caipirinhas_ made with local _cachaça,_ and a few elaborate main courses (eg duck confit in dark-beer reduction) make for rustic-romantic encounters.
6Drinking & Nightlife
Nightlife is hopping along Av Jorge Schimmelpfeng, where you'll find breezy beer gardens, lively outdoor patio bars and hot-to-trot nightclubs that stay open late. Good outdoor drinking dens include Rafain Chopp (www.facebook.com/rafainchopp; Av Jorge Schimmelpfeng 450; h4pm-2am Mon-Thu, from 2pm Fri-Sun; W) and Capitão Bar (www.capitao.bar; Av Jorge Schimmelpfeng 288; h11:30am-2am; W).
Zeppelin Old BarLive Music
(www.facebook.com/ZeppelinFoz; Raul Mattos 222; h7pm-midnight Tue, 9pm-2am Wed-Sat; W)
Outstanding bar serving up excellent cocktails and live music that spans the gamut from grunge to reggae. The beautiful people congregate from Thursday onwards.
8GETTING THERE & AWAY
Daily flights link **Foz do Iguaçu/Cataratas International Airport** (IGU; %45 3521-4200; Hwy BR-469, Km 16.5) to Lima and several major Brazilian cities.
Bus 120 'Aeroporto/Parque Nacional' runs to the Brazilian side of the waterfalls from Foz do Iguaçu's TTU bus station every 22 to 30 minutes from 5:25am to midnight (R$3.55).
For Parque Nacional Iguazú on the Argentine side, catch a Puerto Iguazú bus in Foz do Iguaçu (R$5/AR$20, one hour) on **Rua Mem de Sá** (Mem de Sá) across from the local bus terminal or along Av das Cataratas every 30 minutes or so between 7am and 7:15pm (less on Sunday).
Itaipú Binacional
With a capacity of 14 million kilowatts, this binational dam is the world's second-largest hydroelectric power station, and the one that produces the most electricity per year. The impressive structure, at some 8km long and 200m high, is a memorable sight, especially when the river is high and a vast torrent of overflow water cascades down the spillway.
The visitor center ( %45 3576-7000; www.turismoitaipu.com.br; Tancredo Neves 6702; h8am-6pm Sun-Thu, to 9pm Fri & Sat) is 12km north of Foz. From here, regular tours (visita panorâmica) depart every 20 minutes between 8:30am to 4:45pm. A variety of other attractions within the complex include a museum, wildlife park and river beaches, and a sunset catamaran tour.
The project was controversial: it plunged Brazil deeply into debt and necessitated the large-scale destruction of rainforest and the displacement of 10,000 people. But it cleanly supplies 75% of Paraguay's energy needs, and 17% of Brazil's.
To get to the Itaipú Dam, catch bus 101 (Vila C Norte-Sohab) or 102 (Vila C Sul-Profilurb), among others (R$3.45, 30 minutes). These leave every 30 minutes from Foz do Iguaçu's Urban Bus Terminal (TTU) (Terminal de Transporte Urbano; %2105-1385; Av Juscelino Kubitschek 1385).
MARKUS DANIEL / GETTY IMAGES ©
## Puerto Iguazú, Argentina
At the end of the road in Argentina, Puerto Iguazú sits at the confluence of the Ríos Paraná and Iguazú and looks across to Brazil and Paraguay. There's little feeling of community: everyone is here to see the falls or to make a buck out of them, and planning laws seem nonexistent as hotels go up on every street. Still, it's not unattractive and is quiet and safe, with good transportation connections. There are also many excellent places to stay and eat.
1Sights
Casa Ecológica de BotellasArchitecture
(<http://lacasadebotellas.googlepages.com>; RN 12, Km 5; AR$100; h9am-6:30pm) S
About 300m off the falls road, this fascinating place is well worth a visit. The owners have taken used packaging materials – plastic bottles, juice cartons and the like – to build not only an impressive house, but furnishings and a bunch of original handicrafts that make unusual gifts. The guided visit talks you through their techniques.
Güirá OgaAnimal Sanctuary
(www.guiraoga.com.ar; RN 12, Km 5; adult/child AR$200/150; h9:30am-6pm, last entry 4:45pm) S
On the way to the falls, this is an animal hospital and center for rehabilitation of injured wildlife. It also carries out valuable research into the Iguazú forest environment and has a breeding program for endangered species. You get walked around the jungly 20-hectare park by one of the staff, who explains about the birds and animals and the sad stories of how they got there. The visit takes about 90 minutes.
Full Moon Over the Falls
For five consecutive nights per month, these guided walks ( %03757-491469; www.iguazuargentina.com/en/luna-llena) visit the Garganta del Diablo. There are three departures nightly. The first, at 8pm, offers the spectacle of the inflated rising moon; the last, at 9:30pm, sees the falls better illuminated. The price (AR$1100) includes admission and a drink; dinner is extra (AR$500). Book ahead as numbers are limited.
Extra buses from Puerto Iguazú cater for moonwalkers. If the weather is bad, the tour is postponed to the following night or a refund is offered, so allow some flexibility in your plans.
PICTUREGARDEN / GETTY IMAGES ©
2Activities & Tours
Numerous local operators offer day tours to the Brazilian side of the Iguazú Falls, some taking in the Itaipú Dam and Paraguay shopping as well. Many have offices at the bus terminal. At the port, there are various options for boat trips around the junction of the Paraná and Iguazú rivers. Boats leave every 30 minutes during peak periods. Expect to pay around AR$150 for a one-hour cruise in a small boat.
Iguazú Bike ToursCycling
( %03757-15-678220; www.iguazubiketours.com.ar)
These guys run anything from gentle jaunts through the nearby forest to long vehicle-supported rides through the lesser-visited corners of Parque Nacional Iguazú.
Iguazú KayakKayaking
( %03757-549930; www.iguazukayak.com; AR$700)
Take a paddle between three countries along the Río Iguazú and down to the Río Paraná on this three-hour, 8km kayaking trip.
Cruceros IguazúBoating
( %03757-421111; www.crucerosiguazu.com; Zona Puerto; 2hr cruise AR$390)
Daily two-hour late-afternoon cruises on the Paraná and Iguazú rivers in a large air-conditioned catamaran with a bar and live music on board. You can buy snacks but nothing gourmet.
5Eating
FeriaMarket $
(Feirinha; cnr Av Brasil & Félix de Azara; picadas AR$250-350; h8am-midnight)
A really nice place to eat or have a beer is this market in the north of town. It's full of stalls selling Argentine wines, sausages, olives and cheese to visiting Brazilians, and several of them put out _picadas,_ grilled meats, other simple regional dishes and cold beer. There's folk music some nights and a good evening atmosphere.
BocamoraArgentine $$
( %03757-420550; www.bocamora.com; Av Costanera s/n; mains AR$145-250; hnoon-11pm Tue-Sun; W)
A romantic location overlooking two rivers and three nations is reason enough to come to this place just down the hill from the Argentine border marker. It specializes in grilled meats and well-prepared plates of river fish; the food is decent, service is very hospitable and the view is just breathtaking.
La RuedaArgentine $$$
( %03757-422531; www.larueda1975.com; Av Córdoba 28; mains AR$245-480; h5:30pm-midnight Mon-Tue, from noon Wed-Sun; W)
A mainstay of upmarket eating in Puerto Iguazú, this culinary heavyweight still packs a punch. The salads are inventive and delicious, as are the imaginative river-fish (mostly pacú and surubí) creations. Meat with a variety of sauce options is reliably good; the homemade pasta is cheaper but doesn't disappoint. Service is good but slow. The wine list has a high flagfall.
6Drinking & Nightlife
Quita PenasBar
( %03757-458223; Av Brasil 120; h6pm-late)
A happening open-air bar in the middle of Puerto Iguazú's little nightlife strip, Quita Penas has an elevated deck and a variety of other spaces below. It serves good food and is a fine place to sink some beers and watch the action. Often has live music.
8GETTING THERE & AWAY
**Aerolíneas Argentinas** ( %03757-420168; www.aerolineas.com.ar; Av Victoria Aguirre 295; h8am-noon & 3-7pm Mon-Fri, 8am-1pm Sat) flies from Iguazú to both Buenos Aires airports, Mendoza via Córdoba or Rosario, Salta and El Calafate (via BA). LAN (www.lan.com) and Andes (www.andesonline.com) also fly the Buenos Aires route. **Four Tourist Travel** ( %03757-420681, 03757-422962; www.ftt.tur.ar) runs an airport shuttle for AR$130 per person that meets most flights.
The **bus terminal** (cnr Avs Córdoba & Misiones) has departures for all over the country.
8GETTING AROUND
Parque Nacional Iguazú is 20km southeast of Puerto Iguazú. From Puerto Iguazú's bus terminal, buses leave every 20 minutes for the park (AR$75, 40 minutes) between 7:20am and 7:20pm, with return trips between 7:50am and 8pm. The buses make flag stops along the road.
A taxi from town to the park entrance is AR$300 (AR$550 return).
8BORDER CROSSING
If traveling by bus, at Brazilian immigration in either direction, most bus drivers won't wait around while you finish visa formalities. Officially, you should get a pass from the driver, get your passport stamped, then wait and reboard the next bus _from the same bus company_.
On the Argentine side, drivers wait, so with luck and patience you can do this for R$5/AR25.
It's important you pay attention as drivers ask if anyone needs to stop at immigration on the Brazilian side – but in Portuguese (or Spanish), if at all. Many travelers miss it and end up with serious immigration hassles later (ie hefty fines).
At Argentine immigration, the bus always stops and usually waits for everyone to be processed. Both borders are open 24 hours but bus service ends around 7:15pm.
Many nationalities, including US citizens, require visas to enter Brazil. It's best to arrange your visa in your home country before coming to South America.
# Buenos Aires, Argentina
#### Tango in Buenos Aires
#### Plaza de Mayo
#### Cementerio de la Recoleta
#### Parrillas
#### Walking Tour: Historical Saunter
#### Sights
#### Shopping
#### Eating
#### Drinking & Nightlife
#### Entertainment
#
Buenos Aires, Argentina
_Buenos Aires combines faded European grandeur with Latin passion. Sexy and alive, this city gets under your skin. It's also undeniably beautiful, with grand French- and Italian-style palaces, as well as a vibrant street-art scene._
_BA's food scene is increasingly dynamic, but for many it's the city's carnivorous pleasures that shine. _Parrillas_ (steakhouses) sit on almost every corner, and make a fine prelude to the city's relentless nightlife. A diverse range of bars, clubs and live-music venues has everything from DJs spinning electronica to live jazz sets, plus plenty of venues for honing those tango moves._
La Boca | ELXENEIZE / SHUTTERSTOCK ©
Two Days in Buenos Aires
Spend day one strolling the colonial, cobbled streets of San Telmo. Have dinner and catch a tango show at El Viejo Almacén. On day two, explore the gardens and blockbuster museums of Palermo. Have a decadent meat-centric dinner at the _parrilla_ La Carnicería, followed by cocktails at Verne.
Four Days in Buenos Aires
On day three, explore the historic heart of Buenos Aires in and around the Plaza de Mayo. Later that night, catch a performance at the iconic Teatro Colón. On the next day, explore Recoleta's famous cemetery, have afternoon tea at L'Orangerie, and spend the evening enjoying top-notch tango at Café de los Angelitos.
Arriving in Buenos Aires
BA's main airport is **Ezeiza airport** (EZE; officially Aeropuerto Internacional Ministro Pistarini). A good way to get downtown is with a shuttle service, such as **Manuel Tienda León** (www.tiendaleon.com).
To catch a taxi, walk past the touts, and you'll see the freestanding city taxi stand, with a blue sign saying **Taxi Ezeiza** (www.taxiezeiza.com.ar). A taxi from the airport to the city center should cost about AR$1290.
Where to Stay
Buenos Aires' hostels range from basic no-frills deals to beautiful, multiperk buildings more luxurious than your standard cheap hotel. Hotels vary from utilitarian to luxurious five-star hotels. Palermo has dozens of boutique hotels; most are pricey but beautiful, with a handful of hip, elegant rooms and decent service. For a quick overview of the pros and cons of BA's diverse neighborhoods, see Where to Stay.
TOP EXPERIENCE
# Tango in Buenos Aires
Buenos Aires' famous dance is possibly the city's greatest contribution to the world: a steamy strut featuring grand feats of athleticism. You'll find supremely entertaining tango shows as well as milongas (dance salons) and dance schools.
Great For...
drh
yDon't Miss
Tango performances at the legendary Esquina Carlos Gardel.
8Need to Know
For discount tickets, show and venue descriptions, and some reviews, check out www.tangotix.com.
5Take a Break
A San Telmo gem, La Poesía (map; %011-4300-7340; Chile 502; mains AR$57-200; h8am-2am Sun-Thu, to 4am Fri & Sat; bLínea C Independencia) is a vintage 19th-century cafe with good snacks and live music.
oTop Tip
For a unique outdoor experience, head to the bandstand at the Barrancas de Belgrano, where there's live tango on Saturday and Sunday evenings around 7pm.
### Café de los Angelitos
This cafe (map Google map; %011-4952-2320; www.cafedelosangelitos.com; Av Rivadavia 2100; show from US$90, show & dinner from US$130; hcafe 8am-midnight; bLínea A Congreso) puts on one of the best tango shows in Buenos Aires. The performers dress in top-notch costumes and use interesting props, such as drapes and moving walls. They also dance to modern tunes, including those by local band Bajofondo, and despite a nightclub feel at times – especially due to the lighting – it's all tastefully and creatively done.
### El Beso
This small, intimate dance salon (map Google map; %011-4953-2794; <http://elbeso.com.ar>; Riobamba 416, 1st fl; bLínea B Callao) hosts various different _milongas_ ; the most popular is Milonga Cachirulo on Tuesdays from 9pm to 5am. The Cachirulo attracts some very good dancers – you should be confident of your skills if you plan to take to the floor.
### Tango Porteño
Staged in a renovated art-deco theater, this tango show (map Google map; %011-4124-9400; www.tangoporteno.com.ar; Cerrito 570; show from US$28, show & dinner from US$120; bLínea D Tribunales) features snippets of old footage interspersed with plenty of athletic (and at times sensual) dancing. There's an interesting blindfold number and the orchestra is excellent. There's a complimentary tango class before the show.
Tango show at El Viejo Almacén | LOOK / ALAMY STOCK PHOTO ©
### Tango Trips
Tango Trips ( %011-5235-4923; www.tangotrips.com; tour for 1/2/3 people US$160/190/240) offers private tours to _milongas_ with experienced and passionate _tangueros_ (tango dancers). Start with a private tango lesson to gain confidence before hitting the salons; if you're not a dancer, just sit back and watch tango danced in its most authentic form.
### Salón Canning
Some of BA's finest dancers (no wallflowers here) grace this traditional venue (map; %011-15-5738-3850; www.parakultural.com.ar; Av Scalabrini Ortiz 1331; g141, 140, 15) with its great dance floor. Well-known tango company Parakultural stages good events on Monday, Tuesday and Friday, involving orchestras, singers and dance performances; check the website for class times and the milonga musical lineup. Expect big crowds and plenty of tourists.
### La Viruta
La Viruta (map Google map; %011-4774-6357; www.lavirutatango.com; Armenia 1366; g141, 15, 110) is a popular basement venue in Palermo. Good beginner tango classes are available before the _milongas_ – translating into many inexperienced dancers on the floor earlier on – so if you're an expert get here late (after 2am).
### Esquina Carlos Gardel
One of the fanciest tango shows in town plays at this impressive 430-seat theater (map Google map; %011-4867-6363; www.esquinacarlosgardel.com.ar; Carlos Gardel 3200; show from US$96, show & dinner from US$140; bLínea B Carlos Gardel), where Carlos Gardel once hung out. This fine show highlights passionate, top-notch musicians and performers in period costumes, though a modern segment involving a skin suit is cutting-edge, athletic and memorable.
Esquina Carlos Gardel | WIM WISKERKE / ALAMY STOCK PHOTO ©
### DNI Tango
This excellent tango school (map Google map; %011-4866-6553; <https://dni-tango.com>; Bulnes 1101; group class per person AR$130; bLínea B Medrano) offers group and private classes in English and Spanish for all levels. For those starting out, the Saturday afternoon práctica is a friendly place to dance with different partners without the pressure of taking to the floor at a more formal milonga.
### El Viejo Almacén
One of BA's longest-running shows (since 1969), this venue (map Google map; %011-4307-7388; www.viejoalmacen.com; Balcarce 799; show from US$90, show & dinner from US$140; g29, 24, 111) is a charming old building from the 1800s. Dinner is served at a multistory restaurant in the main building, then everyone heads across the street to the small theater with an intimate stage, highly athletic dancers and plenty of glitz.
### Complejo Tango
For those who wish to not only watch tango but also experience it, there's this classy venue ( %011-4941-1119; www.complejotango.com.ar; Av Belgrano 2608; show US$55, dinner & show from US$125; bLínea A Plaza Miserere) in Once. The first hour is a free beginners' tango lesson, which you can follow with a tasty dinner, then an excellent tango show – be warned that performers go around toward the end, picking out audience members to dance with them (usually badly).
yWhat's On
Hoy Milonga (www.hoy-milonga.com) is a useful website listing the day's _milonga_ schedule.
### Rojo Tango
This sexy performance is the tango show (map Google map; %011-4952-4111; www.rojotango.com; Martha Salotti 445, Faena Hotel; show US$220, show & dinner US$290; g111, 43, 143) to top all others, though it comes with a hefty price tag. Offering only 100 seats, the Faena's cabaret room is swathed in blood-red curtains and gilded furniture. The show itself loosely follows the history of the tango, starting from its cabaret roots to the modern fusions of Ástor Piazzolla.
### La Ventana
This long-running basement venue (map Google map; %011-4334-1314; www.laventanaweb.com; Balcarce 431; show from US$70, show & dinner from US$120; bLínea E Belgrano) is located in an old converted building with rustic brick walls in San Telmo. The tango show includes a folkloric segment with Andean musicians and a display of _boleadores_ (gaucho hunting weapons). There's also a patriotic tribute to Evita, and the dinner offers a wide variety of tasty main dishes – unusual for tango shows.
### Maldita Milonga
Held on Wednesday at Buenos Ayres Club, Maldita Milonga (map Google map; %011-15-2189-7747; www.facebook.com/malditamilonga1; Perú 571, Buenos Ayres Club; hclass 9pm, milonga 10:30pm Wed; bLínea E Belgrano) is a well-run and popular event, and it's one of the best places to see tango being danced by real couples. The highlight of the night is when the dynamic orchestra El Afronte plays at 11pm; at midnight there's a professional dance demonstration.
z Tango Festival
If you're in Buenos Aires in mid- to late August, don't miss the celebrated Tango BA Festival y Mundial.
TOP EXPERIENCE
# Plaza de Mayo
Plaza de Mayo is the city's political, social and symbolic center. Surrounded by the Casa Rosada, the Cabildo and the city's main cathedral, the plaza has borne witness to many key events from Argentina's turbulent history.
Great For...
vrA
yDon't Miss
The striking Pirámide de Mayo at the center of the plaza and one of the icons of Buenos Aires.
8Need to Know
map; cnr Av de Mayo & San Martín; bLínea A Plaza de Mayo
5Take a Break
For modern Argentine food washed down with exquisite malbec, head to Aldo's Restoran & Vinoteca.
oTop Tip
Come on Sunday, when the Feria de San Telmo stretches all the way along Calle Defensa from Av San Juan to Plaza de Mayo.
### The City's Historic Heart
When Juan de Garay refounded Buenos Aires in 1580, he laid out the large Plaza del Fuerte (Fortress Plaza) in accordance with Spanish law. Later called the Plaza del Mercado (Market Plaza), then the Plaza de la Victoria (after victories over British invaders in 1806 and 1807), the plaza acquired its present name of Plaza de Mayo after the date Buenos Aires declared independence from Spain: May 25, 1810.
At the center of the plaza is the **Pirámide de Mayo** , a white obelisk built to mark the first anniversary of independence from Spain. Looming on the plaza's northern side is the headquarters of Banco de la Nación (map Google map; Bartolomé Mitre 326; bLínea D Catedral), the work of famed architect Alejandro Bustillo. Most other public buildings in this area belong to the late 19th century, when the Av de Mayo first connected the Casa Rosada with the **Plaza del Congreso** , obliterating much of the historic and dignified Cabildo (map Google map; %011-4342-6729; <https://cabildonacional.cultura.gob.ar>; Bolívar 65; h10:30am-5pm Tue, Wed & Fri, to 8pm Thu, to 6pm Sat & Sun; bLínea A Perú) F in the process.
### Madres de Plaza de Mayo
Plaza de Mayo has long been the preferred site of many civil protests; note the unsightly barricades separating the plaza in two, meant to discourage large numbers of _piqueteros_ (picketers) from congregating. But these barricades haven't prevented the **Madres de la Plaza de Mayo** – the mothers of the 'disappeared,' those abducted by the state during the military dictatorship of 1976–1983, from gathering in the plaza every Thursday afternoon at 3:30pm since 1977, and circling the pyramid holding photographs of their missing children. To this day they march on as a reminder of the past – and for other social justice causes.
Plaza de Mayo and Casa Rosada | JEREMY WOODHOUSE / GETTY IMAGES ©
### Casa Rosada
On the eastern side of Plaza de Mayo stands the Casa Rosada, named for its distinctive color. It was from the balcony here that Eva Perón famously addressed the throngs of impassioned supporters packed into Plaza de Mayo. The building houses the Argentine president's offices; the presidential residence is in the suburb of Olivos, north of the center. Free hourlong guided tours are given at weekends and must be booked online in advance; bring ID.
The building occupies the site where colonial riverbank fortifications once stood; today, however, after repeated landfills, the palace stands more than 1km inland. The Museo Casa Rosada (map Google map; %011-4344-3802; cnr Av Paseo Colón & Hipólito Yrigoyen; h10am-6pm Wed-Sun; bLínea A Plaza de Mayo) F is located behind the palace.
TOP EXPERIENCE
# Cementerio de la Recoleta
Wander for hours in this city of the dead, where countless 'streets' are lined with impressive statues and marble sarcophagi. Peek into the crypts, check out the dusty coffins and try to decipher the history of its inhabitants.
Great For...
Avh
yDon't Miss
The striking tomb of Evita (Eva Perón), the most visited landmark in the cemetery.
8Need to Know
map Google map; %0800-444-2363; visitasguiadasrecoleta@buenosaires.gob.ar; Junín 1760; h7am-5:30pm; bLínea H Las Heras; F
5Take a Break
Go for a meal at the traditional neighborhood restaurant Rodi Bar, which serves classic Argentine fare.
oTop Tip
For an overlook of the cemetery, head to the cloisters museum at the Basílica de Nuestra Señora del Pilar.
### Illustrious Inhabitants
Originally the vegetable garden of the monastery next door, Recoleta cemetery was created in 1822. It covers four city blocks and contains about 4800 mausoleums decorated in many architectural styles, including art nouveau, art deco, classical, Greek, baroque and neo-Gothic. Popular motifs include crosses of all kinds, marble angels, stone wreaths, skulls and crossbones, draped urns, winged hourglasses and the occasional gargoyle. All decorate the final resting places of past presidents, military heroes, influential politicians, famous writers and other very noteworthy personages.
The most impressive tomb is not Evita's, though it's certainly the most visited. Get a good map and look for other sarcophagi; interesting stories, odd facts and myths abound. Also note the cemetery's rough edges – the cobwebs and detritus inside many of the tombs, the vegetation growing out of cracks, the feral cats prowling the premises. All add to the charm.
### Tours
Free tours are offered in Spanish at 11am and 2pm from Tuesday to Sunday and at 11am and 3pm on Saturday and Sunday (weather permitting). For a great map and information, order Robert Wright's PDF guide (www.recoletacemetery.com); touts also sell maps at the entrance.
### Basílica de Nuestra Señora del Pilar
A short stroll from the cemetery is this gleaming white colonial church (map; %011-4806-2209; www.basilicadelpilar.org.ar; Junín 1904; museum adult/child AR$25/free; hmuseum 10:30am-6pm Mon-Sat, 2:30-6pm Sun; bLínea H Las Heras), built by Franciscans in 1732. Its centerpiece is a Peruvian altar adorned with silver from Argentina's northwest. Inside, head to the left to visit the small but historic cloisters museum; it's home to religious vestments, paintings, writings and interesting artifacts, and there are good views of Recoleta cemetery.
On the left-hand side of the courtyard as you enter the church, look for a ceramic tiled artwork depicting Buenos Aires as it was in 1794, back when the church stood in open countryside outside the town.
### Centro Cultural Recoleta
Before or after visiting the cemetery, be sure to head to the nearby Centro Cultural Recoleta (map Google map; %011-4803-1040; www.centroculturalrecoleta.org; Junín 1930; bLínea H Las Heras) F. Part of the original Franciscan convent and alongside its namesake church and cemetery, this excellent cultural center houses a variety of facilities, including art galleries, exhibition halls and a cinema.
Tomb of Rufina Cambaceres | MANVMEDIA / SHUTTERSTOCK ©
TOP EXPERIENCE
# Parrillas
Argentines have perfected the art of grilling beef on the _asado_ _(barbecue). Succulent char-grilled meats, matched with rich red wines, are served up in hallowed_ _parrillas(steakhouses) all across the city_.
Great For...
krb
yDon't Miss
A sizzling plate of _bife de chorizo_ (sirloin), a popular thick and juicy cut.
8Need to Know
Parrillas are generally open daily from noon to 3:30pm for lunch and 8pm to midnight or 1am for dinner.
5Take a Break
For dining in a hurry, stop by the _parrilla_ Lo de Freddy (map Google map; Bolívar 950; choripanes AR$50, vaciopanes AR$100; h1:30-5:30pm & 8-11:30pm Mon-Fri, 1-11:30pm Sat & Sun; g29) for delectable _choripanes_ (sausage sandwiches).
yDon't Miss
_Chimichurri,_ a tasty sauce made with olive oil, garlic and parsley. It adds a tantalizing spiciness.
### Don Julio
Classy service and a great wine list add an upscale bent to this traditional – and very popular – corner steak house (map; %011-4832-6058; www.parrilladonjulio.com.ar; Guatemala 4691; mains AR$296-657; hnoon-4pm & 7:30pm-1am; bLínea D Plaza Italia). The _bife de chorizo_ (sirloin steak) is the main attraction here, but the baked goat cheese provolone, _bondiola de cerdo_ (pork shoulder) and gourmet salads are a treat as well, and portions are large. Reserve ahead.
Don Julio | CAVAN / ALAMY STOCK PHOTO ©
### La Carnicería
The menu at this modern parrilla (map Google map; %011-2071-7199; www.facebook.com/xlacarniceriax; Thames 2317; mains AR$360-380; h8pm-midnight Tue-Fri, 1-3:30pm & 8pm-midnight Sat & Sun; bLínea D Plaza Italia) is limited, but everything on it is spectacular, from the crispy _provoleta_ (barbecued cheese), to the homemade chorizo and morcilla, to the tenderloin and rib cuts. This is a place for serious meat lovers who won't be put off by the butcher-themed decor. Portions are huge. Reserve ahead.
### La Cabrera
La Cabrera (map Google map; %011-4832-5754; www.lacabrera.com.ar; José Antonio Cabrera 5099; mains AR$388-562; h12:30-4:30pm & 8:30pm-midnight Sun-Thu, to 1am Fri & Sat; g140, 34) is hugely popular for grilling up some of BA's most sublime meats, so soft they can be cut with a spoon. Steaks weigh in at 400g or 800g and arrive with many little complimentary side dishes. Come at 7pm for happy hour, when everything is 40% off – arrive early to score a table; reservations are taken from 8:30pm only.
_Chimichurri_ (herb sauce) | ILDI PAPP / SHUTTERSTOCK ©
### La Boca
The same family has been running El Obrero ( %011-4362-9912; Agustín R Caffarena 64; mains AR$175-360; hnoon-4pm & 8pm-midnight Mon-Sat; g130, 168, 29) since 1954, and a number of famous people have passed through over the years, including Bono and Robert Duvall (check out the photos on the walls). You'll also see old Boca Juniors jerseys, antique furniture, old tile floors and chalkboards showing the day's specials and standard parrilla fare. Take a taxi.
### Parrilla Tour
On this foodie tour ( %011-15-4048-5964; www.parrillatour.com; per person US$85; hPalermo tour noon Tue, Fri & Sat, San Telmo tour noon Mon & Wed), you'll meet your knowledgeable guide at a restaurant for a _choripán_ , then an empanada. You'll finish at a local _parrilla_.
Central Buenos Aires
1Sights
1Banco de la NaciónE5
2Basílica de Nuestra Señora del PilarB1
3CabildoD6
4Cementerio de la RecoletaB2
5Centro Cultural KirchnerE5
6Centro Cultural RecoletaB1
7El Zanjón de GranadosE7
Feria de San Telmo(see 16)
8Manzana de las LucesD6
9Mercado de San TelmoE8
10Museo Casa RosadaE6
11Museo de Arte Contemporáneo Buenos AiresE8
12Museo de Arte Moderno de Buenos AiresE8
13Museo Nacional de Bellas ArtesA1
14Pasaje de la DefensaE8
15Plaza de MayoE6
16Plaza DorregoE8
17Teatro ColónC4
7Shopping
18Arte y EsperanzaE6
19El Ateneo Grand SplendidA3
20Feria Artesenal Plaza FranciaB2
21Materia UrbanaE7
5Eating
22Aldo's Restoran & VinotecaE6
23Café San JuanE8
24D'OroD6
25El Refuerzo Bar AlmacenD8
26La PoesiaE7
Lo de Freddy(see 9)
27L'OrangerieB2
28Rodi BarB2
6Drinking & Nightlife
29Bar Plaza DorregoE8
30Florería AtlánticoD2
31Gran Bar DanzónC3
3Entertainment
32Café de los AngelitosA6
33El BesoA5
34El Viejo AlmacénE7
35La VentanaE6
36Maldita MilongaD7
37Rojo TangoF7
38Tango PorteñoC4
1Sights
### 1City Center
Centro Cultural KirchnerCultural Center
(map Google map; %0800-333-9300; www.culturalkirchner.gob.ar; Sarmiento 151; h1-8pm Wed-Sun; bLínea B Alem) F
It was former president Néstor Kirchner who, in 2005, first proposed turning the abandoned former central post office into a cultural center. He died in 2010 before the project was completed, and the breathtaking cultural center was named in his honor. Within the vast beaux-arts structure – which stands eight stories tall and takes up an entire city block – are multiple art galleries, events spaces and auditoriums. The highlight, however, is the Ballena Azul, a concert hall with world-class acoustics that seats 1800.
Centro Cultural Kirchner | DIEGO GRANDI / SHUTTERSTOCK. CCK ARCHITECTS ARE ENRIQUE, FEDERICO AND NICOLáS BARES ©
Manzana de las LucesNotable Building
(map Google map; Block of Enlightenment; %011-4342-6973; Perú 222; h10am-7:30pm Mon-Fri, 2-8pm Sat & Sun; bLínea E Bolívar) F
In colonial times, the Manzana de las Luces was Buenos Aires' most important center of culture and learning, and today the block still symbolizes education and enlightenment. Two of the five original buildings remain; Jesuit defensive tunnels were discovered in 1912. Free tours in Spanish are given at 2pm from Monday to Friday, but you can go inside and see the main patio area without taking a tour.
Teatro ColónTheater
(map Google map; %011-4378-7100; www.teatrocolon.org.ar; Tucumán 1171, Cerrito 628; tours AR$500-600; htours 9am-5pm; bLínea D Tribunales)
This impressive seven-story building is one of BA's most prominent landmarks. It's the city's main performing-arts venue and a world-class forum for opera, ballet and classical music, with astounding acoustics. Occupying an entire city block, the Colón can seat 2500 spectators and provide standing room for another 500. The theater's real beauty lies on the inside, so if you can't get hold of tickets to a performance, take one of the frequent 50-minute backstage tours to view the stunning interior.
Feria de San Telmo
On Sundays, San Telmo's main drag (map Google map; Defensa; h10am-6pm Sun; g10, 22, 29, 45, 86) is closed to traffic and the street is a sea of both locals and tourists browsing craft stalls, waiting at vendors' carts for freshly squeezed orange juice, poking through the antique glass ornaments on display on Plaza Dorrego, and listening to street performances by myriad music groups. Runs from Avenida San Juan to Plaza de Mayo. It's a tight and crowded scene, so be prepared to bump into people and watch your bag carefully.
Street performers | TERRY CARTER/LONELY PLANET ©
### 1La Boca
El CaminitoStreet
(map; Av Don Pedro de Mendoza, near Del Valle Iberlucea; g33, 64, 29, 168, 53) F
La Boca's most famous street and 'open-air' museum is a magnet for visitors, who come to see its brightly painted houses and snap photographs of the figures of Juan and Eva Perón, Che Guevara and soccer legend Diego Maradona, who wave down from balconies. (Expect to pay a few pesos to take pictures of tango dancers or pose with props.) Sure, it could be called a tourist trap, but don't let that put you off.
Museo Benito Quinquela MartínMuseum
(map; %011-4301-1080; www.buenosaires.gob.ar/museoquinquelamartin; Av Don Pedro de Mendoza 1835; AR$40; h10am-6pm Tue-Fri, 11:15am-6pm Sat & Sun; g33, 64, 29)
Once the home and studio of painter Benito Quinquela Martín (1890–1977), this fine-arts museum exhibits his works and those of other Argentine artists. Quinquela Martín used silhouettes of laboring men, smokestacks and water reflections as recurring themes, and painted with broad, rough brushstrokes and dark colors. Don't miss the colorful tiles of his former kitchen and bathroom, his hand-painted piano and the sculptures on the rooftop terraces; the top tier has awesome views of the port.
Palermo
1Sights
1Museo de Arte Latinoamericano de Buenos AiresD2
2Parque 3 de FebreroC1
2Activities, Courses & Tours
3DNI TangoC5
7Shopping
4Elementos ArgentinosB4
5Eating
5Don JulioB3
6Il MatterelloA4
7La CabreraA4
8La CarniceríaB3
9ProperB4
6Drinking & Nightlife
10On TapA3
11VerneC4
3Entertainment
12Esquina Carlos GardelD6
13La VirutaA4
14Salón CanningB4
### 1Palermo
Museo de Arte Latinoamericano de Buenos AiresMuseum
(map Google map; MALBA; %011-4808-6500; www.malba.org.ar; Av Figueroa Alcorta 3415; adult/student Thu-Mon AR$140/70, Wed AR$70/free; hnoon-8pm Thu-Mon, to 9pm Wed; g102, 130, 124)
Sparkling inside its glass walls, this airy modern art museum is one of BA's most impressive. It displays the fine collection of Latin American art of millionaire and philanthropist Eduardo Costantini, including works by Argentines Xul Solar and Antonio Berni, as well as pieces by Mexicans Diego Rivera and Frida Kahlo. The temporary exhibitions here are usually world class so it's worth checking what's currently on offer. A cinema screens art-house films.
Museo Nacional de Bellas ArtesMuseum
(map Google map; %011-5288-9900; www.mnba.gob.ar; Av del Libertador 1473; h11am-8pm Tue-Fri, 10am-8pm Sat & Sun; g130, 92, 63) F
This is Argentina's most important fine art museum and contains many key works by Benito Quinquela Martín, Xul Solar, Eduardo Sívori and other Argentine artists, including a whole room of works by Antonio Berni. There are also pieces by European masters such as Cézanne, Degas, Picasso, Rembrandt, Toulouse-Lautrec and Van Gogh.
Parque 3 de FebreroPark
(map Google map; cnr Avs del Libertador & de la Infanta Isabel; g10, 34, 130)
Also known as Bosques de Palermo (Palermo Woods), this sweeping open parkland abounds with small lakes and pretty gazebos. Stands rent bikes and in-line skates, and joggers and power walkers circle the ponds – if you don't have the energy to join them, lie back under a tree and people-watch. There's also a monument to literary greats called El Jardín de los Poetas (the Garden of Poets), and the exquisite Rosedal (rose garden).
7Shopping
Arte y EsperanzaArts & Crafts
(map Google map; %011-4343-1455; www.arteyesperanza.com.ar; Balcarce 234; h10am-6pm Mon-Fri; bLínea A Plaza de Mayo)
This store sells fair-trade, handmade products that include many made by Argentina's indigenous craftspeople. Shop for silver jewelry, pottery, ceramics, textiles, _mate_ gourds, baskets, woven bags, wood utensils and animal masks.
Materia UrbanaArts & Crafts
(map Google map; %011-4361-5265; www.materiaurbana.com; Defensa 702; h11am-7pm Wed-Fri, from 2pm Sat & Tue, 11am-6pm Sun; g29)
This innovative design shop shows the work of over 100 local artists; cool finds include leather animal organizers, retro tote bags, plastic _mate_ sets and jewelry made from metal, wood and leather.
El Ateneo Grand SplendidBooks
(map Google map; %011-4813-6052; www.yenny-elateneo.com/local/grand-splendid; Av Santa Fe 1860; h9am-10pm Mon-Thu, to midnight Fri & Sat, noon-10pm Sun; bLínea D Callao)
This glorious bookstore in a converted theater continues to flourish in the age of the Kindle. The Grand Splendid theater opened in 1919 and was converted into a bookstore in 2000. Most of the seating was replaced with bookshelves, but the original features have been preserved, including the beautiful painted cupola and balconies.
El Ateneo Grand Splendid | FABRIZIO248 / SHUTTERSTOCK ©
Feria Artesanal Plaza FranciaMarket
(map; www.feriaplazafrancia.com; Plaza Intendente de Alvear; h11am-8pm Sat & Sun; bLínea H Las Heras)
Recoleta's popular artisan fair has dozens of booths and a range of creative, homemade goods. Hippies, mimes and tourists mingle. It's at its biggest on weekends, though there are usually a few stalls open during the week. Despite its name, the market is located just outside the Cementerio de la Recoleta in Plaza Intendente de Alvear.
Elementos ArgentinosHomewares
(map Google map; %011-4832-6299; www.elementosargentinos.com.ar; Gurruchaga 1881; h11am-7pm Tue-Sat; bLínea D Plaza Italia) S
The high-quality carpets, rugs and blankets sold here are hand-dyed, hand-woven on a loom, and fair trade; the owners work with cooperatives and NGOs to help the communities in northwestern Argentina where the textiles are produced. Larger items can be shipped home, or pick up a super-soft llama-wool blanket to squeeze into your suitcase.
5Eating
### 5City Center
D'OroItalian $$
(map Google map; %011-4342-6959; Perú 159; mains AR$190-300; hnoon-4pm Mon-Fri; bLínea E Bolívar)
This tiny, popular lunch spot is a serious Italian wine bar and restaurant to rival others in more gastronomically famous neighborhoods. Come for thin, crispy, oven-baked pizzas, mushroom risotto, fettuccine with shellfish and garlic-topped focaccia.
Aldo's Restoran & VinotecaArgentine $$$
(map Google map; %011-4334-2380; www.aldosvinoteca.com; Moreno 372; mains AR$360-490; hnoon-midnight Sun-Thu, to 1am Fri, 7pm-1am Sat; bLínea A Plaza de Mayo)
This restaurant and wine shop is an upscale eatery serving a small but tasty menu of meat, seafood and pasta dishes, all amid walls lined with wine. Look out for wine tastings held on Tuesday evenings, or ask your server to recommend one of the nearly 600 bottles in stock.
Proper Eating
Bringing new life to the BA food scene, the young chefs behind Proper (map Google map; %011-4831-0027; www.properbsas.com.ar; Aráoz 1676; small plates AR$85-320; h8pm-midnight Tue-Sat; W; g111, 160, 141) pack flavor into seasonal produce cooked in a wood-fired oven at their roughly finished restaurant in a former car-repair shop. Order several small plates to share and delight in dishes such as perfectly cooked lamb chops or sweet potato with Patagonian blue cheese, fried almonds and kale.
### 5La Boca
Il MatterelloItalian $$
(map Google map; %011-4307-0529; Martín Rodríguez 517; mains AR$200-260; h12:30-3pm & 8:30pm-midnight Tue-Sat, 12:30-3pm Sun; g29, 53, 152)
This Genovese trattoria serves up some of the best pasta in town, including perfectly al dente _tagliatelle alla rúcola_ (tagliatelle with arugula), and _tortelli bianchi con burro foso al aglio_ (pasta pillows stuffed with chard and Parmesan in a burned garlic sauce). For dessert there's a great tiramisu. Take a taxi here at night. Also in Palermo (map; %011-4831-8493; Thames 1490; mains AR$230-290; hnoon-4pm & 8pm-1am Tue-Sun; g140, 39).
### 5Recoleta & Barrio Norte
Rodi BarArgentine $$
(map Google map; %011-4801-5230; Vicente López 1900; mains AR$200-450; h8am-1am Mon-Sat; W; bLínea H Las Heras)
A great option for well-priced, unpretentious food in upscale Recoleta. This traditional neighborhood restaurant with a fine old-world atmosphere and extensive menu offers something for everyone, from inexpensive combo plates to relatively unusual dishes such as marinated beef tongue.
L'OrangerieFrench $$$
(map Google map; %011-4808-2949; Alvear Palace Hotel, Av Alvear 1891; full tea Mon-Fri AR$450, Sat & Sun $490; hafternoon tea 4:30-7pm Mon-Sat, 5-7pm Sun; g130, 92, 63)
Afternoon tea at the Alvear Palace Hotel's beautiful, flower-filled Orangerie is a classic Recoleta experience for a special occasion. The formal tea, served from 4:30pm (from 5pm on Sunday), offers an endless array of exquisite cakes, sandwiches and pastries as well as a selection of loose-leaf teas and background piano music.
L'Orangerie | KRZYSZTOF DYDYNSKI / LONELY PLANET ©
### 5Palermo
La Mar CebicheriaPeruvian $$$
( %011-4776-5543; <http://lamarcebicheria.com.ar>; Arévalo 2024; mains AR$320-520; h8pm-midnight Mon, 12:30pm-midnight Tue-Thu & Sun, to 12:30am Fri & Sat; W; bLínea D Ministro Carranza)
For lovers of fresh fish, this upmarket Peruvian _cebichería_ is a welcome addition to the meat-dominated BA food scene. The team at La Mar travel across Argentina to source their ingredients, and the quality of the fish is evident in the eating. The menu is lengthy, with a choice of Peruvian dishes, and there's plenty of outdoor seating.
### 5San Telmo
Café San JuanInternational $$$
(map Google map; %011-4300-1112; www.facebook.com/CafeSanJuanrestaurant; Av San Juan 450; mains AR$300-390; h12:30-4pm & 8pm-midnight Sun & Tue-Thu, to 1am Fri & Sat; bLínea C San Juan)
Having studied in Milan, Paris and Barcelona, TV-chef Leandro Cristóbal now runs the kitchen at this renowned San Telmo bistro. Start with fabulous tapas, then delve into the grilled Spanish octopus (AR$1100), _molleja_ (sweetbreads) cannelloni and amazing pork _bondiola_ (deliciously tender after nine hours' roasting). Reserve ahead for lunch and dinner.
El Refuerzo Bar AlmacenArgentine $$$
(map Google map; %011-4361-3013; www.facebook.com/elrefuerzobaralmacen; Chacabuco 872; mains AR$230-450; h10am-2am Tue-Sun; bLínea C Independencia)
The small dining room fills up quickly at this _almacén_ -style restaurant. There's an excellent wine list to match the menu of top-notch dishes written on blackboards on the walls – think cured meats, cheeses, homemade pastas and bistro-style casseroles. It's a casual, friendly place that's popular with locals.
6Drinking & Nightlife
Florería AtlánticoCocktail Bar
(map Google map; %011-4313-6093; <http://floreriaatlantico.com.ar>; Arroyo 872; h7pm-2am Mon-Wed, to 2:30am Thu, to 4am Fri, 8pm-4am Sat, 8pm-2am Sun)
This basement speakeasy is located within a flower shop, which adds an air of mystery and is likely a key reason for its success. Hipsters, artists, chefs, businesspeople and foreigners all flock here for the excellent cocktails, both classic and creative, accompanied by delicious tapas.
Florería Atlántico | MICHAEL THOMAS / ALAMY STOCK PHOTO ©
UptownBar
( %011-2101-4897; www.uptownba.com; Arévalo 2030; h8:30pm-2am Tue & Wed, to 3am Thu-Sat; bLínea D Ministro Carranza)
Descend the graffiti-strewn stairwell and step aboard the carriage of an actual New York subway train to reach BA's hottest bar, a cavernous underground space with DJs and dancing, and a smaller, cozier room styled as a pharmacy. It's a popular place, so call ahead to put your name on the guest list.
VerneCocktail Bar
(map Google map; %011-4822-0980; <http://vernecocktailclub.com>; Av Medrano 1475; h9pm-2am Sun-Tue, 8pm-2am Wed, 8pm-3am Thu, 8pm-4am Fri, 9pm-4am Sat; g160, 15)
This upscale yet casual bar has a vague Jules Verne theme. Cocktails are the house specialty, whipped up by one of BA's best bartenders, Fede Cuco. A few tables, some cushy sofas and an airy outdoor patio offer a variety of seating options, but plant yourself at the bar to watch the mixologists work their magic.
Gran Bar DanzónBar
(map Google map; %011-4811-1108; www.granbardanzon.com.ar; Libertad 1161; h7pm-2am Mon-Fri, from 8pm Sat & Sun; bLínea D Tribunales)
Upscale restaurant-wine bar with a good selection of wines by the glass as well as fresh fruit cocktails, exotic martinis and Euro- and Asian-inspired dinner selections. It's very popular, so come early and snag a good seat on a sofa.
On TapCraft Beer
(map Google map; %011-4771-5424; www.ontap.com.ar; Costa Rica 5527; h5pm-midnight Mon-Wed & Sun, to 1am Thu-Sat)
This popular pub has 20 Argentine microbrews on tap, including IPAs, pilsners, stouts, wheat porters and honey beers. It's more of a place to enjoy beers than to hang out – there's only counter seating and a communal table, though a few burgers and other pub food are available. Bring a growler for refills; happy hour runs from 5pm to 8:30pm.
3Entertainment
Usina del ArteConcert Venue
(www.buenosaires.gob.ar/usinadelarte; Agustín R Caffarena 1; g130, 86, 8) F
This former power station has been transformed into a spectacular concert venue in an effort to regenerate a somewhat sketchy area of La Boca. It's a gorgeous red-brick building complete with scenic clock tower, and the two concert halls have top-notch acoustics. Nearly all the art exhibitions, concerts and dance performances here are free; check the website for upcoming events.
Usina del Arte | BERNARDO GALMARINI / ALAMY STOCK PHOTO ©
8INFORMATION
There are several tourist offices and kiosks in Buenos Aires. Staff speak English and can provide maps and information about free guided walks and other activities.
**Ezeiza airport** ( %011- 4480-0224; Terminal A arrivals, 1st fl, Ezeiza Airport; h8:15am-7:15pm)
**Florida** (map; cnr Florida & Roque Sáenz Peña; h9am-6pm; bLínea D Catedral)
**La Boca** (<https://turismo.buenosaires.gob.ar>; Av Don Pedro de Mendoza 1901; h9am-6pm; W; g33, 64, 29)
**Plaza San Martín** (map; cnr Av Florida & Marcelo T de Alvear; h9am-6pm; bLínea C San Martín)
La Bombonera Stadium
Seeing Boca Juniors play at La Bombonera ( %011-5777-1200; www.bocajuniors.com.ar; Brandsen 805; g29, 53, 152) is one of the world's top spectator sports experiences, especially if the game happens to be the 'superclasico' derby match against River. Tickets are hard to come by – it's best to go via an agent like LandingPadBA (<http://landingpadba.com>). You can see the stadium during a visit to the Museo de la Pasión Boquense.
SUNSINGER / SHUTTERSTOCK ©
8GETTING THERE & AWAY
AIR
Most international flights arrive at Buenos Aires' Aeropuerto Internacional Ministro Pistarini, about 35km south of the Center. Ezeiza is a modern airport with ATMs, restaurants, a pharmacy and duty-free shops.
BUS
Buenos Aires' modern **Retiro bus terminal** (Retiro; map; %011-4310-0700; www.retiro.com.ar; Av Antártida Argentina; bLínea C Retiro) has buses to practically anywhere in Argentina and departures are fairly frequent to the most popular destinations.
8GETTING AROUND
BA's public transportation is efficient. Use Como Llego (<http://comollego.ba.gob.ar>), the city government website, to plot your journey, or download the free app.
**Bus** The city has hundreds of bus lines that can take you within a few blocks of any destination.
**Subte** BA's underground, or subway, is not difficult to figure out. It's a quick way to get around – though it gets hot and very crowded during rush hour.
**Taxi** Black-and-yellow street taxis are ubiquitous and generally fine.
# iWhere To Stay
_You'll find a wide range of places to rest your head here, from hostels to boutique hotels, guesthouses, rental apartments and international five-star hotels. Just remember to book ahead – or pay in cash – for the best deals._
Neighbourhood | Atmosphere
---|---
**The Center** | Great transportation options; fairly close to all neighborhoods except Palermo; limited eating, shopping and nightlife options; noisy and crowded during the day.
**Congreso & Tribunales** | Reasonably central, with traditional theater and cultural options, but limited shopping and dining; certain sections are desolate and less safe at night.
**San Telmo** | Endearing traditional atmosphere, reasonable shopping and nightlife, a good range of restaurants and many decent hostels; far from Palermo; some areas can be edgy at night; public transportation is somewhat limited.
**Retiro** | Beautiful upscale neighborhood within walking distance of Recoleta and the Center; convenient for public transportation; very expensive; limited accommodations options; not many affordable restaurants or shops.
**Recoleta & Barrio Norte** | Buenos Aires' most upscale neighborhood; gorgeous architecture, good transportation options and fairly safe; most accommodations, restaurants and shopping are very expensive.
**Palermo** | Many boutique hotels to choose from; the city's widest range of interesting restaurants; great shopping and nightlife; a bit of a trek to the Center and San Telmo; might be too touristy for some.
# Southern Patagonia, Argentina
#### Glaciar Perito Moreno
#### Hiking the Fitz Roy Range
#### El Calafate
#### El Chaltén
#
Southern Patagonia, Argentina
_On South America's southern frontier, nature grows wild, barren and beautiful. Spaces are large, as are the silences that fill them. For the newly arrived, such emptiness can be as impressive as the sight of Patagonia's jagged peaks, pristine rivers and dusty backwater oases. In its enormous scale, Patagonia offers a wealth of potential experiences and landscapes._
_Though now mostly paved, lonely RN 40 remains the iconic highway that stirred affection in personalities as disparate as Butch Cassidy and Bruce Chatwin. On the eastern seaboard, paved RN 3 shoots south, connecting oil boomtowns with ancient petrified forests, Welsh settlements and the incredible Península Valdés._
Glaciar Perito Moreno | AILTONSZA / SHUTTERSTOCK ©
Two Days in Southern Patagonia
Spend day one taking in the magnificent sight of Glaciar Perito Moreno on a cruise with Southern Spirit or on a horseback riding excursion with Cabalgatas del Glaciar. In the evening have a memorable meal at Pura Vida in El Calafate. On day two, head out on a four-hour hike with Glaciar Sur.
Four Days in Southern Patagonia
On your third day set out on a scenic full-day hike in the Fitz Roy Range: the Laguna de Los Tres trek is spectacular if you're fit. On day four, go rock-climbing with Patagonia Mágica. At trip's end treat yourself to a massage at Spa Yaten.
Arriving in Southern Patagonia
The modern **Aeropuerto El Calafate** (www.aeropuertoelcalafate.com/en/) is 23km east of town and has flights to Buenos, Bariloche and other destinations.
El Chaltén is 220km from El Calafate via smooth paved roads. Car hire is available in El Calafate; or you can take a bus (AR$600, 3½ hours, three daily in summer).
Where to Stay
For visiting the Glaciar Perito Moreno, the village of El Calafate has ample lodging options. The town of El Chaltén, near the Fitz Roy Range, likewise offers abundant lodging for all price points. Wherever you stay, book well ahead. If you bring a sturdy tent, however, there's always space in the campgrounds.
TOP EXPERIENCE
# Glaciar Perito Moreno
Among the Earth's most dynamic and accessible ice fields, Glaciar Perito Moreno is the stunning centerpiece of the southern sector of Parque Nacional Los Glaciares.
Great For...
gAf
yDon't Miss
The sight (and sound!) of huge icebergs calving into the Iceberg Channel.
8Need to Know
Glaciar Perito Moreno is inside Parque Nacional Los Glaciares (www.parquesnacionales.gob.ar/areas-protegidas/region-patagonia-austral/pn-los-glaciares; adult/child AR$600/150; h8am-6pm Sep-Easter, 9am-4pm Apr-Aug).
5Take a Break
There are no restaurants on-site. Bring a picnic from the nearby town of El Calafate.
oTop Tip
Visit late in the afternoon on sunny days for the most dramatic ice cracking.
### The Experience
The glacier measures 30km long, 5km wide and 60m high, but what makes it exceptional in the world of ice is its constant advance – up to 2m per day, causing building-sized icebergs to calve from its face.
Glaciar Perito Moreno is as much an auditory as a visual experience when huge icebergs calve and collapse into the Canal de los Témpanos (Iceberg Channel). This natural-born tourist attraction at Península de Magallanes is close enough to guarantee great views, but far enough away to be safe.
AGE FOTOSTOCK / ALAMY STOCK PHOTO ©
### Vantage Points
A series of steel catwalks (almost 4000m total) and vantage points allow visitors to see, hear and photograph the glacier. Sun hits its face in the morning and the glacier's appearance changes as the day progresses and shadows shift. A closed _refugio_ with glass walls allows for glacier viewing in bad weather.
### Cruises
Excursions by boat allow you to sense the magnitude of Glaciar Perito Moreno, still from a safe distance.
Southern Spirit (map Google map; %02902-491582; www.southernspiritfte.com.ar; Libertador 1319, El Calafate; 1hr cruise AR$1800; h9am-1pm & 4-8pm) has one- to three-hour cruises on Lago Argentino in southern Parque Nacional Los Glaciares to see Glaciar Perito Moreno. An active alternative includes hiking.
Solo Patagonia (map Google map; %02902-491115; www.solopatagonia.com; Libertador 867, El Calafate; h9am-12:30pm & 4-8pm) offers the Ríos de Hielo Express (AR$1350) from Puerto Punta Bandera, visiting Glaciar Upsala, Glaciar Spegazzini and Glaciar Perito Moreno. Weather may alter the route.
Mar Patag (map Google map; %02902-492118; www.crucerosmarpatag.com; Libertador 1319, El Calafate; day cruise AR$2550; h7am-9pm) has luxury cruises with a chef serving gourmet meals. The day trip leaves from the private port of La Soledad and visits Glaciar Upsala, with a four-course meal served on board. The two-day cruise (from US$1050 per person) leaves five times a month and also visits the Glaciares Mayo and Perito Moreno.
### Cabalgatas del Glaciar
Cabalgatas del Glaciar ( %02902-495447; www.cabalgatasdelglaciar.com; full day AR$2250) runs one-day and multiday riding or trekking trips with glacier panoramas to Lago Roca and Paso Zamora on the Chilean border. Tours include transfer from El Calafate and a steak lunch.
TOP EXPERIENCE
# Hiking the Fitz Roy Range
The Fitz Roy Range – with its rugged wilderness and shark-tooth summits – is the de facto trekking and climbing capital of Argentina. Numerous well-marked trails offer spectacular scenery.
Great For...
gcf
yDon't Miss
The view of the wildly beautiful glacial Laguna de los Tres.
8Need to Know
The Fitz Roy Range is in the northern swath of the Parque Nacional Los Glaciares.
5Take a Break
After a long hike, treat yourself to a pint and comfort fare at La Cervecería ( %02962-493109; Av San Martín 320; mains AR$100-190; hnoon-midnight).
oTop Tip
Before heading out on hikes, stop by El Chaltén's Park Ranger Office for updated trail conditions.
### Laguna de Los Tres
This hike to a high alpine tarn provides one of the most photogenic spots in Parque Nacional Los Glaciares. It's somewhat strenuous (10km and four hours one way) and best for those in good physical shape. Exercise extra caution in foul weather as trails are very steep.
Cerro Fitz Roy | JUDYTA JASTRZEBSKA / SHUTTERSTOCK ©
### Laguna Torre
Views of the stunning rock needle of Cerro Torre are the highlight of this 18km round-trip hike. If you have good weather – ie little wind – and clear skies, make this hike (three hours one way) a priority, since the toothy Cerro Torre is the most difficult local peak to see on normal blustery days.
### Loma del Pliegue Tumbado & Laguna Toro
Heading southwest from El Chaltén's Park Ranger Office, this trail (10km and four to five hours one way) skirts the eastern face of Loma del Pliegue Tumbado going toward Río Túnel, then cuts west and heads to Laguna Toro. It's less crowded than the main routes. The hike is gentle, but prepare for strong winds and carry extra water.
Hiking in Parque Nacional Los Glaciares | HAGEPHOTO / GETTY IMAGES ©
### Piedra del Fraile
This 16km round-trip (three hours one way) follows the Valle Río Eléctrico. There are some stream crossings with sturdy tree trunks to cross on and one bridge crossing; all are well-marked. From Hostería El Pilar walk 1km northeast on the main road to hit the signposted trailhead for Piedra del Fraile, near a big iron bridge.
### Guided Trips
Many companies offer multiday treks, guided climbs and kayaking trips.
Fitzroy Expediciones (Adventure Patagonia; %02962-436110; www.fitzroyexpediciones.com.ar; Av San Martín 56, El Chaltén; h9am-1pm & 2-8pm) runs trekking excursions, ice trekking on Glaciar Cagliero, kayaking and a five-day itinerary that includes trekking in the Fitz Roy and Cerro Torre area. Note that Fitzroy Expediciones does accept credit cards.
Chaltén Mountain Guides ( %02962-493329; www.chaltenmountainguides.com; Río de las Vueltas 212, El Chaltén) do ice-field traversing, trekking and mountaineering. Rates decrease significantly with group size. The office is in the Kaulem hotel.
The Casa de Guías ( %02962-493118; www.casadeguias.com.ar; Lago del Desierto 470, El Chaltén; h10am-1pm & 4:30-9pm) is friendly and professional, with English-speaking guides certified by the Argentine Association of Mountain Guides (AAGM). It specializes in small groups.
## El Calafate
Named for the berry that, once eaten, guarantees your return to Patagonia, El Calafate hooks you with another irresistible attraction: Glaciar Perito Moreno, 80km away in Parque Nacional Los Glaciares. This magnificent must-see has converted once-quaint El Calafate into a chic fur-trimmed destination. With a range of traveler services, it's still a fun place to be.
El Calafate
1Sights
1Centro de Interpretacíon HistoricoC1
2Activities, Courses & Tours
2Always GlaciersD3
3Chaltén TravelC3
4Glaciar SurC3
5Hielo y AventuraD3
Mar Patag(see 7)
6Solo Patagonia SAC3
7Southern SpiritB3
5Eating
8Buenos CrucesC2
9Mi RanchoC2
10OliviaC2
11Pura VidaA3
12Viva la PepaD3
6Drinking & Nightlife
13Borges y Alvarez Libro-BarC3
14La ZorraD3
3Entertainment
15Don Diego de la NocheB3
16La TolderíaC3
1Sights
Reserva Natural Laguna NimezWildlife Reserve
(AR$150; hdaylight hours)
Reserva Natural Laguna Nimez is a prime avian habitat alongside the lakeshore north of El Calafate, with a self-guided trail and staffed Casa Verde information hut with binoculars rental. It's a great place to spot flamingos – but watching birds from El Calafate's shoreline on Lago Argentino can be just as good.
Centro de Interpretación HistoricoMuseum
(map Google map; %02902-497799; www.museocalafate.com.ar; Brown & Bonarelli; AR$170; h10am-8pm Sep-May, 11am-5pm Jun-Aug)
Small but informative, with a skeleton mold of _Austroraptor cabazai_ (found nearby) and Patagonian history displays. The friendly host invites museum-goers for a post-tour _mate_ (a bitter ritual tea).
Kayaking near Glaciar Perito Moreno | YMGERMAN / GETTY IMAGES ©
TTours
Some 40 travel agencies arrange excursions to Glaciar Perito Moreno and other local attractions, including fossil beds and stays at regional _estancias,_ where you can hike, ride horses and relax. Tour prices for Glaciar Perito Moreno don't include the park entrance fee. Ask agents and other travelers about added benefits, such as extra stops, boat trips, binoculars or multilingual guides.
Glaciar SurAdventure
(map Google map; %02902-495050; www.glaciarsur.com; 9 de Julio 57; per person US$250; h10am-8pm)
Get glacier-stunned _and_ skip the crowds with these recommended day tours to the unexplored end of Parque Nacional Los Glaciares. Small groups drive to Lago Roca with an expert multilingual guide to view Glaciar Frías. The adventure option features a four-hour hike; the culture option includes a traditional _asado_ (barbecue grill) and off-hour visits to Glaciar Perito Moreno.
Trips run October through April, weather permitting.
Hielo y AventuraOutdoors, Cruise
(map Google map; %02902-492094, 02902-492205; www.hieloyaventura.com; Libertador 935)
Hielo y Aventura's conventional cruise Safari Nautico (AR$500, one hour) tours Brazo Rico, Lago Argentino and the south side of Canal de los Témpanos. Catamarans crammed with up to 130 passengers leave hourly between 10:30am and 4:30pm from Puerto Bajo de las Sombras. During busy periods buy tickets in advance for afternoon departures.
To hike on Glaciar Perito Moreno, try minitrekking (AR$2700, under two hours on ice), or the longer and more demanding Big Ice (AR$5200, four hours on ice). Both involve a quick boat ride from Puerto Bajo de las Sombras, a walk through lenga forests, a chat on glaciology and then an ice walk using crampons. Children under eight are not allowed; reserve ahead and bring your own food. Don't forget rain gear: it's often snowing around the glacier and you might quickly become wet and cold on the boat deck. Transfers cost extra (AR$1000).
Chaltén TravelTours
(map Google map; %02902-492212; www.chaltentravel.com; Libertador 1174; h9am-9pm)
Recommended tours to Glaciar Perito Moreno, stopping for wildlife-viewing (binoculars provided); also specializes in RN 40 trips. It outsources some excursions to Always Glaciers (map Google map; %02902-493861; www.alwaysglaciers.com; Libertador 924).
Glaciarium: The World of Ice
Unique and exciting, this gorgeous museum (map; %02902-497912; www.glaciarium.com; adult/child AR$480/200; h9am-8pm Sep-May, 11am-8pm Jun-Aug) illuminates the world of ice. Displays and bilingual films show how glaciers form, along with documentaries on continental ice expeditions and stark meditations on climate change. Adults suit up in furry capes for the _bar de hielo_ (AR$240 including drink), a blue-lit below-zero club serving vodka or fernet and cola in ice glasses.
The gift shop sells handmade and sustainable gifts crafted by Argentine artisans. Glaciarium also hosts international cinema events. It's 6km from Calafate toward Parque Nacional Los Glaciares. To get here, take the free hourly shuttle from 1 de Mayo between Av Libertador and Roca.
MICHELE FALZONE / GETTY IMAGES ©
5Eating
OliviaCafe $
(map Google map; %02902-488038; 9 de Julio 187; snacks AR$90-180; h11am-8pm Mon-Sat, 2-8pm Sun; W)
This adorable coffee shop does _croque monsieurs_ (grilled ham and cheese), fresh donuts and espresso drinks in a loungy pastel setting. It also uses whole-bean Colombian coffee. Want to take the chill off? Try the cheese scones served hot with cream.
Viva la PepaCafe $
(map Google map; %02902-491880; Amado 833; mains AR$125-390; h11am-11pm Mon-Sat)
Decked out in children's drawings, this cheerful cafe specializes in crepes but also offers great sandwiches with homemade bread (try the chicken with apple and blue cheese), fresh juice and gourds of _mate_.
Buenos CrucesArgentine $$
(map Google map; %02902-492698; www.facebook.com/BuenosCrucesRestaurante; Espora 237; mains AR$150-310; h12:30-3pm & 7-11pm Mon-Sat; c)
This dynamic family-run enterprise brings a twist to Argentine classics. Start with a warm beet salad with balsamic reduction. The nut-crusted trout is both enormous and satisfying, as is the guanaco meatloaf or the baked ravioli crisped at the edge and bubbling with Roquefort cheese. The service is excellent and there's a play area for children.
Pura VidaArgentine $$
(map Google map; %02902-493356; Libertador 1876; mains AR$130-290; h7:30-11:30pm Thu-Tue; v)
Featuring the rare treat of Argentine home cooking, this offbeat, low-lit eatery is a must. Its longtime owners are usually found cooking up buttery spiced chicken potpies and filling wine glasses. For vegetarians, brown rice and wok veggies or salads are satisfying. Servings are huge. Don't skip the decadent chocolate brownie with ice cream and warm berry sauce. Reserve ahead.
Mi RanchoArgentine $$$
(map Google map; %02902-490540; Moyano 1089; mains AR$180-310; hnoon-3:30pm & 8pm-midnight)
Inspired and intimate, Mi Rancho serves up oversized osso buco, delicious braided pastas stuffed with king crab, divine salads and sweetbreads with wilted spinach on toast. For dessert, chocolate fondant or passion-fruit semifreddo are both worth the calorie hit, and more. Located in a tiny brick pioneer house with space for few.
La Vinería
Transplanted from Alaska, this tiny, congenial wine bar ( %02962-493301; Lago del Desierto 265; h2:30pm-3am Oct-Apr) offers a long Argentine wine list accompanied by 70 craft-beer options and standout appetizers. With 50 wines sold by the glass and an entire gin menu, enthusiasts might be tempted to sabotage their next day on the trail.
AFRICA STUDIO / SHUTTERSTOCK ©
6Drinking & Nightlife
La ZorraMicrobrewery
(map Google map; %02902-490444; www.facebook.com/cervezalazorra; Av San Martín s/n; h6pm-2am Tue-Sun)
Long, skinny tables fill with both locals and travelers to quaff what we consider to be the best artisan beer in Patagonia, La Zorra. The smoked porter and double IPA do not disappoint. There's also pub fare such as fries and sausages.
Borges y Alvarez Libro-BarBar
(map Google map; Libertador 1015; hnoon-9pm Mon-Thu, 3pm-midnight Fri & Sat; W)
Upstairs in the gnome village shopping complex, this hip bookstore-bar serves coffee, artisan beers and pricey cocktails. Peruse the oversized photography books on Patagonian wildlife or bring your laptop and take advantage of the free wi-fi. There's also 2nd-story deck seating.
La TolderíaLive Music
(map Google map; %02902-491443; www.facebook.com/LaTolderia; Libertador 1177; hnoon-4am Mon-Thu, to 6am Fri-Sun)
This petite storefront opens its doors to dancing and live acts at night; it's probably the best spot to try if you're feeling boisterous.
Don Diego de la NocheLive Music
(map Google map; Libertador 1603; h8pm-late)
This perennial favorite serves dinner and features live music such as tango, guitar and _folklórica_ (folk music).
## El Chaltén
This colorful village overlooks the stunning northern sector of Parque Nacional Los Glaciares. Every summer thousands of trekkers explore the world-class trails that start right here.
Outdoor cafe in El Chaltén | H. MARK WEIDMAN PHOTOGRAPHY / ALAMY STOCK PHOTO ©
1Sights & Activities
Capilla de los EscaladoresChapel
A simple chapel of Austrian design memorializes the many climbers who have lost their lives to the precarious peaks since 1953.
Spa YatenHealth & Fitness
( %02962-493394; spayaten@gmail.com; Av San Martín 36; 1hr massage AR$1750; h10am-10pm)
Spa Yaten has showers, robes and slippers, so sore hikers can come straight here off the trail. There are various therapies, massage, dry sauna and Jacuzzi tubs in a communal room. Reserve massages in advance.
TTours
El RelinchoHorseback Riding
( %02962-493007, in El Calafate 02902-491961; www.elrelinchopatagonia.com.ar; Av San Martín 505)
Outfitter El Relincho takes riders to the pretty valley of Río de las Vueltas (three hours) and also offers more challenging rides up the Vizcacha hill followed by a barbecue on a traditional ranch. Cabin-style accommodations are also available through the company.
Zona AustralTours
( %02902-489755; <http://zonaaustralturismo.com>; Av MM de Güemes 173; half-day/full-day tour US$15.50/20)
Offers sea kayaking and the Glaciar Vespignani tour at Lago del Desierto.
Patagonia MágicaOutdoors
( %02962-486261; www.patagoniamagica.com; Fonrouge s/n)
The friendly Patagonia Mágica runs one-day rock-climbing workshops for beginners on the natural rock-climbing walls near El Chaltén. Experienced climbers can go on expeditions with certified guides.
7Shopping
ChalteñosChocolate
(Av San Martín 249; h10am-1pm & 4-9pm)
Bring home Chalteños' handmade _alfajores_ filled with _dulce de leche_ or homemade jams and your loved ones will forgive all your wanderings.
Viento OesteBooks
( %02962-493200; Av San Martín 898; h10am-11pm)
Sells books, maps and souvenirs and also rents a wide range of camping equipment, as do several other sundries shops around town.
5Eating
CúrcumaVegan $$
( %02902-485656; Av Rojo 219; mains AR$320-380; h10am-10pm; v)
With an avid following, this vegan, gluten-free cafe does mostly takeout, from adzuki-bean burgers to whole-wheat pizzas, stuffed eggplant with couscous and arugula. Salads, coconut-milk risottos and smoothies are as rare as endangered species in Patagonia – take advantage. Hikers can reserve a lunch box a day in advance.
MaffíaItalian $$
(Av San Martín 107; mains AR$180-360; h11am-11pm)
Bring your appetite. In a gingerbread house, this pasta specialist makes delicious stuffed _panzottis_ and _sorrentinos,_ with creative fillings like trout, eggplant and basil or fondue. There are also homemade soups and garden salads. Service is professional and friendly. For dessert, the oversize homemade flan delivers.
EstepaArgentine $$
( %02962-493069; cnr Cerro Solo & Av Antonio Rojo; mains AR$120-350; h11:30am-2pm & 6-11pm)
Local favorite Estepa cooks up consistent, flavorful dishes such as lamb with calafate sauce, trout ravioli or spinach crepes. Portions are small but artfully presented, with veggies that hail from the on-site greenhouse. For a shoestring dinner, consider its rotisserie takeout service.
La TaperaArgentine $$$
( %02962-493195; Antonio Rojo 74; mains AR$260-330; hnoon-3pm & 6:30-11pm Oct-Apr)
With tender steak in balsamic-reduction sauce, ultra-fresh trout from Lago del Desierto and red-wine glasses as big as your head, it's hard to go wrong at Chipo's place, reminiscent of a log cabin with an open fireplace. Vegetarian options invite less enthusiasm – best to try elsewhere. Otherwise, service is snappy, portions generous and there are wonderful wine options.
6Drinking
La ChocolateríaCafe
( %02962-493008; Lago del Desierto 105; h11am-9pm Mon-Fri, 9am-9pm Sat & Sun Nov-Mar)
This irresistible chocolate factory tells the story of local climbing legends on the walls. It makes for an intimate evening out, with options ranging from spirit-spiked hot cocoa to wine and fondue. Chocolate and coffee drinks are AR$90.
Laguna los TresBar
(Trevisán 42; h6pm-2am)
For a dose of rock or reggae with ping-pong on the side, this disheveled bar will do nicely. There's live music on weekends; check the Facebook page for events.
# Santiago, Chile
#### Cerro San Cristóbal
#### Maipo Valley Wineries
#### Cajón del Maipo Outdoor Activities
#### Sights
#### Shopping
#### Eating
#### Drinking & Nightlife
#
Santiago, Chile
_Surprising, cosmopolitan, energetic, sophisticated and worldly, Santiago is a city of syncopated cultural currents, madhouse parties, expansive museums and top-flight restaurants._
_It's a wonderful place for strolling, and each neighborhood has its unique flavor and tone. Head out for the day to take in the museums, grand architecture and pedestrian malls of the Centro, before an afternoon picnic in one of the gorgeous hillside parks that punctuate the city's landscape. Nightlife takes off in the sidewalk eateries, cafes and beer halls of Barrios Brasil, Lastarria and Bellavista._
Plaza de Armas and the Catedral Metropolitana de Santiago | EMPERORCOSAR / SHUTTERSTOCK ©
Two Days in Santiago
Start off at the iconic Museo Chileno de Arte Precolombino, then delve into Chile's dark past at the Museo de la Memoria y los Derechos Humanos. Have a long and lazy dinner at Peumayen.
Start day two at Pablo Neruda's house, La Chascona, then take in great views atop Cerro San Cristóbal. That night catch a show at Centro Gabriela Mistral.
Four Days in Santiago
Spend your third day touring the Maipo Valley vineyards. Have a picnic lunch at Viña Santa Rita and make your own blend at Viña de Martino.
Get active on day four in the Cajón del Maipo. Go rafting with Rutavertical or take a scenic walk in the Monumento Natural El Morado. Afterwards soak in the rejuvenating Termas Valle de Colina.
Arriving in Santiago
Aeropuerto Internacional Arturo Merino Benítez (Santiago) Centropuerto and Turbus Aeropuerto run buses to the city center (40 minutes). A taxi to Centro costs CH$18,000; Transvip runs shared shuttles (from CH$7000).
**Terminal de Buses Sur** The city's main bus station is connected to the metro at the Universidad de Santiago station. From here, take trains east on Línea 1 to reach the Centro.
Where to Stay
Santiago's unique neighborhoods provide the backdrop for your stay. For easy access to museums and restaurants, consider the Centro, budget-friendly Barrio Brasil or nightlife districts like classy Barrio Lastarria and raucous Bellavista. For fancier digs and sophisticated dining – but limited access to most major sights – head to leafy Providencia, trendy Barrio Italia or well-heeled Las Condes.
TOP EXPERIENCE
# Cerro San Cristóbal
The best views over Santiago are from the peaks and viewpoints of the Parque Metropolitano, better known as Cerro San Cristóbal. At 722 hectares, the park is Santiago's largest green space.
Great For...
gbr
yDon't Miss
The marvelous view from the _teleférico_ as you ascend to the hilltop.
8Need to Know
map Google map; www.parquemet.cl; Pio Nono 450; funicular adult/child 1-way from CH$1500/1000; hfunicular 10am-6:45pm Tue-Sun, 1-6:45pm Mon; c; mBaquedano
5Take a Break
Enjoy a drink and a snack with unrivaled views from Terraza Bellavista.
oTop Tip
The park has entrances in both Bellavista and Providencia neighborhoods.
### Virgen de la Inmaculada Concepción
A snowy white 14m-high statue of the Virgin of the Immaculate Conception towers atop the _cumbre_ (summit) at the Bellavista end of the park. The benches at its feet are the outdoor church where Pope John Paul II said Mass in 1984.
Virgen de la Inmaculada Concepción | JOPSTOCK / GETTY IMAGES ©
### Skybound Journey
A funicular carries you between different landscaped sections on one side, while a _teleférico_ (cable car) swoops you away on the other.
To go by foot you can take a steep switchbacked dirt trail from the Plaza Caupolicán. This is also the start of the funicular (<http://funicularsantiago.cl>; Barrio Bellavista; adult/child 1-way from CH$1500/1000, round-trip from CH$2000/1500; h10am-6:45pm Tue-Sun, 1-6:45pm Mon; mBaquedano), and the site of a tourist info kiosk. Alternatively, enter the park from Pedro de Valdivia and board the teleférico (www.parquemet.cl; Parque Metropolitano; weekdays/weekends 1-way' CH$1910/2290; h10am-7pm Tue-Sun; mBaquedano).
### Family Fun
Other attractions on the hillside include the Zoológico Nacional (map Google map; National Zoo; %2-2730-1368; www.parquemet.cl/zoologico-nacional; Parque Metropolitano; adult/child CH$3000/1500; h10am-5pm Tue-Sun; mBaquedano); the Jardín Botánico Mapulemu, a botanical garden; the child-oriented Plaza de Juegos Infantiles Gabriela Mistral, featuring attractive wooden playground equipment and an interactive water fountain; and two huge public swimming pools, the Piscina Tupahue ( %2-2730-1331; Cerro San Cristóbal s/n, Parque Metropolitano; adult/child CH$6000/3500; h10am-6:30pm Tue-Sun Nov-Mar; mBaquedano) and Piscina Antilén (map Google map; %2-2730-1331; Cerro San Cristóbal s/n, Parque Metropolitano; adult/child CH$7500/4000; h10am-6:30pm Tue-Sun Nov-Mar; mBaquedano). The small but perfectly landscaped Jardín Japonés (Japanese Garden) is just above the Pedro de Valdivia entrance.
_Teleféricos_ (cable cars) and views of Santiago | TIFONIMAGES / GETTY IMAGES ©
### Terraza Bellavista
Near the top of the funicular is the Terraza Bellavista (map Google map; Parque Metropolitano; mBaquedano), where there are a few snack stands and extraordinary views across the city.
TOP EXPERIENCE
# Maipo Valley Wineries
When you've had your fill of Santiago's museums and plazas, head south of the city center to check out the gorgeous vineyards of the Maipo Valley, home to some outstanding big-bodied reds.
Great For...
ugA
yDon't Miss
The chance to blend your own wine at Viña de Martino.
8Need to Know
If you plan on visiting two or more wineries, you're better off renting a car.
5Take a Break
Viña Santa Rita ( %2-2362-2520; www.santarita.com; Camino Padre Hurtado 0695, Alto Jahuel; tours CH$14,000-40,000; htours 10am-5pm Tue-Sun) has its own on-site restaurant, and can also prepare picnic baskets.
oTop Tip
Book your Viña Santa Rita bike or wine trip through Turistik (map Google map; %2-2820-1000; <https://turistik.com>; Municipal Tourist Office, Plaza de Armas s/n, Centro; day pass from CH$23,000; h10am-6pm Mon-Fri; mPlaza de Armas).
### Organized Tours
You can go it alone as many wineries are within 1½ hours of the city center on public transportation. But if you'd like to hit the wine circuit with a knowledgeable guide, try the specialized tours at Uncorked Wine Tours ( %2-2981-6242; www.uncorked.cl; full-day tour US$195). An English-speaking guide will take you to three wineries, and a lovely lunch is included.
Also recommended is the winery bike tour with La Bicicleta Verde (map Google map; %2-2570-9939; <https://labicicletaverde.com>; Loreto 6, Barrio Recoleta; bike tours from CH$25,000, rentals half-/full-day from CH$5000/9000; mBellas Artes).
Enotour (map Google map; %2-2481-4081; www.enotourchile.com; Luis Thayer Ojeda 0130, Oficina 1204, Providencia; tours from CH$37,000; mTobalaba) is another curated wine-tour outfit. Most wine tours require advance reservations.
### Viña Santa Rita
Famous for the premium Casa Real Cabernet, Santa Rita offers bike and wine trips, as well as picnics, tastings and tours of its stunning winery. There's also a jaw-dropping pre-Columbian art collection on display at the on-site Museo Andino, with pottery, textiles and gold Incan jewelry.
### Viña Cousiño Macul
A historic winery ( %2-2351-4100; www.cousinomacul.com; Av Quilín 7100, Peñalolen; tours CH$14,000-24,000; hEnglish-language tours 11am, 12:15pm, 3pm & 4:15pm Mon-Fri, 10:15am & 11:30am Sat) set in Santiago's urban sprawl. Most of the vineyards are now at Buin, but tours take in the production process and underground _bodega_ , built in 1872. La Bicicleta Verde runs frequent bike and wine tours here.
Viña Cousiño Macul | JOHN WARBURTON-LEE PHOTOGRAPHY / ALAMY STOCK PHOTO ©
### Viña Undurraga
The subterranean _bodegas_ at Undurraga ( %2-2372-2850; www.undurraga.cl; Camino a Melipilla, Km34, Talagante; tours from CH$12,000; htours at 10:15am, noon & 3:30pm daily) date from 1885. Come for tours or to try wines by the glass.
TINAFIELDS / GETTY IMAGES ©
### Viña de Martino
Reserve ahead for personalized tours and tastings in a Tuscan-style manor ( %2-2577-8037; www.demartino.cl; Manuel Rodríguez 229, Isla de Maipo; tastings from CH$14,000, tours from CH$17,500; h9am-1:30pm & 3-6pm Mon-Fri, 10:30am-1pm Sat), including one where you can blend your own wine. Note: you will need a car to visit this winery.
TOP EXPERIENCE
# Cajón del Maipo Outdoor Activities
Rich greenery lines the steep, rocky walls of this stunning gorge of the Río Maipo. It's popular on weekends with Santiaguinos, who come here to camp, hike, climb, cycle, raft and ski.
Great For...
gfn
yDon't Miss
Abundant bird species, foxes and rodents, and the endangered Chilean iguana at Reserva Nacional Río Clarillo.
8Need to Know
The Cajón del Maipo starts 25km southeast of Santiago, though it's about 100km to Termas Valle de Colina.
5Take a Break
Pizzería y Cervecería Jauría (www.cervezajauria.cl; Bernardo O'Higgins
18; pizzas CH$9000; h5pm-midnight Fri, 1:30pm-1am Sat, 1:30pm to 8pm Sun) serves creative pizzas and satisfying IPAs.
oTop Tip
November to March is peak rafting season as glacier meltwater brings Class III or IV rapids to the Río Maipo.
### Rafting
Rutavertical Rafting ( %cell 9-9435-3143; www.rutavertical.cl; Camino al Volcán 19635, San José de Maipo; trips CH$17,000; hdaily departures at 11am, 2pm & 4:30pm) has one-hour rafting trips led by enthusiastic multilingual guides that take in some lovely gorges before ending up back in San José de Maipo. Allot about 2½ hours in total for the briefing, outfitting and drive upriver to the starting point. Helmets, wetsuits and lifejackets are provided, and there are lockers to store your belongings.
Termas Valle de Colina | DORLING KINDERSLEY / GETTY IMAGES ©
### Monumento Natural El Morado
At Baños Morales you'll find the entrance to Monumento Natural El Morado (www.conaf.cl/parques/monumento-natural-el-morado; adult/child CH$5000/2500; hmust enter 8:30am-1pm & leave by 6pm Oct-Apr, enter 8:30am-12:30pm & leave by 5:30pm May-Sep), a small national park. From the banks of sparkling Laguna El Morado there are fabulous views of Glaciar San Francisco and the 5000m summit of Cerro El Morado. It takes about two hours to reach the lake on the well-marked 6km trail from the Conaf post.
### Reserva Nacional Río Clarillo
A mix of Andean forest and scrubland make up this hilly, 100-sq-km nature reserve (www.conaf.cl/parques/reserva-nacional-rio-clarillo; Camino a Reserva Nacional Río Clarillo s/n, Pirque; adult/child CH$6000/3000; h8:30am-6pm) in a scenic tributary canyon of the Cajón del Maipo, 18km southeast of Pirque.
Two short, clearly labeled trails start near the Conaf rangers office, 300m after the entrance: Quebrada Jorquera takes about half an hour; Aliwen Mahuida takes 1½ hours. The rangers give advice on longer hikes along the river, but plan on starting early as camping is not allowed here.
### Cascada de las Animas
Organized activities are the only way to visit this private nature reserve ( %2-2861-1303; www.cascadadelasanimas.cl; Camino al Volcan 31087), which takes its name from a stunning waterfall reached by the shortest walk offered (CH$7000); there are also guided half-day hikes into the hills (CH$12,000) and rafting trips (CH$18,000). Horseback riding is the real specialty, however – indeed, the reserve is also a working ranch.
Cascada de las Animas | LEON WERDINGER / ALAMY STOCK PHOTO ©
### Termas Valle de Colina
At Termas Valle de Colina ( %2-2985-2609; www.termasvalledecolina.com; entrance incl camping adult/child CH$8000/4000), scolding hot pools of milky blue water overlook a stark Andean valley. There's a well-organized camping ground with showers, but be sure to bring food and supplies.
Santiago Centro
1Sights
1Barrio París-LondresC6
2Centro Cultural Palacio La MonedaB6
3Centro Gabriela MistralF5
4Cerro San CristóbalG2
5Cerro Santa LucíaE5
6La ChasconaG2
7La Vega CentralC2
8Londres 38C6
9Museo Chileno de Arte PrecolombinoB5
10Museo de Artes VisualesE4
11Museo Nacional de Bellas ArtesE4
12Terraza BellavistaG1
13Zoológico NacionalG2
2Activities, Courses & Tours
14La Bicicleta VerdeE3
15TuristikC4
7Shopping
Artesanías de Chile(see 2)
5Eating
16Mercado CentralC3
17PeumayenG3
18Salvador Cocina y CaféB5
19Vietnam DiscoveryE2
6Drinking & Nightlife
20BocanárizE4
21Chipe LibreE4
22Restobar KYF1
1Sights
### 1Centro
Museo Chileno de Arte PrecolombinoMuseum
(map Google map; Chilean Museum of Pre-Columbian Art; %2-2928-1500; www.precolombino.cl; Bandera 361, Centro; CH$6000; h10am-6pm Tue-Sun; mPlaza de Armas)
Exquisite pottery from most major pre-Columbian cultures is the backbone of Santiago's best museum, the Museo Chileno de Arte Precolombino. As well as dozens of intricately molded anthropomorphic vessels, star exhibits include hefty Maya stone columns, towering Mapuche totems and a fascinating Andean textile display.
Museo Chileno de Arte Precolombino | LUCAS VALLECILLOS / ALAMY STOCK PHOTO ©
Barrio París-LondresArea
(map Google map; cnr París & Londres; mUniversidad de Chile)
This pocket-sized neighborhood developed on the grounds of the Franciscan convent of Iglesia de San Francisco is made up of two intersecting cobblestone streets, París and Londres, which are lined by graceful European-style town houses built in the 1920s. Look for the memorial at Londres 38 (map Google map; www.londres38.cl; Barrio Paris-Londres; h10am-1pm & 3-6pm Tue-Fri, 10am-2pm Sat; mUniversidad de Chile) F, a building that served as a torture center during Augusto Pinochet's government.
Centro Cultural Palacio La MonedaArts Center
(map Google map; %2-2355-6500; www.ccplm.cl; Plaza de la Ciudadanía 26, Centro; exhibitions from CH$3000; h9am-9pm, exhibitions to 7:30pm; c; mLa Moneda)
Underground art takes on a new meaning in one of Santiago's newer cultural spaces: the Centro Cultural Palacio La Moneda beneath Plaza de la Ciudadanía. A glass-slab roof floods the vaultlike space with natural light, and ramps wind down through the central atrium past the Cineteca Nacional, a state-run art-house movie theater, to two large temporary exhibition spaces that house some of the biggest touring shows to visit Santiago.
### 1Barrio Lastarria & Barrio Bellas Artes
Centro Gabriela MistralArts Center
(map Google map; GAM; %2-2566-5500; www.gam.cl; Av O'Higgins 227, Barrio Lastarria; hplazas 8am-10pm, exhibition spaces 10am-9pm Tue-Sat, from 11am Sun; mUniversidad Católica) F
This striking cultural and performing-arts center – named for Chilean poet Gabriela Mistral, the first Latin American woman to win the Nobel Prize in Literature – is an exciting addition to Santiago's art scene, with concerts and performances most days. Drop by to check out the rotating art exhibits on the bottom floor, the iconic architecture that vaults and cantilevers on the inside and looks like a giant rusty cheese grater from the street, the little plazas, murals, cafes and more.
Cerro Santa LucíaPark
(map Google map; entrances cnr Av O'Higgins & Santa Lucía, cnr Santa Lucía & Subercaseaux, Bellas Artes; h9am-6:30pm Mar-Sep, to 8pm Oct-Feb; mSanta Lucía) F
Take a break from the chaos of the Centro with an afternoon stroll through this lovingly manicured park. It was just a rocky hill until 19th-century mayor Benjamín Vicuña Mackenna had it transformed into one of the city's most memorable green spaces.
Museo Nacional de Bellas ArtesMuseum
(map Google map; National Museum of Fine Art; %2-2499-1600; www.mnba.cl; José Miguel de la Barra 650, Barrio Bellas Artes; h10am-6:45pm Tue-Sun; mBellas Artes) F
This fine art museum is housed in the stately neoclassical Palacio de Bellas Artes, built as part of Chile's centenary celebrations in 1910. The museum features an excellent permanent collection of Chilean art. There are free guided tours starting at 10:30am daily (except January and February, when they begin at noon).
Museo de Artes VisualesMuseum
(map Google map; MAVI, Visual Arts Museum; %2-2664-9337; www.mavi.cl; Lastarria 307, Plaza Mulato Gil de Castro, Barrio Lastarria; CH$1000, Sun free; h11am-7pm Tue-Sun; mUniversidad Católica)
Exposed concrete, stripped wood and glass are the materials local architect Cristián Undurraga chose for the stunningly simple Museo de Artes Visuales. The contents of the four open-plan galleries are as impressive as the building: top-notch modern engravings, sculptures, paintings and photography form the regularly changing temporary exhibitions.
Parque Bicentenario
This gorgeous urban oasis (map Google map; Bicentennial Park; Bicentenario 3236, Vitacura; mTobalaba) was created, as the name suggests, in celebration of the Chilean bicentennial. In addition to more than 4000 trees, a peaceful location alongside the Río Mapocho and access to city bike paths, the park features inviting chaise lounges and sun umbrellas, plus state-of-the-art playground equipment for kids.
It's a quick taxi ride from the Tobalaba metro station, or hop on bus 405 and get off at Av Alonso de Córdova (three blocks east of the park).
DFLC PRINTS / SHUTTERSTOCK ©
### 1Bellavista
La ChasconaHistoric Building
(map Google map; %2-2777-8741; www.fundacionneruda.org; Fernando Márquez de La Plata 0192, Barrio Bellavista; adult/student CH$7000/2500; h10am-7pm Tue-Sun Jan & Feb, to 6pm Tue-Sun Mar-Dec; mBaquedano)
When poet Pablo Neruda needed a secret hideaway to spend time with his mistress Matilde Urrutia, he built La Chascona, the name loosely translated as 'Messy Hair' and inspired by her unruly curls. Neruda, of course, was a great lover of the sea, so the dining room is modeled on a ship's cabin and the living room on a lighthouse.
### 1Barrio Brasil & Barrio Yungay
Museo de la Memoria y los Derechos HumanosMuseum
(Museum of Memory & Human Rights; map; %2-2597-9600; www.museodelamemoria.cl; Matucana 501, Barrio Yungay; h10am-8pm Tue-Sun Jan & Feb, to 6pm Tue-Sun Mar-Dec; mQuinta Normal) F
Opened in 2010, this striking museum isn't for the faint of heart: the exhibits expose the terrifying human rights violations and large-scale 'disappearances' that took place under Chile's military government between 1973 and 1990.
Las Condes & Providencia
1Sights
1Costanera CenterC3
2Parque BicentenarioD2
Sky Costanera(see 1)
2Activities, Courses & Tours
3EnotourC3
4Piscina AntilénB2
7Shopping
5Galería DrugstoreB4
5Eating
6BoragóD1
7El HuertoB4
8HolmB5
6Drinking & Nightlife
9Santo RemedioA6
### 1Las Condes, Barrio El Golf & Vitacura
Museo de la ModaMuseum
(Museum of Fashion; map; %2-2219-3623; www.museodelamoda.cl; Av Vitacura 4562, Vitacura; adult/student & senior/child CH$3000/1500/free; h10am-6pm Tue-Fri, 11am-7pm Sat & Sun; mEscuela Militar)
This slick, privately operated fashion museum comprises a vast and exquisite permanent collection of Western clothing – 20th-century designers are particularly well represented.
Costanera CenterNotable Building
(map Google map; %2-2916-9226; www.costaneracenter.cl; Av Andrés Bello 2425, Providencia; h10am-10pm; mTobalaba)
The four skyscrapers that make up Costanera Center include **Gran Torre Santiago** , the tallest building in Latin America (300m). The complex also contains offices, a high-end hotel and the largest shopping mall in South America. Head to Sky Costanera (map Google map; www.skycostanera.cl; adult/child CH$15,000/10,000; h10am-10pm; mTobalaba) for panoramic views from the top of Gran Torre.
7Shopping
Artesanías de ChileArts & Crafts
(map Google map; %2-2697-2784; www.artesaniasdechile.cl; Plaza de la Ciudadanía 26, Centro Cultural Palacio La Moneda, Centro; h9:30am-7:30pm Mon-Fri, 10:30am-7pm Sat & Sun; mLa Moneda) S
Not only do this foundation's jewelry, wood carvings, ceramics and naturally dyed textiles sell at reasonable prices, most of what you pay goes directly to the artisans who made them. Look for other locations at Los Dominicos and the airport, as well as towns throughout Chile.
Pueblito Los DominicosArts & Crafts
(Los Dominicos Handicraft Village; www.plosdominicos.cl; Av Apoquindo 9085, Las Condes; h10:30am-7pm Tue-Sun; mLos Dominicos)
The best place in Santiago to buy quality gifts that were actually made in Chile. This mock village houses dozens of small stores, art galleries and traditional cafes. Look for lapis lazuli jewelry, Andean textiles, carved wooden bowls and ceramics with indigenous motifs.
Galería DrugstoreFashion & Accessories
(map Google map; www.drugstore.cl; Av Providencia 2124, Providencia; hshops 11am-8pm Mon-Fri, to 6:30pm Sat; mLos Leones)
Head to this cool three-story independent shopping center for clothes no one back home will have – it has tiny boutiques of several up-and-coming designers, arty bookstores and cafes.
5Eating
Mercado CentralSeafood $
(map Google map; Central Market; www.mercadocentral.cl; cnr 21 de Mayo & San Pablo, Centro; hfood stands & restaurants 9am-5pm Mon-Fri, 7am-3:30pm Sat & Sun; mPuente Cal y Canto)
Santiago's wrought-iron fish market is a classic for seafood lunches (and hangover-curing fish stews like the tomato- and potato-based _caldillo de congrio,_ Pablo Neruda's favorite). Skip the touristy restaurants in the middle and head for one of the tiny low-key stalls around the market's periphery.
_Paila marina_ (seafood soup), Mercado Central | DORADALTON / GETTY IMAGES ©
Salvador Cocina y CaféChilean $$
(map Google map; www.facebook.com/Salvador CocinaYCafe; Bombero Ossa 1059, Centro; mains CH$7700; h8am-8:30pm Mon-Fri; mUniversidad de Chile)
This no-frills two-story lunch spot packs a surprising punch with market-focused menus that change daily and highlight unsung dishes (and exotic meats) from the Chilean countryside. Chef Rolando Ortega won Chile's coveted chef of the year award in 2015 and the tables have been packed ever since.
La DianaChilean $$
( %2-2632-8823; www.ladiana.cl; Arturo Prat 435, Centro; mains CH$6000-8500; h1pm-12:30am Tue-Sat, to 6:30pm Sun; mUniversidad de Chile)
Attached to a children's arcade of the same name and built within the walls of an old monastery, La Diana defies easy description. Its ceilings are adorned with as many potted plants as chandeliers, its tables are a mishmash of found furnishings, and its menu is as notable for grilled seafood as it is for seafood-packed pizzas.
SilabarioChilean $$
( %2-2502-5429; www.facebook.com/Silabarioficial; Lincoyan 920, Barrio Italia; mains CH$6000-9500; h7-11:30pm Tue-Fri, 1-4:30pm & 7pm-midnight Sat, 1-4:30pm Sun; mIrarrázaval)
Tucked away in an old home just south of Barrio Italia's main drag, this intimate restaurant re-envisions staples from the Chilean countryside as gourmet dishes. From northern quinoa salads to hearty Mapuche stews from the south, each culinary journey ends with a complimentary homemade _bajativo_ (digestif).
HolmCafe $$
(map Google map; %cell 9-4227-4411; <https://holmcomidafeliz.cl>; Padre Mariano 125, Providencia; meals CH$5000-7500; h9am-10pm Mon-Fri, 10am-4pm Sat & Sun; v; mPedro de Valdivia)
Santiaguinos flock to this homey Providencia cafe each weekend for the best brunches in town. Fresh-baked breads and jams, yogurt and granola or scrambled eggs with crispy bacon are but a few of the offerings. Salads, sandwiches and fresh fruit juices round out the midweek fare.
El HuertoCafe $$
(map Google map; %2-2231-4443; www.elhuerto.cl; Orrego Luco 054, Providencia; mains CH$6000-8000; hnoon-11pm Mon-Sat, 12:30-4:30pm Sun; v; mPedro de Valdivia)
This earthy restaurant's healthy, vegetarian fare is a big hit with both hip young things and ladies who lunch. Come for seaweed ceviches, fresh fruit juices, quinoa salads and wonderfully rich desserts.
PeumayenChilean $$$
(map Google map; www.peumayenchile.cl; Constitución 134, Barrio Bellavista; tasting menu CH$12,500; h1-3pm & 7pm-midnight Tue-Sat, 1-4pm Sun; mBaquedano)
Without a doubt one of the most unusual culinary experiences in Chile, this upstart is innovating Chilean cuisine by looking back to the culinary roots of the Mapuche, Rapa Nui and Atacameños.
BoragóChilean $$$
(map Google map; %2-2953-8893; www.borago.cl; Av Nueva Costanera 3467, Vitacura; tasting menu from CH$50,000; h7-11:15pm Mon-Sat)
Chef Rodolfo Guzman earned a coveted spot among the World's 50 Best Restaurants by elevating Chilean cuisine to new heights at this Vitacura restaurant, whose minimalist design forces you to focus on the food. The multicourse tasting menus, which include little-known endemic ingredients, sweep you away on an unforgettable culinary adventure from the Atacama to Patagonia. Reserve well in advance.
El Huaso Enrique
Going strong in Barrio Yungay for almost 60 years, El Huaso Enrique ( %2-2681-5257; www.elhuasoenrique.cl; Maipú 462, Barrio Yungay; cover CH$2500-3000; h7pm-2am Wed-Sun; mQuinta Normal) is the best place to see Chileans performing their national dance, _la cueca_. You can watch proud locals hit the dance floor while you feast on hearty regional dishes like _pastel de choclo_ (a casserole-like dish consisting of baked corn, meat and onions). The place comes alive on weekends with live music. If you really want to get into the spirit, El Huaso Enrique also offers _cueca_ lessons; check out the website for details.
6Drinking & Nightlife
BocanárizWine Bar
(map Google map; %2-2638-9893; www.bocanariz.cl; Lastarria 276, Barrio Lastarria; h12:30pm-midnight Mon-Sat, 7-11:30pm Sun; mBellas Artes)
You won't find a better wine list anywhere in Chile than at this homey restobar whose servers are trained sommeliers. Try creative wine flights (themed by region or style) or sample several top bottles by the glass. There are also meat and cheese plates, as well as hearty Chilean dishes (mains CH$8000 to CH$12,000). Reservations recommended.
Chipe LibreCocktail Bar
(map Google map; %2-2664-0584; Lastarria 282, Barrio Lastarria; h12:30pm-12:30am Mon-Wed, to 1am Thu-Sat; mBellas Artes)
Learn about the big sour over _pisco_ – and who made it first – at the only bar in Santiago dedicated to the South American brandy. There are as many _piscos_ from Peru as Chile on the menu and you can try them in flights of three or within an array of flavored sours. Reserve ahead for tables on the interior patio.
Santo RemedioCocktail Bar
(map Google map; www.santoremedio.cl; Román Díaz 152, Providencia; h1-3:30pm & 6pm-2am Mon-Fri, 6pm-2am Sat; mManuel Montt)
Strictly speaking, this low-lit and spectacularly funky old house is a restaurant, and an aphrodisiacal one at that. But it's the bar action people really come for: powerful, well-mixed cocktails and regular live DJs keep the 20- and 30-something crowds happy.
Barrio Recoleta
Bustling Korean eateries and Middle Eastern takeout counters, a happening marketplace overflowing with ripe fruit, a colorful jumble of street vendors, an achingly hip cocktail lounge – this burgeoning barrio just west of Bellavista is a slight detour off the beaten path. Here are a few spots you shouldn't miss.
Restobar KY (map Google map; %2-2777-7245; www.restobarky.cl; Av Perú 631, Barrio Recoleta; h8pm-2am Tue-Sat; mCerro Blanco) Set in a rambling old house, this stunning cocktail bar is a mix of glowing Chinese lanterns, antique chandeliers, carved wooden furnishings and vibrant artwork.
La Vega Central (map Google map; www.lavega.cl; cnr Nueva Rengifo & Antonia López de Bello, Barrio Recoleta; h6am-6pm Mon-Sat, 7am-2pm Sun; mPatronato) Raspberries, quinces, figs, peaches, persimmons, custard apples...if it grows in Chile you'll find it at this buzzing market.
Vietnam Discovery (map Google map; %2-2737-2037; www.vietnamdiscovery.cl; Loreto 324, Barrio Recoleta; mains CH$7000-11,000; h1pm-midnight Mon-Sat, to 4pm Sun; v; mPatronato) Run by a hip French-Vietnamese couple, this lavishly designed eatery serves creative Thai and Vietnamese dishes. Book ahead, using the online form, or you'll never see the inside.
La Vega Central | JEREMYRICHARDS / SHUTTERSTOCK ©
8INFORMATION
**Municipal Tourist Office** (map; www.santiagocapital.cl; Plaza de Armas s/n, Centro; h9am-6pm Mon-Fri, to 4pm Sat & Sun; mPlaza de Armas) Well-meaning but under-resourced staff provide basic maps and information. There's also a small gallery and a shop with Chilean products.
Plaza de Armas metro station | DIEGO GRANDI / SHUTTERSTOCK ©
8Getting There & Away
AIR
Chile's main air hub for both national and domestic flights is Aeropuerto Internacional Arturo Merino Benítez, 16km west of central Santiago.
BUS
Bus services leave from four different terminals, though you'll find countrywide services at these terminals:
**Terminal de Buses Sur** (Terminal Santiago; Av O'Higgins 3850, Barrio Estación Central; mUniversidad de Santiago) Santiago's largest terminal (aka Terminal Santiago) serves destinations south of Santiago as well as a few northbound buses, and many international services.
**Terminal de Buses Alameda** (cnr Av O'Higgins & Jotabeche, Barrio Estación Central; mUniversidad de Santiago) Next door to Terminal de Buses Sur, buses here run to destinations all over Chile.
8GETTING AROUND
The city's ever-expanding **metro** (www.metro.cl; per ride from CH$630; h6am-11pm Mon-Fri, from 6:30am Sat, from 8am Sun) is a clean and efficient way of getting about. Services on its six interlinking lines are frequent, but often painfully crowded.
# The Lakes District, Chile
#### Adrenalin Activities
#### Visiting Huilliche Communities
#### Osorno
#### Puerto Varas
#### Cochamó
#### Frutillar
#### Puerto Octay
#
The Lakes District, Chile
_The Lakes District – named 'Los Lagos' for its myriad glacial lakes that dot a countryside otherwise characterized by looming, snowcapped volcanoes, otherworldly national parks and serene lakeside villages – is one of Chile's most picturesque regions. Outdoor adventurers congregate around pretty Puerto Varas, the region's most touristy town and the jumping-off point for most of the area's attractions. Go horseback riding or rock climbing in the Río Cochamó Valley, linger around Lagos Llanquihue, Puyehue or Todos los Santos, or flashpack through any number of impressive national parks._
Puerto Varas with views of Lago Llanquihue and Volcán Osorno | JOSE L. STEPHENS / SHUTTERSTOCK ©
Two Days in the Lakes District
Start off with a whitewater rafting adventure on Río Petrohué with Al Sur. In the evening enjoy a first-rate meal and microbrews in Puerto Varas.
On day two, hit the trails on a memorable mountain-bike ride that takes in volcanic scenery and lake views with La Comarca Puelo Adventure.
Four Days in the Lakes District
On days three and four, make a trip to the Huilliche Communities on the Osorno Coast. Spend the first night in the picturesque Altos de Pichi Mallay in San Juan de la Costa, then head to gorgeous Caleta Cóndor for outdoor adventures (and a memorable homestay) with Caleta Cóndor Expediciones.
Arriving in the Lakes District
Puerto Montt's **Aeropuerto El Tepual** (www.aeropuertoeltepual.cl), located 22km southwest of Puerto Varas, is Sur Chico's busiest airport and its main gateway for those arriving by air.
Puerto Montt's **bus terminal** (www.terminalpm.cl) is also the region's busiest and biggest, serving major Chilean cities. The city also has a seaport for destinations deeper into Patagonia.
Where to Stay
With such an abundance of gorgeous lakes, one might expect lakeside accommodations to be the main event in Los Lagos, but there are plenty more options from which to choose. Puerto Varas leads the way – from excellent hostels to high-design boutique hotels, there is something for everyone in PV. Elsewhere you'll find family-run guesthouses, adventure-centric camping and _cabañas_.
TOP EXPERIENCE
# Adrenalin Activities
You'll find adventure aplenty in Chile's stunning Lakes District. You can go rafting, kayaking, mountain biking, volcano trekking, horseback riding and skiing amid some of the country's most majestic scenery.
Great For...
cfo
yDon't Miss
Whitewater rafting Class III and IV rapids on the Río Petrohué.
8Need to Know
Puerto Varas is the region's most touristy town and the jumping-off point for many adventures.
5Take a Break
After a day of adventure, kick back with a microbrew at Chester Beer Brewing Company.
oTop Tip
Be sure to bring a waterproof and windproof jacket, and bring ample layers for cooler weather.
### Climbing Volcanoes
Summiting Osorno or Calbuco volcanoes on a tour runs CH$200,000 for one person, but it drops to CH$180,000 per person thereafter and includes transport, all meals, overnight in a mountain retreat, all technical equipment, insurance and bilingual guide. No experience is necessary.
Moyca Expediciones (map Google map; %cell 9-7790-5679; www.facebook.com/moycaexpediciones; San José 192, oficina 203; h10am-2pm & 3-7pm, shorter hr winter) is the go-to outfitter in Puerto Varas, but the three companies that climb all share the same 10 or so guides. Independent climbers must obtain Conaf ( %65-248-6115; www.conaf.cl; Ochagavía 458; h9am-1pm & 2:30-5:45pm Mon-Thu, to 4:30pm Fri) permission, but this is not recommended unless you are a highly trained technical mountaineer.
### Mountain Biking
La Comarca Puelo Adventure ( %cell 9-9799-1920; www.pueloadventure.cl; Av Vicente Pérez Rosales 1621) S specializes in dramatic and custom-tailored adventure trips to less-explored areas of the Río Puelo Valley, Río Cochamó Valley and beyond. Highlights include extensive mountain-bike and road-cycling holidays in Chile and Argentina (including an epic 12-day single-track ride from Bariloche to Puerto Varas); beer-sluggin' and biking; and a wine-and-cheese excursion to the 900m viewpoint at Arco Iris in La Junta.
The Bike & Beer tour is a good-fun 30km bike ride along the lake, culminating in craft-brew tasting at Chester Beer. Groups are never more than 12 strong and everyone here is dedicated to giving travelers a unique off-the-beaten-path experience. Also rents road bikes, full-suspension mountain bikes and e-bikes.
### Rafting & Kayaking
Al Sur (map Google map; %65-223-2300; www.alsurexpeditions.com; cnr Aconcagua & Imperial) S specializes in rafting – with an exclusive base camp on the banks of the Río Petrohué – and also does high-end, multiday kayaking trips within the fjords of Parque Pumalín, all with a heavy focus on environmental NGOs. Also inquire about their epic one-month Top to Bottom trip, a trekking, rafting, fishing and driving tour from San Pedro de Atacama to Torres del Paine.
Ko'Kayak (map Google map; %65-223-3004; www.kokayak.cl; San Pedro 311; h8:30am-7pm) is another long-standing favorite for half-day rafting trips, as well as one-and two-day sea kayaking trips.
Yak Expediciones ( %cell 9-8332-0574; www.yakexpediciones.cl) is a fan favorite for both short and multiday sea-kayaking trips on Lago Todos Los Santos, Reloncaví fjord and beyond.
Kayaking on Futaleufú River | GUAXINIM / SHUTTERSTOCK ©
### Other Adventures
A bit of a renegade cowboy, guide-of-all-trades Guillermo Duarte ( %cell 9-7952-4279; www.facebook.com/guillermoduartetravel) can get you on the trails, horseback, up the volcano or on cross-country skies, but where he really excels is with bespoke tours – no adventure is too outrageous. Prices run around CH$160,000 per day for his services plus vehicle – the itinerary is left to your wildest dreams. Spanish, English, French and Portuguese spoken.
OpenTravel ( %65-226-0524; www.opentravel.cl) offers off-the-beaten-path trekking and horseback riding to remote areas in northern Patagonia and across the Andes to Argentina, including remote Argentine-French retreats on Isla Las Bandurrias in Lago Las Rocas and multiday horseback-riding/cultural-farmstay trips between Argentina and Chile.
Hiker in Puerto Montt | BENJAMIN WU / EYEEM / GETTY IMAGES ©
### Trekking
Co-run by a young, enthusiastic climber from Osorno, Patagonia Expeditions ( %cell 9-9104-8061; www.anticura.com; Ruta Internacional 215, Km90) operates the concession on Centro Turístico Anticura. Short hikes – trail fee CH$1000 – from the visitors center include Salto de Princesa, Salto del Indio – where, according to legend, a lone Mapuche hid to escape _encomienda_ (colonial labor system) service in a nearby Spanish gold mine – and Repucura, which ends back up on Ruta 215 (buses come careening down the highway; walk on the opposite side).
There's also a 4km steep hike up to a lookout point.
Excursions from here include climbing Volcán Casablanca (1960m; CH$40,000); Volcán Puyehue (2240m; CH$55,000, or CH$90,000 along with the 2011 Puyehue eruption crater), nocturnal waterfall visits and multiday treks. A restaurant serves three meals a day (CH$5000 to CH$9000).
Río Petrohué and Volcán Osorno in the background | MONA F / GETTY IMAGES ©
### Saltos del Petrohué
Six kilometers southwest of Petrohué, the Saltos del Petrohué (www.conaf.cl/parques/parque-nacional-vicente-perez-rosales; adult/child Chilean CH$2000/1000, foreigner CH$4000/2000; h9am-6pm) is a rushing, frothing waterfall raging through a narrow volcanic rock canyon carved by lava. Anyone wondering why the rafting trips don't start from the lake will find the answer here, although experienced kayakers have been known to take it on. Parking is CH$1000.
An excellent new visitors center houses a cafe and handicrafts shops. Try to arrive in the sweet spot between 9am and 9:15am before the TurisTour day-tour bus unloads the masses.
yWhen to Go
Hiking and aquatic sports are best from December to March, while June to mid-October is the skiing season.
### Skiing
If you come from mid-June to mid-October, you can hit the ski slopes! The Centro de Esquí Antillanca ( %64-261-2070; www.antillanca.cl; Ruta U-485; lift tickets CH$35,000, full rentals adult/child CH$33,000/22,000) has five surface lifts and 460m of vertical drop. The resort includes a pricey hotel (two buildings, Hotel Eduardo Meyer and Refugio Carlos) open year-round with typical ski-resort trimmings. There is a full-service restaurant at the resort's lodge (mains CH$6000 to CH$12,000).
Further south, the Centro de Ski y Montaña Volcán Osorno ( %cell 9-9158-7337; www.volcanosorno.com; Ruta V-555, Km12, Volcán Osorno; half-/full-day lift tickets CH$20,000/26,000; h10am-5:30pm) has two lifts for skiing and sightseeing and has recently undergone an expansion of its restaurant and rental shop. It has ski and snowboard rentals (full packages CH$20,000) and food services on the mountain year-round. There's also a tubing park – fun for kids and adults alike (from CH$15,000).
Expanded summer options include taking the ski lift up for impossibly scenic views at 1420m (CH$12,000) or 1670m (CH$16,000). You can descend a little faster via zip lines (CH$12,000).
oBariloche Bound
You can make an adventurous crossing by buses and boats from Puerto Varas to Bariloche, Argentina, with Cruce Andino (www.cruceandino.com).
### Stand-up Paddling
Stand-up paddle boarding (SUP) enthusiast Eduardo offers sunset excursions on Laguna Escondida, advanced trips on the Río Petrohué, an eight-day paddling extravaganza through the Río Puelo Valley, SUP surf trips on the coast), classes, yoga retreats and rentals.
TOP EXPERIENCE
# Visiting Huilliche Communities
Along Osorno's gorgeous coast, you can immerse yourself in indigenous Huilliche communities. On multiday treks, you can take in stunning beaches and Valdivian forest treks, while lodging in rural homestays.
Great For...
rgp
yDon't Miss
A trekking, kayaking or horseback riding adventure with Caleta Cóndor Expediciones.
8Need to Know
The easiest way to get to Caleta Cóndor is via two-hour boat ride from Bahía Mansa.
5Take a Break
Enjoy scrumptious seafood with bay views from the outdoor patio at Glorymar ( %cell 9-8299-8587; Costanera Bahía Mansa s/n, Bahía Mansa; mains CH$5900-12,900; h10am-9pm, to 8pm Sun, shorter hr winter).
oTop Tip
If arriving by boat, be sure to take preventatives for seasickness!
### San Juan de la Costa
In San Juan de la Costa, a series of five magnificent _caletas_ (coves) – Pucatrihue, Maicolpué, Bahía Mansa, Manzano and Tril-Tril are the jumping off point for Caleta Cóndor.
Popular hikes include the 10km round-trip jaunt from Maicolpué to Playa Tril-Tril (also reachable by car) and, from Pucatrihue, a 16km round-trip to Caleta Manzano (reachable within 1km, by 4WD only). Boat trips (CH$10,000 per person, minimum three) to visit a penguin colony are also offered. At sunset, surfers even hit the waves in Maicolpué.
Maicolpué on Osorno's coast | SRREPHO / SHUTTERSTOCK ©
### Caleta Cóndor
Caleta Cóndor, part of the protected indigenous zone of Territorio Mapa Lahual, is an impossibly gorgeous bay almost completely off the grid. If the weather is clear, arriving here is miraculous. As you enter via the scenic and translucent Río Cholguaco from the Pacific, you are greeted with idyllic moss-strewn riverbanks, horses and seabirds, and hillsides peppered with arrayán trees, with a small community (20 or so families – all running on solar power) living in this piece of paradise. It all culminates at the river's end on a sublime ocean beach.
### Caleta Cóndor Expediciones
Besides gawking at scenery, kayaking, diving, trekking, and horseback riding are the main activities in Caleta Cóndor. Locally owned and operated Caleta Cóndor Expediciones ( %cell 9-9773-6383; www.caletacondorexpediciones.cl) is the go-to agency to set things up and connect travelers with providers.
### Rural Homestays
Perched high above Maicolpué beach in Maicolpué Río Sur, the Altos de Pichi Mallay ( %64-255-4165; raicesrestaurant1@gmail.com; Maicolpué Río Sur camino a Tril Tril, Maicolpué Río Sur; campsites/r per person CH$5000/25,000; p W) is family-run paradise with 12 cozy guest rooms, most with panoramic sea views. There's also a six-person, wood-fired hot tub nestled in the forest with a jaw-dropping ocean view.
Imagine free-range horses, sheep and cows grazing in front of one of South America's most perfect Pacific beaches and you have an idea what's in the front yard of the purpose-built Hostal Caleta Cóndor ( %9-9382-4035; www.caletacondorexpediciones.cl; Caleta Cóndor; r per person without bathroom incl breakfast CH$18,000, meals CH$5000). Sleeps 13.
## Osorno
Osorno is a bustling place and the commercial engine for the surrounding agricultural zone. Though it's an important transportation hub on the route between Puerto Montt and Santiago and the Huilliche communities of the Osorno coast, most visitors spend little time here, though Osorno now harbors a handful of cool cafes, good restaurants and quality brewpubs.
5Eating
Mercado MunicipalChilean $
(cnr Prat & Errázuriz; h6am-9pm)
Large and modern Mercado Municipal has an array of _cocinerías_ (lunch stalls) serving good and inexpensive food.
Café CentralCafe $
(O'Higgins 610; sandwiches CH$1950-6500; h8am-midnight Mon-Sat, noon-10pm Sun)
This bi-level plaza hot spot is more or less a Chilean diner, crowded for its decent coffee, colossal burgers, ridiculous sandwiches and hot dogs _completísimo_ (sauerkraut, avocado, mayo). There's also a counter – convenient for solo travelers.
PancaPeruvian $$
( %64-223-2924; Rodríguez 1905; mains CH$5500-12,000; h12:30-3pm & 8-11:30pm Mon-Sat, 12:30-3:30pm Sun; W)
You'll be as surprised as we were about how authentic the classic Peruvian dishes at this hip newcomer are, considering the Peruvian side of this Chilean-Peruvian partnership handles the finances, not the food. Spicy ceviche, _chaufa_ (Peruvian-Chinese fried rice), _ají de gallina_ (spicy creamed chicken), _lomo saltado_ (marinated steak) – it's all pretty perfect.
El GalpónSteak $$$
( %64-223-4098; www.hotelwaeger.cl; Cochrane 816; steaks CH$13,000-18,500; h12:30-3pm & 7:30-11pm Mon-Sat; W)
Tucked away around the side of Hotel Waeger, you'll find dark flooring and a rustic, barnyard vibe (old-school metallic buckets as lampshades) complementing the main event: perfectly grilled steaks for devout carnivores, on a _parrilla_ piled high with wood, like a fireplace. The fillets are superthick and the potent _pisco_ sours often outrun the wine.
Parque Nacional Vicente Pérez Rosales
In this park of celestial lakes and soaring volcanoes, **Lago Todos Los Santos** and **Volcán Osorno** may be the standouts, but they're actually just part of a crowd. One lake leads to the next and volcanoes dominate the skyline on all sides of this storied pass through the Andes range. The needlepoint of **Volcán Puntiagudo** (2493m) lurks to the north and craggy **Monte Tronador** (3491m) marks the Argentine border to the east.
Established in 1926, the 2510-sq-km Pérez Rosales was Chile's first national park. Get park info and pick up maps from Conaf's visitors center ( %65-221-2036; www.conaf.cl; Laguna Verde; h8:30am-5:30pm Sun-Thu, to 4:30pm Fri).
6Drinking & Nightlife
Cervecería Artesanal Armin SchmidMicrobrewery
( %cell 9-8294-1818; Ruta 215, Km12; beer CH$2300-3300, pizzas CH$8100; h1:30-10:30pm Tue-Sat)
If you're within 100km of Osorno and a beer lover, you'll want to readjust your itinerary to visit this improvised temple of suds. It's the south's most interesting craft brewery, located 12km outside of Osorno on Ruta 215 toward Entre Lagos and the border with Argentina.
GallardíaGastropub
(www.facebook.com/gallardia.sg; O'Higgins 1270; h6pm-1am Mon, 1pm-1am Tue-Sat, 1-6pm Sun; W)
Occupying a traditional Osorno residence, Gallardía is a rooster-adorned enclave for craft beers (Kross, along with Belgian, Dutch, Spanish and German selections), cocktails made with Patagonia's very own hipster craft spirit Trä·kál (born in Osorno; CH$5000) and a battery of slang-named gourmet sandwiches that hit the spot when buzzed (CH$5000 to CH$7300). It's where the cool kids go.
Taberna PirataBar
(www.facebook.com/tabernapirata; MacKenna 1873; h6pm-3am Mon-Sat; W)
The quirkily themed Pirata is vaguely divey and catering to bohemian locals in the know. It's the city's best for craft beer, with 10 taps flowing in summer (weekends only otherwise), which are a mix of house brews and Chilean micro/nanobrews from Valdivia and Puerto Varas (along with a West Coast USA–heavy bottle list).
Chilean blankets and ponchos | HOLGER LEUE / GETTY IMAGES ©
## Puerto Varas
Two menacing, snowcapped volcanoes, Osorno and Calbuco, stand sentinel over picturesque Puerto Varas and its scenic Lago Llanquihue like soldiers of adventure, allowing only those on a high-octane quest to pass. Just 23km from Puerto Montt but worlds apart in charm, scenery and options for the traveler, Puerto Varas is the region's destination for outdoor adventure sports.
Puerto Varas
2Activities, Courses & Tours
1Al SurC3
2Ko'KayakB2
3Moyca ExpedicionesB2
7Shopping
4Fundación Artesanías de ChileB3
5Eating
5CostumbristaB2
6El HumedalB1
6Drinking & Nightlife
7Caffé El BaristaB2
1Sights
Paseo PatrimonialArchitecture
Many notable constructions in town are private houses from the early 20th century. Grab a city map at the tourist office to follow the **Paseo Patrimonial** , a walking tour of historic homes listed as Monumentos Nacionales. Several of these houses serve as _hospedajes_ , including the 1941–42 **Casa Schwerter** (Del Carmen 873), the 1930 **Casa Hitschfeld** (Prat 107) and the 1930 **Casa Wetzel** (O'Higgins 608).
7Shopping
Fundación Artesanías de ChileArts & Crafts
(map Google map; www.artesaniasdechile.cl; Del Salvador 109; h9am-9pm Mon-Sat, 10:30am-7pm Sun) S
A not-for-profit foundation offering beautiful Mapuche textiles as well as high-quality jewelry and ceramics from all over southern Chile.
5Eating
CostumbristaChilean $
(map Google map; %cell 9-6237-2801; Del Salvador 547B; mains CH$5000-6000; h1-4pm & 7-9:30pm Mon-Sat; W)
This little, unassuming Chiloé-inspired eatery is Puerto Varas' secret gourmet _cocinería_ (kitchen). It's a two-man, eight-table show among aqua clapboard walls with a limited menu of fish and heartier dishes – a perfect pork chop, osso buco, a beautiful salmon or _meluza_ (hake) – and is executed with skill and results that completely outshine its extraordinary-value price range.
La GringaAmerican $
(Imperial 605; mains CH$4800-11,900; h8am-11pm Mon-Fri, 9am-11pm Sat; W) S
Evoking the rainy-day cafes of the American Pacific Northwest, this charming spot run by an adorable Seattleite dishes up scrumptious house-produced bread and pastries, hearty sandwiches (such as pulled pork with cabbage) and refreshing main courses (barbecue baby back ribs with spicy austral-pepper mash) along with standout CH$7300 _menús_ (fixed-price menus; 1pm to 4pm except February).
El HumedalFusion $$
(map Google map; %65-223-6382; www.humedal.cl; Turismo 145; mains CH$7000-11,000; h12:30-11pm Tue-Sat, to 5pm Sun; W)
In an adorable and cozy home perched on a hilltop over town, El Humedal (oddly, it means 'Wetland') serves up one of the best lunches in Puerto Varas with its interpretations of Asian curries and stir-fries, Mexican enchiladas and burritos, ramen bowls, and fish and chips, among others. Desserts are scrumptious as well.
Enchiladas | JOSIE GRANT / SHUTTERSTOCK ©
6Drinking & Nightlife
Chester Beer Brewing CompanyMicrobrewery
(www.chesterbeer.cl; Línea Nueva 93, Campo Molino Viejo; pints CH$2500; h10am-5pm Mon-Fri, noon-7pm Sat)
American nanobrewer 'Chester' (real name Derek Way) has been at it since 2006, long before 'craft' was a thing in Chile. His rustic but epic countryside brewery – fashioned from shipping containers – is a true makeshift beervana, with four taps dedicated to experimental one-offs and all of his staples in bottles (IPA, summer ale, APA and stout).
Brewing espresso | BARANQ / SHUTTERSTOCK ©
Caffé El BaristaCafe
(map Google map; www.elbarista.cl; Martínez 211; coffee CH$1400-3200; h8am-1am Mon-Thu, to 2am Fri & Sat, noon-1pm Sun, shorter hours winter; W)
This stylish Italian-style coffeehouse serves Sur Chico's best espresso from its La Marzocco machine, and draws a healthy breakfast and lunch crowd for cast-iron eggs, excellent CH$6800 _menús del día_ and a selection of tasty sandwiches. As night falls, it morphs into a fine spot for a drink as well.
## Cochamó
The Chilote-style, alerce-shingled **Iglesia Parroquial María Inmaculada** stands picturesque and proud against a backdrop of milky-blue water along the road to Cochamó, forming one of the most stunning spots throughout the region and the gateway to the upper Río Cochamó Valley. In addition to its made-over _costanera_ , some vibrant new accommodation choices are now vying for your attention, doing their best to graduate Cochamó to more than just the spot where you put your kayaks in the water on a day trip from Puerto Varas.
Río Cochamó Valley | MANON VAN GOETHEM / SHUTTERSTOCK ©
5Eating
El FaroChilean $
(Costanera s/n; mains CH$3700-8000; h9am-10pm, shorter hr winter)
Probably your best bet for a decent home-cooked meal in Cochamó, El Faro does ceviches, fresh fish such as _merluza_ (hake) and congrio (conger eel) a number of ways, and other seafood delights. It's right on the water, so views are expectedly dramatic.
## Frutillar
Frutillar, right up the coastline of Lago Llanquihue from Puerto Varas, has an attractive pier, a long lakeside beach and quaint German architecture, with a soundtrack provided by buff-necked ibises, who cackle from rooftops all over town. Though it can at times feel touristy, Frutillar remains a serene spot that makes for a pleasant alternative to staying in busier Puerto Varas.
Frutillar Pier | YURI BARICHIVICH / GETTY IMAGES ©
1Sights
Museo Histórico AlemánMuseum
(www.museosaustral.cl; cnr Pérez Rosales & Prat; CH$2500; h9am-5:30pm)
The Museo Histórico Colonial Alemán was built with assistance from Germany and is managed by the Universidad Austral. It features nearly perfect reconstructions of a water-powered mill, a blacksmith's house and a farmhouse and belfry set among manicured gardens. It is considered the best museum on German colonialism in the region.
7Shopping
Vipa & Co.Food, Cosmetics
(www.vipaonline.cl; Av Philippi 989; h11am-7:50pm)
A great shop for picking up artisanal foodstuff and biodegradable cosmetics, including high-quality brands such as Melí (spices, salsas and mustards) and Agua de Patagonia (soaps, shampoos, aromatherapy).
5Eating
Lavanda Casa de TéTeahouse $$
( %cell 9-9458-0804; www.lavandacasadete.cl; Camino a Quebrada Honda, Km1.5; menú CH$15,000; h1-8pm Dec-Mar)
On a lavender farm just outside town, this is a lakeside favorite for tea, gourmet lavender products and farm-fresh lunches. Make a reservation for afternoon tea (CH$9800) or a lingering lunch. Minimum consumption is CH$6000.
Se CocinaChilean $$$
( %cell 9-8972-8195; www.secocina.cl; Camino a Quebrada Honda, Km2; mains CH$10,500, 3-course meal CH$21,200; h1-3pm & 8-10:30pm Tue-Sun Jan-Feb, closed Mon & Tue Dec-Mar)
Se Cocina is hit or miss, but even when it misses, this beautiful 1850s farmstead 2km from Frutillar is one of the most interesting foodie destinations on the lake. The daily changing menu marries Nueva Chilena cuisine with a modern atmosphere housed inside a historically protected farm.
## Puerto Octay
Cute and quaint, Puerto Octay (ock- _tie_ ) isn't heavily visited, but it's actually one of the more charming towns on Lago Llanquihue. The tranquil streets, perched on a hillside above the lake, yield interesting 1800s German-settler architectural treasures around every turn, making for a nice tour of historic homes and buildings and giving the town a supremely sedate and picturesque colonial air. It is the oldest town on the lake settled by Germans, and, of course, it has a requisite **Oktoberfest** (www.oktoberfestpuertooctay.cl).
1Sights
Museo de Puerto OctayMuseum
(Independencia 591, 2nd fl; CH$1000; h10am-1pm & 3-5pm)
A small but well-done museum inside the historic 1920 Casa Niklitschek telling the story of Puerto Octay via antiques.
Teatro del Lago
This amazing 12-years-in-the-making, US$25-million world-class performing-arts center (Teatro del Lago; %65-242-2900; www.teatrodellago.cl; Av Philippi 1000; hbox office 10am-6pm) opened in 2010, and has single-handedly put Frutillar on the global cultural map. The striking copper-roofed structure is a thing of beauty in itself, flanked against the lake with postcard views of four volcanoes. Daily 45-minute tours start at noon throughout the year (CH$4500).
Inside, it houses a state-of-the-art 1178-seat concert hall – acoustically insured by beautiful beechwood walls – and a second 278-seat amphitheater, as well as a pizzeria and lakeside cafe. It currently hosts a wealth of cultural events, including the music festival, and attracts internationally known orchestras and artists in all genres.
Alfresco dining at Teatro del Lago | DIEGO GRANDI / SHUTTERSTOCK ©
5Eating
Rancho EspantapájarosChilean $$$
( %65-233-0049; www.espantapajaros.cl; Quilanto, Camino Puerto Octay–Frutillar Km5; buffet CH$17,000, sandwiches CH$3500-6500; hnoon-10pm Jan-Feb, noon-8pm Mar, noon-5:30pm Mon-Thu, to 10pm Fri & Sat, to 7:30pm Sun Apr-Nov; W)
The most famous restaurant on the lake is 6km outside Puerto Octay on the road to Frutillar. It packs in the crowds for the main attraction, succulent _jabalí_ (wild boar, fatty but fantastic) cooked on 3.5m spits across a giant _fogón_ behind the buffet.
# Torres Del Paine, Chile
#### Hiking the Parque Nacional Torres del Paine
#### Puerto Natales
#### Sights
#### Activities
#### Shopping
#### Eating
#### Drinking
#### Entertainment
#
Torres del Paine, Chile
_Soaring almost vertically above the Patagonian steppe, the granite pillars of Torres del Paine (Towers of Paine) dominate the landscape of South America's finest national park. Part of Unesco's Biosphere Reserve system since 1978, this 1810-sq-km park is, however, much more than its one greatest hit. Its landscapes range from teal and azure lakes to emerald forests, roaring rivers and that one big, radiant blue glacier. Guanacos roam the vast open steppe, while Andean condors soar alongside looming peaks._
Mirador Las Torres, Parque Nacional Torres del Paine | DAVID IONUT / SHUTTERSTOCK ©
Two Days in Parque Nacional Torres del Paine
On the first day, set out early for a rewarding day hike along the Río Pingo up to a lookout with views of Glaciar Zapata. For your second day, admire the amazing scenery while kayaking on the iceberg dotted Lago Grey on an excursion with Big Foot Patagonia.
Four Days in Parque Nacional Torres del Paine
On day three, head off for some uncommon adventures–either ice trekking on Glaciar Grey or horseback riding along Laguna Azul.
On day four take in a bit of rural life (including herding-dog and sheep-shearing demonstrations) at Estancia La Península. In the evening treat yourself to a Chilean feast at Singular Restaurant.
Arriving in Parque Nacional Torres del Paine
Most visitors access the region by road via Punta Arenas (300km south of Torres del Paine), though there are some flights into Puerto Natales (60km south of Torres del Paine). There's frequent bus service between Parque Nacional Torres del Paine and Puerto Natales, but for more remote locations visitors may need a rental car. High-end lodgings provide transfers.
Where to Stay
Forget roughing it. You can hike the whole 'W' while sleeping in beds, eating hot meals, taking showers and toasting your day with a pisco sour. Among the lodging options are campsites, dormitory-style _refugios_ (trailside cabins with four to eight bunks per room), _domos_ (yurt-like 'domes') and high-end hotels. Reservations are essential.
TOP EXPERIENCE
# Hiking the Parque Nacional Torres del Paine
Torres del Paine's 2800m granite peaks inspire a mass pilgrimage of hikers from around the world. Most go for the Paine Circuit or the 'W' to soak in these classic panoramas.
Great For...
cgf
yDon't Miss
The remarkable view over the jagged landscape from the Mirador Las Torres.
8Need to Know
www.parquetorresdelpaine.cl; high/low season CH$21,000/11,000
5Take a Break
Meal plans are available at _refugios_ , though high demand means not all last-minute requests can be satisfied.
oTop Tip
Weather can present four seasons in a day, with sudden rainstorms and knock-down gusts like a hearty Patagonian handshake.
### The 'W'
Hiking west to east provides superior views of Los Cuernos, especially between Lago Pehoé and Valle Francés. Most hikers take the catamaran across Lago Pehoé and head to Mountain Lodge Paine Grande. Going in this direction, the hike is roughly 71km in total. It's also possible to take a cruise from Hotel Lago Grey to Refugio Grey to avoid some backtracking.
In clear weather the 10km stretch between Valle Francés to Los Cuernos is magnificent. The route runs from 3050m Cerro Paine Grande in the west to the lower but still spectacular Torres del Paine and Los Cuernos to the east, with glaciers hugging the trail.
### The Paine Circuit
For solitude, stellar views and bragging rights over your compadres doing the 'W,' this longer trek is the way to go. This loop takes in the 'W,' plus the backside between Refugio Grey and Refugio Las Torres; the total distance is roughly 112km. The landscape is desolate yet beautiful. **Paso John Gardner** (the most extreme part of the trek) sometimes offers knee-deep mud and snow.
Guanacos in Parque Nacional Torres del Paine | EKATERINA POKROVSKY / SHUTTERSTOCK ©
### Other Overnight Hikes
From Guardería Lago Grey, a four-hour trail follows Río Pingo to Conaf's Camping Zapata, from where hikes (another 1½ to two hours) continue to a lookout with impressive views of **Glaciar Zapata** and **Lago Pingo**. Because of ongoing studies of wildlife and fossil beds, hiking in this pristine area is authorized only for groups traveling with a Conaf-approved guide.
From Guardería Laguna Amarga a four-hour hike leads to **Laguna Azul**. The camping area on the northeastern shore closed after a wildfire; check with Conaf about its current status. After another two-hour hike north the trail reaches **Lago Paine**. Accessibility to meet up with the Paine Circuit trail near the other side of the lake is made impossible by the river.
From the Administración, the three-hour hike to Hostería Pehoé is an easy, mainly flat trail with great views. For more solitude and bird-watching, a four-hour hike branches east after crossing Río Paine, zigzags up the skirt of the Sierra del Toro to access a string of lakes, ending with **Laguna Verde**. There is no camping, but those inclined could splurge for a night at Hostería Mirador del Payne.
### Day Hikes
Walk from Guardería Pudeto, on the main park highway, to **Salto Grande** , a powerful waterfall between Lago Nordenskjöld and Lago Pehoé. Another easy hour's walk leads to Mirador Nordenskjöld, an overlook with superb views of the lake and mountains.
For a more challenging day hike with tranquility and gorgeous scenery, try the four-hour trek to **Lago Paine** ; its northern shore is only accessible from Laguna Azul.
Hiker near Salto Grande waterfall | SHARPTOYOU / SHUTTERSTOCK ©
### Kayaking & Boating
Big Foot Patagonia ( %61-241-4611; www.bigfootpatagonia.com; kayaking CH$66,000, ice hike CH$105,000; hOct-Apr) leads 2½-hour tours of the iceberg-strewn Lago Grey several times daily in summer; this is a great way to get up close to glaciers. A more demanding five-hour tour (CH$160,000) starts at Río Pingo to paddle the river toward Glaciar Grey, surrounded by icebergs, ending at Río Serrano.
From October through April, catamaran _Grey III_ does Navegación Glaciar Grey (Glaciar Grey Cruise; %61-271-2100; www.lagogrey.com; adult/child round-trip CH$80,000/40,000, one-way CH$70,000/35,000; hOct-Apr), a three-hour cruise to take in the massive glacier up close, with a stop to let hikers on and off. In high season there are four daily departures; the last two do glacier viewing before stopping at the trail for passengers.
Family-oriented floating trips that take rafts down the mild Río Serrano are run by Fantástico Sur ( %61-261-4184; www.fantasticosur.com; Esmeralda 661; h9am-1pm & 3-6pm Mon-Fri).
Kayaking on Lago Grey | MARCO SIMONI / GETTY IMAGES ©
### Horseback Riding
Baqueano Zamora ( %61-261-3530; www.baqueanozamora.cl; Baquedano 534; h10am-1pm & 3-7pm) runs excursions to Laguna Azul, Valle Francés, Dickson glacier and more remote locations, with one-day and multiday options.
Hotel Las Torres ( %61-261-7450; www.lastorres.com; booking address Magallanes 960, Punta Arenas; s/d incl breakfast from US$382/437; hJul-May; W) S offers full-day horseback-riding trips around Lago Nordenskjöld and beyond.
### Ice Trekking
A fun walk through a sculpted landscape of ice, and you don't need experience to go. Antares' Big Foot Patagonia is the sole company with a park concession for ice hikes on Glaciar Grey, using the Conaf house (former Refugio Grey) as a starting point. The five-hour excursion is available from October to April, in high season three times per day.
8Reserving Ahead
Park regulations require hikers doing the Circuit or the 'W' to reserve all lodgings ahead; this includes both _refugios_ and camping.
### Wildlife
The park is home to flocks of ostrich-like rhea (known locally as the ñandú), Andean condor, flamingo and many other bird species. Its star success in conservation is undoubtedly the guanaco, which grazes the open steppes where pumas cannot approach undetected. After more than a decade of effective protection from poachers, these large, growing herds don't even flinch when humans or vehicles approach. The puma population is also growing, and huemul (an endangered Andean deer) have been spotted in Valle Francés.
### Tours
Tour operators in Puerto Natales offer guided treks, which include all meals and accommodations at _refugios_ (rustic shelters) or hotels. Per person rates decrease significantly in groups.
Guided day trips on minibuses from Puerto Natales are possible, but allow only a glimpse of what the park has to offer.
Patagonia Bagual ( %cell 9-5325-1266; <http://patagoniabagual.cl>; Laguna Azul; CH$135,000) takes hikers into the most pristine part of Torres del Paine to hike cross country to observe wild horses. Nothing about it is a canned experience. Guides are experienced and multilingual. Includes 4WD transfer but park entrance is not included.
Hotel Las Torres ( %61-271-0050; www.lastorres.com; Estancia Cerro Paine) controls the eastern area of the park and runs day and multiday tours with hiking and horseback riding; see the activities desk at the hotel for information if you are already in the park.
oBeating the Crowds
Most hikers go up to the Torres around 8am and down at 4pm. With full summer light, you can beat the crowds by starting a couple of hours earlier or later.
## Puerto Natales
A formerly modest fishing port on Seno Última Esperanza, Puerto Natales has blossomed into a Gore-Tex mecca. The gateway to Parque Nacional Torres del Paine, this town feeds off tourism, an all-you-can-eat feast with unwavering demand. Boutique beers and wine tastings have overtaken tea time, and gear shops have replaced the yarn sellers. While there are growing services that cater to international tastes, there's appeal in Natales' corrugated-tin houses strung shoulder to shoulder and cozy granny-style lodgings.
1Sights
Museo HistóricoMuseum
( %61-241-1263; Bulnes 285; CH$1000; h8am-7pm Mon-Fri, 10am-1pm & 3-7pm Sat & Sun, reduced hours May-Nov)
Worth a quick visit, this is a crash course in local history, with archaeological artifacts, a Yaghan canoe, Tehuelche bolas (throwing weapon) and historical photos.
2Activities
Estancia La PenínsulaOutdoors
( %cell 9-6303-6497; www.estanciaspatagonia.com; Península Antonio Varas; day tour CH$130,000)
Run by a family with pioneer roots in the region, this classic _estancia_ across the water offers day visits that include hiking or riding, herding-dog and sheep-shearing demonstrations and an awesome barbecue lunch of spit-roasted lamb. There are also excellent multiday hiking options with a remote feel. The meeting place is the dock at Singular Hotel.
Antares/Big Foot PatagoniaAdventure
( %61-241-4611; www.antarespatagonia.com; Costanera 161, Av Pedro Montt; Lago Grey kayaking CH$66,000)
Specializing in Torres del Paine, Antares can facilitate climbing permits and made-to-order trips. Its sister company Big Foot has the park concession for Lago Grey activities, including Glacier Grey ice-trekking and kayak trips, with a base in the park.
Chile NativoAdventure
( %61-241-1835, cell 9-9078-9168; www.chilenativo.cl; Eberhard 230, 2nd fl)
Links visitors with local gauchos, organizes photo safaris and can competently plan your tailor-made dream adventures.
Mirador DoroteaHiking
(Ruta 9; CH$5000)
A day hike through a lenga forest on private land to splendid views of Puerto Natales and the glacial valley. Less than 10km from Natales. Dorotea is the large rocky outcrop just off Ruta 9.
Turismo 21 de MayoTours
( %61-261-4420; www.turismo21demayo.com; Eberhard 560; h8am-10pm Oct-Mar)
Organizes day-trip cruises and treks to the Balmaceda and Serrano glaciers (CH$90,000) and horseback riding on Cerro Dorotea (CH$30,000), just outside Puerto Natales.
Mandala AndinoSpa
( %cell 9-9930-2997; mandalaandino@yahoo.com; Bulnes 301; massages from CH$25,000; h10am-9pm Nov-Mar)
A recommended full-service wellness center with spot-on massages, tub soaks and various pampering treatments, including cannabis-oil massages.
Patagom LilaYoga
( %cell 9-6140-7857; www.yogapatagomlila.com; Galvarino 345) S
Wonderful yoga teacher Susanne offers classes in English, German and Spanish in both a downtown house and a spectacular rural dome with views of the Seno Última Esperanza, where you will also find permaculture courses, yoga vacations and Thai and singing bowl massages. She also brings alternative therapies into the local community.
7Shopping
OneacoSports & Outdoors
(cnr Eberhard & Magallanes; h10am-9pm Mon-Sat, 11am-2pm & 4-8:30pm Sun)
Down jackets, hiking boots and mountain equipment from international brands are sold here. Ticket prices may be double those back home, but for those waiting on lost luggage it's a lifesaver.
Wine & MarketWine
( %61-269-1138; www.wmpatagonia.cl; Magallanes 46; tasting CH$20,000; h10am-10pm Mon-Sat)
Picnickers should pop in here for a range of tasty gourmet products from all over Chile, and a great wine and craft-beer selection. If your trip doesn't take you to wine country, it's well worth attending one of its daily tastings, with a sommelier presenting four classic wines.
5Eating
La GuanacaPizza $$
( %61-241-3245; Magallanes 167; mains CH$5000-16,000; h12:30-10pm Mon-Sat; v)
From crisp oven-fired pizzas to crepes and marinated mushroom appetizers, this homespun restaurant delivers warming and satisfying meals. Oversized salads, like the quinoa with roasted vegetables, are abundant and varied. There's craft beer and several wines to choose from.
La AldeaMediterranean $$
( %cell 9-6141-4027; www.aldearestaurant.cl; Barros Arana 132; mains CH$8000-16,000; h7-11pm Wed-Mon)
Chef Pato changes the offerings daily, but the focus is fresh and Mediterranean, with a nod to local ingredients. Think grilled clams, lamb tagine and quinoa dishes with an elegant presentation. Don't skip the decadent _tres leches_ (three milks) cake for dessert.
SantollaSeafood $$$
( %61-241-3493; Magallanes 77; mains CH$15,000-22,000; h7-11pm Mon-Sat)
For worthwhile upscale seafood, look no further than this cozy container restaurant attended by the owner. Feast on gorgeous salads and local king crab prepared several ways; we liked it with _merken_ (smoked chili), white wine and parsley. Nonseafood eaters have options such as steak or rabbit in black-truffle sauce.
Singular RestaurantChilean $$$
( %61-272-2030; Puerto Bories; mains CH$12,000-18,000; h8am-11pm)
The perfect port in a storm, part supper club of yore, part modern bistro, with exquisite food and attentive service. Leather sofas and polished wood meet bare beams and stark views of the sound. Chef Hernan Vaso reinvigorates local ingredients: the freshest ceviche, tender lamb medallions and lovely salads come with original sides and fine Chilean wines. Vegetarian options excel.
6Drinking
Last HopeDistillery
( %cell 9-7201-8585; www.lasthopedistillery.com; Esmeralda 882; h5pm-2am Wed-Sun Dec-Mar, 8pm-2am Sat & Sun Apr-Nov)
Two Australians on vacation tossed in their day jobs to distill whiskey and gin at the end of the world. With bonhomie to spare, their bar caters to locals and travelers alike, with a rotating menu of gorgeous cocktails. The signature drink is a calafate berry gin and tonic. It's tiny and the overflow waits outside – wear your down jacket.
BagualesMicrobrewery
(www.cervezabaguales.cl; Bories 430; h6pm-2:30am Mon-Sat; W)
Climber friends started this microbrewery as a noble quest for quality suds and the beer (crafted on-site) does not disappoint. A 2nd-floor addition seeks to meet the heavy demand. The gringo-style bar food is just so-so.
3Entertainment
Centro Cultural Galpon PatagoniaCultural Center
(<http://galponpatagonia.cl>; Pedro Montt 16; h10am-1pm & 3-7pm Tue-Sun)
This cultural center and teahouse occupies a revamped 1920 warehouse with exposed beams and worn floorboards. Features art exhibits, theater, dance and music.
# The Atacama Desert, Chile
#### Visiting the Desert
#### San Pedro de Atacama
#### Sights
#### Activities
#### Shopping
#### Eating
#
The Atacama Desert, Chile
_The Atacama is home to northern Chile's most spectacular scenery. Amid a parched, untrammeled landscape, you'll find vast salt flats, russet-colored lunar-like ridges and topaz lakes sparkling against a backdrop of soaring mountain peaks. The region is also home to the ruins of pre-Columbian peoples, as well as gurgling geyser fields, towering sand dunes (ideal for sandboarding) and some of the clearest night skies on the planet. Given the great variety on offer, you could easily spend a week in the area and still not see everything._
Valle de la Luna |a PHILIP LEE HARVEY / LONELY PLANET ©
Two Days in the Atacama Desert
On the first day head to the Valle de la Muerte for a walk amid the Mars-like scenery or to sandboard down massive dunes. Later, watch the sunset over the spectacular Valle de la Luna.
On day two rise early for the trip to the bubbling El Tatio Geysers. In the evening, treat yourself to dinner at Baltinache.
Four Days in the Atacama Desert
Visit pre-Columbian ruins on a guided visit to Aldea de Tulor. Later, soak in the Termas de Puritama. At night go stargazing with Una Noche con las Estrellas.
In the morning, check out the salt lake of Laguna Cejar, then spy flamingos at Laguna Chaxa. In the afternoon, take a walk with Tours 4 Tips followed by craft shopping around town.
Arriving in the Atacama Desert
The nearest airport, Aeropuerto El Loa, is 100km northwest of San Pedro de Atacama and has daily flights to Santiago. Several agencies offer transfer services to/from Calama airport including **Tour Magic** (www.tourmagic.cl; CH$8000 per person, 1½ hours).
San Pedro's bus terminal is 1km southeast of the plaza, with regular departures to Calama where you can connect to all major destinations in Chile.
Where to Stay
The village of San Pedro de Atacama is the obvious base for exploring the desert attractions nearby. You'll find a dizzying number of accommodations, particularly hostels. Midrange hotels are in short supply at this backpackers' haven, but many budget hostels have midrange rooms. Top-end hotels have sprouted around San Pedro, though many are outside the center.
TOP EXPERIENCE
# Visiting the Desert
Just outside the small village of San Pedro there are intriguing archaeological ruins, steaming geyser fields and stunning landscapes. One of the big must-see attractions is here: the Valle de la Muerte, with its otherworldly scenery and a big dune that's perfect for sandboarding.
Great For...
gAf
yDon't Miss
Seeing the myriad desert colors come to life while the sun sets over the Valle de la Luna.
8Need to Know
You can visit all of the sights on guided tours from San Pedro de Atacama.
5Take a Break
After a day in the desert, refresh with artisanal ice cream at Babalú.
oTop Tip
Be sure to pack a jacket for those chilly desert nights.
### Valle de la Muerte
Around 3.5km west of San Pedro, this striking valley (map; off Ruta 23; CH$3000; h9am-8pm) should figure high on any itinerary to the region, with jagged rocks, a towering sand dune and dramatic viewpoints of the distant cordillera. The name Valle de la Muerte (Death Valley) is actually a linguistic distortion of Valle de Marte (Mars Valley), which more accurately represents its red rock features and otherworldly beauty.
Tour groups typically come in the afternoon before heading over to the nearby Valle de la Luna for sunset. It's an easy bike ride (or long walk) here, and also accessible by your own vehicle. The tall sand dune is a prime destination for sandboarding, with outfits like Sandboard San Pedro ( %cell 9-8135-1675; www.sandboardsanpedro.com; Caracoles 362-H) offering both morning and afternoon excursions.
Valle de la Muerte | IGNACIO PALACIOS / GETTY IMAGES ©
### El Tatio Geysers
One of the must-see sights of Atacama is this sputtering geyser field (map; 95km north of town; CH$10,000; h6am-6pm), located at an altitude of more than 4300m, some 90km north of San Pedro de Atacama. Fringed by rolling peaks, the landscape churns with bubbling pools and steaming columns wafting into the air. Tiny eruptions from more than 60 gurgling geysers pierce the alpine calm. Tours here typically depart at 4am or 5am to arrive in time for sunrise, which is the most dramatic time to see Tatio.
El Tatio Geysers | IGNACIO PALACIOS / GETTY IMAGES ©
### Valle de la Luna
This valley (map; CH$3000; h9am-6pm) owes its name to the striking lunar-like landscape, which is located in the Cordillera de la Sal. Over the eons, its chiseled rock formations have been carved by the desert winds, leaving behind a handful of caverns. See the desert don its surrealist cloak as you stand atop a giant sand dune, with the sun slipping below the horizon and multicolored hues bathing the sands, all with a backdrop of distant volcanoes and the rippling Cordillera de la Sal.
### Laguna Cejar
San Pedro has its own mini version of the Dead Sea, a mere 22km south of the village. This topaz-colored lake (map; CH$10,000-15,000; h9am-5pm) allows you to float effortlessly because of its high salt content – a fine place to contemplate the mountainous horizon. Cejar is just one of three lakes here; the other two (Laguna Piedra and Laguna Baltinache) do not allow bathing, though you can often see flamingos feeding here.
### Laguna Chaxa
The jagged crust of the **Salar de Atacama** looks for all the world like God went crazy with a stippling brush. But in the midst of these rough lifeless crystals is an oasis of activity: the pungent Laguna Chaxa (map; CH$2500; h8am-8pm), about 25km southwest of Toconao and 65km from San Pedro, the Reserva Nacional Los Flamencos' most easily accessible flamingo-breeding site.
Three of the five known species (James, Chilean and Andean) can be spotted at this salt lake, as well as plovers, coots and ducks: bring zoom lenses and snappy reflexes. Sunrise is feeding time for the birds, though the park doesn't open until 8am. It's also gorgeous at sunset.
Andean flamingos, Laguna Chaxa | DELPIXART / GETTY IMAGES ©
### Lagunas Miscanti & Miñiques
Shimmery high-altitude lakes dot the altiplano and make for worthwhile excursions from San Pedro. The glittery-blue sweet-water lakes, Miñiques (map; incl entrance to Laguna Miscanti CH$3000; h9am-6pm) and Miscanti (map; incl entrance to Laguna Miñiques CH$3000; h9am-6pm) are watched over by snow-touched volcanoes. The smaller Laguna Miñiques is the largest breeding site for the horned coot on the western side of the Andes, and visitors are kept at bay when the birds are breeding.
### Termas de Puritama
These idyllic volcanic hot springs (map; <https://termasdepuritama.cl>; adult/child CH$19,500/9100; h9:30am-5:15pm) puddle together in a box canyon, about 30km northeast of San Pedro en route to El Tatio. Their temperature is about 33°C (91°F), and there are several falls, pools and changing rooms on-site. Few tours stop here because of the hefty admission charged, but taxis will take you or you can drive yourself. The springs are a 20-minute walk from the parking lot.
Bring food, water and sunblock. Go early in the day to beat the crowds. Prices are cheaper Monday to Friday after 2pm.
### Pukará de Quitor
Dominating a curvaceous promontory over the Río San Pedro, this crumbling 12th-century pukará (map; CH$3000; h8:30am-7pm; fort) was one of the last bastions against Pedro de Valdivia and the Spanish in northern Chile. The indigenous forces fought bravely, but were overcome and many were promptly beheaded. About 100 defensive enclosures hug the slopes here, like big stone bird's nests. The hilltop commands an impressive view of the oasis.
The fort is just 3.5km northwest of San Pedro, and easily accessible on foot, by bike or by vehicle. Note that the _mirador_ (lookout) closes at 6pm.
oDid You Know?
The Atacama Desert is one of the driest regions on earth; some areas have not seen rainfall for over 400 years.
### Aldea de Tulor
Circular adobe structures huddle together like muddy bubblewrap in the ruins of Tulor (map; CH$3000; h9am-7pm), the oldest excavated village in the region. It's an interesting diversion 11km southwest of San Pedro. You can get there by your own vehicle, driving along sandy tracks, or by riding a mountain bike. However, you'll get more out of the experience if you go on a good guided tour.
If you go alone, there's often a Spanish-speaking guide on hand who can fill in some of the historical details (included with admission).
### Trekking & Biking
Around San Pedro rise immense volcanoes, a few of them active, and begging to be climbed. If climbing isn't your cup of tea, consider a more active trekking or biking trip to the usual suspects in the area, such as Valle de la Luna. Bikes are available for rent at several agencies and hotels in San Pedro, for about CH$6000 per day; try Km O (Caracoles 282B; half-/full day CH$3500/6000; h9am-9pm).
yUyuni Connection
From San Pedro de Atacama, several agencies, including Cordillera Traveler ( %55-320-5028; www.cordilleratraveller.com; Tocopilla 429-B; h9am-9pm), run multiday trips to the surreal Salar de Uyuni in Bolivia.
## San Pedro de Atacama
It is said that the high quantities of quartz and copper in the region give its people positive energy, and the good vibes of northern Chile's number-one tourist draw, San Pedro de Atacama, are sky high. The village itself is small and compact, with almost everything of interest within easy strolling distance of the plaza. Many buildings now have street numbers, although many still do without.
1Sights
Iglesia San PedroChurch
(Le Paige s/n) F
The recently restored Iglesia San Pedro is a delightful little colonial church built with indigenous or artisanal materials: chunky adobe walls and roof, a ceiling made from _cardón_ (cactus wood) resembling shriveled tire tracks and, in lieu of nails, hefty leather straps. The church dates from the 17th century, though its present walls were built in 1745, and the bell tower was added in 1890.
2Activities
Tours 4 TipsWalking
(www.tours4tips.com; Plaza de Armas; htours 10am & 3pm)
For a deeper understanding of San Pedro, take an edifying stroll with this friendly outfit. On two-hour walks around the village and its outskirts, enthusiastic guides relate fascinating episodes from San Pedro's past, touching on indigenous beliefs and symbols, desert plants and hallucinogens, and even a bit of celestial mythology. Tours are offered in Spanish and English.
Rancho La HerraduraHorseback Riding
( %55-285-1956; www.atacamahorseadventure.com; Tocopilla 406; h9am-8pm)
Sightseeing from the saddle is available from several places, including Rancho La Herradura. Tours vary from two hours for CH$23,000 to epic 10-day treks with camping in the desert. English-, German- and French-speaking guides are available.
Vulcano ExpedicionesAdventure
( %cell 9-5333-6021; www.vulcanochile.com; Caracoles 317; 10am-8pm Tue-Sun)
Runs treks to volcanoes and mountains, including day climbs to Sairecabur (5971m, CH$110,000), Lascar (5592m, CH$85,000) and Tocco (5604m, CH$670,000). Two-day climbs take in Licancábur (CH$250,000) and other mountains. It can also hook you up with motorbike tours offered by On Safari.
Desert AdventureTours
( %cell 9-9779-7211; www.desertadventure.cl; cnr Caracoles & Tocopilla; h9:30am-9pm)
Has the full spectrum of tours and bilingual guides. Unique offerings include an ethnocultural 'Ancestral Caravan' tour that features two hours of llama trekking along routes used by Atacameños in centuries past (CH$25,000).
On SafariTours
( %cell 9-7215-3254; www.onsafariatacama.com; 4hr motorbike tour CH$170,000)
Offers motorbike, all-terrain-vehicle (ATV), 4WD, mountain-biking and mountaineering tours in the Atacama region and further afield. Other offerings include tours with an emphasis on photography, bird-watching or astronomy.
7Shopping
Shaded Paseo Artesanal, a poker-straight alley squeezing north from the plaza, is the place to hunt down novel _cardón_ carvings, llama and alpaca woolens and other curious trinkets. More artisanal outlets are strewn throughout town.
For something unique, visit the handicraft shops by the bus terminal. You'll find beautifully crafted copper jewelry, one-of-a-kind alpaca clothing, handworked leather and other gear.
5Eating
BabalúIce Cream $
(Caracoles 140; ice creams CH$1900-3900; h10am-10pm)
One of several ice-cream shops on the main street, Babalú serves up rich flavors you won't find at home. Try ice creams made from desert fruits like _chañar_ or _algarrobo,_ and sample _pisco_ sour, _hoja de coca_ (coca leaf) and delightful surprises such as quinoa. You can't go wrong.
FranchuteriaBakery $
(www.lafranchuteria.com; Le Paige 527; croissants CH$1100-2500; h7am-8:30pm)
About 500m east of the plaza, you'll find San Pedro's best bakery. Run by a talented young Frenchman, Franchuteria has beautifully baked goods, including perfect baguettes with rare combinations of fillings like fig and Roquefort cheese, or goat's cheese and oregano, and buttery-rich croissants – also available stuffed with unusual things like _manjar_ ( _dulce de leche_ ).
Las Delicias de CarmenChilean $$
(www.lasdeliciasdecarmen.cl; Calama 370; lunch specials CH$4000-7000, mains CH$8000-14,000; h8:30am-10:30pm; W v)
Great breakfasts, delicious cakes and empanadas, brick-oven pizzas (choose your own toppings) and different dishes daily are churned out at this light-flooded restaurant with leafy views. Daily specials – such as _cazuela_ (stew) or carrot-ginger soup – and a filling _menú del día_ (three-course lunch) always bring in the crowds.
La CasonaChilean $$
( %55-285-1337; Caracoles 195; mains CH$9000-14,000; hnoon-midnight Wed-Mon; W)
A high-ceilinged dining room with dark wood paneling and an adobe fireplace in the middle, classic La Casona serves up sizzling _parrilladas_ (mix of grilled meats) and Chilean staples such as _pastel de choclo_ (maize casserole). There's a long list of Chilean wines and a small patio for alfresco lunches.
AdobeInternational $$$
( %55-285-1132; Caracoles 211; mains CH$11,000-14,000; h11am-1am; W)
Popular with travelers for its studied rusticity, rock-art decor, bench-like seating and smoky fire in the alfresco dining room. Adobe serves tasty but pricey dishes such as mushroom quinoa risotto or lamb with tabouli and hummus; it's also a good spot for a drink.
BaltinacheChilean $$$
( %cell 9-3191-4225; Atienza; 3-course menu CH$15,000; h1-4pm & 7:30-10pm)
A short walk south from busy Caracoles, Baltinache has some of San Pedro's best cuisine. Thick adobe walls hung with geoglyph-inspired artwork and flickering candles set the scene at this elegantly understated gem. The menu changes by night and features high-quality local products like river trout, vegetable soup with grated goat's cheese, rabbit and desserts made from desert fruits.
Stargazing in the Atacama
Climatic conditions in the Atacama Desert make it an ideal location for stargazing. This is not only thanks to cloudless desert nights, but also the predictable winds that blow steadily in from the Pacific Ocean, causing minimal turbulence – a crucial requirement for observatories to achieve optimal image quality.
If all that whets your appetite for astronomy, consider taking a 'Tour of the Night Sky' from San Pedro, where there's a range of astronomical tours available, including with reputable outfits like San Pedro de Atacama Celestial Explorations ( %55-256-6278; www.spaceobs.com; Caracoles 400-2; 2½hr tours CH$25,000; h11am-9pm Dec-Mar, to 7pm Apr-Nov) and Una Noche con las Estrellas ( %cell 9-5272-2201; www.unanocheconlasestrellas.cl; Calama 440; astronomy tour CH$20,000; h10am-11pm).
The Milky Way above the Atacama Desert | TJALEX / GETTY IMAGES ©
# La Paz, Bolivia
#### Markets
#### Tiwanaku
#### Sights
#### Tours
#### Shopping
#### Eating
#### Drinking & Nightlife
#
La Paz, Bolivia
_La Paz is a mad carnival of jostling pedestrians, honking, diesel-spewing minivans, street marches and cavalcades of vendors that seems to reinvent itself at every turn – a jaw-dropping subway in the sky brings you from the heights of El Alto to the depths of Zona Sur in the blink of an eye._
_Beneath the blinding altiplano sun, sharp-suited business people flank machine-gun-toting bank guards and balaclava-camouflaged shoeshine boys. Lung-busting inclines terminate in peaceful plazas. This maze of contradictions, where cobblestones hit concrete, and Gothic spires vie with glassine hotels, will amaze all who enter._
Bolivian woman on Calle Jaén | MATTHEW WILLIAMS-ELLIS / GETTY IMAGES ©
Two Days in La Paz
Stroll the cobblestone streets around Calle Jaén, home to the wonderful Calle Jaén Museums, then visit the fascinating Mercado de las Brujas. Dine that evening in award-winning Gustu.
On the second day, hop aboard Mi Teleférico up to El Alto for memorable views. Have dinner in upmarket Sopocachi then catch traditional music at a _peña_.
Four Days in La Paz
On your third day, take a day trip out of La Paz to Tiwanaku to explore pre-Columbian ruins. Then browse for crafts at Walisuma.
On your fourth day, head out for some adventure, on an adrenaline-fueled ride with Gravity Assisted Mountain Biking. In the evening check out La Paz' cocktail scene at the speakeasy-style La Costilla de Adán.
Arriving in La Paz
Minibus 212 runs frequently between El Alto International Airport and Plaza Isabel la Católica between around 7am and 8pm (B$3.80). Look for the sign that says 'Minibus Cotranstur' just outside arrivals. Heading into town from the airport, this service will drop you anywhere along El Prado.
Radio taxis charge around B$70 to the centre.
Where to Stay
The area around the Mercado de las Brujas (Witches' Market; between Illampu, Av Mariscal Santa Cruz and Sagárnaga) is a true travelers' ghetto. To be closer to a wider array of restaurants and a bar or two, consider Sopocachi. For more upmarket luxury, look along the lower Prado and further south in the Zona Sur.
TOP EXPERIENCE
# Markets
La Paz' frenetic markets are easily the highlight of any trip, where modern commerce and culture collide in a wonderful riot of honks, shouts and smells.
Great For...
rzk
yDon't Miss
The strange charms, herbs and amulets of the Mercado de las Brujas.
8Need to Know
You'll find open-air markets all across town, though the **Rosario neighborhood** has a notable concentration.
5Take a Break
Mercado Camacho (map Google map; cnr Av Camacho & Bueno, Casco Viejo; snacks B$5-25; h7am-9pm Mon-Fri, to 6pm Sat & Sun) makes an ideal lunch stop with stands selling empanadas, fruit juices and chorizo sandwiches.
oTop Tip
Every day is a de facto market day in La Paz, but Saturdays are particularly fun.
### Mercado Negro
The narrow cobblestone streets off Max Paredes – the Mercado Negro (map Google map; Black Market; h6am-8pm) – are a good place to start the market experience. Especially interesting are the sections near Graneros ('designer' clothes), Tumusla and Isaac Tamayo (everything and anything), and between Santa Cruz and Sagárnaga (tools and building materials). The best place for electronics is along Eloy Salmón. Be especially careful when wandering around this part of town: it's notorious for light fingers. It's best to take a taxi here at night.
### Mercado Lanza
North of Plaza San Francisco, on Calle Figueroa, the Mercado Lanza (map Google map; Rosario; snacks B$5-25; h6am-8pm) is one of La Paz' main food markets. It sells all manner of fruits, vegetables, juices, dairy products, breads and canned foods, and there are numerous stalls where you can pick up a sandwich, soup, _salteña_ (filled pastry shells), empanada or full meal. It also houses the splendid **Flower Market**.
Street market in La Paz | JOSEMAR FRANCO / SHUTTERSTOCK ©
### Mercado de las Brujas
The famed Mercado de las Brujas (map Google map; Witches' Market; La Hechicería) is chock-a-block with stores selling mysterious potions for lovemaking, llama fetuses and Aymará good-luck charms, including frogs. Artisan stores in the area sell _oriente_ wood carvings and ceramics, and Potosí silver. Others deal in rugs, wall hangings, woven belts and pouches. Amid the lovely weavings and other items of exquisite craftsmanship, you'll find plenty of tourist kitsch, an art form unto itself: Inca-themed ashtrays, fake Tiwanaku figurines, costume jewelry and mass-produced woolens.
Souvenirs on display, Mercado de las Brujas | SAIKO3P / SHUTTERSTOCK ©
### Mercado 16 de Julio
To visit the biggest market in Bolivia – some say it's the largest in South America – you'll need to hop in a cable car and head up to El Alto, where each Thursday and Sunday the massive Mercado 16 de Julio (El Alto; h6am-3pm Thu & Sun) completely absorbs dozens of city blocks. You can buy everything from new cars to animals, textiles and firearms. For the most part, however, it's a colossal flea market, the likes of which you won't soon forget.
### Mercado Uruguay
On the far side of El Prado, Mercado Uruguay (map Google map; off Max Paredes, Rosario; snacks B$5-25; h7am-8pm) is purely the domain of adventurous eaters after tiny _ispi_ fish from Lake Titicaca or stews made with unidentifiable offal.
TOP EXPERIENCE
# Tiwanaku
The ruins of Tiwanaku make for a good day trip from La Paz for those who want to view a few carved monoliths, archways and arcades, and two decent museums. History buffs will love diving into the myths and mysteries of this lost civilization.
Great For...
vhg
yDon't Miss
The many carved faces along the walls of the Templete Semisubterráneo.
8Need to Know
Ticket Office (Av Puma Punku s/n; B$100; htickets 9am-4pm, site 9am-5pm)
5Take a Break
Restaurante Cabaña del Puma (Av Puma Punku s/n; almuerzo B$25; h9am-6pm) serves decent Bolivian fare next to the ruin entrance.
oTop Tip
Guided tours ( %7724-9572; walipini.tiwanacu@gmail.com; Av Ferrocarril s/n; tour for up to 6 people in Spanish/English/French B$150/180/180) are available in English, French and Spanish, and are highly recommended.
### Museo Cerámico
Near the ticket office, this gallery showcases a small collection of the ceramics found at the site, as well as a ceremonially deformed cranium and artifacts from the Chiripa and Wankarani cultures.
### Museo Lítico Monumental
The star of the show at this Tiwanaku museum is the massive 8m Monolito Bennett Pachamama, rescued in 2002 from its former smoggy home at the outdoor Templete Semisubterráneo in La Paz. You'll also find a basic collection of other monoliths and artifacts dug up on-site here. Labeling is in Spanish.
Much of the collection is currently mothballed, as the roof of the relatively new museum is already collapsing.
### Megaphones
At the entrance to the Tiwanaku site there are two stone blocks that can be used as megaphones. Entertain yourself for a minute or two with this interesting pre-Columbian, pre-iPod technology.
### Akapana Pyramid
Climb the hill up to Tiwanaku's most outstanding structure, the partially excavated Akapana pyramid, which was built on an existing geological formation. At its base this roughly square, 16m hill covers a surface area of about 200 sq meters. In the center of its flat summit is an oval-shaped sunken area, which some sources attribute to early, haphazard, Spanish excavation. The presence of a stone drain in the center, however, has led some archaeologists to believe it was used for water storage.
Recent findings include craniums, assumed to be war trophies, suggesting the pyramid may have been a ceremonial temple. Others think it was used for the study of astronomy.
### Kalasasaya
North of the Akapana Pyramid is Kalasasaya, a partially reconstructed 130m-by-120m ritual-platform compound with walls constructed of huge blocks of red sandstone and andesite. The blocks are precisely fitted to form a platform base 3m high. Monolithic uprights flank the massive entrance steps up to the restored portico of the enclosure, beyond which is an interior courtyard and the ruins of priests' quarters. Note the size of the top stair – a massive single block.
The Monolito Ponce monolith, with his turban (no doubt covering up his deformed cranium), mask, ceremonial vase and walking stick, sits at the center of the first platform. Some say the stick and the vase are symbolic of the dualism of Andean culture (nature versus nurture).
Other stairways lead to secondary platforms, where there are other monoliths including the famous El Fraile (priest).
### Templete Semisubterráneo
East of the main entrance to Kalasasaya, a stairway leads down into the Templete Semisubterráneo, an acoustic, red-sandstone pit structure measuring 26m by 28m, with a rectangular sunken courtyard and walls adorned with 175 crudely carved stone faces. In the 1960s archaeologists tried to rebuild these and used cement between the stones.
Carved face on the wall of Templete Semisubterráneo | DANIEL NISTTAHUZ / 500PX ©
### Putuni
West of Kalasasaya is a 55m-by-60m rectangular area known as Putuni (Palacio de los Sarcófagos, Palace of the Sarcophagi). It is surrounded by double walls and you can see the foundations of several tombs. About 90% of the artifacts collected by amateur enthusiast Fritz Buck in the early 20th century from these tombs are found in La Paz' Museo de Metales Preciosos.
### Puerta del Sol
At the far northwest corner of Kalasasaya is Tiwanaku's best-known structure, the 10-ton Puerta del Sol (Gateway of the Sun). This megalithic gateway was carved from a single block of andesite, and archaeologists assume that it was associated with the sun deity. The surface of this fine-grained, gray volcanic rock is ornamented with low-relief designs on one side and a row of four deep niches on the other.
The gateway was most likely originally located in the center of Kalasasaya Platform and was used as a calendar, with the sun striking specific figures on the solstice and equinox.
There's a smaller, similar gateway carved with zoomorphic designs near the western end of the site that is informally known as the Puerta de la Luna.
### Kantatayita
The heap of rubble at the eastern end of the Tiwanaku site is known as Kantatayita. Archaeologists are still trying to deduce some sort of meaningful plan from these well-carved slabs; one elaborately decorated lintel and some larger stone blocks bearing intriguing geometric designs are the only available clues.
oTours From La Paz
Diana Tours (map; www.diana-tours.com) and Vicuña Travel (map; %228-0140; <http://vicuna-travel.com>) have round-trip guided trips to Tiwanaku from La Paz, leaving daily at 8:30am and returning around 4pm.
### Puma Punku
Across the railway line southwest of the Tiwanaku site, you'll see the excavation site of Puma Punku (Gateway of the Puma). In this temple area megaliths weighing more than 130 tons have been discovered. Like Kalasasaya and Akapana, there is evidence that Puma Punku was begun with one type of material and finished with another; part was constructed of enormous sandstone blocks and, during a later phase of construction, notched and jointed basalt blocks were added.
Note also, in the distance of the site's northern boundary, the _sukakollo,_ a highly sophisticated system of terraced irrigation.
Monolith | SL-PHOTOGRAPHY / SHUTTERSTOCK ©
### Aymará New Year
The biggest celebration of the year happens during the Aymará New Year (Machaq Mara; hMar 21, Jun 21, Sep 21). Locals don colorful ceremonial dress and visitors are invited to join the party, drink _singani_ (distilled grape liquor), chew coca and dance until dawn at celebrations that take place on fall and spring equinox, when the rays of the rising sun shine through the temple entrance on the eastern side of Kalasasaya. The big event happens on June 21, when as many as 5000 people, including a large contingent of New Agers, arrive from all over the world. Artisans hold crafts fairs to coincide with the annual celebrations.
Celebration of Aymará New Year | WIN-INITIATIVE / GETTY IMAGES ©
oDid You Know?
Nobody knows why Tiwanaku's population disappeared by 1200, though most archaeologists point to climate change as the likely cause of the civilization's rapid decline.
Central La Paz
1Sights
1Calle Jaén MuseumsE2
Casa de Murillo(see 1)
2Catedral MetropolitanaF4
3Iglesia de San FranciscoE4
4Mercado de las BrujasD5
5Mercado NegroC5
Museo Costumbrista Juan de Vargas(see 1)
6Museo de Etnografía y FolkloreF4
Museo de Metales Preciosos(see 1)
Museo del Litoral(see 1)
7Museo Nacional del ArteF4
2Activities, Courses & Tours
Climbing South America(see 8)
8Gravity Assisted Mountain BikingD5
9HanaqPacha TravelE3
5Eating
10Ali PachaG5
11Mercado CamachoG6
12Mercado LanzaD3
13Mercado UruguayA5
14Popular Cocina BolivianaD4
6Drinking & Nightlife
15Hb Bronze CoffeebarH5
3Entertainment
16JallallaE3
17Peña HuariD5
18Peña JamuyC5
1Sights
Calle Jaén MuseumsMuseum
(map Google map; Calle Jaén, Casco Viejo; 4 museums B$20; h9am-12:30pm & 3-7pm Tue-Fri, 9am-1pm Sat & Sun)
La Paz's best-preserved colonial street is home to four small museums. They are all clustered together and can generally be bundled into one visit. Buy tickets at the Museo Costumbrista (map; cnr Jaén & Sucre; h9am-12:30pm & 3-7pm Tue-Fri, 9am-1pm Sat & Sun) and continue to the Museo de Metales Preciosos (map Google map; Museum of Precious Metals; Jaén 777; h9am-12:30pm & 3-7pm Tue-Fri, 9am-1pm Sat & Sun), Museo del Litoral (map Google map; Museo de la Guerra del Pacífico; Jaén 798; h9am-12:30pm & 3-7pm Tue-Fri, 9am-1pm Sat & Sun) and Casa de Murillo (map Google map; Jaén 790; h9am-12:30pm & 3-7pm Tue-Fri, 9am-1pm Sat & Sun).
Museo de Etnografía y FolkloreMuseum
(map Google map; Ethnography & Folklore Museum; %240-8640; www.musef.org.bo; cnr Ingavi & Sanjinés, Casco Viejo; B$20, with photography B$40; h9am-12:30pm & 3-7pm Mon-Fri, 9am-4:30 Sat, 9am-12:30pm Sun)
Anthropology buffs should check out this museum, one of the city's best. The building, itself a real treasure, was constructed in 1720 and was once the home of the Marqués de Villaverde. Highlights include an awe-inspiring collection of ritualistic masks and an exhibition of stunning weavings from around the country. A guided tour is available by calling ahead.
Mountain biking on Yungas Road | FILROM / GETTY IMAGES ©
Museo Nacional del ArteMuseum
(map Google map; National Art Museum; www.facebook.com/museonacionaldeartebolivia; cnr Comercio & Socabaya, Casco Viejo; B$20; h9:30am-12:30pm & 3-7pm Tue-Fri, 10am-5:30pm Sat, 10am-1:30pm Sun)
This colonial building was constructed in 1775 of pink sandstone and has been restored to its original grandeur, in _mestizo_ (mixed) baroque and Andino baroque styles. In the center of a huge courtyard, surrounded by three stories of pillared corridors, there is a lovely alabaster fountain. The various levels are dedicated to different eras, with an emphasis on religious themes.
Catedral MetropolitanaCathedral
(map Google map; Plaza Murillo, Casco Viejo)
Although it's a relatively recent addition to La Paz's religious structures, the 1835 cathedral is impressive – mostly because it is built on a steep hillside. The main entrance is 12m higher than its base on Calle Potosí. The cathedral's sheer immensity, with its high dome, hulking columns, thick stone walls and high ceilings, is overpowering, but the altar is relatively simple.
Iglesia de San FranciscoChurch
(map Google map; Plaza San Francisco, Rosario)
The hewed stone basilica of San Francisco was founded in 1548 by Fray Francisco de los Ángeles. The original structure collapsed under heavy snowfall around 1610, but it was rebuilt between 1743 and 1772. The second building is made of stone quarried at nearby Viacha. The facade is decorated with carvings of natural themes such as _chirimoyas_ (custard apples), pine cones and tropical birds.
SopocachiArea
(map Google map)
Sopocachi has some of La Paz' best restaurants and nightspots. You can spend a few hours people-watching on **Plaza Eduardo Avaroa** , before hoofing up to the wonderful views from **Montículo Park**.
Mi Teleférico
At 30km-long and growing, Mi Teleférico (Aerial Cable Car System; www.miteleferico.bo; ticket B$3, plus B$2 per line transfer; h6am-11pm Mon-Sat, 7am-9pm Sun) is easily the world's longest aerial cable-car system. Riders can hop between lines for an additional B$2 per segment (pay in advance), creating endless combinations of ways to travel across the city. One popular trip takes you from the Zona Sur to El Alto via the yellow and green lines.
The thrill of riding above La Paz' swirling traffic and deep canyons is undeniably cool. There's more than enough time to cruise down from the center to the Zona Sur for lunch or dinner, and back. Not for the faint of heart.
SAIKO3P / GETTY IMAGES ©
TTours
La Paz on FootEcotour
(map Google map; %cell 7154-3918; www.lapazonfoot.com; Av Ecuador 2022, Sopocachi; h10:30am-6:30pm Mon-Fri)
This tip-top operation, run by the passionate English-speaking ecologist Stephen Taranto, offers a range of activities, including walks in and around La Paz, Apolobamba, the Yungas, Chulumani, Madidi and Titicaca. The interactive La Paz urban treks (half-day or full-day, fee depending on group size) venture from the heights of El Alto to the depths of the historic center.
Gravity Assisted Mountain BikingMountain Biking
(map Google map; %231-0218, cell 7721-9634; www.gravitybolivia.com; Linares 940, Rosario; h9am-7pm Mon-Fri, 10am-3pm Sat, 2-6pm Sun)
This knowledgeable, highly regarded and professional outfit has an excellent reputation among travelers and tip-top Kona downhill bikes. Their Dangerous Road Trip (B$850 per person) ends with hot showers, an all-you-can-eat buffet and an optional tour of the Senda Verde animal refuge (www.sendaverde.org; Yolosa; B$100, bear visit extra B$20; h10am-4pm).
Climbing South AmericaClimbing
(map Google map; %cell 7190-3534; www.climbingsouthamerica.com; Linares 940, 2nd fl, Rosario; h9am-6:30pm Mon-Fri, 10am-3pm Sat)
A reputable English-speaking operator for climbing, mountaineering and trekking in the nearby mountains. Also sells great topography and trekking maps.
HanaqPacha TravelTours
(map Google map; %cell 6980-3602; www.hanaqpachatravel.com; Jaén 765, Casco Viejo; h9am-6:30pm)
Runs recommended daily tours to El Alto (B$140) to step inside the prismatic _cholets_ of Aymará architect Freddy Mamani. Also has a daily Tastes of Bolivia tour (B$140) to learn about typical food. Uyuni, Tiwanaku and Rurrenabaque are the focus of longer tours.
7Shopping
WalisumaArts & Crafts
(www.facebook.com/walisuma.org; Aliaga 1231, San Miguel; h10am-8pm Mon-Fri, to 7pm Sat) S
For a one-of-a-kind souvenir head to Walisuma, which works with 59 Bolivian artisans. Star items include gorgeous (and ultrasoft) alpaca and vicuña textiles made with natural dyes. There are also quinoa soaps, flavored Uyuni salts and designer sweaters. Prices match the quality.
Bolivian Peñas
Typical of La Paz (and most of Bolivia) are folk-music venues known as _peñas_. These present traditional Andean music, rendered on _zampoñas_ (pan flutes), _quenas_ (cane flute) and _charangos_ (ukulele-style instrument) _,_ as well as guitar shows and song recitals. Most shows happen on Friday and Saturday nights, starting at 9pm or 10pm and running into the wee hours.
Jallalla (map Google map; Indaburo 710, cnr Jaén, Casco Viejo; cover incl cocktail B$30-70; h9pm-1am Tue, Thu, Fri & Sat) Just above the Mamani Mamani Gallery, this is one art-filled _peña_ you won't want to miss. The top-tier live music goes nicely with authentic Bolivian tapas and creative cocktails.
Peña Huari (map Google map; %231-6225; Sagárnaga 339, Rosario; cover B$105; hshow 8pm) The city's best-known _peña_ draws tourists and Bolivian business-people. The attached restaurant specializes in Bolivian fare.
Peña Jamuy (map Google map; %cell 7676-7817; www.facebook.com/jamuybolivia; Max Paredes, near Sagárnaga, Rosario; cover B$25-50; h9pm-7am Fri & Sat) Locals flock to this fun two-story Andean-themed venue. It hosts all-night ragers each weekend.
Dancers at Peña Huari | BRENT WINEBRENNER / GETTY IMAGES ©
5Eating
Namas TéVegetarian $
(www.namastebolivia.com; Zoilo Flores 1334, San Pedro; mains B$12-30; h8:30am-7pm Mon-Fri, to 4pm Sat; v)
Tea lovers take note: the tea menu at this lovable lime-green veggie restaurant is a staggering four pages long! There's also plenty of quinoa in all forms (falafel, soup, tabbouleh salad) and even a raved-about tofu pad thai. Smoothies, juices and sandwiches round out the well-priced menu.
Los QñapésBolivian $
(www.facebook.com/losqnapes; René Moreno 1283, San Miguel; snacks B$6-15; h3:30-10pm) S
Snack on Bolivian favorites like _cuñapé_ (a cheesy yuca bread), _humitas_ (a steamed corn pie) and _masacos_ (plantains or yucca mashed with meat or cheese) at this always-busy cafe. All of the ingredients are organic and come from within the country.
Popular Cocina BolivianaBolivian $$
(map Google map; www.facebook.com/popularlapazbolivia; Murillo 826, Rosario; 3-course lunch B$50; h12:30-2:30pm Mon-Sat)
The concept of waiting in line for a restaurant doesn't exist in La Paz, but that's exactly what you'll need to do to get into Popular. Seasonal three-course menus put a gourmet spin on the city's humble lunch spots. Ingredients come fresh from the market and the plates are true works of art. Did we mention that it's ridiculously affordable?
MagicK Cafe CulturalInternational $$
(map Google map; www.cafemagick.com; Presbítero Medina 2526, Sopocachi; mains B$30-55; h4-11:30pm Tue-Sat; W v)
This funky pescatarian restaurant serves up fig-and-blue-cheese pizza, quinoa tabbouleh and pasta with smoked trout in a lovingly converted Sopocachi home. Vegan and gluten-free options abound, as do good tunes and chill vibes.
TogaAsian $$
(map Google map; %mobile 7650-4643; Av Sánchez Lima 2235, Sopocachi; 4-course meal B$35; h10am-11pm Mon-Sat)
There's no fixed menu here so you'll have to put your trust in chef Rubén Gruñeiro as he takes you on a four-course culinary journey mixing Asian flavors with Bolivian produce. With just four tables and an open kitchen, it's an incredibly intimate experience.
Ali PachaVegetarian $$$
(map Google map; %220-2366; www.alipacha.com; Colón 1306, Casco Viejo; 3/5/7 courses B$100/150/200; hnoon-3pm Tue-Sat & 7-10pm Wed-Sat; W v)
Locals thought it absurd on so many levels to open a high-end vegetarian restaurant with degustation menus in La Paz's downtrodden Casco Viejo neighborhood. And it is absurd. Fantastically so! Even carnivores will swoon over the creative plant-based creations and herbaceous cocktails. You're guaranteed to taste the flavors of Bolivia like never before.
GustuBolivian $$$
( %211-7491; www.gustu.bo; Calle 10 No 300, Calacoto; almuerzo B$95, dinner tasting menu B$430-560, à la carte mains B$95-130; hnoon-3pm & 6:30-11pm; p W)
Credited with sparking La Paz' culinary renaissance, and launched by the Danish culinary entrepreneur Claus Meyer (of Noma fame), this groundbreaking restaurant works to both rescue and showcase underutilized Bolivian ingredients. It's located in a gorgeous building rich with Andean textiles and offers everything from Andean grains to caiman from the Amazon. Even the wine pairings come from within Bolivia.
Sopocachi
1Sights
1SopocachiC4
2Activities, Courses & Tours
2La Paz on FootB3
5Eating
3MagicK Cafe CulturalC4
4TogaC3
6Drinking & Nightlife
5La Costilla de AdánB6
6Reineke FuchsC3
6Drinking & Nightlife
Hb Bronze CoffeebarCoffee
(map Google map; <http://hb-bronze.com>; Plaza Tomás Frías 1570, Casco Viejo; h8:30am-midnight Mon-Sat, 1-9pm Sun; W)
Sleek, earthy and architecturally inspiring – coffee shops don't get much cooler than Hb Bronze! This sorely needed addition to the Casco Viejo neighborhood offers the strongest brews in the city, and it doubles as a bar by night with 100% Bolivian cocktails, beers and wines. The food is equally memorable.
La Costilla de AdánCocktail Bar
(map Google map; %cell 7207-4518; Armaza 2974, Sopocachi; h9pm-4am Wed-Sat)
Take a dive down the rabbit hole into the mind of owner Roberto Cazola at this supremely surreal speakeasy. Prepare yourself for hundreds of creepy dolls, a dozen hanging bicycles and a night full of wonder next to the roaring fire. There's no sign out front. Simply ring the bell, wait, and let the magic begin.
Reineke FuchsBeer Hall
(map Google map; www.reinekefuchs.com; Pasaje Jáuregui 2241, Sopocachi; hnoon-3pm & 7pm-late Mon-Fri, 7pm-late Sat)
This woodsy Sopocachi _brewhaus_ features imported German beers, _schnappsladen_ and hearty sausage-based fare. They also make their own Pilsner, dunkel and amber ales, based on centuries-old Deutsch traditions – heady concoctions, indeed.
8INFORMATION
**Tourist Information** (map; cnr Av Mariscal Santa Cruz & Colombia, Prado; h8:30am-7pm Mon-Fri) Stop by to grab some maps and get detailed information. English is spoken by some staff.
**Tourist Information** (map; www.facebook.com/Gamlpturismo; Plaza del Estudiante, Prado; h8:30am-7pm Mon-Fri) Maps, flyers and some English-speaking staff.
8Getting There & Away
AIR
**El Alto International Airport** (LPB; Héroes Km 7, El Alto) is 10km via toll road from the city center on the altiplano. At 4062m, it's the world's highest international airport.
BUS
La Paz has three bus terminals/bus areas. You can use the **main bus terminal** (Terminal de Buses; map; cnr Avs Perú & Uruguay, Challapampa) for most national and international destinations.
8GETTING AROUND
MICRO & MINIBUS
La Paz' buses charge from B$2 per trip. Look for your destination on a signboard posted in the front window.
TAXI & RIDE SHARE
Radio taxis (with roof bubbles advertising their telephone numbers) are recommended.
A newer solution to safe taxis is to use the **Easy Taxi app** (www.easytaxi.com/bo, in Spanish). **Uber** (www.uber.com) is also available in La Paz.
# Salar De Uyuni, Bolivia
#### Visiting the Salt Flats
#### Uyuni
#### Sights
#### Tours
#### Shopping
#### Eating
#### Drinking & Nightlife
#
Salar de Uyuni, Bolivia
_An evocative and eerie sight, the world's largest salt flat measures 12,106 sq km and sits at 3653m (11,984ft). When the surface is dry, the _salar_ is a pure white expanse of the greatest nothing imaginable – just blue sky, white ground and you. When there's a little water, the surface perfectly reflects the clouds and the blue altiplano sky, and the horizon disappears. If you're driving across the surface at such times, the effect is surreal; it's hard to believe that you're not flying through the clouds._
Salar de Uyuni | SARA WINTER / SHUTTERSTOCK ©
Two Days in Salar de Uyuni
On day one, check out the locomotive graveyard of the Cementerio de Trenes. Then head to Colchani village to learn about salt extraction. End the day with a sunset at Isla Incahuasi.
On day two continue south through the Desierto de Siloli, stopping at the strange Árbol de Piedra and flamingo-studded Laguna Colorada.
Four Days in Salar de Uyuni
Rise early for a sunrise visit to the Sol de Mañana geyser field, then soak in the mineral-rich waters of Termas de Polques. Later photograph the striking Desierto de Dalí and emerald-hued Laguna Verde.
On day four make a day trip from Uyuni up to the semi-ghost town of Pulacayo. In the evening dine at first-rate Tika.
Arriving in Salar de Uyuni
Aeropuerto Joya Andina is 1km north of the town of Uyuni. **Amaszonas** (www.amaszonas.com) and **Boliviana de Aviación** (www.boa.bo) operate two flights a day to La Paz (from B$700, one hour).
The safest, most comfortable ground transport from La Paz is with **Todo Turismo** (www.todoturismo.bo), which runs a heated bus service.
Where to Stay
When you're on a salt flat, what better place to spend the night than in a salt hotel? The simple salt hotels in Coquesa, Chuvica and San Juan are nearly identical, with salt floors, furniture and walls, and common dining rooms where you can eat dinner (and shiver). There are also more luxurious salt hotels just outside of Uyuni.
TOP EXPERIENCE
# Visiting the Salt Flats
Bolivia's most awe-inspiring site is the blindingly white Salar de Uyuni, a vast (12,106 sq km) salt flat that sits at an elevation of 3653m (11,984ft). Tours here also take in surreal rock formations, otherworldly lakes (that are home to several flamingo species) and bizarre desert landscapes.
Great For...
gAf
yDon't Miss
Hiking amid the strange salt formations and cacti on the top of Isla Incahuasi.
8Need to Know
Tours to Salar de Uyuni cost B$800 to B$1200 for three days at a standard agency, and B$1200 to B$1500 at a high-end operation.
5Take a Break
There are simple restaurants in the villages of Colchani, Coquesa and Chuvica on the edge of the _salar._
oTop Tip
Bring several liters of water, snacks, a headlamp, sunscreen, sunglasses, a hat, warm clothes, a camera, and a plastic dinosaur (for photos, of course).
### Isla Incahuasi
One of the highlights of a Salar de Uyuni tour is a hike around the spectacular Isla Incahuasi (map), otherwise known as Inkawasi. It's located in the heart of the _salar_ , 80km west of Colchani. This hilly outpost is covered in Trichocereus cactus and surrounded by a flat white sea of hexagonal salt tiles.
It was once a remarkably lonely, otherworldly place but since the advent of _salar_ tours it receives large numbers of visitors every day. Nonetheless, it's still a beautiful sight if you forget the crowds.
You have to pay an entry fee to climb the hill (B$30), and tour groups clamber over the hiking trails chasing the perfect photo of cacti and salt. It's a 15-minute walk to the top of the island, with a trail that loops back, but it's worth it. Note that during the wet season when the _salar_ is flooded, the island is inaccessible.
Flag Monuments | ICHYWONG / SHUTTERSTOCK ©
### Playa Blanca Salt Hotel
Although the **salt hotel** (map) is now closed to overnight visitors, you can still check out the salt sculptures inside and the Dakar Rally and Flag Monuments outside (add your flag if you've brought one). Find your own isolated piece of salt desert to enjoy and take out your props – it's here that plastic dinosaurs come out, as photographers play with the bizarre perspective caused by the bright blue skies and superflat landscape. Many tour groups stop for lunch here.
Bolivian cacti | GALYNA ANDRUSHKO / SHUTTERSTOCK ©
### Colchani
Located right on the edge of Salar de Uyuni, Colchani is the easiest place to access the great salt flat and the place to go if you just want a glimpse of the _salar_. Most salt-flat tours stop here to visit the souvenir stalls and small salt museum, which is built from salt and contains salt sculptures (admission is free). Colchani is 22km north of Uyuni; buses to Oruro pass the village (B$5, 40 minutes).
The families here make their living from salt extraction as part of a cooperative; you can see small pyramids of salt draining on the _salar_ , close to the village.
### Isla Cáscara de Huevo
The small 'Eggshell Island' (map) was named for the broken shells of birds' eggs that litter it. It's located near the southern end of Salar de Uyuni and is visited mainly for the strange patterns of salt crystallization in the area, some of which resemble roses. However, due to nearby lithium mines and restricted access to the surrounding area, Cáscara de Huevo is no longer included on tour itineraries.
### Coquesa
At the village of Coquesa on the northern edge of the _salar_ , take time to explore the **ruined ancient villages** and **burial grounds** nearby. Ceramic, gold and copper artifacts, and articles of clothing have been discovered at some of the sites, indicating the presence of an advanced but little-known culture. The Museo Coquesa (per 4WD B$20; hhours vary) houses a collection of mummies and ceramics.
Unfortunately, the sites' remoteness has left the sites vulnerable to amateur treasure hunters who have plundered several items of archaeological value.
Coquesa is also the starting point for climbs up **Volcán Tunupa**. Local guides charge B$500 to take you up.
### Árbol de Piedra
This much-photographed, 5m-tall **rock** (map) in Desierto Siloli resembles a tree. Most tours stop here on the way to Laguna Colorada.
Árbol de Piedra | BOTOND HORVATH / SHUTTERSTOCK ©
### Laguna Colorada
Laguna Colorada (map; 4278m) is a rusty-burnt-orange-hued lake that covers approximately 60 sq km and reaches a depth of just 80cm. The rich red coloration on the lake is derived from algae and plankton that thrive in the mineral-rich water, and the shoreline is fringed with brilliant white deposits of sodium, magnesium, borax and gypsum. More apparent are the flamingos that breed here; all three South American species are present.
The lake sediments are rich in diatoms (tiny microfossils used in the production of fertilizer, paint, toothpaste and plastics, and as a filtering agent for oil, pharmaceuticals, aviation fuel, beer and wine). The clear air is bitterly cold and winter nighttime temperatures can drop below –20°C.
### Sol de Mañana
Most tour groups wake at dawn to visit the large geyser field dubbed Sol de Mañana (map). This 4850m-high geyser basin has bubbling mud pots, hellish fumaroles and a thick and nauseating aroma of sulfur fumes. Approach the site cautiously; any damp or cracked earth is potentially dangerous and cave-ins do occur, sometimes causing serious burns.
### Desierto de Dalí
Where the route splits about 20km south of Sol de Mañana, the more scenic left fork climbs up and over a 5000m pass, then up a hillside dotted with enormous rocks, which look like they were meticulously placed by the surrealist master Salvador Dalí.
### Termas de Polques
At the foot of Cerro Polques lies Termas de Polques (map; B$6), a small 29.4°C hot-spring pool, and an absolute paradise after the chilly _salar_ nights. Although the mineral-rich waters are not boiling by any means, they're suitable for bathing and are thought to relieve the symptoms of arthritis and rheumatism.
### Laguna Verde
The stunning blue-green **Laguna Verde** (map; 4400m) is tucked into the southwestern corner of Bolivian territory, 52km south of Sol de Mañana. The incredible green color comes from high concentrations of lead, sulfur, arsenic and calcium carbonates. Most tours visit this in the morning, but it's at its most dramatic during the afternoon when incessant icy winds have whipped the water into a brilliant green-and-white froth.
This surface agitation, combined with the high mineral content, means that it can remain liquid at temperatures as low as –21.2°C. Behind the lake rises the cone of Volcán Licancabur (5960m), whose summit is said to have once sheltered an ancient Inca crypt.
### Volcán Licancabur
Though not included on the standard tours, there is plenty of scope for hiring your own driver and climbing some volcanoes.
The most frequently climbed is Volcán Licancabur (5960m); it takes about eight hours to climb to the summit and two to get down. Several Uyuni and Tupiza agencies can include a guided climb of the volcano in a Southwest Circuit route, adding an extra day to the trip.
The climb can be done comfortably (if you can handle the altitude) in one day. As the volcano is sacred to the locals, the guides usually perform a ritual for Pachamama, asking the earth goddess for her permission to climb.
## Uyuni
Standing in defiance of the desert-like landscape that surrounds it, Uyuni occupies a desolate corner of southwestern Bolivia. Mention Uyuni to a Bolivian and they will whistle and emphasize _harto frío_ (extreme cold). Yet despite the icy conditions, the town has a cheerful buzz about it, with hundreds of travelers passing through every week to kick off their tour of Salar de Uyuni or the Southwest Circuit.
1Sights
Cementerio de TrenesHistoric Site
(Train Cemetery) F
The only real attraction in Uyuni, Cementerio de Trenes is a rusty collection of historic steam locomotives and rail cars dating back to the 19th century, when there was a rail-car factory here. Today they sit decaying in the yards about 3km southwest of the modern-day station along Av Ferroviaria. They're fun to climb on, and it's a nice walk from town to keep you warm. Most tours visit the train cemetery as a first or last stop on the three-day _salar_ circuit.
TTours
Quechua ConnectionTour
( %693-3923; www.quechuaconnection4wd.com; cnr Bolívar & Cabrera; h7:30am-noon & 3-8pm)
This company offers salt-flat tours with English-speaking guides. Its best sellers are tours that include the chance to cycle 3km across the salt flats.
Salty Desert AventoursTour
( %7237-0444; www.saltydesert-uyuni.com; Av Ferroviaria, btwn Av Arce & Bolívar; h10am-12:30pm & 2:30-7:30pm)
This operator gets good reviews. Tours with an English-speaking guide are available for an extra cost.
Hidalgo ToursTours
( %693-2989; www.salardeuyuni.net; Av Potosí 113, Hotel Jardines de Uyuni; h7am-7:30pm)
This upscale agency offers high-end, private tours with accommodations at luxury salt hotels.
Andes Salt ExpeditionsTour
( %591 7241 4748; www.andes-salt-uyuni.com.bo; Av Ferroviaria 56)
This agency offers a range of tours at various price points. Its founder, ex-miner Raul Braulio Mamani, worked as the guide for _The Devil's Miner_ documentary, a 2005 film about two brothers working in mines near Potosí.
Cordillera ToursTours
( %693-3304; www.cordilleratraveller.com; Av Ferroviaria 314; h7:30-11am & 1:30-7pm)
Offers the usual salt-flat tours as well as transfers to Chile (from B$350 per person).
7Shopping
MarketMarket
(Av Potosí; hSun & Thu)
The big market day in Uyuni is Thursday when Av Potosí gets taken over by stalls selling anything from arts and crafts to television sets; Sunday is a smaller market day.
5Eating
Mercado CampesinoMarket $
(Avaroa, cnr Av Potosí; meals B$5-10; h6am-noon)
For quick eats, cheap meals are on offer at the market's _comedor_ (dining hall) and nearby street-food stalls.
TikaBolivian $$
( %693-2989; www.tikarestaurante.com.bo; Av Potosí 113; mains B$40-92; h4-10pm; W)
Tika's chic, modern dining room is an appealing setting to sample contemporary takes on traditional Bolivian dishes, such as _charque lipeño_ (sun-dried llama meat with potatoes, cheese and corn) and _k'alaphurka_ , a gurgling corn soup served with a hot volcanic stone.
Minuteman Revolutionary PizzaPizza $$
( %693-3186; Av Ferroviaria 60; mains B$55-65; h8-10am & 5-9pm)
This convivial spot inside Toñito Hotel is run by Chris from Boston and his Bolivian wife Sussy, and is a travelers' favorite. Sample the best pizzas in town as well as tasty gourmet salads. It's also a cozy spot for a beer, glass of Tarija wine or a hearty breakfast (B$20 to B$50).
Lithium ClubBolivian $$$
( %693-3399; Av Potosí 24; mains B$70-120; h10:30am-10pm Mon-Sat, 4-10pm Sun)
This upper-end choice has international takes on traditional Bolivian dishes like _charque de llama_ (llama jerky) and _pailita de llama_ (llama stew), bringing together authentic flavor combinations with a smidgen of European styling.
Ghostly Pulacayo
At this virtual ghost town (B$10), brilliantly colored rocks rise beside the road and a mineral-rich stream reveals streaks of blue, yellow, red and green. The silver mines north of the village closed in 1959 and today only a few hundred hardy souls remain.
Also worth seeing here is the mansion of the 22nd president of Bolivia, Aniceto Arce Ruíz. Nearby is a collection of decaying steam locomotives that were originally imported to transport ore. They include Bolivia's first steam engine, El Chiripa, and the train that was robbed by legendary bandits Butch Cassidy and the Sundance Kid, including a wooden rail car that bears the bullet holes from the attack.
Take any bus to Potosí and ask to be dropped off at Pulacayo (B$5, 30 minutes) or book a tour. There is a fee of B$10 to enter the village.
6Drinking & Nightlife
Extreme Fun PubPub
(La Llamita; Sucre 23; h2:30pm-1am)
For a night of extreme fun head to Uyuni's top drinking spot. The extensive cocktail list features concoctions such as Sexy Llama Bitch and Sensual Llama's Navel, as well as hot cocktails to warm you up on cold nights. There are salt floors, drinking games, friendly service and beautiful _salar_ photos. Guitars are on hand for impromptu music sessions.
8INFORMATION
Infotur ( %693-3666; cnr Avs Potosí & Arce; h8am-noon & 2:30-6:30pm Mon-Fri) Well stocked with leaflets on Uyuni and the rest of Bolivia. At the time of research, the office was located in the bus terminal while the office on Avenida Arce was being renovated.
Office of Reserva Nacional de Fauna Andina Eduardo Avaroa (REA; %693-2225; www.boliviarea.com; cnr Colón & Avaroa; h9am-6pm Mon-Fri) Administrative office for the park in Uyuni. You can buy your park entry (B$150) here if going on your own.
8GETTING THERE & AWAY
AIR
The quickest way to get to Uyuni is by flying direct from La Paz to Aeropuerto Joya Andina, 1km north of town.
BUS
Uyuni's gleaming new bus terminal (Avaroa) has a choice of companies to most destinations, so ask around to get the best price, time and service.
Potosí buses leave hourly; for Sucre it's easiest to head to Potosí and change there.
TRAIN
Uyuni has a modern, well-organized train station ( %693-2153; www.fca.com.bo; Av Ferroviaria s/n). Trains take you north to Oruro and south to Villazón. Seats often sell out, so buy your ticket several days in advance or get an agency to do it for you.
# Lake Titicaca, Peru
#### Islas Uros
#### Chullpas
#### Sillustani
#### Cutimbo
#### Isla Taquile
#### Puno
#
Lake Titicaca, Peru
_In Andean belief, Titicaca is the birthplace of the sun. In addition, it's the largest lake in South America and the highest navigable body of water in the world. Enthralling and in many ways singular, the shimmering deep-blue Lake Titicaca is the longtime home of highland cultures steeped in the old ways. Pre-Inca Pukara, Tiwanaku and Collas all left a mark on the landscape._
_Today, the region is a mix of crumbling cathedrals, desolate altiplano and checker-board fields backed by high Andean peaks. In this world, ancient holidays are marked with riotous celebrations where elaborately costumed processions and brass bands start a frenzy that lasts for days._
Peruvian women on Islas Uros SAIKO3P / SHUTTERSTOCK ©
Lake Titicaca in Two Days
Spend the day exploring the extraordinary floating Islas Uros. Overnight in the excellent homestay of Cristina Suaña on Isla Khantati.
On day two visit the impressive funerary towers of Sillustani, keeping an eye out for abundant birdlife in the area. In the evening, return to Puno for a fantastic Peruvian meal at Mojsa.
Lake Titicaca in Four Days
Head to traditional Isla Taquile. Climb the island's many steps, admire the fine stonework and shop for locally made crafts.
On day four, explore the sights of Puno. See pre-Columbian artifacts at Museo Carlos Dreyer, learn about coca and traditional dress at the Museo de la Coca y Costumbres and dine at La Table del Inca.
Arriving in Lake Titicaca
**Aeropuerto Internacional Inca Manco Cápac** The region's only airport is one hour from Puno, the gateway to Titicaca.
**Bus & Car** Overland crossings from the southeast side of the lake are usually through Copacabana in Bolivia to the border post at Yunguyo.
Where to Stay
In Puno, pedestrianised Calle Lima between Parque Pino and Plaza de Armas is the most active and safest street. Staying anywhere within three blocks of here is the most convenient and secure. Calle Lima and Jirón Arequipa can suffer from nighttime bar or traffic noise, so try to avoid street-facing rooms.
There are homestays on many of the islands, which include the option of a home-cooked meal while staying with a local family.
TOP EXPERIENCE
# Islas Uros
These extraordinary floating islands are Lake Titicaca's top attraction. Their uniqueness is due to their construction, created entirely with the buoyant totora reeds that grow abundantly in the shallows of the lake.
Great For...
hrA
yDon't Miss
A taste test of the reeds, which resemble sugarcane without any sweetness.
8Need to Know
Boat tours take you to the islands, 7km east of Puno.
5Take a Break
Snacks and drinks are sold by island inhabitants, but there are no restaurants.
oTop Tip
Handicrafts sold on the islands are an important livelihood for inhabitants, who sometimes see little tour money.
The lives of the Uros people are interwoven with these reeds. Partially edible (tasting like nonsweet sugarcane), the reeds are also used to build homes, boats and crafts. The islands are constructed from many layers of the _totora_ , which are constantly replenished from the top as they rot from the bottom, so the ground is always soft and springy.
Some islands also have elaborately designed versions of traditional tightly bundled reed boats on hand and other whimsical reed creations, such as archways and even swing sets. Be prepared to pay for a boat ride (S10) or to take photographs.
Intermarriage with the Aymará-speaking indigenous people has seen the demise of the pure-blooded Uros, who nowadays all speak Aymará. Always a small tribe, the Uros began their unusual floating existence centuries ago in an effort to isolate themselves from the aggressive Collas and Incas.
The popularity of the islands has led to aggressive commercialization in some cases. The most traditional reed islands are located further from Puno through a maze of small channels, only visited by private boat. Islanders there continue to live in a relatively traditional fashion and prefer not to be photographed.
Sculpture and boat made of _totora_ reeds | AKUGASAHAGY / SHUTTERSTOCK ©
### Homestays
There are now more than a handful of upmarket homestays on Islas Uros, ranging from elegant huts to luxury rooms with floor-to-ceiling windows to maximize lake views. Check if boat transfer and any cultural activities are included in rates.
An outstanding option is staying in the reed huts of Isla Khantati with effervescent Cristina Suaña ( %951-472-355, 951-695-121; uroskhantati@hotmail.com; Isla Khantati; per person full board S180), an Uros native who has built a number of impeccable semi-traditional huts (with solar power, outhouses and shady decks), which occupy half the tiny island. The hyper-relaxed pace means a visit here is not ideal for those with little time on their hands. The tariff is steep but includes top-notch accommodations, three meals, fishing and ample cultural activity.
### Getting There & Away
Getting to the Uros is easy – there's no need to go with an organized tour, though you'll miss out on the guides' history lesson. Ferries leave from the port for Uros (return trip S10) at least once an hour from 6am to 4pm. The community-owned ferry service visits two islands, on a rotation basis. Ferries to Taquile and Amantaní can also drop you off in the Uros.
TOP EXPERIENCE
# Chullpas
The ancient Colla people who once dominated the Lake Titicaca area buried their nobility in imposing chullpas (funerary towers), which can be seen scattered widely around the hilltops of the region.
Great For...
vAg
yDon't Miss
Lago Umayo (partially encircling Sillustani), home to a wide variety of plants and Andean waterbirds.
8Need to Know
Sillustani tours leave at 2:30pm daily (from S30; taxi S80); Cutimbo tours cost from US$59 (taxi S30).
5Take a Break
Tours often include visiting local families and eating boiled potato dipped in _arcilla_ (edible clay).
oTop Tip
A _turismo vivencia_ (homestay) means you can help your host family with farming and visit lesser-known archaeological sites.
The Colla people were a warlike, Aymará-speaking tribe, who later became the southeastern group of the Incas. Their _chullpas_ housed the remains of complete family groups, along with plenty of food and belongings for their journey into the next world. Their only opening was a small hole facing east, just large enough for a person to crawl through, which would be sealed immediately after a burial. Nowadays, nothing remains of the burials, but the _chullpas_ are well preserved.
## Sillustani
The most impressive towers are at Sillustani (adult/child S15/2), where the tallest reaches a height of 12m. The afternoon light is the best for photography, though the site can get busy at this time.
The walls of the towers are made from massive coursed blocks reminiscent of Inca stonework, but are considered to be even more complicated. Carved but unplaced blocks, and a ramp used to raise them, are among the site's points of interest, and you can also see the makeshift quarry. A few of the blocks are decorated, including a well-known carving of a lizard on one of the _chullpas_ closest to the parking lot.
Sillustani is partially encircled by the sparkling Lago Umayo (3890m), which is home to a wide variety of plants and Andean waterbirds, plus a small island with vicuñas (threatened, wild relatives of alpacas). Bird-watchers take note: this is one of the best sites in the area.
Tours to Sillustani leave Puno at around 2:30pm daily and cost from S30.
_Chullpas_ , Sillustani | SAIKO3P / SHUTTERSTOCK ©
## Cutimbo
Just over 20km from Puno, this dramatic site has an extraordinary position upon a table-topped volcanic hill surrounded by a fertile plain. Its modest number of well-preserved _chullpas_ , built by the Colla, Lupaca and Inca cultures, come in both square and cylindrical shapes. You can still see the ramps used to build them. Look closely and you'll find several monkeys, pumas and snakes carved into the structures. Go in a group and keep an eye out for muggers.
_Combis_ (minibuses) en route to Laraqueri (S3, 30 minutes) leave from the Terminal Zonal in Puno.
Prehistoric Inca ruin, Cutimbo | RAFAL CICHAWA / SHUTTERSTOCK ©
TOP EXPERIENCE
# Isla Taquile
In the strong island sunlight, the deep, red-colored soil of Taquile contrasts with the intense blue of the lake and the glistening backdrop of Bolivia's snowy Cordillera Real.
Great For...
fgh
yDon't Miss
Fiesta de San Diego – a celebration with dancing and music from July 25 until early August.
8Need to Know
Ferries (round-trip S25; island admission S5) leave from the Puno port for Taquile from 7:35am.
5Take a Break
Consider eating in the Restaurante Comunál, Taquile's only community-run food outlet.
oTop Tip
There are no roads or streetlights, so bring a flashlight for an overnight stay.
Inhabited for thousands of years, Isla Taquile is a tiny 7-sq-km island with a population of about 2000 people. Taquile's lovely scenery is reminiscent of the Mediterranean, with several hills that boast Inca terracing on their sides and small ruins on top.
### Taquile's People
Quechua-speaking islanders are distinct from most of the surrounding Aymará-speaking island communities and maintain a strong sense of group identity. They rarely marry non-Taquile people.
The view from Isla Taquile | NEALE COUSLAND / SHUTTERSTOCK ©
### Handicrafts
Taquile has a fascinating tradition of handicrafts, and the islanders' creations are made according to a system of deeply ingrained social customs. Men wear tightly woven woolen hats that resemble floppy nightcaps, which they knit themselves – only men knit, learning from the age of eight. These hats are closely bound up with social symbolism: men wear red hats if they are married and red-and-white hats if they are single, and different colors can denote a man's current or past social position.
Taquile women weave thick, colorful waistbands for their husbands, which are worn with roughly spun white shirts and thick, calf-length black pants. Women wear eye-catching outfits comprising multilayered skirts and delicately embroidered blouses. These fine garments are considered some of the most well-made traditional clothes in Peru, and can be bought in the cooperative store on the island's main plaza.
Woman weaving on Isla Taquile | HADYNYAH / GETTY IMAGES ©
### Visiting
Visitors are free to wander around, explore the ruins and enjoy the tranquility. The island is a wonderful place to catch a sunset and gaze at the moon, which looks twice as bright in the crystalline air, rising over the breathtaking peaks of the Cordillera Real.
A stairway of more than 500 steps leads from the dock to the center of the island. The climb takes a breathless 20 minutes if you're acclimatized – more if you're not.
Make sure you already have lots of small bills in local currency, as change is limited and there's nowhere to exchange dollars. You may want to bring extra money to buy some of the exquisite crafts sold in the cooperative store.
The _hospedajes_ (small, family-owned inns) on Taquile offer basic accommodation for around S20 a night. Meals are additional (S10 to S15 for breakfast, S20 for lunch). Options range from a room in a family house to small guesthouses. As the community rotates visitors to lodgings, there is little room for choosing.
## Puno
With a regal plaza, concrete block buildings and crumbling bricks that blend into the hills, Puno has its share of both grit and cheer. It serves as the jumping-off point for Lake Titicaca and is a convenient stop for those traveling between Cuzco and La Paz. But it may just capture your heart with its own rackety charm.
1Sights
Casa del CorregidorHistoric Building
( %051-35-1921; www.casadelcorregidor.pe; Deustua 576; h9am-9pm Mon-Sat) F
An attraction in its own right, this 17th-century house is one of Puno's oldest residences. The former community center now houses a small fair-trade arts-and-crafts store and a cafe.
Museo Carlos DreyerMuseum
(Conde de Lemos 289; admission with English-speaking guide S15; h9am-7pm Mon-Fri, to 1pm Sat)
This small museum houses a fascinating collection of Puno-related archaeological artifacts and art from pre-Inca, Inca, colonial and Republic periods. Upstairs there are three mummies and a full-scale fiberglass _chullpa_ (funerary tower). It's around the corner from Casa del Corregidor. Guides tend to leave an hour before closing.
Museo de la Coca y CostumbresMuseum
(Museum of Coca and Customs; %051-977-0360; www.museodelacoca.com; Ilave 581; admission S10; h9am-7pm Mon-Sat, 3-7pm Sun)
Tiny and quirky, this museum offers lots of interesting information – historical, medicinal, cultural – about the coca plant and its many uses. Presentation isn't that interesting, though: reams of text (in English only) are stuck to the wall and interspersed with photographs and old Coca-Cola ads. The display of traditional costumes is what makes a visit here worthwhile.
Though the relation between traditional dress and coca is unfathomable, it's a boon for making sense of the costumes worn in street parades.
TTours
Edgar AdventuresCultural
( %051-35-3444; www.edgaradventures.com; Lima 328; hoffice 7am-8pm)
Longtime agency with positive community involvement. More unusual activities include kayaking on Lake Titicaca and visiting remote areas.
All Ways TravelCultural
( %051-35-3979; www.titicacaperu.com; Deustua 576, 2nd fl)
Offers both classic and 'nontourist' tours of Islas Uros and Taquile, Sillustani and rural communities in Llachón. They aim to be socially conscious.
TransturinBoat
( %051-35-2771; www.transturin.com; Ayacucho 148; 2-day tour US$181; hdepartures 6:30am)
There are no passenger ferries across Lake Titicaca from Puno to Bolivia, but you can get to La Paz via the lake in one or two days on high-class catamaran tours with Transturin, visiting Isla del Sol and other sights along the way.
7Shopping
FeriaArts & Crafts
(Av Costanera, Puno port; h7am-5pm)
This craft market sells llama toys, rugs, alpaca sweaters, masks from Puno's La Virgen de la Candelaria festival and other handicrafts you'll see elsewhere in town and on the islands, but at prices more open to haggling. The dozens of nearly identical stalls are at the port entrance, at the end of Av Del Puerto.
5Eating
PushkaPeruvian $
(Grau 338; mains S8-15 ménus S8; c)
It's hard to imagine that just a couple blocks from Puno's busy tourist street there is a large beer-garden-style restaurant. Yet here it is, with play equipment for kids, astroturf and excellent-value _menús_ (set meals) of Peruvian dishes. The food is of the same quality you'll find back on Calle Lima, but without the frills.
MojsaPeruvian $$
( %051-36-3182; Lima 635; mains S22-30; hnoon-9.30pm; W v)
The go-to place for locals and travelers alike, Mojsa overlooks the plaza and lives up to its name, which is Aymará for 'delicious.' It has a thoughtful range of Peruvian and international food, including innovative trout dishes and a design-your-own salad option. All meals start with fresh bread and a bowl of local olives. In the evening crisp brick-oven pizzas are on offer.
La Table del IncaFusion $$
( %994-659-357; www.fb.me/latabledelinca; Ancash 239; mains S26-40, 3 courses with wine S80; hnoon-2pm & 6-9.30pm Mon-Thu, 6-9.30pm Sat & Sun; W)
If you need a reason to dress up, this fusion restaurant, a little away from the noise, shows off paintings by local artists on its walls, with colorful plating. Peruvian dishes like _lomo saltado_ (stir-fried beef with potatoes and chili) hold their own against Euro-Peruvian twists such as quinoa risotto, alpaca carpaccio with _huacatay_ (a local aromatic herb) and French desserts.
TulipansPizza $$
( %051-35-1796; Lima 394; mains S15-30, menús S20; h10am-10pm; W)
Highly recommended for its yummy sandwiches, big plates of meat and piled-high vegetables, this cozy spot is warmed by the pizza oven in the corner. It also has a selection of South American wines. The courtyard patio is attractive for warm days – whenever those happen! Pizzas are only available at night. Tulipans is inside La Casona Parodi.
La CasonaPeruvian $$
( %051-35-1108; <http://lacasona-restaurant.com>; Lima 423, 2nd fl; mains S22-45; hnoon-9.30pm)
A solid choice for upscale _criollo_ (spicy Peruvian fare with Spanish and African influences) and international food, even if portions are on the small side. Trout comes bathed in garlic or chili sauce. There's also pasta, salad and soup.
Exploring Ichu
Ten kilometers out of Puno, this rural community, spread across a gorgeous green valley, is home to little-known Inca ruins – Centro Ceremonial Tunuhuire. It has superb views, so it's a great place for a hike.
Leave the Panamericana at Ichu's second exit (after the service station) and head inland past the house marked 'Villa Lago 1960.' Walk 2km, bearing left at the junction, aiming for the two small, terraced hills you can see in the left of the valley. Bear left at a second junction (you'll pass the school if you miss it), and the road will take you between the two hills. Turn left again and head straight up the first one. Fifteen minutes of stiff climbing brings you to the top, where you'll be rewarded with the remains of a multilayered temple complex, and breathtaking 360-degree views.
This can be done as an easy half-day trip from Puno, arranged by private tour. Take plenty of water and food as there's no store.
6Drinking
Kamizaraky Rock PubPub
(Grau 158; h5pm-midnight)
With a classic-rock soundtrack, grungy cool bartenders, cocktails, pizzas and liquor-infused coffee drinks essential for staying warm during Puno's bone-chilling nights, this may be a hard place to leave.
# Cuzco & The Sacred Valley, Peru
#### Festivals & Events
#### Pisac Ruins
#### Ollantaytambo Ruins
#### Walking Tour: Central Cuzco
#### Cuzco
#### Pisac
#### Ollantaytambo
#
Cuzco & the Sacred Valley, Peru
_For the Incas, Cuzco was the belly button of the world. A visit to this city and its nearby ruins tumbles you back into the cosmic realm of ancient Andean culture – knocked down and fused with the colonial imprint of Spanish conquest, only to be repackaged as a thriving tourist center. The capital of Cuzco is only the gateway. Beyond lies the Sacred Valley, Andean countryside dotted with villages, high-altitude hamlets and ruins linked by trail and railway tracks to the continent's biggest draw – Machu Picchu._
Agricultural terraces, the Sacred Valley TETYANA DOTSENKO / SHUTTERSTOCK ©
Two Days in Cuzco & the Sacred Valley
Spend two full days exploring the city of Cuzco. On day one visit the Museo Machu Picchu and Qorikancha and enjoy decadent Andean fare at Chicha.
On day two, browse the wares at the Mercado San Pedro and the Center for Traditional Textiles of Cuzco. Take the Marcelo Batata Cooking Class and hear some live music at Centro Qosqo de Arte Nativo.
Four Days in Cuzco & the Sacred Valley
Head out early for a trip to the impressive Pisac Ruins. Take a taxi to the top, then make the walk down to Pisac for a visit to the famed Mercado de Artesanía.
On day four, catch a bus to ancient, cobbled Ollantaytambo and hike through the spectacular ruins above town. End the day with a meal at El Albergue Restaurante.
Arriving in Cuzco & the Sacred Valley
Cuzco's Aeropuerto Internacional Alejandro Velasco Astete, 6km south of the center, receives national and international flights. The _combi_ lines Imperial and C4M (S0.80, 20 minutes) run from just outside the airport to Av El Sol. An official radio taxi from within the airport costs S40.
Where to Stay
The region offers a range of lodging options from hostels to luxury hotels. Cuzco has hundreds of lodgings of all types and prices, with some of Peru's highest room rates. Booking ahead is recommended in high season (June to August). Rural areas may have lodging without adequate heating – ask ahead if you're concerned.
TOP EXPERIENCE
# Festivals & Events
Cuzco and the surrounding highlands celebrate many lively fiestas and holidays. Between them they provide riveting manifestations of both Andean and Catholic culture.
Great For...
qrh
yDon't Miss
June's Inti Raymi, the drawcard festival of Cuzco.
8Need to Know
National holidays and local festivals are the most crowded times – book accommodations well ahead.
5Take a Break
Feasts are integral to most festivals in Cuzco, so expect plentiful, delicious street food.
oTop Tip
Book a seat for the Inti Raymi ceremony (though it can be more fun to stand with the locals).
### El Señor de los Temblores
This procession through the Plaza de Armas takes place on Holy Monday, the Monday before Easter. It dates to the earthquake of 1650. El Señor de los Temblores' crucifix of the savior (now charred with soot) is considered the patron saint of Cuzco, responsible for saving the city from further earthquake damage.
### Crucifix Vigil
On May 2–3, a Crucifix Vigil is held on all hillsides with crosses atop them.
### Corpus Christi
Held on the ninth Thursday after Easter, Corpus Christi usually occurs in early June and features fantastic religious processions and celebrations in the cathedral.
### The Q'oyoriti Pilgrimage
Incredibly elaborate costumes, days of dancing, repetitive brass-band music, fireworks and sprinklings of holy water: welcome to one of Peru's lesser known, but most intense, festivals, Q'oyoriti (Star of the Snow).
Held at the foot of Ausangate the Tuesday before Corpus Christi, in late May or early June, this is a dizzy, delirious spectacle, yet no alcohol is involved or even allowed. Offenders are whipped by anonymous men dressed as _ukukus_ (mountain spirits) with white masks that hide their features.
At 6384m, Ausangate is the Cuzco department's highest mountain and the most important _apu_ (sacred deity) in the area. The subject of countless legends, it's the _pakarina_ (mythical place of sacred origin) of llamas and alpacas, and controls their health and fertility.
Q'oyoriti is a pilgrimage – the only way in is by trekking three or more hours up a cold mountain, arriving around dawn. The sight of a solid, endless line of people quietly wending their way up or down the track and disappearing around a bend in the mountain is unforgettable, as is Q'oyoriti's eerie, otherworldly feel.
Many _cuzqueños_ (inhabitants of Cuzco) believe that if you attend Q'oyoriti three times, you'll get your heart's desire.
### Inti Raymi
Cuzco's most important festival, the 'Festival of the Sun' is held on June 24. Visitors from throughout Peru and the world join the whole city celebrating in the streets with dancing and parades. The festival culminates in a re-enactment of the Inca winter-solstice festival at Sacsaywamán. Despite its commercialization, it's still worth seeing the pageantry in the city and at Sacsaywamán.
Inti Raymi festival | HUGHES HERVé / HEMIS.FR / GETTY IMAGES ©
TOP EXPERIENCE
# Pisac Ruins
A truly awesome site with relatively few tourists, this hilltop Inca citadel lies high above the village on a triangular plateau with a plunging gorge on either side.
Great For...
vgA
yDon't Miss
Taking a taxi up to the ruins and walking the 4km back to town.
8Need to Know
_boleto turístico_ adult/student S130/70; h7am-6pm
5Take a Break
Vendors sell drinks and snacks at the top.
oTop Tip
Note that organized tours, while often informative, are more rushed than private tours.
### Terracing
The most impressive feature is the agricultural terracing, which sweeps around the south and east flanks of the mountain in huge and graceful curves, almost entirely unbroken by steps (which require greater maintenance and promote erosion). Instead, the terracing is joined by diagonal flights of stairs made of flagstones set into the terrace walls. On the terraces, the Incas could grow food crops that would feed much of the mountaintop city; many terraces are still planted today. Above the terraces are cliff-hugging footpaths, watched over by caracara falcons and well defended by massive stone doorways, steep stairs and a short tunnel carved out of the rock.
FRANCESCO82BALESTRI / SHUTTERSTOCK ©
### Temples
This dominating site guards not only the Urubamba Valley below, but also a pass leading into the jungle to the northeast. Topping the terraces is the site's ceremonial center, with an _intihuatana_ (literally 'hitching post of the sun'; an Inca astronomical tool), several working water channels, and some painstakingly neat masonry in the well-preserved temples.
A path leads up the hillside to a series of ceremonial baths and around to the military area. Looking across the Kitamayo Gorge from the back of the site, you'll also see hundreds of holes honeycombing the cliff wall. These are Inca tombs that were plundered by _huaqueros_ (grave robbers), and are now completely off-limits to tourists.
No one knows the exact date of construction, but it was likely built no earlier than the early 15th century because no pre-Inca elements have been found. It probably served as a defensive outpost at the southern end of the valley, though some archaeologists believe it may have been constructed to celebrate victory over the Cuyos, a group living north of the Sacred Valley.
### Hiking
At the time of writing, the trail starting above the west side of the church in town was closed. Check for updates in Pisac. When it's open, it's a two-hour climb and 1½ hours return. Worthwhile but grueling, it's good training for the Inca Trail! The footpath has many crisscrossing trails, but if you aim toward the terracing, you won't get lost. To the west, or the left of the hill as you climb up on the footpath, is the Río Kitamayo Gorge; to the east, or right, is the Río Chongo Valley.
TOP EXPERIENCE
# Ollantaytambo Ruins
Both fortress and temple, these spectacular Inca ruins rise above Ollantaytambo, making a splendid half-day trip.
Great For...
vcg
yDon't Miss
Hiring a local guide is worthwhile to demonstrate how to turn on the faucet of the royal baths.
8Need to Know
_boleto turístico_ adult/student S130/70; h7am-5pm
5Take a Break
There are bars and local restaurants just around the corner from the ruins.
oTop Tip
The _boleto turístico_ tourist card, used for admission, is valid for 10 days and for 16 other sites across the region.
### Inca Victory
The huge, steep terraces that guard Ollantaytambo's spectacular Inca ruins mark one of the few places where the Spanish conquistadores lost a major battle.
The rebellious Manco Inca had retreated to this fortress after his defeat at Sacsaywamán. In 1536, Hernando Pizarro, Francisco's younger half-brother, led a force of 70 cavalrymen to Ollantaytambo, supported by large numbers of indigenous and Spanish foot soldiers, in an attempt to capture Manco Inca.
The conquistadores, showered with arrows, spears and boulders from atop the steep terracing, were unable to climb to the fortress. In a brilliant move, Manco Inca flooded the plain below the fortress through previously prepared channels. With Spaniards' horses bogged down in the water, Pizarro ordered a hasty retreat, chased down by thousands of Manco Inca's victorious soldiers.
Yet the Inca victory would be short lived. Spanish forces soon returned with a quadrupled cavalry force and Manco fled to his jungle stronghold in Vilcabamba.
Inca storehouses, Ollantaytambo | IZABELA23 / SHUTTERSTOCK ©
### Construction
Though Ollantaytambo was a highly effective fortress, it also served as a temple. A finely worked ceremonial center is at the top of the terracing. Some extremely well-built walls were under construction at the time of the conquest and have never been completed. The stone was quarried from the mountainside 6km away, high above the opposite bank of the Río Urubamba. Transporting the huge stone blocks to the site was a stupendous feat. The Incas moved the blocks across the river by carting them to the riverside and then diverting the entire river channel around them.
### Hiking & Views
The 6km hike to the Inca quarry on the opposite side of the river is a good walk from Ollantaytambo. The trail starts from the Inca bridge by the entrance to the village. It takes a few hours to reach the site, passing several abandoned blocks known as _piedras cansadas_ – tired stones.
Looking back towards Ollantaytambo, you can see the enigmatic optical illusion of a pyramid in the fields and walls in front of the fortress. A few scholars believe this marks the place where the legend says the original Incas first emerged from the earth.
Cuzco
1Sights
1Iglesia del TriunfoD4
2Iglesia San FranciscoA5
3Iglesia y Convento de Santa ClaraB6
4Iglesia y Monasterio de Santa CatalinaD5
5La CatedralD4
6Museo de Arte PrecolombinoD3
7Museo de Arte ReligiosoE4
8Museo Histórico RegionalC5
9Museo Machu PicchuE5
10Plaza de ArmasC4
11QorikanchaE6
2Activities, Courses & Tours
12Alpaca ExpeditionsC5
13Apus PeruD8
Marcelo Batata Cooking Class(see 18)
7Shopping
14Center for Traditional Textiles of CuzcoF8
15Mercado San PedroA7
5Eating
16ChichaB5
17CicciolinaE4
18Marcelo BatataE4
19Monkey CafeE2
6Drinking & Nightlife
20LimbusF3
21Museo del PiscoE5
22Republica de PiscoC4
3Entertainment
23Centro Qosqo de Arte NativoE8
24Ukuku'sC4
## Cuzco
1Sights
Cusco PlanetariumMuseum
( %974-877-776, 084-23-1710; www.planetariumcusco.com; Carr Sacsayhuamán, Km 2; per person with transport S75; hpresentations 6pm)
An excellent way to explore the fascinating Inca cosmovision. They defined constellations of darkness as well as light, used astronomy to predict weather patterns, and designed Cuzco's main streets to align with constellations at key moments. After an indoor presentation in English and Spanish there's high-powered telescope viewings outside. Reservations are essential; price varies with group size, and includes pickup and drop-off. The planetarium van picks up visitors at 5:40pm from Plaza Regocijo.
Museo de Arte PrecolombinoMuseum
(map Google map; %084-23-3210; www.map.museolarco.org/museo.html; Plazoleta Nazarenas 231; S20; h8am-10pm)
Inside a Spanish colonial mansion with an Inca ceremonial courtyard, this dramatically curated pre-Columbian art museum showcases a stunningly varied, if selectively small, collection of archaeological artifacts previously buried in the vast storerooms of Lima's Museo Larco. Dating from between 1250 BC and AD 1532, the artifacts show off the artistic and cultural achievements of many of Peru's ancient cultures, with exhibits labeled in Spanish, English and French.
La CatedralChurch
(map Google map; Plaza de Armas; adult/student S25/12.50; h10am-5:45pm)
A squatter on the site of Viracocha Inca's palace, the cathedral was built using blocks pilfered from the nearby Inca site of Sacsaywamán. Its construction started in 1559 and took almost a century. It is joined by the 1536 Iglesia del Triunfo (map Google map; Triunfo s/n) to its right and the 1733 **Iglesia de Jesús María** to the left.
La Catedral | ARTMARIE / GETTY IMAGES ©
QorikanchaRuins
(map Google map; %084-24-9176; Plazoleta Santo Domingo; admission S15 or boleto turístico; h8:30am-5:30pm Mon-Sat, 2-5pm Sun)
If you visit only one Cuzco site, make it these Inca ruins forming the base of the colonial church and convent of Santo Domingo. Once the richest temple in the Inca empire, all that remains today is the masterful stonework. The temple was built in the mid-15th century during the reign of the 10th Inca _,_ Túpac Yupanqui. Postconquest, Francisco Pizarro gave it to his brother Juan who bequeathed it to the Dominicans, in whose possession it remains.
CCourses
Marcelo Batata Cooking ClassCooking
(map Google map; %984-384-520; www.cuzcodining.com; Calle Palacio 135; 4hr course S297; h2pm)
If you've fallen for Peruvian cooking, this four-hour course is a worthwhile foray. A fully stocked market pantry demystifies the flavors of the region and the kitchen setup is comfortable. Includes appetizers, a pisco tasting and a main course. In English, Spanish or Portuguese. Accommodates vegetarians, and there's a private-course option.
TTours
Alpaca ExpeditionsHiking
(map Google map; %084-25-4278; www.alpacaexpeditions.com; Heladeros 157, piso 2 No 24; h9am-7:30pm Mon-Fri, 4:30-7:30pm Sat & Sun) S
A popular outfitter for the Inca Trail, Sacred Valley treks, Salkantay and Choquequirao, this is one of the few companies to prioritize hiring female guides and porters. Also uses portable bathrooms, plants trees and participates in trail cleanup.
Apus PeruHiking
(map Google map; %084-23-2691; www.apus-peru.com; Cuichipunco 366; h9am-1pm & 3-7pm Mon-Sat)
A recommended outfitter for the Inca Trail and others, also offering conventional tours. Responsible and popular with travelers. The company joins the Choquequirao trek with the Inca Trail for a total of nine days of spectacular scenery and an ever-more-impressive parade of Inca ruins, culminating in Machu Picchu.
Museo Machu Picchu
This newish museum (map Google map; Casa Concha; %084-25-5535; Santa Catalina Ancha 320; adult/child S20/10; h8am-7pm Mon-Fri, 9am-5pm Sat) exhibits 360 pieces from Machu Picchu taken by Hiram Bingham's expeditions and recently returned by Yale University, including lithic and metals, ceramics and bones. The collection shows the astounding array of fine handicrafts and ceramics acquired from throughout the vast Incan empire. There's also good background on the Bingham expeditions with informative documentaries (subtitled). Signs are in English and Spanish.
7Shopping
Center for Traditional Textiles of CuzcoArts & Crafts
(map Google map; %84-228-117; Av El Sol 603; h7:30am-8pm)
This nonprofit organization, founded in 1996, promotes the survival of traditional weaving. You may be able to catch a shop-floor demonstration illustrating different weaving techniques in all their finger-twisting complexity. Products for sale are high end.
A Quechua woman demonstrates traditional weaving | ALLEN.G / SHUTTERSTOCK ©
Mercado San PedroMarket
(map Google map; Plazoleta San Pedro; h6am-7pm)
Cuzco's central market is a must-see. Pig heads for _caldo_ (soup), frogs (to enhance sexual performance), vats of fruit juice, roast _lechón_ (suckling pig) and tamales are just a few of the foods on offer. Around the edges are typical clothes, spells, incense and other random products to keep you entertained for hours.
Mercado San Pedro | AGUSTINA CAMILION / SHUTTERSTOCK ©
5Eating
Monkey CafeCafe $
(map Google map; %084-59-5838; Tandapata 300; mains S15-20; h8am-8pm Wed-Mon)
Cuzco's finest coffee shop is shoehorned into a tiny locale at the top of San Blas hill. All espresso drinks feature double shots made with Peruvian-origin roasts. There are also very tasty sweets and hearty breakfasts ranging from healthy to heart-stopping.
CicciolinaInternational $$
(map Google map; %084-23-9510; www.cicciolinacuzco.com; Triunfo 393, 2nd fl; mains S38-59; h8am-11pm)
On the 2nd floor of a lofty colonial courtyard mansion, Cicciolina may be Cuzco's best restaurant. The eclectic, sophisticated food is divine, starting with house-marinated olives, and continuing with crisp polenta squares with cured rabbit, huge green salads, charred octopus and satisfying mains like red trout in coconut milk, beetroot ravioli and tender lamb. With impeccable service and warmly lit seating.
Marcelo BatataPeruvian $$
(map Google map; %084-22-2424; www.cuscodining.com/marcelo-batata; Palacio 121; mains S43-56; h12:30-11pm)
A sure bet for delectable Andean cuisine with a twist. Marcelo Batata innovates with traditional foods to show them at their best – like the humble _tarwi_ pea, which makes a mean hummus. The chicken soup with _hierba Luisa_ (a local herb) is exquisite, alongside satisfying beet _quinotto_ (like risotto), tender alpaca and twice-baked Andean potatoes that offer crispy-creamy goodness.
ChichaPeruvian $$$
(map Google map; %084-24-0520; Regocijo 261, 2nd fl; mains S30-65)
A Gastón Acurio venture serving up haute versions of Cuzco classics in an open kitchen. Its riff on _anticuchos_ (beef skewers) is a delectable barbecued octopus with crisp herbed potato wedges. Other contenders include _papas rellenas_ (stuffed potatoes), curried alpaca with quinoa, and _chairo_ (lamb and barley soup) served in a clay pot.
6Drinking & Nightlife
LimbusRooftop Bar
(map Google map; %084-43-1282; www.limbusrestobar.com; Pasñapakana 133; h8am-1am Mon-Sat, noon-midnight Sun)
Billed as the best view in Cuzco, it's all that (even after climbing to the top of San Blas). Don't worry, if you come during peak hours you'll have plenty of time to catch your breath while you queue to get in. With gorgeous cocktails and glass-walled panoramas, this was the hottest city spot when we visited.
Republica de PiscoBar
(map Google map; %084-24-4111; www.facebook.com/republicadelpiscocusco; Plateros 354; h5pm-2am)
A wonderful, elegant bar with attentive bartenders and drinks that merit seconds. It's popular with locals and travelers alike.
Museo del Pisco
When you've had your fill of colonial religious art, investigate this pisco museum (map Google map; %084-26-2709; www.museodelpisco.org; Santa Catalina Ancha 398; hnoon-1am), where the wonders of the national drink are extolled, exalted and – of course – sampled. Opened by an enthusiastic expat, this museum-bar is Pisco 101, combined with a tapas lounge. Grab a spot early for show-stopping live music (9pm to 11pm nightly).
Pisco sour cocktails | VIENNETTA / SHUTTERSTOCK ©
3Entertainment
Ukuku'sLive Music
(map Google map; %084-24-2951; Plateros 316; h6pm-2am)
The most consistently popular nightspot in town, Ukuku's plays a winning combination of crowd-pleasers – Latin and Western rock, reggae, _reggaetón_ , salsa and hip-hop – and often hosts live bands. Usually full to bursting after midnight with as many Peruvians as foreign tourists, it's good, sweaty, dance-a-thon fun. Happy hour is 8pm to 10:30pm.
Centro Qosqo de Arte NativoPerforming Arts
(map Google map; %084-22-7901; www.centroqosqodeartenativo.com; Av El Sol 604)
Has live nightly performances of Andean music and folk dancing at 6:45pm.
8INFORMATION
**iPerú** (www.peru.travel; %084-59-6159; Portal de Harinas 177; h9am-7pm Mon-Fri, to 1pm Sat) has excellent source for tourist information for both the region and entire country.
8GETTING THERE & AWAY
AIR
Cuzco's **Aeropuerto Internacional Alejandro Velasco Astete** (CUZ; %084-22-2611) has regular flights to Lima, Bogota and La Paz.
BUS
All international and most long-distance buses depart from the **Terminal Terrestre** ( %084-22-4471; Vía de Evitamiento 429), about 2km out of town toward the airport. Take a taxi (S30) or walk via Av El Sol.
TRAIN
Cuzco has two train stations. **Estación Huanchac** (Wanchaq; %084-58-1414; Av Pachacutec s/n; h7am-5pm Mon-Fri, to midnight Sat & Sun), near the end of Av El Sol, serves Juliaca and Puno on Lake Titicaca. **Estación Poroy** (Calle Roldan s/n, Carr Cuzco-Urubamba), east of town, serves Ollantaytambo and Machu Picchu.
8GETTING AROUND
TAXI
There are no meters in taxis, but there are set rates. At the time of research, trips within the city center cost S8 and destinations further afield, such as El Molino, cost S12.
## Pisac
It's not hard to succumb to the charms of sunny Pisac, a bustling and fast-growing colonial village with a fabulous market at the base of a spectacular Inca fortress perched on a mountain spur. Located just 33km northeast of Cuzco by a paved road, it's the most convenient starting point to the Sacred Valley.
1Sights
Mercado de ArtesaníaMarket
(Plaza Constitución; h8am-4pm)
Pisac is known far and wide for its market, by far the biggest and most touristy in the region. While there are still some local arts and crafts of note, watch out for mass-produced goods invading from as far as Colombia. Its success has it filling the Plaza Constitución and surrounding streets every day.
2Activities & Tours
Parque de la PapaEcotour
( %084-24-5021; www.ipcca.info/about-parque-de-la-papa; Pisac)
Day treks and cooking workshops are some of the offerings of this wonderful nonprofit, which promotes potato diversity and communal farming.
5Eating
Doña ClorindaPeruvian $$
( %084-20-3051; Urb San Luis, La Rinconada; mains S18-35; h9am-5pm)
In a lovely colonial home, this longtime Pisac mainstay serves up hearty Andean fare. Order some homemade _chicha morada_ (blue-corn juice) to go with heaping plates of _arroz chaufa_ (Peruvian fried rice), trout, beef and _rocoto relleno_ (stuffed peppers) with _kapchi_. A classic.
8GETTING THERE & AWAY
Minibuses to Pisac (S5, one hour) leave from terminals at Tullumayo 207 and Puputi 208 in Cuzco.
## Ollantaytambo
Dominated by two massive Inca ruins, the quaint village of Ollantaytambo, also called Ollanta, is the best surviving example of Inca city planning, with narrow cobblestone streets that have been continuously inhabited since the 13th century. After the hordes passing through on their way to Machu Picchu die down around late morning, Ollanta is a lovely place to be. It's perfect for wandering the mazy, narrow byways, past stone buildings and babbling irrigation channels, pretending you've stepped back in time. It also offers access to excellent hiking and biking.
Local woman in Ollantaytambo | DONYANEDOMAM / GETTY IMAGES ©
TTours
Coffee & distillery toursFood
( %084-20-4014; www.elalbergue.com; Estación de Tren, El Albergue; tours S50)
Led by El Albergue B&B, these tours offer a fascinating behind-the-scenes look at a small-scale coffee roaster and distiller of _cañazo, a_ sugarcane alcohol that's the oldest spirit in the Americas. Brought by the Spanish colony, the rustic Andean digestif is now taking on new dimensions as a high-end spirit. Participants get a free coffee or cocktail.
7Shopping
AwamakiArts & Crafts
( %84-43-6744; www.awamaki.org; Ventiderio s/n; h9am-6pm Mon-Fri, 10am-5pm Sat & Sun)
A nonprofit boutique selling gorgeous locally woven sweaters, hats and gloves made with organic dyes, as well as handmade leather handbags and wallets. Products are high quality and design conscious; they're wonderful gifts for those back home. The foundation also runs worthwhile tours to nearby weaving villages.
5Eating
El Albergue RestauranteInternational $$
( %084-20-4014; Estación de Tren; mains S29-45; h5:30-10am, noon-3pm & 6-9pm; v) S
This whistle-stop cafe serves elegant and well-priced Peruvian fare. It's inviting, with an open kitchen bordered by heaping fruit bowls and candles adorning linen-topped tables. Start with the _causas_ (potato dish) or organic greens from the garden. Lamb medallions with _chimichurri_ (herb sauce) are a standout, as is as the molle-pepper steak spiced from the tree outside. Access via the train platform.
8GETTING THERE & AWAY
Minibuses to Ollantaytambo (S12, two hours) leave from near the Puente Grau in Cuzco.
There are also several trains daily between Cuzco and Ollantaytambo.
# Machu Picchu, Peru
#### Visiting Machu Picchu
#### The Inca Trail
#### Aguas Calientes
#### Sights
#### Activities
#### Eating
#
Machu Picchu, Peru
_For many visitors to Peru and even South America, a visit to the Inca city of Machu Picchu is the long-anticipated highpoint of their trip. In a spectacular location, it's the best-known archaeological site on the continent._
_This awe-inspiring ancient city was never revealed to the conquering Spaniards and was virtually forgotten until the early part of the 20th century. Now, in the high season from late May until early September, 2500 people arrive daily. Despite this great tourist influx, the site manages to retain an air of grandeur and mystery, and is a must for all visitors to Peru._
Aerial view of Machu Picchu | HANNAHWADE / GETTY IMAGES ©
Two Days in Machu Picchu
Machu Picchu is huge, so it's best to spend at least two days, using Aguas Calientes as a base camp to allow ample time to explore the ruins. Staying in town also allows you to get up early and beat the midday crowds at Machu Picchu. If you are also climbing Wayna Picchu or Cerro Machu Picchu, you may need even more time.
Four Days in Machu Picchu
Put aside at least four days to hike the Inca Trail, with time to return and recover from the strenuous trek. The ancient trail was laid by the Incas, from the Sacred Valley to Machu Picchu. You will push yourself up and down across mountains, and past rivers and lakes, and feel proud to reach Machu Picchu on day four.
Arriving in Machu Picchu
Unless you are taking a trek on the Inca Trail, Aguas Calientes is the entry point to Machu Picchu. Cuzco is the launching point to Aguas Calientes.
**Bus** From Aguas Calientes, the only option up to Machu Picchu is a 20-minute bus ride.
**Air** The nearest airport is in Cuzco, which only serves Bolivia or domestic flights. Entry to Peru is always via Lima.
Where to Stay
Lodgings in Aguas Calientes are consistently overpriced – probably costing two-thirds more than counterparts in less-exclusive locations.
TOP EXPERIENCE
# Visiting Machu Picchu
A sublime stone citadel. A staggering cloud-forest perch. And a backstory that's out of a movie. Machu Picchu is an extraordinary Inca settlement and Unesco World Heritage Site.
Great For...
vhA
yDon't Miss
Museo de Sitio Manuel Chávez Ballón by Puente Ruinas at the base of the climb to Machu Picchu.
8Need to Know
Machu Picchu Historical Sanctuary (www.machupicchu.gob.pe; adult/student S152/77; h6am-6pm)
5Take a Break
Bring drinking water. Bringing food is not officially allowed.
oTop Tip
Try to visit outside peak times (between 10am and 2pm); June through August are the busiest months.
Unless you arrive via the Inca Trail, you'll officially enter the ruins through a ticket gate on the south side of Machu Picchu. About 100m of footpath brings you to the mazelike main entrance of Machu Picchu proper, where the ruins lie stretched out before you, roughly divided into two areas separated by a series of plazas.
Entrance tickets often sell out: buy them in advance in Cuzco. The site is limited to 5940 visitors daily, with 400 paid spots for hiking Wayna Picchu and Cerro Machu Picchu. Visitation is limited to a morning or afternoon ticket: morning tickets are valid between 6am and noon while afternoon tickets are valid between noon and 5:30pm.
Local guides (S150 per person, S30 for groups of six to 10) are readily available for hire at the entrance. Their expertise varies, but look for one wearing an official guide ID from Dircetur.
### Buying Machu Picchu Tickets
You would think accessing the continent's number-one destination might be easier. Get ready. Currently, Machu Picchu tickets can be purchased online (www.machupicchu.gob.pe), though not all foreign credit cards go through. If you reserve online, can't get your card to work and happen to be in Cuzco, you can deposit the amount due at a Banco de la Nación outlet within a three-hour window; later check in via the website to print your ticket.
In Cuzco, you can also purchase tickets from the Dirección Regional de Cultura Cusco (DIRCETUR; map; %084-58-2030; www.dirceturcusco.gob.pe; Maruri 340; h7:15am-6:30pm Mon-Sat) or the Dircetur outlet (map; %084-58-2030, ext 2000; www.dirceturcusco.gob.pe; Garcilaso s/n, Museo Histórico Regional; h7am-7:30pm Mon-Sat) in the Museo Histórico. Both outlets accept Peruvian soles, Visa or Mastercard. If you want to risk waiting, you can also purchase them from the Centro Cultural in Aguas Calientes, but only in Peruvian soles. Note that Aguas Calientes ATMs frequently run out of cash. Student tickets must be purchased in person with valid photo ID from the institution.
For a reasonable fee, travel agencies can also obtain tickets, which some readers recommend.
Entry to Machu Picchu requires a valid photo ID.
Lastly, ticketing procedures can change, but iPerú can offer the latest updates. Good luck.
Travelers exploring Machu Picchu | ALEX ROBINSON / GETTY IMAGES ©
### Inside the Complex
Hut of the Caretaker of the Funerary RockRuins
An excellent viewpoint to take in the whole site. It's one of a few buildings that has been restored with a thatched roof, making it a good shelter in the case of rain. The Inca Trail enters the city just below this hut. The carved rock behind the hut may have been used to mummify the nobility, hence the hut's name.
Hut of the Caretaker of the Funerary Rock | BRENT WINEBRENNER / GETTY IMAGES ©
Ceremonial BathsRuins
If you head straight into the ruins from the main entry gate, you pass through extensive terracing to a beautiful series of 16 connected ceremonial baths that cascade across the ruins, accompanied by a flight of stairs.
Temple of the SunRuins
Just above and to the left of the ceremonial baths is Machu Picchu's only round building, a curved and tapering tower of exceptional stonework. This structure is off-limits and best viewed from above.
Royal TombRuins
Below the Temple of the Sun, this almost hidden, natural rock cave was carefully carved by Inca stonemasons. Its use is highly debated; though known as the Royal Tomb, no mummies were actually ever found here.
Sacred PlazaPlaza
Climbing the stairs above the ceremonial baths, there is a flat area of jumbled rocks, once used as a quarry. Turn right at the top of the stairs and walk across the quarry on a short path leading to the four-sided Sacred Plaza. The far side contains a small viewing platform with a curved wall, which offers a view of the snowy Cordillera Vilcabamba in the far distance and the Río Urubamba below.
Temple of the Three WindowsRuins
Important buildings flank the remaining three sides of the Sacred Plaza. The Temple of the Three Windows features huge trapezoidal windows that give the building its name.
Principal TempleRuins
The 'temple' derives its name from the massive solidity and perfection of its construction. The damage to the rear right corner is the result of the ground settling below this corner rather than any inherent weakness in the masonry itself.
House of the High PriestRuins
Little is known about these mysterious ruins, located opposite the Principal Temple.
SacristyRuins
Behind and connected to the Principal Temple lies this famous small building. It has many well-carved niches, perhaps used for the storage of ceremonial objects, as well as a carved stone bench. The Sacristy is especially known for the two rocks flanking its entrance; each is said to contain 32 angles, but it's easy to come up with a different number whenever you count them.
IntihuatanaRuins
This Quechua word loosely translates as the 'Hitching Post of the Sun' and refers to the carved rock pillar, often mistakenly called a sundial, at the top of the Intihuatana hill. The Inca astronomers were able to predict the solstices using the angles of this pillar. Thus, they were able to claim control over the return of the lengthening summer days. Its exact use remains unclear, but its elegant simplicity and high craftwork make it a highlight.
Central PlazaPlaza
The plaza separates the ceremonial sector from the residential and industrial areas.
Prison GroupRuins
At the lower end of this area is the Prison Group, a labyrinthine complex of cells, niches and passageways, positioned both under and above the ground.
Temple of the CondorRuins
This 'temple' is named for a carving of the head of a condor with rock outcrops as outstretched wings. It is considered the centerpiece of the Prison Group.
### Ruins Outskirts
IntipunkuGate
( hcheckpoint closes around 3pm)
The Inca Trail ends after its final descent from the notch in the horizon called Intipunku (Sun Gate). Looking at the hill behind you as you enter the ruins, you can see both the trail and Intipunku. This hill, called Machu Picchu (Old Peak), gives the site its name.
Access from the Machu Picchu ruins may be restricted. It takes about an hour to reach Intipunku. If you can spare at least a half-day for the round-trip, it may be possible to continue as far as Wiñay Wayna (Huiñay Huayna). Expect to pay S15 or more as an unofficial reduced-charge admission fee to the Inca Trail, and be sure to return before 3pm, which is when the checkpoint typically closes.
### Inca Drawbridge
A scenic but level walk from the Hut of the Caretaker of the Funerary Rock takes you right past the top of the terraces and out along a narrow, cliff-clinging trail to the Inca drawbridge. In under a half-hour's walk, the trail gives you a good look at cloud-forest vegetation and an entirely different view of Machu Picchu. This walk is recommended, though you'll have to be content with photographing the bridge from a distance; someone crossed the bridge some years ago and tragically fell to their death.
Inca Drawbridge | TIMOTHY MULHOLLAND / ALAMY STOCK PHOTO ©
### Wayna Picchu
Wayna Picchu is the small, steep mountain at the back of the ruins. Wayna Picchu is normally translated as 'Young Peak,' but the word _picchu,_ with the correct glottal pronunciation, refers to the wad in the cheek of a coca-leaf chewer. Access to Wayna Picchu is limited to 400 people per day – the first 200 in line are let in at 7am, and another 200 at 10am. A ticket (S48), which includes a visit to the Moon Temple, may only be obtained when you purchase your Machu Picchu entrance ticket. These spots sell out a week in advance in low season and a month in advance in high season, so plan accordingly.
At first glance, it would appear that Wayna Picchu is a challenging climb but, although the ascent is steep, it's not technically difficult. However, it is not recommended if you suffer from vertigo. Hikers must sign in and out at a registration booth located beyond the central plaza between two thatched buildings. The 45- to 90-minute scramble up a steep footpath takes you through a short section of Inca tunnel.
Take care in wet weather as the steps get dangerously slippery. The trail is easy to follow, but involves steep sections, a ladder and an overhanging cave, where you have to bend over to get by. Partway up Wayna Picchu, a marked path plunges down to your left, continuing down the rear of Wayna Picchu to the small **Temple of the Moon**. From the temple, another cleared path leads up behind the ruin and steeply onward up the back side of Wayna Picchu.
The descent takes about an hour, and the ascent back to the main Wayna Picchu trail longer. The spectacular trail drops and climbs steeply as it hugs the sides of Wayna Picchu before plunging into the cloud forest. Suddenly, you reach a cleared area where the small, very well-made ruins are found.
Cerro Machu Picchu (S48) is a very good alternative if you miss out on Wayna Picchu tickets.
Descending steps at Wayna Picchu | MINKIMO / ALAMY STOCK PHOTO ©
yDon't Miss
Hike an hour to Intipunku (Sun Gate) for a different angle overlooking Machu Picchu.
oTop Tip
There are no signposts here – it's not a museum – so read up or hire a guide.
### The Mystery of Machu Picchu
Machu Picchu is not mentioned in any of the chronicles of the Spanish conquistadores. Nobody apart from local Quechua people knew of Machu Picchu's existence until American historian Hiram Bingham was guided to it by locals in 1911.
Despite scores of more recent studies, knowledge of Machu Picchu remains sketchy. Even today archaeologists are forced to rely heavily on speculation and educated guesswork as to its function. Some believe the citadel was founded in the waning years of the last Incas as an attempt to preserve Inca culture or rekindle their predominance, while others think that it may have already become an uninhabited, forgotten city at the time of the conquest.
A more recent theory suggests that the site was a royal retreat or the country palace of Pachacutec, abandoned at the time of the Spanish invasion. The site's director believes that it was a city, a political, religious and administrative center. Its location, and the fact that at least eight access routes have been discovered, suggests that it was a trade nexus between Amazonia and the highlands.
It seems clear from the exceptionally high quality of the stonework and the abundance of ornamental work that Machu Picchu was once vitally important as a ceremonial center. Indeed, to some extent, it still is: Alejandro Toledo, the country's first indigenous Andean president, impressively staged his inauguration here in 2001.
yNeed to Know
Drones, tripods and backpacks over 20L are not allowed into the ruins. Walking sticks are allowed.
oTop Tip
For really in-depth explorations, take along a copy of _Exploring Cuzco_ by Peter Frost.
TOP EXPERIENCE
# The Inca Trail
The views of snowy mountain peaks, distant rivers and cloud forests are stupendous – and walking from one cliff-hugging pre-Columbian ruin to the next is a mystical and unforgettable experience.
Great For...
cgv
yDon't Miss
Hot springs in the towns along the way will help weary hiking legs to recover.
8Need to Know
Most trekking agencies run buses to the start of the trail, also known as Piscacucho or Km 82.
5Take a Break
Most tours include meals and snacks. Bring a refillable water bottle.
oTop Tip
The Inca Trail is best visited in the dry season, April to October, and is closed in February.
The most famous hike in South America, the four-day Inca Trail is walked by thousands every year. Although the total distance is only about 24 miles, the ancient trail laid by the Incas from the Sacred Valley to Machu Picchu winds its way up and down and around the mountains, snaking over three high Andean passes en route, which have collectively led to the route being dubbed 'the Inca Trial.'
Llamas | AAABBBCCC / SHUTTERSTOCK ©
### Booking Your Trip
It is important to book your trip at least six months in advance for dates between May and August. Outside these months, you may get a permit with a few weeks' notice, but it's very hard to predict. Only licensed operators can get permits, but you can check general availability at www.camino-inca.com.
Consider booking a five-day trip to lessen the pace and enjoy more wildlife and ruins. Other positives include less-crowded campsites and being able to stay at the most scenic one – Phuyupatamarka (3600m) – on the third evening.
Make sure you have international travel insurance that covers adventure activities.
Inside Machu Picchu | AGATA B / SHUTTERSTOCK ©
### Regulations & Fees
The Inca Trail is the only trek in the Cuzco area that cannot be walked independently – you must go with a licensed operator. Treks start at US$595 and can cost up to US$6000 or more.
Only 500 people each day (including guides and porters) are allowed to start the trail. You must go through an approved Inca Trail operator. Permits are issued to them on a first-come, first-served basis. You will need to provide your passport number to get a permit and carry the passport with you to show at checkpoints along the trail. Be aware that if you get a new passport but had applied with your old, it may present a problem.
Permits are nontransferrable: name changes are not allowed.
### Choosing an Operator
While it may be tempting to quickly book your trek and move onto the next item on your to-do list, it's a good idea to examine the options carefully before sending that deposit. If price is your bottom line, keep in mind that the cheapest agencies may cut corners by paying their guides and porters lower wages. Other issues are substandard gear (ie leaky tents) and dull or lackadaisical guiding.
Yet paying more may not mean getting more, especially because international operators take their cut and hire local Peruvian agencies. Talk with a few agencies to get a sense of their quality of service. You might ask if the guide speaks English (fluently or just a little), request a list of what is included and inquire about group size and the kind of transportation used. Ensure that your tour includes a tent, food, a cook, one-day admission to the ruins and the return train fare.
Porters who carry group gear – tents, food etc – are also included. You'll be expected to carry your own personal gear, including sleeping bag. If you are not an experienced backpacker, it may be a good idea to hire a porter to carry your personal gear; this usually costs around US$50 per day for about 10kg.
If you prefer more exclusive services, it's possible to organize private trips with an independent licensed guide (US$1250 to US$2000 per person).
For a list of agencies and guides based in Cuzco.
Warmiwañusca (Dead Woman's Pass; | MIKAEL KVIST / GETTY IMAGES ©
### The Two-Day Inca Trail
This 10km version of the Inca Trail gives a fairly good indication of what the longer trail is like. It's a real workout, and passes through some of the best scenery and most impressive ruins and terracing of the longer trail.
It's a steep three- or four-hour climb from Km 104 to Wiñay Wayna, then another two hours or so on fairly flat terrain to Machu Picchu. You may be on the trail a couple of hours longer, just to enjoy the views and explore. We advise taking the earliest train possible from Cuzco or Ollantaytambo.
The two-day trail means overnighting in Aguas Calientes, and visiting Machu Picchu the next day, so it's really only one day of walking. Prices range from US$400 to US$535.
### What to Expect
Even if you are not carrying a full backpack, this trek requires a good level of fitness. In addition to regularly exercising, you can get ready with hikes and long walks in the weeks before your trip (also a good time to test out your gear). Boots should be already worn in by the time you go. On the trail, you may have to deal with issues such as heat and altitude. Just don't rush it; keep a reasonable pace and you should do fine.
8Need to Know
Take cash (in Peruvian soles) for tipping; an adequate amount is S100 for a porter and S200 for a cook.
oTop Tip
On the trail, get your next day's water hot in a well-sealed bottle; you can use it as a sleeping bag warmer and it will be cool to drink by the time you're hiking.
### Day One
After crossing the Río Urubamba (2600m) and taking care of registration formalities, you'll climb gently alongside the river to the trail's first archaeological site, **Llactapata** (Town on Top of the Terraces), before heading south down a side valley of the Río Cusichaca. (If you start from Km 88, turn west after crossing the river to see the little-visited site of **Q'ente** , about 1km away, then return east to Llactapata on the main trail.)
The trail leads 7km south to the hamlet of **Wayllabamba** (Grassy Plain; 3000m), near which many tour groups will camp for the first night. You can buy bottled drinks and high-calorie snacks here, and take a breather to look over your shoulder for views of the snow-capped **Nevado Verónica** (5750m).
### Day Two
Wayllabamba is situated near the fork of Ríos Llullucha and Cusichaca. The trail crosses the Río Llullucha, then climbs steeply up along the river. This area is known as **Tres Piedras** (Three Stones; 3300m), though these boulders are no longer visible. From here it is a long, very steep 3km climb through humid woodlands.
The trail eventually emerges on the high, bare mountainside of **Llulluchupampa** (3750m), where water is available and the flats are dotted with campsites, which get very cold at night. This is as far as you can reasonably expect to get on your first day, though many groups will actually spend their second night here.
From Llulluchupampa, a good path up the left-hand side of the valley climbs for a two- to three-hour ascent to the pass of **Warmiwañusca** , also colorfully known as 'Dead Woman's Pass.' At 4200m above sea level, this is the highest point of the trek, and leaves many a seasoned hiker gasping. From Warmiwañusca, you can see the Río Pacamayo (Río Escondido) far below, as well as the ruin of Runkurakay halfway up the next hill, above the river.
The trail continues down a long and knee-jarringly steep descent to the river, where there are large campsites at **Paq'amayo**. At an altitude of about 3600m, the trail crosses the river over a small footbridge and climbs toward **Runkurakay** ; at 3750m this round ruin has superb views. It's about an hour's walk away.
### Day Three
Above Runkurakay, the trail climbs to a false summit before continuing past two small lakes to the top of the second pass at 3950m, which has views of the snow-laden Cordillera Vilcabamba. You'll notice a change in ecology as you descend from this pass – you're now on the eastern, Amazon slope of the Andes and things immediately get greener. The trail descends to the ruin of **Sayaqmarka** , a tightly constructed complex perched on a small mountain spur, which offers incredible views. The trail continues downward and crosses an upper tributary of the Río Aobamba (Wavy Plain).
The trail then leads on across an Inca causeway and up a gentle climb through some beautiful cloud forest and an **Inca tunnel** carved from the rock. This is a relatively flat section and you'll soon arrive at the third pass at almost 3600m, which has grand views of the Río Urubamba Valley, and campsites where some groups spend their final night, with the advantage of watching the sun set over a truly spectacular view, but with the disadvantage of having to leave at 3am in the race to reach the Sun Gate in time for sunrise. If you are camping here, be careful in the early morning as the steep incline makes the following steps slippery.
Just below the pass is the beautiful and well-restored ruin of **Phuyupatamarka** (City Above the Clouds), about 3570m above sea level. The site contains six beautiful ceremonial baths with water running through them. From Phuyupatamarka, the trail makes a dizzying dive into the cloud forest below, following an incredibly well-engineered flight of many hundreds of Inca steps (it's nerve-racking in the early hours, use a headlamp). After two or three hours, the trail eventually zigzags its way down to a collapsed red-roofed white building that marks the final night's campsite.
A 500m trail behind the old disused pub leads to the exquisite little Inca site of **Wiñay Wayna** , which is variously translated as 'Forever Young,' 'To Plant the Earth Young' and 'Growing Young' (as opposed to 'growing old'). Peter Frost writes that the Quechua name refers to an orchid _(Epidendrum secundum)_ that blooms here year-round. The semitropical campsite at Wiñay Wayna boasts one of the most stunning views on the whole trail, especially at sunrise. For better or worse, the famous pub located here is now deteriorated and no longer functioning. A rough trail leads from this site to another spectacular terraced ruin, called **Intipata** , best visited on the day you arrive to Wiñay Wayna (consider coordinating it with your guide if you are interested).
Wiñay Wayna | NIARKRAD / SHUTTERSTOCK ©
### Day Four
From the Wiñay Wayna guard post, the trail winds without much change in elevation through the cliff-hanging cloud forest for about two hours to reach **Intipunku** (Sun Gate) – the penultimate site on the trail, where it's tradition to enjoy your first glimpse of majestic Machu Picchu while waiting for the sun to rise over the surrounding mountains.
The final triumphant descent takes almost an hour. Trekkers generally arrive long before the morning trainloads of tourists, and can enjoy the exhausted exhilaration of reaching their goal without having to push past enormous groups of visitors.
oTop Tip
As a courtesy, don't occupy the dining tent until late if it's where the porters sleep.
8 Recommended Reading
Mark Adams' _Turn Right at Machu Picchu_ (2011) is a humorous and fascinating travelogue about visiting the famous Lost City.
## Aguas Calientes
Also known as Machu Picchu Pueblo, this town lies in a deep gorge below the ruins. A virtual island, it's cut off from all roads and enclosed by stone cliffs, towering cloud forest and two rushing rivers. Despite its gorgeous location, Aguas Calientes has the feel of a gold-rush town, with a large itinerant population, slack services that count on one-time customers and an architectural tradition of rebar and unfinished cement. With merchants pushing the hard sell, it's hard not to feel overwhelmed. Your best bet is to go without expectations.
1Sights
Museo de Sitio Manuel Chávez BallónMuseum
(admission S22; h9am-5pm)
This museum has superb information in Spanish and English on the archaeological excavations of Machu Picchu and Inca building methods. Stop here before or after the ruins to get a sense of context (and to enjoy the air-conditioning and soothing music if you're walking back from the ruins after hours in the sun).
There's a small botanical garden with orchids outside, down a cool if nerve-testing set of Inca stairs. It's by Puente Ruinas, at the base of the footpath to Machu Picchu.
Las TermasHot Springs
(admission S20; h5am-8:30pm)
Weary trekkers soak away their aches and pains in the town's hot springs, 10 minutes' walk up Pachacutec from the train tracks. These tiny, natural thermal springs, from which Aguas Calientes derives its name, are nice enough but far from the best in the area, and get scummy by late morning.
Towels can be rented cheaply outside the entrance.
2Activities
PutucusiHiking
This jagged minimountain sits directly opposite Machu Picchu. Parts of the walk are up ladders, which get slippery in the wet season. The view across to Machu Picchu is worth the trek. Allow three hours. Follow the railway tracks about 250m west of town and you'll see a set of stairs, the start of a well-marked trail.
Porter Welfare on the Inca Trail
In the past, Inca Trail porters have faced excessively low pay, enormous carrying loads and poor working conditions. Relatively recent laws now stipulate a minimum payment of S170 to porters, adequate sleeping gear and food, and treatment for on-the-job injuries. At checkpoints on the trail, porter loads are weighed (each is allowed 20kg of group gear and 5kg of their own gear).
Yet there is still room for improvement and the best way to help is to choose your outfitter wisely. A quality trip will set you back at least US$500. The cheaper trips cut costs and often affect porter welfare.
Porters on the Inca Trail | CHUNCHANG WU / SHUTTERSTOCK ©
5Eating
Restaurants range from basic eateries to fine dining. Touts standing in the street will try to herd you into their restaurant, but take your time making a selection. Standards are not very high in most restaurants – if you go to one that hasn't been recommended, snoop around to check the hygiene first. Since refrigeration can be a problem, it's best to order vegetarian if you're eating in low-end establishments.
Indio FelizFrench $$
( %084-21-1090; Lloque Yupanqui 4; mains S34-48; h11am-10pm)
Hospitality is the strong suit of French cook Patrik at this multi-award-winning restaurant, but the food does not disappoint. Start with _sopa a la criolla_ (mildly spiced, creamy noodle soup with beef and peppers). There are also nods to traditional French cooking – like Provençal tomatoes, crispy-perfect garlic potatoes and a delicious apple tart.
Tree HouseFusion $$
( %084-21-1101; www.thetreehouse-peru.com; Huanacaure s/n; mains S38-60; h4:30am-10pm)
The rustic ambience of Tree House provides a cozy setting for its inviting fusion menu served alongside South American wines, craft beers and cocktails. Dishes like stuffed wontons with tamarind sauce, alpaca tenderloin and crisp quinoa-crusted trout are lovingly prepared. For dessert, lip-smacking chocolate mousse. There are raw and vegan options. Reserve ahead. It's part of the Rupa Wasi hotel.
MapachoCafe $$
( %984-759-634; Av Imperio de los Incas 614; mains S20-48; h10am-10pm)
This friendly streetside cafe is popular with the backpacking set. Perhaps it's all the craft beer and burgers on offer. It's worth checking out the _arroz chaufa_ (fried rice) and _lomo saltado (_ strips of beef stir-fried with onions, tomatoes, potatoes and chili).
ChullpiPeruvian $$$
( %914-169-687; Av Imperio de los Incas 140; mains S49-69; h11:30am-11pm)
Touting _'cocina de autor,'_ this stylish shoebox of a restaurant serves beautifully plated food, but in its mass production it might not quite live up to gourmet standards. The fare is classic Andean – including grilled trout, shredded pork and _wallpa chupe_ (a chicken-tomato stew). There are also soups and _causas_ (creamy potato dishes). Tour groups have a tendency to take over – reserve ahead.
Café InkaterraPeruvian $$$
( %084-21-1122; Machu Picchu Pueblo Hotel; menú lunch/dinner S93/122; h11:30am-4pm & 6-10pm; v c)
Upstream from the train station, this tucked-away riverside restaurant is housed in elongated thatched rooms with views of water tumbling over the boulders. There's a set menu (starter, main dish and dessert), with gluten-free and vegetarian options, and a decent kids menu to entice the young ones. The _lomo saltado_ bursts with flavor.
8INFORMATION
BCP (Av Imperio de los Incas s/n) ATM.
Centro Cultural (Machu Picchu Tickets; %084-21-1196; Av Pachacutec s/n; h5:30am-8:30pm) This is the only spot in town selling Machu Picchu entrance tickets. It's cash only and never a sure thing last minute.
iPerú ( %084-21-1104; Av Pachacutec, cuadra 1; h9am-6pm Mon-Sun) A helpful information center for everything Machu Picchu.
8Getting There & Away
TRAIN
All train companies have ticket offices in the train station.
To Cuzco (three hours), **Peru Rail** (map; %084-58-1414; www.perurail.com; Estación Poroy; h7am-5pm Mon-Fri, to noon Sat) has service to Poroy and taxis connect to the city, another 20 minutes away.
To Ollantaytambo (two hours), both Peru Rail and **Inca Rail** (map; %084-25-2974; www.incarail.com; Portal de Panes 105, Plaza de Armas; 1 way S231-330; h8am-9pm Mon-Fri, 9am-7pm Sat, to 2pm Sun) S provide service. Inca Rail offers a ticket with connecting bus service to Cuzco.
To **Santa Teresa** (via Hidroelectrica Station, 45 minutes), Peru Rail travels at 6:45am, 1:30pm and 3:40pm daily.
BUS
There is no road access to Aguas Calientes. The only buses go from the **bus stop** (where you can purchase tickets) up the hill to Machu Picchu (round-trip S80, 25 minutes) from 5:30am to 4pm; buses return until 6pm.
# The Cordilleras, Peru
#### Cordillera Blanca
#### Chavín de Huántar
#### Huaraz
#### Sights
#### Activities & Tours
#### Shopping
#### Eating
#### Drinking & Nightlife
#
The Cordilleras, Peru
_Ground zero for outdoor-adventure worship in Peru, the Cordilleras are one of the preeminent hiking, trekking and backpacking spots in South America. Here are some of Peru's most majestic vistas, with glaciated white peaks razoring their way through expansive lime-green valleys, which hold scores of pristine jade lakes, ice caves and torrid springs._
_Huaraz is a hotbed of hiking inspiration. New adventurers mix it up with experienced climbers to share their recent thrills and show off their snaps. Land here first to fill up on good food and plan your next adventure in the Cordilleras._
Basecamp at Laguna Jahuacocha, Cordillera Huayhuash JOERG STEBER / SHUTTERSTOCK ©
Four Days in the Cordilleras
On your third day, head off on a two-wheeled outing with Mountain Bike Adventures. Recover that night over craft beers and snacks at Los 13 Buhos. The next morning set out for Chavín de Huántar and spend the day exploring the fascinating archaeological site and its top-notch museum.
Two Days in the Cordilleras
Spend your first day in Huaraz (elevation 3091m) acclimatizing. Visit the Wari ruins of Monumento Nacional Wilkahuaín north of town, check out the Museo Regional de Ancash and eat well at Manka. Get an early start the next day for a memorable visit to the stunning Laguna Parón, either going solo or by organized tour from Huaraz.
Arriving in the Cordilleras
**Comandante FAP Germán Arias Graziani Airport** The Huaraz airport is actually at Anta, 23km north of town. A taxi will cost about S40.
**Bus** Buses from Lima and most other destinations arrive in central Huaraz.
Where to Stay
Hotel prices can double during holiday periods and rooms become very scarce. Better hotels, at the southeast fringe of central Huaraz, are often perched higher, making for better views of Huascarán.
TOP EXPERIENCE
# Cordillera Blanca
One of the most breathtaking parts of the continent (both figuratively and literally), the Cordillera Blanca is the world's highest tropical mountain range and encompasses some of South America's highest mountains.
Great For...
cgf
yDon't Miss
Lakes, ruins and hot springs en route – it's not just about seeing the (spectacular) mountains.
8Need to Know
Even experienced mountaineers would do well to add a local guide to their trekking group.
5Take a Break
Ask in Huaraz what amount of food and water is required for your trek, or bring a cook.
oTop Tip
The dry season, May to September, offers the best trekking conditions.
### Parque Nacional Huascarán
This 3400-sq-km park encompasses practically the entire area of the Cordillera Blanca above 4000m, including more than 600 glaciers and nearly 300 lakes, and protects such extraordinary and endangered species as the giant _Puya raimondii_ plant, the spectacled bear and the Andean condor.
Visitors to the park can register (bring your passport) and pay the park fee at the park office in Huaraz, although most of the main entrances to the park also sell tickets. Fees are S30 per person for a day visit, S60 for a three-day visit and S150 for a month.
### Laguna Parón
Silent awe enters people's expressions when they talk of Laguna Parón (map; S5). Nestled at 4185m above sea level, along a bumpy road 25km east of Caraz, and surrounded by spectacular snow-covered peaks, many claim this to be the most beautiful lake in the Cordillera Blanca. It is certainly the largest, despite its water levels being lowered from 75m to 15m in the mid-1980s to prevent a collapse of Huandoy's moraine.
Ringed by formidable peaks, Parón offers close-up views of **Pirámide de Garcilaso** (5885m), **Huandoy** (6395m), **Chacraraju** (6112m) and several 1000m granite rock walls. The challenging rock-climbing wall of Torre de Parón, known as the Sphinx, is also found here.
Most people see the lake as part of an organized tour out of Huaraz or Caraz (from S50). Going solo, you can organize a taxi in Caraz for around S150 round-trip with wait. The journey from Caraz takes 1½ hours on an unpaved road.
Laguna Parón and Pirámide de Garcilaso | AGUSTAVOP / GETTY IMAGES ©
### Hiking
Laguna ChurupHiking
If overnight trekking isn't your bag, but you'd like to experience the sight of some of the area's extravagant high-altitude lakes, this one-day hike is for you. It begins at the hamlet of Pitec (3850m), just above Huaraz, and takes you to the emerald green **Laguna Churup** (map; 4450m), at the base of Nevado Churup. Note the altitudes and the ascent (it's a steep 600m straight up). The walk takes roughly six hours and is a good acclimatization hike.
Laguna 69Hiking
This vivid blue **lake** (map) surrounded by snow-covered peaks is the jewel of the Cordilleras. Set at 4600m, it's a challenging acclimatization hike (we recommend working up to it with an easier hike at altitude). 'Sixty-nine' is most commonly visited as a day trip from Huaraz and has become increasingly popular in recent years. Don't expect to have the place to yourself. Swimming in the lake is prohibited.
Hiking in Parque Nacional Huascarán | BYELIKOVA OKSANA / SHUTTERSTOCK ©
### Acclimatize!
It is important that trekkers freshly arrived in the Cordilleras spend several days in Huaraz acclimatizing before sallying forth into the mountains. Huaraz is located at an altitude of 3091m – a sharp jump in elevation if you're arriving from sea level – but most Cordillera hikes go much higher, climbing to altitudes between 3500m and 5000m.
Rush into the high country and it's possible you'll end up suffering from dizziness, nausea or worse, putting a premature end to your trekking trip or, at best, turning the whole thing into one long miserable pain-fest.
On your first day, take it easy by pursuing some gentle, flat walks around town. On day two, depending on how you feel (everyone is different), you might want to try the Laguna Wilkacocha hike (map), one of the few trails around Huaraz that doesn't climb above 4000m. On day three, well-trained hikers may feel strong enough to break the 4000m barrier by shinning it up to Laguna Churup.
TOP EXPERIENCE
# Chavín de Huántar
Chavín de Huántar is the most intriguing of the many relatively independent, competitive ceremonial centers constructed throughout the central Andes.
Great For...
vnA
yDon't Miss
The carved tenon heads at the Museo Nacional de Chavín.
8Need to Know
admission S15; h9am-4pm Tue-Sun
5Take a Break
Buongiorno (Calle 17 de Enero Sur s/n; mains S20-35; h7am-7pm Tue-Sun) A sophisticated Peruvian-Italian restaurant in a pleasant garden setting right next to the ruins.
oTop Tip
Stay overnight in under-appreciated Chavín town and have the ruins to yourself in the morning.
The quintessential site of Peru's Mid–Late Formative Period (c 1200–500 BC), Chavín de Huántar is a phenomenal achievement of ancient construction, with large temple-like structures above ground and labyrinthine (now electronically lit) under-ground passageways. Although not as initially impressive as sites like Machu Picchu and Kuélap, Chavín tells an engrossing story together with its excellent affiliated museum.
### Edificio A
The largest and most important building, Edificio A, has withstood some mighty earthquakes over the years. Built on three different levels of stone-and-mortar masonry, the walls here were at one time embellished with tenon heads (blocks carved to resemble human heads with animal or perhaps hallucinogen-induced characteristics backed by stone spikes for insertion into a wall). Only one of these remains in its original place, although around 30 others may be seen in the local museum.
Underground passageway | KLUBLU / SHUTTERSTOCK ©
### Lanzón de Chavín
A series of tunnels underneath the Castillo are an exceptional feat of engineering, comprising a maze of complex corridors, ducts and chambers. In the heart of this complex is an exquisitely carved, 4.5m monolith of white granite known as the Lanzón de Chavín. In typical terrifying Chavín fashion, the low-relief carvings on the Lanzón represent a person with snakes radiating from his head and a ferocious set of fangs, most likely feline. The Lanzón, almost certainly an object of worship given its prominent, central placement in this ceremonial center, is sometimes referred to as the Smiling God – although its appearance seems anything but friendly.
Ancient stone tenon head | KLUBLU / SHUTTERSTOCK ©
### Museo Nacional de Chavín
This outstanding museum ( %043-45-4011; 17 de Enero s/n; h9am-5pm Tue-Sun) F, funded jointly by the Peruvian and Japanese governments, houses most of the intricate tenon heads carved with horror-stricken expressions from Chavín de Huántar, as well as the magnificent Tello Obelisk, another stone object of worship with low relief carvings of a caiman and other fierce animals.
The museum is located around 2km from the ruins on the north side of town – an easy 25-minute walk.
## Huaraz
Huaraz is the restless capital of this Andean adventure kingdom and its rooftops command exhaustive panoramas of the city's dominion: one of the most impressive mountain ranges in the world. Nearly wiped out by the earthquake of 1970, Huaraz isn't going to win any Andean-village beauty contests anytime soon, but it does have personality – and personality goes a long way.
This is first and foremost a trekking metropolis. During high season (May to September) the streets buzz with hundreds of backpackers and adventurers freshly returned from arduous hikes or planning their next expedition as they huddle in one of the town's many fine watering holes. Dozens of outfits help plan trips, rent equipment and organize a list of adventure sports as long as your arm.
Huaraz
1Sights
1Museo Regional de AncashA2
2Activities, Courses & Tours
2Andean KingdomB2
3Eco Ice PeruC2
4Mountain Bike AdventuresB1
5Pablo ToursB1
6QuechuandesD3
7Shopping
7Feria Artesanal La PlazaB2
8Montaña MagicaB2
5Eating
9La Casona HuaracinaC3
10MankaB2
11Rinconcito MineiroB1
12TrivioB2
6Drinking & Nightlife
13Los 13 BuhosB2
14Tio EnriqueB1
1Sights
Monumento Nacional WilkahuaínRuins
(adult/student S5/2; h9am-5pm Tue-Sun)
This small Wari ruin about 8km north of Huaraz is remarkably well preserved, dating from about AD 600 to 900. It's an imitation of the temple at Chavín done in the Tiwanaku style (square temples on raised platforms). Wilkahuaín means 'grandson's house' in Quechua. The three-story temple has seven rooms on each floor, each originally filled with bundles of mummies. The bodies were kept dry using a sophisticated system of ventilation ducts. A one-room museum gives some basic background information in English and Spanish.
To get here by foot take a taxi to 'El Pinar' (S7) from where there are two paths leading to the ruins – a direct route via the main road (6km) or a longer but more scenic route via Marian. Alternatively, a taxi direct to the ruins will set you back around S25.
There are actually two sets of ruins. Buy your ticket at the lower complex. It's a 10-minute walk along a dirt road to the smaller second complex (Wilkahuaín Pequeño).
Avoid taking the path from the ruins down to the baths at Monterrey as robberies have been reported on this stretch.
Museo Regional de AncashMuseum
(map Google map; Plaza de Armas; adult/child S5/1; h8:30am-5:15pm Tue-Sat, 9am-2pm Sun)
The Museo Regional de Ancash houses one of the most significant collections of ancient stone sculptures in South America lined up in a garden out back. Most of them are from the Recuay culture (400 BC–AD 600) and the Wari culture (AD 600–1100). Otherwise the collection is limited to a few mummies and some trepanned skulls.
Jirón José OlayaArchitecture
On the east side of town, Jirón José Olaya is the only street that remained intact through the earthquakes and provides a glimpse of what old Huaraz looked like; go on Sunday when a street market sells regional foods.
2Activities & Tours
QuechuandesTrekking
(map Google map; %943-386-147; www.quechuandes.com; Santa Gadea 995; h9am-8pm Mon-Sat, from 11am Sun)
A very well organized agency that gets rave reviews for its quality guides and ethical approach to treks. Management will assess your level before sending you out into the mountains or renting gear, to ensure you are up to the task. In addition to offering treks, summit expeditions and mountaineering courses, its staff are experts in rock climbing and bouldering.
Owners Marie and David researched and wrote the Cordillera's definitive climbing guidebook. Prices vary depending on activity, number of people and type of trip. Contact them via their website for a quote.
Eco Ice PeruTrekking
(map Google map; www.ecoice-peru.com; Figueroa 1185; 3- to 4-day treks from US$240; h8am-6pm)
Run by a gregarious and passionate young guide, this agency gets top reviews from travelers for its customer service, guides, _arrieros_ (mule drivers) and food. Treks often end with a dinner at the owner's pad in Huaraz.
Andean KingdomAdventure
(map Google map; %944-913-011; www.andeankingdomhuaraz.com; Parque Ginebra; climbing trips from S120; h9am-9pm Mon-Sat)
A laid-back but enthusiastic agency offering day courses for aspiring climbers, logistical support for experts and the usual day excursions, with an obvious bias toward climbing trips (Los Olivos and Hatun Machay feature highly).
Skyline AdventuresMountaineering
( %043-42-7097; www.skyline-adventures.com; Pasaje Industrial 137)
Based just outside Huaraz, this high-end operator comes highly recommended and provides guides for treks and mountain climbs. Leads six- and 12-day mountaineering courses (from US$1000).
Pablo ToursSightseeing Tours
(map Google map; %043-42-1145; www.pablotours.com; Luzuriaga 501; h9am-8pm)
An agency selling the standard Huaraz day tours to places like Chavín de Huántar (S40), Lagunas Llanganuco (S40) and Laguna 69 (S45).
Mountain Biking
Mountain Bike Adventures (map Google map; %972-616-008; www.chakinaniperu.com; Lúcar y Torre 530, 2nd fl; 2-day tours from US$380; h9am-1pm & 3-8pm) has been in business for well over a decade and receives repeated visits from mountain bikers for its decent selection of bikes, knowledgeable and friendly service, and good safety record. It offers guided tours, ranging from an easy five-hour cruise to 12-day circuits around the Cordillera Blanca.
Involved owner, Julio, is a lifelong resident of Huaraz who speaks English and will tailor-make a trip for your specific requirements. No one knows the region's single-track possibilities better than he does.
Mountain biking in the Cordillera Blanca | PAWEL CEBO / SHUTTERSTOCK ©
7Shopping
Feria Artesanal La PlazaArts & Crafts
(map Google map; Plaza de Armas; h10am-10pm)
Huaraz' main craft market is made up of an association of local artisans and has sprawling digs next to the unfinished cathedral in the Plaza de Armas. Here you can buy everything from alpaca wool jumpers to brightly colored canvas bags. It's a great place for browsing aimlessly.
Montaña MagicaSports & Outdoors
(map Google map; %949-680-107; Parque Ginebra 25; h10am-2pm & 4-8pm Mon-Sat)
Forget your kit? Head to Mountain Magic where you can stock up on a full gamut of decent trekking and mountaineering gear from rain jackets to camping stoves.
5Eating
MankaPeruvian, Italian $
(map Google map; %043-23-4306; Bautista 840; menú S10; h8:30am-11pm)
If you were curious about what happens when Peruvian _cocina_ (cuisine) collides with Italian _cucina_ then, let us tell you, it's a taste worth savoring. For proof, head straight to this simply decorated restaurant whose mix-and-match menu can deliver bruschetta for starters, _lomo saltado_ (strips of beef stir-fried with onions, tomatoes, potatoes and chili) for a main and a delectable chocolate mousse for desert.
TrivioInternational $$
(map Google map; %043-22-0416; Parque del Periodista; mains S21-37; h8am-midnight) S
Cementing a three-way marriage of craft beer, micro-roasted coffee and food made with local ingredients, Trivio joins a few Huaraz restaurants that wouldn't be alien in Lima. The decor is North America hip, the clientele predominantly gringo, and the food clever enough to excite the taste buds but filling enough to cover the hole left by your recent four-day trek.
Mi ComediaItalian $$
( %043-58-7954; Centenario 351; mains S22-38; h5-11pm Mon-Sat)
Many restaurants claim great pizzas with some even uttering the word 'Naples' blasphemously in the description. But at Mi Comedia the Italian boasts are no exaggeration. This is about as Neapolitan as a pizza can get in Peru.
Rinconcito MineiroPeruvian $$
(map Google map; Morales 757; menú S8-16, mains S12-35; h7am-11pm; W)
This popular place is _the_ spot to tuck into homey and cheap Peruvian daily _menús_ (set meals). The daily blackboard of 10 or so options includes an excellent _lomo saltado_ , plus grilled trout, _tacu-tacu_ (a Peruvian fusion dish of rice, beans and a protein) and the like.
La Casona HuaracinaPeruvian $$$
(map Google map; %43-39-6420; Campos 735; mains S22-42; h11am-10pm)
The colonial architectural style is noticeably absent in Huaraz until you step into the sparkling new Huaracina, the clever designers of which have drawn inspiration from the graceful buildings of Arequipa. The business is split into three interconnecting spaces: a smart lounge bar, a _pastelería_ (pastry shop) and the restaurant, all decked out with astute attention to detail (glass lampshades, elaborate balustrades, bookshelves).
6Drinking & Nightlife
Los 13 BuhosBar
(map Google map; Parque Ginebra; h11am-2am)
Halfway up some monstrous Andean pass with a 15kg pack on your back, it's not uncommon to start dreaming of 13 Buhos with its Luchos craft beer, pool table and delectable afternoon 'snacks' (waffles anyone?).
Tio EnriqueBar
(map Google map; Bolivar 572; h4-11pm)
In an energy-sapping mountain town like Huaraz, this cozy Swiss-themed drinking hole with a long bar and communal pine tables is just what the doctor ordered. Popular with hardcore climbers, it serves around three dozen varieties of imported beers from the UK, Belgium and Germany as well as sausages grilled at the door by the charismatic apron-toting owner.
8INFORMATION
English newspaper the _Huaraz Telegraph_ (www.thehuaraztelegraph.com) is a good source of information about the region.
**iPerú** ( %043-42-8812; iperuhuaraz@promperu.gob.pe; Pasaje Atusparia, Oficina 1, Plaza de Armas; h9am-6pm Mon-Sat, to 1pm Sun) Has general tourist information but little in the way of trekking info.
**Parque Nacional Huascarán Office** ( %043-42-2086; www.sernanp.gob.pe; Sal y Rosas 555; h8:30am-1pm & 2:30-6pm Mon-Fri) Staff have some limited information about visiting the park. You can also pay your park fees here.
8GETTING THERE & AWAY
AIR
**LC Perú** ( %043-42-4734; www.lcperu.pe; Luzuriaga 904; h9am-7pm Mon-Fri, to 6pm Sat) is currently the only company offering service, with flights to/from Lima (US$120, one hour) on Tuesdays, Thursdays and Saturdays.
BUS
Buses leave from different company offices: most are located in and around Raimondi and Bolívar streets a few blocks north of the Plaza de Armas.
A plethora of companies have departures for Lima. The top four for comfort and reliability are Cruz del Sur, Oltursa, Linea and Movil Tours. Most depart midmorning or late evening.
# Quito & the Central Highlands, Ecuador
#### Quito's Old Town
#### Adventures in Baños
#### Parque Nacional Cotopaxi
#### Quito
#### Baños
#
Quito & the Central Highlands, Ecuador
_Quito is dramatically situated, squeezed between high Andean peaks whose greenery is concealed by the afternoon mist. The city's crown jewel is its 'Old Town,' a Unesco World Heritage Site packed with colonial monuments and architectural treasures. There is no sterile museum mile: here everyday life pulses along the handsomely restored blocks with 17th-century facades, picturesque plazas and magnificent art-filled churches._
_Quito is also the launchpad for attractions in the Andean highlands, including adventure activities in Baños and stunning alpine scenery in the Parque Nacional Cotopaxi._
Plaza Grande, Quito's Old Town JESS KRAFT / SHUTTERSTOCK ©
Two Days in Quito
Spend a full day in Quito's Old Town. Visit the decadent Iglesia de la Compañía de Jesús, take in pre-Columbian treasures at the Casa del Alabado and admire the view from the Plaza San Francisco.
On day two head to Parque Nacional Cotopaxi. Check into a hacienda, take short walks, admire the views and adjust to the serious elevation!
Four Days in Quito
On day three, rise early for a sunrise in the park. then go horseback riding or trekking amid the breathtaking scenery.
On the fourth day continue to Baños for a bit of adventure. Make the (mostly downhill) mountain-biking trip towards Puyo, stopping at waterfalls along the way. At day's end soak your weary gams in the Termas de la Virgen.
Arriving in Quito
Quito's modern Aeropuerto Internacional Mariscal Sucre is 37km northeast of the city in a broad valley near Tababela. There is a tourist information booth in the arrivals hall.
Taxi prices into the city center are fixed at $26 to Mariscal Sucre or the Old Town (allow 45 to 75 minutes travel time).
Where to Stay
In Quito, most travelers tend to stay near the Mariscal, a guesthouse- and hostel-packed district. The quieter neighborhood of La Floresta is a pleasant alternative and begins only a few blocks away. There's a good range of accommodations in the Old Town, including most of Quito's best top-end hotels. You'll also find atmospheric haciendas near Cotopaxi and ample guesthouses in Baños.
TOP EXPERIENCE
# Quito's Old Town
With its narrow streets, restored colonial architecture and lively plazas, Quito's Centro Histórico is a marvel to wander. Built centuries ago by indigenous artisans and laborers, Quito's churches, convents, chapels and monasteries are cast in legend and steeped in history. It's bustling but magical: the more you look, the more you find.
Great For...
hvr
yDon't Miss
The Last Supper painting with subtle Ecuadorian elements (including roasted guinea pig) at the Catedral Metropolitana.
oTop Tip
Avoid visiting on Monday when some sights are closed.
### Plaza Grande
The heart of the Old Town is the Plaza Grande (map Google map), a picturesque, palm-fringed square surrounded by historic buildings and bustling with everyday life.The benches are great for soaking up the Andean morning sun and watching the day unfold. On Monday, the changing of the guards takes place on the plaza at 11am.
FOTOS593 / SHUTTERSTOCK ©
### Catedral Metropolitana
On Plaza Grande's southwest side stands Quito's cathedral (map Google map; cathedral adult/child $3/2, cathedral & dome [adults only] $6; h9am-5pm Mon-Sat). Although not the most ornate of the Old Town's churches, it has some fascinating works by artists from the Quito School and houses the tomb of independence hero Antonio José de Sucre. Behind the main altar is a plaque marking where President Gabriel García Moreno died on August 6, 1875; after being slashed with a machete outside the Palacio del Gobierno, he was carried, dying, to the cathedral.
Don't miss the painting of the Last Supper, with Christ and disciples feasting on _cuy_ (guinea pig), and a nativity painting that features a llama and a horse peering over the newborn Jesus. Admission includes a free guided tour in Spanish and entry to the small cathedral museum. For an extra $3 you can climb up a narrow spiral staircase and onto the cathedral's domed roof; the views are impressive but there is some clambering along narrow ledges involved.
### Palacio Arzobispal
On the northeast side of Plaza Grande, this former archbishop's palace (map Google map) is now a colonnaded row of small shops and restaurants, located between Avenidas García Moreno and Venezuela. Concerts are often held on the covered patio on weekends.
Palacio Arzobispal | PETER ADAMS / GETTY IMAGES ©
### Iglesia de la Compañía de Jesús
Capped by green-and-gold domes, La Compañía de Jesús (map Google map; www.fundacioniglesiadelacompania.org.ec; García Moreno & Sucre, adult/student $5/2.50; h9:30am-6:30pm Mon-Thu, to 5:30pm Fri, to 4pm Sat, 12:30-4pm Sun) is Quito's most ornate church and a standout among the baroque splendors of Old Town. Free guided tours in English or Spanish highlight the church's unique features, including its Moorish elements, perfect symmetry (right down to the trompe l'oeil staircase at the rear), symbolic elements (bright-red walls are a reminder of Christ's blood) and its syncretism (Ecuadorian plants and indigenous faces are hidden along the pillars).
oAndean High
Quito's elevation of about 2850m can leave you breathless. To minimize symptoms of altitude sickness, take things easy upon arrival, eat light and lay off the alcohol.
### Museo de la Ciudad
This first-rate museum (map Google map; %02-228-3883; www.museociudadquito.gob.ec; García Moreno S1-47 near Rocafuerte; adult/child $3/1.50; h9am-5pm Tue-Sun) depicts daily life in Quito through the centuries, with displays including dioramas, model indigenous homes and colonial kitchens. The 1563 building itself (a former hospital) is a work of art. There are also a number of temporary exhibitions. Entry is free on the last Sunday of the month.
The museum sits just past the 18th-century arch, Arco de la Reina, built to give shelter to churchgoers.
### Casa del Alabado
Housed in an elegant colonial-era home, this privately owned museum (map Google map; %02-228-0940; www.alabado.org; Cuenca N1-41 near Bolívar; adult/child $6/1; h9:30am-5:30pm), with contemporary displays, showcases an impressive collection of pre-Columbian artifacts. Thematically organized around subjects such as shamans, pigmentation and the afterlife, explanations in English and Spanish (audioguides are available) explore the indigenous beliefs represented by the finely crafted ceramic pieces and jewelry.
8Need to Know
The Old Town tourist office (Corporación Metropolitana de Turismo; map; %02-257-2445; www.quito-turismo.gob.ec; Venezuela near Espejo; h9am-6pm Mon-Sat, 9:30-4:30pm Sun; W) is well located on Plaza Grande.
### La Ronda
La Ronda (map Google map; h24hr) is a completely restored narrow cobblestone lane lined with postcard-perfect 17th-century buildings housing festive restaurants, bars and colorful shops. Sleepy by day, it comes alive on Friday and Saturday nights when _canelazo_ ( _aguardiente_ with hot cider and cinnamon) vendors keep the crowds nice and warm and live music spills outdoors. Placards along the walls describe (in Spanish) some of the street's history and the artists, writers and political figures who once resided here.
### Plaza San Francisco
Walking from the Old Town's narrow colonial streets into this open plaza (map Google map) reveals one of the finest sights in all of Ecuador: a sweeping cobblestone plaza backed by the mountainous backdrop of Volcán Pichincha, and the long, whitewashed walls and twin bell towers of Ecuador's oldest church.
Interior of Iglesia de San Francisco | PHILIP LEE HARVEY / LONELY PLANET ©
### Museo Camilo Egas
Inside this restored colonial home is a small but iconic collection (map Google map; Venezuela 1302 near Esmeraldas; h8:30am-5pm Tue-Fri, 10am-4pm Sat) F of work by painter Camilo Egas (1899–1962), Ecuador's first _indigenista_ (indigenous-led movement) painter. One of the galleries showcases temporary exhibitions by contemporary painters.
### Museo Franciscano
To the right of the Iglesia de San Francisco's main entrance, and within the Convent of St Francis, this museum (map Google map; Museo Fray Pedro Gocial; www.museofraypedrogocial.com; Cuenca 477 near Sucre, Plaza San Francisco; adult/child $3/1; h9am-5:30pm Mon-Sat, to 1pm Sun) contains the church's finest artwork, including paintings, sculpture and 16th-century furniture, some of which is fantastically wrought and inlaid with thousands of pieces of mother-of-pearl. The admission fee includes a guided tour in English or Spanish.
Good guides will point out Mudejar (Moorish) representations of the eight planets revolving around the sun in the ceiling and will explain how the light shines through the rear window during the solstices, lighting up the main altar. They'll also demonstrate an odd confessional technique, where two people standing in separate corners can hear each other while whispering into the walls.
### Cumandá Parque Urbano
Old Town's old bus terminal has been converted into a sparkling covered cultural center ( %02-257-3645; www.facebook.com/quitocumanda; Av 24 de Mayo; h7am-8pm Tue-Fri, 8am-6pm Sat & Sun) F and sports complex with a volleyball court, soccer pitch, climbing wall, yoga studios and several small swimming pools – all free of charge. Temporary art exhibitions, theater performances, live music and other cultural events are also held here. It's accessed from La Ronda.
oEaster Processions
Colorful religious processions are held during Easter Week, the most spectacular being the procession of _cucuruchos_ (penitents wearing purple robes and conical masks) on Good Friday.
TOP EXPERIENCE
# Adventures in Baños
Baños offers unrivaled adventures: you can visit thundering waterfalls, hike down impossibly steep forest-lined gorges and bike all the way to the Amazon Basin. Afterwards, you can rejuvenate in the town's steaming thermal springs.
Great For...
gcf
yDon't Miss
The views over the countryside from Bellavista, high above Baños.
8Need to Know
Near the Parque Central, the tourist office ( %03-274-0483; www.facebook.com/gadbanosdeaguasanta; Halflants near Rocafuerte; h8am-noon & 2-5pm) has lots of info and free maps.
5Take a Break
You'll find filling South American dishes at the lively Cafe Hood.
oTop Tip
Be sure to bring your swimsuit, so you can take a dip in the thermal springs.
### Mountain Biking
The most popular ride is the dramatic descent past a series of waterfalls on the road to Puyo, a jungle town 61km to the east. Various other mountain-biking options are available and the outfitters will be happy to tell you about them.
Many companies around town rent mountain bikes (it's worth paying extra for disk brakes and suspension). From Puyo (or any point on the way), you can simply take a bus back to Baños, putting your bike on the roof (or in the luggage compartment).
Ruta de las Cascadas, Baños | KSENIA RAGOZINA / SHUTTERSTOCK ©
### Hiking
The tourist office provides a useful map showing some of the trails around town. Reports of assaults on nearby hikes have dropped in recent years. Nevertheless, it's advisable to bring just the cash you need and leave the expensive camera at your hotel.
Going south on Maldonado takes you to a path that climbs to **Bellavista** , where a white cross stands high over Baños. The path then continues to the settlement of **Runtún** , some two hours away, where the views are outstanding. From here, you can continue up to the Casa del Arbol ($1; h7am-6:30pm), or loop around and back down to Baños, ending up at the southern end of Mera. This takes you past the statue of **La Virgen del Agua Santa** , about half an hour from town. The whole walk takes four to five hours.
Rainforest zip-lining | AMMIT JACK / SHUTTERSTOCK ©
### Climbing
The climbing conditions on Tungurahua (5016m), an active volcano, are naturally in flux. At the time of research, climbing to the peak was allowed, but only with a licensed guide (six hours for those in optimum condition; others may be well advised to take the two-day approach and sleep at the refuge). You can also climb just to the refuge at 3830m, a steep three- to four-hour climb from the village of Pondoa.
Andean Summit Adventure ( %968-959-533; www.andeansummitadventure.com; Ambato & Reyes) is an experienced and certified Aseguim/IFMGA team of European and South American mountaineers with an excellent success rate for scaling Tungurahua and other Ecuadorian peaks.
### Rafting
The town's tour operators offer guided trips (half-day $30) on the Río Patate and Pastaza. The trips bring you to Class III and IV water (Class IV is enough to really get your heart pumping). Kayak classes are also available ($80).
### Horseback Riding
José & Two Dogs ( %098-420-6966; paulo_climb@hotmail.com; cnr Maldonado & Martínez) offers horseback riding excursions to nearby springs as well as canyoning, zip-lining and salsa classes.
TOP EXPERIENCE
# Parque Nacional Cotopaxi
Covered in a draping glaciated skirt that gives way to sloping gold-and-green páramo, the flanks of Cotopaxi are home to wild horses, llamas and fox, with Andean condors soaring overhead. Whether you come for trekking, horseback riding or simply to admire the view, the setting is spectacular.
Great For...
gfc
yDon't Miss
The sunrise views against the backdrop of majestic Volcán Cotopaxi.
8Need to Know
All of the haciendas provide transportation from Quito, often at an additional cost.
5Take a Break
Within the park itself, Tambopaxi ( %02-600-0365; www.tambopaxi.com; camping per person $16, dm $24, s/d/tr incl breakfast $92/115/140; W) S serves food all day in its cozy dining room with spectacular views of Cotopaxi.
oTop Tip
The skies are clearest around sunrise. Get up early for the best views.
### Climbing Cotopaxi
Although the climb is not technical – save for a few basic crevasse crossings and heart-pounding shimmies up fallen seracs – it is physically demanding, freezing and, for some people, vertigo-inducing.
The ascent starts around midnight from Refugio José Rivas. Even experienced, fit and acclimatized climbers can only reach the summit at dawn about once out of every two tries (no guarantees, baby!). The reward for those who make it to the top (on a clear day) are awesome views of other mountains and a peek at the crater's smoking fumaroles.
Many outfitters lead climbs, including the reputable Quito-based Condor Trekk.
### Mountain Biking
Cruising around the park's circuit of relatively flat dirt roads is popular, as is a descent down from the _refugio_ (mountain refuge) parking lot to the Control Caspi. Tour operators in Latacunga and Quito, such as Biking Dutchman, can arrange trips.
Hikers near Volcán Cotopaxi | ECUADORPOSTALES / SHUTTERSTOCK ©
### Bird-Watching
Keep your eyes peeled for the giant, soaring Andean condor and the Ecuadorian hill star, one of the world's highest-altitude hummingbirds. Andean lapwing, Baird's sandpiper, Andean coot, caracara, Andean teal, Andean gull and solitary sandpiper are common visitors to Laguna Limpiopungo.
Andean condor | JOHN STAGER / SHUTTERSTOCK ©
### Haciendas
The luxurious lodges located in and around Cotopaxi National Park are some of the best in the country, and well worth the extra bucks you pay for the views. You can arrange all manner of activities, and all lodges have restaurants on site.
The modern adobe dwelling of Hacienda Los Mortiños ( %02-334-2520; www.losmortinos.com; dm/s/d/ste incl breakfast $30/80/103/151; W) has jaw-dropping views of the neighboring volcanoes and comfortable rooms.
Located within the national park boundaries, Tambopaxi is perfectly positioned for hikes, though you could spend all day gazing at the volcano, llamas and wild horses through the windows of the stove-heated main lodge.
A lovely rustic property on the way to the northern entrance of Parque Nacional, Secret Garden Cotopaxi ( %099-357-2714; www.secretgardenquito.com/the_secret_garden_cotopaxi; near Santa Ana del Pedregal; incl full board dm $40, d $100-120) S has superb views and loads of activities.
## Quito
In Quito, warm and relaxed, traditional Ecuadorian Sierra culture – overflowing market stands, shamanistic healers, fourth-generation hatmakers – mixes with a vibrant and sophisticated culinary and nightlife scene.
Old Town
1Sights
1Casa del AlabadoA3
2Catedral MetropolitanaB2
3Iglesia de la Compañía de JesúsB2
4La RondaB4
5Museo Camilo EgasD1
6Museo de la CiudadA3
7Museo FranciscanoA2
8Palacio ArzobispalC2
9Plaza GrandeB2
10Plaza San FranciscoA2
2Activities, Courses & Tours
11CarpeDM AdventuresE2
7Shopping
12TianguezA2
5Eating
13Cafetería ModeloB3
14Cafetería Modelo IIB3
15Masaya BistroB4
16Mercado CentralD2
17San AgustínC2
Tianguez(see 12)
18Vista HermosaC2
19Vista Hermosa ItchimbiaF2
6Drinking & Nightlife
20Bandido BrewingD3
1Sights
TelefériQoCable Car
(map; <https://teleferico.com.ec>; Av Occidental near Av La Gasca, New Town; adult/child $8.50/6.50; h9am-8pm Tue-Thu, 8am-8pm Fri-Mon)
For spectacular views over Quito's mountainous landscape, hop aboard this sky tram. This is one of the world's highest aerial lifts and it takes passengers on a 2.5km ride (10 minutes) up the flanks of Volcán Pichincha to the top of Cruz Loma. Once you're at the top (a mere 4100m), you can hike to the summit of Rucu Pichincha (4680m), a 4km (five-hour) round-trip – ask about the safety situation before attempting the climb and bring warm clothes.
TelefériQo's cable cars and a view over Quito's landscape | JESS KRAFT / SHUTTERSTOCK ©
Capilla del HombreGallery
(Chapel of Man; www.guayasamin.org; Calvache E18-94 & Chávez, Bellavista; adult/child incl Casa Museo Guayasamín $8/4; h10am-5pm)
One of the most important works of art in South America, Ecuadorian artist Oswaldo Guayasamín's Capilla del Hombre stands next to the Casa Museo Guayasamín (www.guayasamin.org; Calvache E18-94 & Chávez, Bellavista; adult/child incl Capilla del Hombre $8/4; h10am-5pm). The fruit of Guayasamín's greatest vision, this giant monument-cum-museum is a tribute to humankind, to the suffering of Latin America's indigenous poor and to the undying hope for something better. It's a moving place and tours (in English, French and Spanish, included in the price) are highly recommended. Admission includes entrance to the Casa Museo.
Museo NacionalMuseum
(map Google map; <http://muna.culturaypatrimonio.gob.ec>; cnr Avs Patria & 12 de Octubre, Eugenio Espejo, New Town; h9am-6pm Tue-Sun) F
Located in the circular, glass-plated, landmark building of the Casa de la Cultura is one of the country's largest collections of Ecuadorian art, with magnificent works of pre-Hispanic and colonial religious art. The museum collection includes more than 1000 ceramic pieces dating from 12,000 BC to AD 1534. Highlights are 'whistle bottles' from the Chorrera culture, figures showing skull deformation practiced by the Machalilla culture, wild serpent bowls from the Jama-Coaque and ceramic representations of _tzantzas_ (shrunken heads).
TTours
Quito is one of the easiest places in Ecuador to arrange a guided tour, be it a Galápagos cruise, mountain-climbing, Amazon lodge, biking tour or whitewater rafting.
Happy GringoTours
( %02-512-3486; www.happygringo.com; Aldaz N34-155 near Portugal, Edificio Catalina Plaza, 2nd fl, La Carolina, New Town; h9am-6pm Mon-Fri)
A British- and Dutch-owned company catering to a midrange market, Happy Gringo can organize week- to month-long customized itineraries throughout the country, from the Galápagos to the Amazon. Professionally run with English-speaking guides and private drivers available, it's one of the best all-around tour companies in the city.
CarpeDM AdventuresTours
(map Google map; %02-295-4713; www.carpedm.ca; Antepara E4-70 near Los Rios, San Blas, Old Town; day tours per person $50)
CarpeDM earns high marks for its affordable prices and wide range of tours, though it's the excellent service that makes this agency stand out from many others. Day trips to Cotapaxi, Otavalo and Mindo for those short on time.
Condor TrekkAdventure
(map Google map; %02-222-6004; <https://condortrekkexpeditions.com/>; Reina Victoria N24-281 near Rodriguez, Mariscal Sucre, New Town; h9:30am-6pm Mon-Fri, to noon Sat & Sun)
Reputable climbing operator offering single- and multiday guided climbs and tough hikes up and around most of Ecuador's peaks. It is best to pop into its office to discuss what you are looking for.
Biking DutchmanCycling
(map Google map; %02-256-8323; www.bikingdutchman.com; La Pinta E-731 near Reina Victoria, New Town; 1-day tours from $59; hoffice 9:30am-5:30pm Mon-Fri)
Ecuador's pioneer mountain-biking operator has good bikes and guides, and an outstanding reputation. It offers one- to eight-day tours. The office is just north of Mariscal Sucre.
Biking in Quito
Every Sunday, the entire length of Avenida Amazonas and most of the Old Town closes to cars from 9am to 2pm as thousands of cyclists take to the street for the weekly **ciclopaseo**. The entire ride (some 30km), which you can cycle part or all of, stretches past the old airport, through the Old Town and into the southern reaches of Quito. It's a marvelous way to experience the city. Bikes can be hired along the way for $3 an hour.
Another good place for cyclists to check out is the bicycle-loving cafe La Cleta Bici Café (map Google map; %02-223-3505; www.facebook.com/lacleta.bicicleta; Lugo N24-250 near Guipuzcoa, La Floresta, New Town; pizzas $6-14; h11am-11pm Mon-Thu, to 11:45pm Fri, 5:30-11pm Sat; W).
Cyclists enjoying the weekly ciclopaseo | MARGIE POLITZER / GETTY IMAGES ©
7Shopping
TianguezArts & Crafts
(map Google map; www.tianguez.org; Plaza San Francisco, Old Town; h9am-6pm)
Next to Tianguez cafe, this fair-trade shop sells a wide selection of quality handmade crafts from across Ecuador. Items are arranged by region, with information on the techniques used to produce them.
Galería LatinaArts & Crafts
(map Google map; www.galerialatina-quito.com; Mera N23-69 near General Banquedano, Mariscal Sucre, New Town; h10am-7pm Mon-Sat, 11am-6pm Sun)
One of the finest handicraft and clothing shops in the city, Galería Latina has a huge selection of beautifully made pieces: tagua carvings, colorful Andean weavings, textiles, jewelry, sweaters and handmade items from across Latin America. Prices are high, but so is the craftsmanship.
New Town
1Sights
1Museo NacionalA6
2Activities, Courses & Tours
2Biking DutchmanC3
3Condor TrekkB3
7Shopping
4Galería LatinaA4
5Eating
5Cosa NostraC4
6La Cleta Bici CaféD5
7ZazuC1
6Drinking & Nightlife
8Finn McCool'sB4
9IsveglioD4
3Entertainment
10La JulianaD4
5Eating
Masaya BistroBistro $
(map Google map; www.masaya-experience.com/en/hostel-quito; cnr Venezuela & Rocafuerte, Old Town; mains $3-8; h7am-midnight; W v c)
This large, inviting space with bare brick pillars and long wooden tables spills onto the peaceful enclosed garden of the affiliated Masaya Hostel. Ecuadorian-international food is served up in this casual, relaxed setting, where beef in chimichurri sauce, quinoa croquettes, Ecuadorian-style ceviche, burgers, wraps and pastas are all possibilities.
Cafetería ModeloEcuadorian $
(map Google map; www.facebook.com/cafeteriamodelocentrohistorico; Sucre 391 near García Moreno, Old Town; mains $4-6.25; h8am-7:30pm Mon-Sat, 9am-6pm Sun)
Opened in 1950, Modelo is one of the city's oldest cafes and a great spot to try traditional snacks such as _empanadas de verde_ (empanadas made with plantain dough), _quimbolitos_ (sweet, cake-like corn dumplings) and tamales. It's also a popular spot for ice cream. The fun, slightly kitsch Ecuador-of-yore trappings make the wait for the food interesting.
Nearby Cafetería Modelo II (map Google map; Sucre OE4-48; mains $4-6.25; h9am-6:30pm Mon-Thu, to 7:30pm Fri & Sat, to 5:30pm Sun) offers similar environs plus live music on Friday afternoons.
San AgustínEcuadorian $$
(map Google map; %02-228-5082; <http://heladeriasanagustin.net>; Guayaquil N5-59 near Mejia, Old Town; mains $5.50-12.50; hrestaurant 10am-3pm, ice-cream shop 10am-5pm; W)
Kitschy religious icons and old-fashioned radios decorate this old-school classic serving Ecuadorian fare to workday crowds. Opt for first-rate _seco de chivo_ (goat stew), _corvina_ (sea bass) or _arroz marinero_ (seafood rice) – there are so many options, the menu takes an age to peruse. Follow these up with old-fashioned _helados de paila:_ ice cream handmade in big copper bowls.
Cosa NostraItalian $$
(map Google map; %02-252-7145; <https://pizzeriacosanostra.ec>; cnr Baquerizo Moreno & Almagro, Mariscal Sucre, New Town; mains $11-18; h12:30-3:30pm & 6-11pm Mon-Wed, 12:30-4pm & 6-11pm Thu & Sat, 12:30-4pm & 6-11:30pm Fri, 12:30-3:30pm & 6-10pm Sun; W)
Italian-owned Cosa Nostra has a pleasant front patio, cozy dining room and nearly three-dozen varieties of pizza piled with generous toppings and fired up in a brick oven – we consider it the best in town. It has good gnocchi and other pastas, and tiramisu for dessert.
ZazuFusion $$$
(map Google map; %02-254-3559; www.zazuquito.com; Aguilera 331, La Pradera, New Town; mains $13-25, tasting menu $70; h12:30-3:30pm & 6:30-11pm Mon-Fri, 6:30-11pm Sat)
One of Quito's best restaurants, Zazu serves beautifully prepared seafood dishes, grilled meats and ceviches in a stylish setting of light brick, ambient electronica and an inviting, backlit bar. The menu showcases the best of Ecuadorian cuisine with dishes such as Andean grains and veg, confit guinea pig empanada and seafood tamale.
6Drinking & Nightlife
IsveglioCafe
(map Google map; www.isveglio.com; Isabel La Católica N24-682 near La Coruña, La Floresta, New Town; coffee & snacks $2-6; h9am-8:30pm Mon-Fri, 11am-8:30pm Sat, 2-8pm Sun) S
The standard of artisan coffee has been on an upward trajectory in Quito and Isveglio is at the forefront of the movement. This cafe, coffee education center and barista training school serves up organic brews from Ecuador's various coffee-growing regions, plus some beans from further afield.
Bandido BrewingMicrobrewery
(map Google map; <http://bandidobrewing.com>; Olmedo E1-136 near Cevallos, San Blas, Old Town; h4-11pm Mon-Fri, 2-11pm Sat)
These guys from Oregon produce their own creative brews such as La Gran Calabaza Imperial Pumpkin Ale (made with cinnamon, cloves and pumpkin spice) and La Gua.Pa (an American Pale Ale flavored with _guayusa,_ a tea leaf native to the Ecuadorian Amazon). The bar has a vaguely Gothic feel, with stone walls, wooden benches and even a small chapel.
Finn McCool'sIrish Pub
(map Google map; www.facebook.com/FinnMcCools Quito; Almagro E24-64 near Pinto, Mariscal Sucre, New Town; h1pm-1am Mon-Wed, to 3am Thu, 11am-3am Fri & Sat, 11am-7pm Sun)
Proudly flying the green, white and gold, this Irish-owned bar attracts a mix of locals and foreigners who come for games of pool, table football, pub grub and theme nights (pub quiz on Tuesdays, regular live bands that often play Celtic music, open-mic nights, game nights whenever there's soccer on).
3Entertainment
La JulianaLive Music
(map Google map; www.lajuliana.com.ec; Av 12 de Octubre N24-722 near Coruña, La Floresta, New Town; admission incl drink $25; h9pm-2:30am Fri & Sat)
In an old converted house, La Juliana is a colorfully decorated space with a good mix of bands (rock, salsa, merengue) lighting up the dance floor most weekend nights.
Vista Hermosa
A much-loved spot in El Centro, Vista Hermosa (map Google map; %02-295-1401; <http://vistahermosa.ec>; Mejía 453, 5th fl, Old Town; mains $13-28; h1pm-midnight Mon-Sat, noon-8pm Sun) lives up to its name ('beautiful view') with a magnificent 360-degree panorama over the Old Town from its open rooftop terrace. Ecuadorian specials include _seco de chivo_ (goat stew). Live music on Friday and Saturday from 10pm adds to the magic. Arrive early to beat the crowds.
There is another Vista Hermosa (map; <https://vistahermosa.ec/>; Samaniego s/n, Parque Itchimbia, Old Town; mains $13-28; h3pm-midnight Mon-Sat) up by Parque Itchimbia serving up similar food and impressive views (truth be told, the views are still loftier here).
Quito's Old Town | F11PHOTO / HUTTERSTOCK ©
8INFORMATION
**Quito Tourism** (Corporación Metropolitana de Turismo; %02-257-2445; www.quito-turismo.gob.ec; Venezuela near Espejo, Plaza Grande, Old Town; h9am-6pm Mon-Sat, 9:30-4:30pm Sun; W).
**Tourist Information** ( %02-281-8363; aeropuerto@quito-tourismo.gob.ec; Aeropuerto Internacional Mariscal Sucre) At the airport.
**Tourist Information** ( %02-382-4815; quitumbre@quito-turismo.gob.ec; Terminal Terrestre Quitumbre) At Terminal Terrestre Quitumbre.
**Tourist Information** (map; cnr Reina Victoria & Foch; h9am-6pm Mon-Sat) Small booth on Plaza Foch in Mariscal Sucre.
8Getting There & Away
AIR
Quito's Aeropuerto Internacional Mariscal Sucre is 37km northeast of the city in a broad valley near Tababela.
BUS
Quito has two main bus terminals that are a long way from the center; allow at least an hour by public transport, 30 minutes or more by taxi.
**Terminal Terrestre Quitumbe** (cnr Cóndor Ñan & Sucre), 10km southwest of the Old Town, has buses to Baños and other destinations in the Central and Southern Andes, the coast, and the Oriente.
**Terminal Terrestre Carcelén** (Av Eloy Alfaro s/n), located in the north, services northern destinations including Otavalo, Ibarra, Mindo and Tulcán.
## Baños
Baños is the central highlands' premiere destination for outdoor activities. Not surprisingly, it's often packed with out-of-towners, with abundant guesthouses and diverse restaurants to serve the international crowd.
1Sights & Activities
Termas de La VirgenThermal Baths
(Montalvo; adult/child $2/1, after 6pm $3/1.50; h5am-4pm & 6-9:30pm)
These are the only hot pools in the town proper. This community project built in 1928 was named for the Virgin Mary, who is said to have come here to dip her own feet. Some baths are cold, others warm and one reaches an intense 42°C (118°F). The rooftop pools have lovely views of a nearby waterfall. There are two locations; the 'old' pools at the top of the street, and the newer building below, which is open Wednesday to Sunday ($6/3).
Termas El SaladoThermal Baths
(Salado; adult/child $3/1.50; h5am-8pm)
These wonderful hot springs are located in a verdant canyon, 2.5km from town. There are hot, medium and cool pools surrounded by tree-covered hills, with the soothing sounds of a fast-flowing river close by. You'll need a swimming cap. Buses to here ($0.25, 10 minutes) depart from the stop on Rocafuerte.
7Shopping
Maki AwashkaArts & Crafts
(Montalvo, near Santa Clara; h8am-7pm)
The family from Salasaca who own this shop weave many of the gorgeous wool rugs and tapestries in their workshop upstairs. There's a good selection of artisan crafts for sale here, including tablecloths, scarves and blankets.
5Eating
Cafe HoodInternational $
( %03-274-1609; www.cafehoodecuador.com; Montalvo & Rafael Viera; mains $4-8; hnoon-10pm; W v)
Named for owner Ray Hood, a long-standing gringo-in-residence, this excellent cafe has cheap _almuerzos_ (set lunches) and a menu of Asian and South American dishes. The cafe is a welcoming place to meet with friends or just chill _solito_ (alone). If you don't feel like leaving there's a comfy _hostal_ in the back with nice views.
La Tasca de BañosSpanish $$
(12 de Noviembre, near Montalvo; tapas $3.50-6; h6:30-10:30pm Wed-Fri, 12:30-4:30pm & 6:30-10:30pm Sat & Sun)
It may be difficult to get a table at this tiny tapas restaurant. The selection of small dishes is excellent, ranging from perennial favorites like _tortilla española_ (Spanish omelet) to Andalusian meatballs and shellfish. Order up five to share.
6Drinking & Nightlife
MocamboBar
( %03-274-1329; Alfaro near Ambato; h8pm-midnight)
Tired of the bus _bachata_? Guitar-hungry gringos need look no further than this joint, where the owner's laptop always has a classic playlist cued up, from Guns N' Roses to Metallica, while revisiting some classics from further back. And he takes requests. You'll likely _hear_ Mocambo before you see it. Best bar sign in Ecuador: 'No wi-fi. Talk amongst yourselves.'
Stray DogBrewpub
(www.facebook.com/straydoginbanos; cnr Rocafuerte & Maldonado; h4pm-midnight Mon-Thu, to 2am Fri & Sat)
This brewpub features artisanal offerings like light Llamas' Breath Belgian and bold Stray Dog Stout. There's a small bar menu to balance out the bevvies.
Andes by Rail
After massive investment, the country's train network ( %1-800-873-367; <http://trenecuador.com>; cnr Guayllabamba & Sincholagua, Estación de Ferrocarril Chimbacalle, Chimbacalle) is once again ferrying passengers on slow-motion journeys through breathtaking high-altitude scenery. Routes include the Tren de los Volcanes ( %1-800-873-637; www.trenecuador.com; adult/child $53/37; h8am-5:30pm Fri-Sun), which makes the round-trip to El Boliche in Cotopaxi National Park, and the luxurious Tren Crucero (Train of Wonders; %1-800-873-637; www.trenecuador.com; adult/child $1650/1485; hTue-Fri), a four-day, four-night luxury train tour from Quito to Guayaquil.
Trains depart from Quito's beautifully renovated Estación de Ferrocarril Chimbacalle (Chimbacalle Train Station; cnr Guayllabamba & Sincholagua) 2km south of the Old Town. Book tickets online or at the booking desk at Quito Tourism.
8GETTING THERE & AWAY
From Quito's Terminal Terrestre Quitumbe, Transportes Baños offers frequent buses direct to Quito ($4.45, 3½ hours).
# Galápagos Islands, Ecuador
#### Visiting the Galápagos
#### Puerto Ayora
#### Sights
#### Eating
#### Drinking
#
Galápagos Islands, Ecuador
_The Galápagos Islands may just inspire you to think differently about the world. The creatures that call the islands home, many found nowhere else in the world, act as if humans are nothing more than slightly annoying paparazzi._
_This isolated group of volcanic islands and its fragile ecosystem has taken on almost mythological status as a showcase of biodiversity. Yet you don't have to be an evolutionary biologist or an ornithologist to appreciate one of the few places left on the planet where the human footprint is kept to a minimum._
Galápagos giant tortoise KONSTIK / GETTY IMAGES ©
Two Days in the Galápagos
Starting in Isla San Cristóbal, spy marine iguanas and sea lions at La Lobería. Visit the intriguing Interpretation Center, then stroll up to Cerro de las Tijeretas.
Take a boat to Isla Santa Cruz, and eye giant tortoises at El Chato Tortoise Reserve, followed by a peak through Lava Tunnels. End the day at lovely Tortuga Bay.
Four Days in the Galápagos
On day three take a boat to Isla Isabela and go snorkeling at Los Túneles, then bike out to Muro de las Lágrimas.
Rise early for a visit to Volcán Sierra Negra. Look for abundant birdlife and admire the views from the fumarole-dotted Volcán Chico.
Arriving in the Galápagos
Flights from the mainland arrive at two airports: Isla Baltra just north of Santa Cruz, and Isla San Cristóbal. There are almost an equal number of flights to Baltra and San Cristóbal.
Most flights come via Guayaquil, though there are a few direct options from Quito.
Where to Stay
Whether you're tacking a night on to either end of a cruise or looking for a land-based way to enjoy the Galápagos, you'll find perfect places to sleep on the inhabited islands of Santa Cruz, San Cristóbal, Isabela and Floreana. Options range from bustling, budget-friendly hostels and guesthouses to five-star luxury lodges.
TOP EXPERIENCE
# Visiting the Galápagos
There are three kinds of tours in the Galápagos: the most common and recommended are boat-based trips with nights spent aboard (for their relatively low environmental impact and the exposure to a variety of wildlife and geography). There are also day trips returning to the same island each night and hotel-based trips staying on different islands.
Great For...
fgA
yDon't Miss
The sight of massive tortoises lumbering through the misty highlands at El Chato Tortoise Reserve.
8Need to Know
The 13 major islands of the Galápagos Archipelago straddle the equator about 1000km west of Ecuador's mainland.
5Take a Break
Hit the seafood kiosks along Charles Binford in Puerto Ayora for a satisfying al fresco meal.
oTop Tip
Bring an underwater camera forphotographing/filming sea lions, sea turtles and other aquatic life.
### Isla Santa Cruz
The island of Santa Cruz has the largest and most developed town in the Galápagos. Though many people are just passing through (from airport to cruise ship), the island is a destination in itself, full of visitor sites. It has easily accessible beaches and remote highlands in the interior, making it a base for adventurous activities far from the tourism trail.
Sea lions, Isla Española | GUENTERGUNI / GETTY IMAGES ©
#### El Chato Tortoise Reserve
South of Santa Rosa is **El Chato Tortoise Reserve** (map), where you can observe giant tortoises in the wild. When these virtually catatonic, prehistoric-looking beasts extend their accordionlike necks to feed, it's an impressive sight. The reserve is also a good place to look for short-eared owls, Darwin's finches, yellow warblers, Galápagos rails and paint-billed crakes (these last two are difficult to see in the long grass). The reserve is part of the national park and a guide is required.
#### Rancho Primicias
Next to El Chato is this private ranch, where there are dozens of giant tortoises, and you can wander around at will. The entrance is beyond Santa Rosa, off the main road – ask locals for directions. Remember to close any gates that you go through. There is a cafe selling cold drinks and hot tea, which is welcome if the highland mist has soaked you.
#### Lava Tunnels
These impressive underground **tunnels** (map) southwest of the village of Santa Rosa are more than 1km in length and were formed when the outside skin of a molten-lava flow solidified. When the lava flow ceased, the molten lava inside the flow kept going, emptying out of the solidified skin and thus leaving tunnels. Because they are on private property, the tunnels can be visited without an official guide. The tunnels have electrical lighting (you can also hire flashlights/torches).
Tours to the lava tunnels are offered in Puerto Ayora.
#### Cerro Crocker
A path north from Bellavista leads toward Cerro Crocker (864m) and other hills and extinct volcanoes. This is a good chanceto see the vegetation of the scalesia, miconia and fern-sedge zones and to look for birds such as the vermilion flycatcher, the elusive Galápagos rail and the paint-billed crake. It's around 5km from Bellavista to the crescent-shaped hill of Media Luna, and 3km further to the base of Cerro Crocker. This is a national park, so a guide is required.
#### El Garrapatero Beach
A 30-minute taxi ride from Puerto Ayora through the highlands, plus a 15-minute walk, brings you to this beautiful beach. It has tidal pools that are good for exploring and snorkeling on calm days, and a lagoon with flamingos, white-cheeked ducks and black-necked stilts.
### Isla San Cristóbal
San Cristóbal is the only island with fresh water and an airport in town, and it has several easily accessible visitor sites. It's also the fifth-largest island and home to the second-largest population in the archipelago. The Chatham mockingbird, common throughout the island, is found nowhere else.
#### Cerro Brujo
Possibly one of the nicest beaches in the Galápagos, Cerro Brujo is a huge white expanse found on the west side of the island. The sand here feels like powdered sugar. A colony of sea lions and blue-footed boobies call Cerro Brujo home, and behind the beach is a lagoon where you'll find great egrets and great blue herons. There's also good snorkeling in the turquoise waters.
#### El Junco Lagoon
Around 10km east of El Progreso along the main road, you'll find El Junco Lagoon – a freshwater lake some 700m above sea level. It's one of the few permanent fresh water bodies in the Galápagos. Here you can see frigate birds shower in the freshwater to remove salt from their feathers; white-cheeked pintails and common gallinules; and the typical highland miconia vegetation and endemic tree ferns. The weather is often misty or rainy.
#### La Lobería
Southwest of the town of Puerto Baquerizo Moreno, a road leads 2.5km (about a 30-minute walk) to La Lobería, a rocky beach with a lazy sea-lion colony. It's good for year-round surfing, and there are lots of iguanas along the trail leading past the beach. Bring water and sun protection. Taxis charge about $3 to take you out here and you can walk back. Once there, a cliffside path takes you past marine iguanas, lava lizards and soaring frigate birds. To get here on foot, take Av Alsacio Northia toward the airport, turn left after the stadium (look for the murals) and then take the first right.
ySurfing
There's excellent surfing in the Galápagos, especially in Isla San Cristóbal near Puerto Baquerizo Moreno (with board hire in town).
8Wall of Tears
On Isla Isabela, you can visit the _Muro de las Lágrimas_ (map; Wall of Tears), a 100m-long wall of lava rocks built by convicts when the island was a penal colony.
#### Interpretation Center
This modern and easily digestible center ( %05-252-0021; <https://galapagos.unc.edu/gsc>; Av Alsacio Northia; h8:30am-5:30pm Mon-Sat) F explains the history and significance of the Galápagos better than anywhere else in the country. Exhibits deal with the biology, geology and human history of the islands – it deserves a visit even if you've been inundated with facts from boat guides. From the center there are also many well-marked trails that wind around the cacti and scrub of Cerro de las Tijeretas (Frigate-Bird Hill).
#### Punta Pitt
The northeasternmost point of the island is Punta Pitt, whose volcanic tuff formations are of interest to geologists (and attractive in their own right), but the unique feature of the site is that it's the only one where you can see all of the Galápagos booby species nesting. The walk is a little strenuous, but rewarding.
### Isla Isabela
At 4588 sq km, Isabela is the largest island in the archipelago. Despite its size and imposing skyline of active volcanoes, it's the delicate sights, like frigates flying as high as the clouds or penguins making their way tentatively along the cliffs, that reward visitors.
It's a relatively young island that consists of a chain of five intermittently active volcanoes, including Volcán Sierra Negra, which erupted in late 2005 and sent up a 20km-high smoke column.
Isla Isabela | UWE-BERGWITZ / GETTY IMAGES ©
#### Volcán Sierra Negra
Northwest of the tiny settlement of Tomás de Berlanga lies the massive **Volcán Sierra Negra** (map; 1490m), which last erupted in late 2005. An 8km trail leads around the east side of the volcano. It's possible to walk all the way around the caldera, but the trail peters out.
Galápagos hawks, short-eared owls, finches and flycatchers are among the birds commonly seen on this trip. The summit is often foggy and it is easy to get lost. There are spectacular views from nearby Volcán Chico, a subcrater where you can see fumaroles. Several agencies in Puerto Villamil offer all-day tours.
#### Darwin Lake
A dry landing deposits you at the beginning of a 2km-long trail that brings you past this postcard-perfect saltwater lagoon. It has twice the salinity of the ocean, and is a tuff cone, like a chimney from the main volcano. The trail leads to the lower lava slopes of **Volcán Darwin** (1280m), where various volcanic formations and stunning views of surrounding slopes can be observed. There are some steep sections on this trail. A panga ride along the cliffs to Tagus Cove will enable you to see the historical graffiti and various seabirds, usually including Galápagos penguins and flightless cormorants. There are snorkeling opportunities in the cove.
#### Los Túneles
Around a 30- to 40-minute boat ride from Puerto Villamil is this outstanding spot for snorkeling, formed by convoluted lava formations standing between mangroves and the open sea. Look out for white-tipped sharks, manta rays, eagle rays, sea lions, turtles and even sea horses in the shallows. Tour operators in Puerto Villamil run daily five-hour trips here for around $90.
### Isla Floreana
Floreana is known for the mysterious history of its first residents – a small contingent of European settlers who became entangled in power struggles, peculiar disappearances and alleged murders. It's also home to intensely pink flamingos and top-flight snorkeling sites.
#### Devil's Crown
This ragged semicircle of rocks, poking up out of the ocean a few hundred meters from Punta Cormorant, is one of the most outstanding marine sites in all of the Galápagos. A strong current sweeps snorkelers briskly past thousands of bright tropical fish, a small coral formation, sea lions, marine turtles and the occasional shark. A _panga_ ride around the semi-submerged volcanic cone will give views of red-billed tropic birds, pelicans, herons and lava gulls nesting on the rocks.
Diver viewing a Galápagos green turtle | MICHELE WESTMORLAND / GETTY IMAGES ©
#### Post Office Bay
Most groups spend several perfunctory minutes on the north coast at Post Office Bay, where scraps of wood covered in graffiti surround a a few gone-to-seed barrels. It was a functioning mailbox for American and British whalers from the late 18th century, but these days it's tourists who leave postcards, hoping they will find their way, like a message in a bottle. Actually, it's more prosaic than that: visitors are asked to grab a few to post when they return to their home countries.
### Isla Fernandina
The one visitor site at Punta Espinoza, just across from Tagus Cove on Isabela, is a memorable one. Marine iguanas, too many to count, can be seen sunning themselves on the black-lava formations, a dramatic sight that looks like a museum diorama on dinosaurs come to life. Flightless cormorants nest nearby, hawks soar overhead and Galápagos penguins, turtles and sea lions sometimes frolic in an admirable display of multispecies tolerance in the lagoon near the landing.
### Isla Santiago
Once a hideout for British buccaneers and one of the stops on Darwin's itinerary, Isla Santiago is the fourth largest of the islands. Its terrain of rough lava fields is an example of the island's challenging beauty.
8 Evolution in Action
A fascinating account of the sometimes rapid changes unfolding on the islands is _The Beak of the Finch: A Story of Evolution in Our Time_ by Jonathan Weiner.
y Biking the Galápagos
In the main towns on Isla Santa Cruz and Isla San Cristóbal you can hire bikes for a ride (uphill) to the misty interior.
## Puerto Ayora
This town, the largest in terms of population and size in the Galápagos, is a surprise to most visitors, who don't expect to find anything but plants and animals on the islands. Puerto Ayora looks and feels like a fairly prosperous mainland Ecuadorian coastal town, despite the sea lions and pelicans hanging around the waterfront.
1Sights
MAPRAEMuseum
(Museo de Arte Precolombino de Realidad Aumentada; %05-252-5197; www.maprae.com; Av Darwin & Av Binford; $5-10, includes tablet with headphones; h10am-10pm)
The first of its kind in the world, this museum uses augmented reality to showcase a permanent exhibition of 55 pre-Columbian artifacts. The ancient cultures of Ecuador's Amazon and coastal regions are brought to life as visitors point smart phones or tablets at one of the relics, with historical information and three-dimensional images appearing directly on the devices.
Charles Darwin Research StationWildlife Reserve
( %05-252-6146; www.darwinfoundation.org; Av Darwin; h7:30am-12:30pm & 2-5:30pm) F
Just northeast of Puerto Ayora is this iconic national-park site, where over 200 scientists and volunteers are involved with research and conservation efforts, the most well known of which involves a captive breeding program for giant tortoises. Paths leading through arid-zone vegetation take you past tortoise enclosures, where you can look at these Galápagos giants. There's also a **baby-tortoise house** with incubators (when the tortoises weigh about 1.5kg or are about four years old, they're repatriated to their home islands).
Several of the 11 remaining subspecies of tortoise can be seen here. Other attractions include a small enclosure containing several land iguanas, with explanations in Spanish and English concerning efforts to restore their populations on islands where they've been pushed to the brink of extinction. Follow paths through arid-zone vegetation such as saltbush, mangroves and prickly pear, and see a variety of land birds, including Darwin's finches. The research station is supported by contributions to the Galápagos Conservancy (www.galapagos.org).
Tortuga BayBeach
In terms of sheer white-sand beauty, this beach is the rival of any in South America. You'll find it at the end of a 2.5km paved trail southwest of Puerto Ayora. In addition to swimming (a spit of land provides protection from the strong and dangerous currents on the exposed side), surfing or just sunbathing, you can see sharks, marine iguanas, pelicans and the occasional flamingo. There's no drinking water or other facilities.
It's about a half-hour walk from the start of the path – often used by local runners – where you must sign in between 6am and 6pm.
Playa MansaLagoon
If you walk the length of Tortuga Bay, you'll reach this picturesque lagoon lined with mangroves. Here you can spot marine iguanas, brown pelicans and blue herons, among other species. On the nearby dunes, sea turtles lay their eggs. The placid, shallow water is a great swimming spot for kids. Kayaks (per person per hr $10; h9am-6pm) are available for hire.
Laguna de las NinfasLagoon
( F)
This peaceful lagoon has a short boardwalk path, where you can stop to take in the mangroves while looking for stingrays, baby sharks, sea turtles and other creatures sometimes spotted here.
5Eating
More than a half-dozen popular food kiosks sell inexpensive and hearty meals – mainly seafood – along Charles Binford, just east of Avenida Baltra. It's liveliest at night, particularly on weekends, when there's a festive atmosphere among the outdoor tables set out on the street.
Galápagos DeliDeli, Pizzeria $$
(Berlanga; mains $4.50-10; h7am-9:45pm Tue-Sun; W)
Tired of standard _almuerzos_ (set lunches)? Head to this sleek and modern place for brick-oven pizza (small $6.50 to $9.75) and high-quality deli sandwiches ($4.70 to $8.75), as well as fish and chips, espresso and delicious gelato. Because it's on a block with few pedestrians, it feels like a secret.
AlmarJapanese $$$
(Av Darwin, Red Mangrove Aventura Lodge; mains from $16-24; h8am-10pm; W)
Located inside Red Mangrove Aventura Lodge, Almar prepares its seafood dishes with ingredients sourced from local suppliers. The big draw is sitting on the back waterside deck where sea lions have free rein.
GarrapataEcuadorian $$$
(Av Darwin; mains $15-30; h11am-10:30pm Mon-Sat)
Good tunes, cool breezes and tasty Ecuadorian and international dishes (seared tuna, grilled seafood platters and fish in coconut sauce) draw a lively crowd most nights. It's pricey but casual, with open sides and a pebble floor.
6Drinking
Bongo BarBar
(Av Darwin; h7pm-2am; W)
What nightlife there is in Puerto Ayora centers mainly on this bar, a trendy 2nd-floor spot replete with flat-screen TVs, music and a lubricated mix of hip locals, guides and tourists. Surprisingly, you'll also find some of the best sushi on the islands here (rolls $10 to $14).
Las Grietas
For nice swimming and snorkeling, head to Las Grietas, a water-filled crevice in the rocks. Talented and fearless locals climb the nearly vertical walls to plunge gracefully (and sometimes clumsily) into the water below. Take a water taxi (per person around US$1, from 6am to 7pm) to the dock for the Angermeyer Point restaurant, then walk past the Finch Bay Hotel, then past an interesting salt mine, and finally up and around a lava-rock-strewn path to the water. Good shoes are recommended for the 700m walk from the dock. Keep an eye on any valuables that you leave on the rocks.
Las Grietas, Puerto Ayora | MAREK POPLAWSKI / SHUTTERSTOCK ©
8INFORMATION
**i-Tur** (Av Darwin, entrance to water-taxi pier) A small kiosk with flyers, maps and basic hotel and travel-agency info.
8GETTING THERE & AWAY
Three airlines fly to the Isla Baltra, a small island practically touching the far northern edge of Isla Santa Cruz: **Avianca** ( %1-800-003-434; www.avianca.com; Av Francisco de Orellana), **LAN** ( %05-269-2850; Av Darwin; h8am-6pm Mon-Sat, 10am-1pm Sun) and **TAME** ( %05-252-6527; cnr Av Darwin & Av 12 de Febrero; h8am-noon & 2-5pm Mon-Fri, 8am-1pm Sat).
8GETTING AROUND
_Lanchas_ (speedboats) head daily to the islands of Isabela (two to 2¼ hours) and San Cristóbal (two hours). Boats depart to Isabela/San Cristóbal at 7am and 2pm. There's also boat service to Floreana (1¾ hours) at 8am, but this service is irregular, often running on demand only.
# Cartagena, Colombia
#### Cartagena's Old Town
#### Islas del Rosario
#### Sights
#### Activities & Tours
#### Shopping
#### Eating
#### Drinking & Nightlife
#### Entertainment
#
Cartagena, Colombia
_Cartagena de Indias is a historic city of superbly preserved beauty and the undisputed queen of the Caribbean coast. Cartagena's Old Town – a maze of cobbled alleys, balconies covered in bougainvillea and massive churches that cast their shadows across leafy plazas – is a Unesco World Heritage Site._
_This is a place to drop all sightseeing routines. Instead of trying to tick off all the sights, just stroll through the Old Town day and night. Soak up the sensual atmosphere, pausing to ward off the brutal heat and humidity in one of the city's many excellent bars and restaurants._
Cartagena balconies and Catedral JESS KRAFT / 500PX ©
Two Days in Cartagena
Delve into Cartagena's Old Town, visiting the photogenic Plaza de Bolívar, the chilling Palacio de la Inquisición and the glittering Museo del Oro Zenú. Dine at El Boliche.
On day two, explore the atmospheric Castillo de San Felipe de Barajas and enjoy a memorable view from the Convento de la Popa. Afterwards, have dinner and drinks in Getsemaní.
Four Days in Cartagena
On day three, take a trip out to the Islas del Rosario for a Caribbean escape. Overnight on Isla Grande and enjoy the beach without the daytrippers.
After a morning on the sands, make your way back to Cartagena for a final night on the town. Enjoy dinner at La Cevichería followed by dancing and drinks at Bazurto Social Club.
Arriving in Cartagena
Cartagena's Aeropuerto Internacional Rafael Núñez is just 3km from the city. _Colectivos_ (COP$2000), as well as nicer air-conditioned shuttles called Metrocar, go to the Old Town.
By taxi, expect to pay between COP$10,000 and COP$15,000 to Getsemaní, San Diego and El Centro.
Where to Stay
Cartagena has a huge choice of places to sleep, though you'll pay a pretty penny for anything above a hostel or a very simple midrange hotel. Catering to wealthy Colombian and US weekenders, the town's top-end accommodations have truly stratospheric rates, and there's an enormous number of beautifully restored boutique colonial options to choose from.
TOP EXPERIENCE
# Cartagena's Old Town
The Old Town is packed with perfectly preserved colonial churches, monasteries, plazas, palaces and mansions, with balconies and shady patios that overflow with bright flowers.
Great For...
vhA
yDon't Miss
Pre-Columbian pottery and Spanish colonial torture instruments at the Palacio de la Inquisición.
8Need to Know
Cartagena's old city consists of the historical districts of El Centro in the west and San Diego in the northeast.
5Take a Break
Munch on creative, mouth-watering ceviche at La Cevichería.
oTop Tip
A beautiful walkway alongside the Muelle Turístico de los Pegasos (map Google map) links the neighborhood of Getsemaní with the Old Town.
### Plaza de Bolívar
Formerly the Plaza de Inquisición, this leafy and shaded plaza (map Google map) is surrounded by some of the city's most elegant balconied colonial buildings. It's one of Cartagena's most alluring plazas and offers wonderful respite from the Caribbean heat. A statue of the eponymous Simón Bolívar stands in the middle of the square.
Entrance to the Catedral | HANOHIKI / SHUTTERSTOCK ©
### Catedral
Work on Cartagena's cathedral (map Google map; Calle de los Santos de Piedra) began in 1575, but in 1586, while still under construction, it was partly destroyed by the cannons of Francis Drake. The structure known officially as the Basilica Santa Catalina de Alejandría wasn't completed until 1612, although the distinctive terracotta dome visible all over town was added later. Further alterations were carried out in the early 20th century.
### Museo del Oro Zenú
This museum (map Google map; Plaza de Bolívar; h9am-5pm Tue-Sat, 10am-3pm Sun) F is like a miniature version of Bogotá's world-class gold museum, the Museo del Oro. Though small, it offers a fascinating collection of the gold and pottery of the Zenú people, who inhabited the region of the present-day departments of Bolívar, Córdoba, Sucre and northern Antioquia before the Spanish Conquest. Some pieces are exquisitely detailed.
### Convento & Iglesia de San Pedro Claver
Founded by Jesuits in the first half of the 17th century as Convento San Ignacio de Loyola, this convent (map Google map; %5-664-4991; Plaza de San Pedro Claver; adult/child COP$13,000/8000; h8am-5:30pm) later changed its name to honor Spanish-born monk Pedro Claver (1580–1654), who lived and died here. Called the 'Apostle of the Blacks' or the 'Slave of the Slaves,' the monk spent his life ministering to enslaved people brought from Africa. A series of lucid paintings inside the building relates his life story.
### Palacio de la Inquisición
The Palace of the Inquisition (map Google map; Plaza de Bolívar; adult/child COP$20,000/17,000; h9am-6pm Mon-Sat, 10am-4pm Sun) may today be one of the finest buildings in the city, but in the past it housed the notoriously grisly Inquisition, whose bloody task it was to stamp out heresy in colonial Cartagena. The palace is now a museum, displaying the inquisitors' instruments of torture, some of which are quite horrific. The museum also houses pre-Columbian pottery and plots a historical trajectory of the city using armaments, paintings, furniture and even church bells.
TOP EXPERIENCE
# Islas del Rosario
An archipelago about 35km southwest of Cartagena, these 27 islands are surrounded by coral reefs, where the color of the sea is an incredible combination of cerulean and turquoise.
Great For...
gAn
yDon't Miss
Basking on the pretty sands of Playa Blanca on Isla de Barú.
8Need to Know
Most people visit the islands on organized tours from Cartagena, but you can also arrange private transfers with the hotels on the island.
5Take a Break
Beach shacks serve simple but filling dishes, including fresh seafood.
oTop Tip
Get an early start to make a full day out on the islands, or better yet spend the night.
### Isla Grande
The most developed of the islands, Isla Grande is indeed also the biggest. There's a large lagoon perfect for swimming in the unbelievably blue waters, and several sleeping options.
LUCIANO LEJTMAN / ALAMY STOCK PHOTO ©
### Isla del Rosario
The smaller neighbor to Isla Grande has several stretches of gorgeous beach, a huge lagoon perfect for swimming, and a handful of hotels and guesthouses.
### Tours
The usual way to visit the park is on a one-day boat tour of the islands. Tours depart year-round from the Muelle Turístico de la Bodeguita (map Google map) in Cartagena; contact the cruise office at the _muelle_ (pier; tours from COP$60,000 per person). Boats leave between 8am and 9am daily and return roughly between 4pm and 6pm.
A number of smaller operators at the pier offer cheaper tours. Popular budget hotels in Cartagena sell tours, too, and may offer lower prices – COP$50,000 per person is common.
Tours all take a similar route to the islands, though the trip may differ a little between small and large boats. All vessels go through the Bahía de Cartagena and into the open sea through the Bocachica strait, passing between Batería de San José and, directly opposite, the Fuerte de San Fernando. Boats then cruise among the islands (there's generally Spanish commentary along the way) and stop at the tiny Isla de San Martín de Pajarales. Here there's an aquarium and a shady wooded area to chill out in or a beach to swim at while you wait for the trip to continue. Boats then go to Playa Blanca, on the Isla de Barú, for lunch and two hours or so of free time.
Tours usually include lunch but not port taxes, the national-park entrance fee and aquarium entry; check with your operator to confirm. Some higher-end hotels offer their own boat transfers from Cartagena.
Playa Blanca, Isla de Barú | STREETFLASH / SHUTTERSTOCK ©
### Overnighting on the Islands
It's worth spending more than a day on the Islas del Rosario if you're looking for true escape, peace and quiet, and are happy with slow or nonexistent wi-fi. Most hotels are on Isla Grande and tend to be very rustic no matter how much you pay – and at the top end that won't be an insignificant amount. A few standout options include Coralina Island ( %313-245-9244; www.coralinaisland.com; Isla del Rosario; r from COP$680,000; W) and Eco Hotel Las Palmeras ( %314-584-7358; Isla Grande; hammock/r per person incl full board COP$80,000/120,000) S.
Cartagena Old Town
1Sights
1CatedralA4
2Convento & Iglesia de San Pedro ClaverA5
3GetsemaníD6
4Las BóvedasD1
5Las MurallasB2
6Muelle Turístico de la BodeguitaA6
7Muelle Turístico de los PegasosB5
8Museo del Oro ZenúA5
9Palacio de la InquisiciónA5
10Plaza de BolívarA5
11Plaza de los CochesB5
2Activities, Courses & Tours
12BackpackersC3
13Diving PlanetA3
14SicoC3
7Shopping
15ÁbacoA4
5Eating
16Beer & LaundryD5
17BeiyuD5
18Caffé LunáticoD5
19El BolicheB3
20El SantísimoC2
21Espíritu SantoB4
22InternoC2
23La CevicheríaB3
6Drinking & Nightlife
24AlquímicoB4
25Bazurto Social ClubC5
26Café HavanaD5
27Donde FidelB5
1Sights
GetsemaníArea
(map Google map)
Getsemaní, the outer walled town, is less obviously impressive than Cartagena's old city but has some charming parts and is well worth exploring. In recent years it has been the focus of Cartagena's exploding gentrification and growing hotel industry, and boasts excellent bars, restaurants and cafes, as well as some superb street art on its derelict buildings.
The plaza in front of the Iglesia de la Santísima Trinidad is a massive draw for locals and travelers alike come sundown, when street-food vendors set up their stalls and a crowd of revelers sticks around until well after midnight.
Convento de la PopaChurch
(map; adult/child COP$11,000/8000; h8am-6pm)
On a 150m-high hill, the highest point in Cartagena, stands this convent. The views from here are outstanding and stretch all over the city. The convent's name literally means the 'Convent of the Stern,' after the hill's similarity to a ship's back end. Founded by Augustine fathers in 1607, it was initially just a small wooden chapel, but when the hill was fortified two centuries later it was replaced by a stouter construction.
Las MurallasFortress
(map Google map)
The Old Town is surrounded by Las Murallas, the thick walls built to protect it. Construction was begun toward the end of the 16th century after the attack by Francis Drake; until that time, Cartagena was almost completely unprotected. The project took two centuries to complete, due to repeated storm damage and pirate attacks. It's possible to walk along large stretches of these magnificent structures, with wonderful sea views and a cooling breeze from the the northern and western sides.
Vibrant Getsemaní | GARYTOG / GETTY IMAGES ©
2Activities & Tours
Cartagena has grown into an important scuba-diving center, taking advantage of the extensive coral reefs along its coast. La Boquilla, just outside town, is also popular for kitesurfing.
SicoCycling
(map Google map; %300-339-1728; www.sicobikerental.com; Calle Puntales 37-09; h9am-10pm)
This friendly outfit in the middle of the Old Town offers two-hour guided bicycle tours of the city and its surroundings. It also rents out good-quality hybrids and mountain bikes (COP$24,000 per half-day). Multilingual two-hour city tours depart at 8am and 4:30pm daily.
Diving PlanetDiving
(map Google map; %310-657-4926, 320-230-1515; www.divingplanet.org; Calle Estanco del Aguardiente No 5-09; h8am-6pm Mon-Sat)
This five-star PADI diving school offers two-tank dives in the Islas del Rosario, including transportation, equipment, lunch and instructors, for COP$408,000. Discounts of 10% are available if you pay in cash.
BackpackersTours
(map Google map; %5-664-8594, 300-504-9929; www.backpackersctg.com; cnr Calle de los 7 Infantes & Calle del Pilar; day tours from COP$70,000; h8am-6pm)
Highly recommended English-speaking agency offering a raft of day trips and tours aimed at budget travelers. These include daily departures to Playa Blanca, Volcán de Lodo El Totumo and Islas del Rosario, for which prices are some of the lowest in town. There are also less-standard tours available, such as trips to the Aviario Nacional de Colombia.
7Shopping
Cartagena has a wide range of shops selling crafts and souvenirs, and the quality of the goods is usually high. The biggest tourist shopping center in the walled city is Las Bóvedas (map Google map; Playa del Tejadillo), which offers handicrafts, clothes and kitschy souvenirs. You'll find more interesting things for sale by wandering in Getsemaní, San Diego and El Centro, however.
ÁbacoBooks
(map Google map; %5-664-8338; cnr Calle de la Iglesia & Calle de la Mantilla; h8am-10pm Mon-Sat, 10am-8pm Sun; W)
An erudite-looking bookshop-cafe that's short on space but big on atmosphere, with the obligatory ladder to reach the higher shelves. Search carefully and you'll encounter Cervantes tomes, English-language titles and pretty much everything Gabriel García Márquez ever wrote. There's also Italian beer, Spanish wine and strong espresso.
Castillo de San Felipe de Barajas
The greatest fortress ever built by the Spaniards in any of their colonies, the Castillo de San Felipe de Barajas (map; Av Arévalo; adult/child COP$25,000/10,500; h8am-6pm) still dominates an entire section of Cartagena's cityscape. It should definitely be the first fortress you visit. The original edifice was quite small. It was commissioned in 1630, and construction began in 1657 on top of the 40m-high San Lázaro hill. In 1762 an extensive enlargement was undertaken, which resulted in the entire hill's being covered with this powerful bastion.
It was truly impregnable and was never taken despite numerous attempts to storm it. A complex system of tunnels connected strategic points to allow provisions to be distributed and to facilitate evacuation. Some of the tunnels are lit and are open to visitors – an eerie walk not to be missed. Take an audio tour or hire a guide for the full story.
The fortress is a short walk over the bridge from Getsemaní.
ALEX SEGRE / GETTY IMAGES ©
5Eating
Espíritu SantoColombian $
(map Google map; Calle del Porvenir No 35-60; mains COP$16,000; h11:30am-3:30pm)
In a city full of tourists, Espíritu Santo is where you come to meet the locals – hundreds of 'em in one sitting. (The restaurant is small from the outside but deceptively large within.) Choose your protein plus three sides and, in the true spirit of _comida corriente_ (set menu of the day), it'll arrive in the blink of an eye.
Beer & LaundryPizza $
(map Google map; www.beerandlaundry.com; Calle de la Media Luna No 10-113; pizzas from COP$12,000; h8am-7pm Mon-Sat; W)
It's the bane of every backpacker: where to get your laundry done with minimal fuss. Here's the answer: deposit it with the friendly English-speaking staff at this Getsemaní laundry-bar and enjoy beer and a pizza while you wait (two hours max). It's possibly the best pizza in Cartagena – and definitely the best laundry!
BeiyuCafe $$
(map Google map; Calle del Guerrero No 29-75; breakfast COP$10,000-13,000; h7am-9pm Mon-Sat, 9am-6pm Sun; W v) S
A simple little cafe plying excellent Colombian coffee, fresh juices, full breakfasts and an innovative selection of lunch and dinner dishes, Beiyu is a slice of organic, sustainable heaven in the heart of Getsemaní. It's top pick as the place to slowly eat off the effects of a late night. Portions are supergenerous. Don't miss the açai bowl.
Caffé LunáticoTapas $$
(map Google map; %320-383-0419; Calle Espíritu Santo No 29-184; tapas COP$12,000-20,000, mains COP$26,000-53,000; h11am-3pm & 7-11pm; W v)
There's nothing remotely looney about this artsy little spot in the hosteling hub of Gestemaní that pays homage to Amy Winehouse in a head-and-shoulders mural that takes up a whole wall. The perfect solution to minor lunchtime hunger pangs, midafternoon cake yearnings or early-evening cocktail urges, the Lunático does a full range of small plates and desserts exceedingly well.
El BolicheCeviche $$$
(map Google map; %5-660-0074, 310-368-7908; Cochera del Hobo No 38-17; mains COP$48,000-60,000; h12:30-3pm & 7-11pm Mon-Sat; W)
Small, delightful and not so well known that it's inundated, El Boliche basks in its relative obscurity. If you're reticent about raw fish, Boliche offers hot and cold ceviche daubed with bold and adventurous ingredients such as tamarind, coconut milk and mango. The handsome six-table interior features a glass waterfall and a bar that dispatches spot-on mojitos.
InternoColombian $$$
(map Google map; %310-260-0134, 310-327-3682; www.restauranteinterno.com; Cárcel San Diego, Calle Camposanto; 3-course set meal COP$90,000; h7-11pm Tue-Sun)
Interno is a restaurant inside Cartagena's women's prison that raises money for the rehabilitation of its inmates, who cook for and wait on you. Trained by a top Bogotá chef, the cooks prepare set meals that are delectable takes on modern Colombian cuisine. Meals are served on a gorgeously repurposed open-air patio.
You'll need to make reservations 24 hours ahead with your passport number, and you'll need to bring your passport with you when you come.
La CevicheríaSeafood $$$
(map Google map; %5-664-5255; Calle Stuart No 7-14; mains COP$39,000-120,000; hnoon-11pm Wed-Mon; W)
A once little-known spot prized by locals, La Cevichería was given ample publicity in an episode of Anthony Bourdain's _No Reservations_ in 2008. Cue the crowds but, thankfully, no real slide in the excellent quality of the food. Each dish is prepared with panache; the octopus in peanut sauce is incredible, as is the black-squid-ink rice and Peruvian ceviche.
El SantísimoColombian, Fusion $$$
(map Google map; %5-660-1531, 314-541-2117; www.elsantisimo.com; Calle del Torno No 39-62; mains COP$45,000-60,000; hnoon-11pm)
A family-friendly restaurant that won't scare off romantic couples, Santísimo indulges in a bit of creative recycling. The chandeliers are made from empty wine bottles, while wood from rescued packing cases lines the walls and ceiling. There's nothing secondhand about the food, though; it's a trip through Colombian cuisine, combining ingredients and reinventing classics as it goes.
Street Food
Cartagena is strong on street food: plenty of snack bars all across the Old Town serve local snacks such as _arepas de huevo_ (fried maize dough with an egg inside), _dedos de queso_ (deep-fried cheese sticks), empanadas and _buñuelos_ (deep-fried maize-and-cheese balls). Try the region's sweets at confectionery stands lining El Portal de los Dulces on the Plaza de los Coches (map Google map).
ANAMARIA MEJIA / SHUTTERSTOCK ©
6Drinking & Nightlife
There's a long-standing bar scene centered on the Plaza de los Coches in El Centro for salsa and vallenato, while most of the hotter and hipper action can be found in thumping Getsemaní, where the venues are bigger and the crowd younger.
AlquímicoCocktail Bar
(map Google map; %318-845-0433; www.alquimico.com; Calle del Colegio No 34-24; h5pm-2am Sun-Thu, to 3:30am Fri & Sat)
Anyone remotely hip ultimately gravitates to this Old Town bar with a pharmaceutical theme. Occupying a gorgeous colonial building, the bar is spread over three floors: the ground floor is a sleek, low-lit lounge perfect for an aperitif, while upstairs there's a kitchen and pool table. One more floor up is the always-packed roof-terrace bar, serving imaginative aguardiente cocktails.
Café HavanaClub
(map Google map; cnr Calle del Guerrero & Calle de la Media Luna; cover COP$30,000; h8:30pm-4am Thu-Sat, 5pm-2am Sun)
Havana has it all: live salsa from horn-blowing Cubans, strong drinks, a gorgeous horseshoe-shaped bar surrounded by brilliant eccentrics, wood-paneled walls and a ceiling full of whirring fans. While it's no secret these days, it's still worth a pilgrimage. Expect sweaty crowds and the odd elbow in the ribs when you're trying out your salsa moves at 1am. No shorts.
Bazurto Social ClubClub
(map Google map; www.bazurtosocialclub.com; Av del Centenario No 30-42; cover COP$5000; h8pm-4am Wed-Sat)
Join the crowds at this lively spot where locals dance in unison under an enormous glowing red fish to live _champeta_ music, sip knockout cocktails and catch up on the Getsemaní gossip. The music's great, and after a few drinks you'll find yourself being dragged in, though your ears will likely ring for days.
Donde FidelBar
(map Google map; %5-664-3127; El Portal de los Dulces No 32-09; h11am-2am)
Old salts meet salsa-seeking tourists at this usually packed and always loud Old Town bar characterized by glaring florescent lighting and busy walls covered with photos of owner Fidel posing with a Wikipedia's worth of Latin celebrities from A-list to D-list. Order a bottle of rum along with an ice bucket and some mixers, and hunker down.
3Entertainment
Estadio Olímpico Jaime Morón LeónStadium
(Villa Olímpico)
Cartagena's local football team, Real Cartagena, plays games at this stadium located 5km south of the city.
8INFORMATION
The city's main **tourist office** (Turismo Cartagena de Indias; map; %5-660-1583; Plaza de la Aduana; h9am-noon & 1-6pm Mon-Sat, 9am-5pm Sun) can be found on Plaza de la Aduana. There are also small booths in Plaza de San Pedro Claver and Plaza de los Coches.
8GETTING THERE & AWAY
AIR
All major Colombian carriers operate flights to and from Cartagena's **Aeropuerto Internacional Rafael Núñez** ( %5-693-1351; www.sacsa.com.co; Calle 71 No 8-9), 3km from the city, in Crespo.
BUS
For most destinations, you'll need to head to Cartagena's **Terminal de Transportes de Cartagena** ( %304-577-5743; www.terminaldecartagena.com; Calle 1A No 3-89). It's on the eastern outskirts of the city, around 45 minutes from the center.
8GETTING AROUND
Large green-and-red-signed **Metrocar buses** (map; Av Santander; COP$2000) shuttle between the city and the bus terminal every 15 to 30 minutes (COP$3000, 40 minutes). In the center you can catch them on Av Santander. A taxi between the bus station and El Centro costs COP$15,000, plus COP$5000 after 8pm.
# In Focus
## South America Today
Economic growth is on the rise, though deforestation and corruption remain pervasive problems.
## History
A look at the forces shaping South America, from colonization and slavery to military dictatorships.
## People & Culture
A portrait of the continent, covering multiculturalism, religion, lifestyles and music.
## Outdoor Activities
From trekking and wildlife-watching to skiing and surfing, South America offers an array of captivating activities.
# South America Today
The past decade has witnessed a rising middle class and strong economic growth in large swaths of South America. Poverty rates have fallen, and addressing the large gap between rich and poor seems to be the hot topic nearly everywhere. Big challenges, however, remain. Deforestation is once again on the rise and the political leadership has tacked sharply to the right, threatening to put an end to recent social progress.
Rio de Janeiro, Brazil | BRASIL2 / GETTY IMAGES ©
## Breaking Down Barriers
Machismo has taken a blow, with the first female presidents in South America helping to break down barriers. Cristina Fernández de Kirchner of Argentina, Dilma Rousseff of Brazil and Michelle Bachelet of Chile have all recently served as presidents of some of South America's largest economies. Speaking of historic elections, Evo Morales (now in his third term) also deserves special mention, becoming the first president of Bolivia to hail from an indigenous background. He follows on the heels of Alejandro Toledo, who became Peru's first indigenous president back in 2001. On other fronts, there have been equally dramatic changes in recent years. Gay marriage has been legalized in Argentina, Brazil and Uruguay, and three other countries (Chile, Colombia and Ecuador) have a form of same-sex civil union. Gay marriage is also legal in French Guiana, which is considered part of France.
## Social & Environmental Threats
Despite economic growth in many countries, not all have benefited. Rural poverty remains a gripping problem in every country in South America, with many families still struggling with basic needs: adequate nutrition, health care and clean water. And one in seven still live in extreme poverty, subsisting on less than US$2.50 per day.
When it comes to the environment, there's a mix of good and bad news. On the plus side, there's increased interest in a green future. Sustainable sources meet more than 70% of Brazil's energy needs. Chile has greatly expanded its protected areas, creating five new national parks and expanding three existing parks in the process; in total it has added more than 40,500 sq km – an area larger than Switzerland.
Sadly, deforestation has risen once again in the Amazon. After falling year-on-year from 2005 to 2015, it surged again in the last few years, rising by 38% over the course of 2018. At the same time, oil production will soon begin in Ecuador's Parque Nacional Yasuní, a pristine area of the Amazon that holds one of the country's largest reserves. Petroamazonas began drilling in this Unesco World Heritage Site in early 2016. The building of access roads and pipelines – not to mention the possibility of oil spills – could be devastating for Yasuní.
In Peru, coca and cocaine production not only have serious social repercussions, but also affect Peru's environment through deforestation in remote growing areas and chemical contamination. The Amazon is also now bisected by the Interoceanic Highway, an overland trade route that links Peru to Brazil. Economics aside, there's great concern about the irrevocable impact this could have on the world's most biologically rich rainforest.
## Corruption Scandals
Political corruption remains pervasive in South America. In Brazil, one of the largest scandals in the nation's history (a staggering US$3 billion bribery and money-laundering scheme) brought down many politicians and heads of industry, and led to the impeachment of President Dilma Rousseff in 2016. In Argentina, former president Cristina Kirchner had to deal with damaging allegations that she and her late husband (who was president before her) enriched themselves at taxpayers' expense. Likewise, former Chilean president Michelle Bachelet also found herself briefly embroiled in a corruption scandal involving abuse of power by her son. Her name was ultimately cleared, and after her term ended she was appointed by the UN to the prestigious role of Human Rights High Commissioner.
# History
South America has a long and tumultuous history. It was the birthplace of one of the world's great empires, sadly brought to ruin upon European arrival, and the destination for millions of men, women and children who were enslaved and brought over from Africa. Independence movements freed the continent from foreign rule, but did little to address the yawning divide between rich and poor, which still remains today.
Machu Picchu, Peru | ALEX ROBINSON / GETTY IMAGES ©
## First Peoples
There are various competing theories about how the indigenous peoples arrived in the Americas. Until recently, it was generally believed that early inhabitants traveled from present-day Siberia to Alaska over a land bridge across the Bering Strait. Some scholars estimate this epic migration occurred around 14,000 years ago. In the last several decades, new evidence of older sites in the southern reaches of South America has challenged the land-bridge theory. Early humans may have arrived by a combination of foot and boat travel following the coastline south as early as 23,000 years ago. In Monte Verde, Chile, scientists have discovered some of the oldest undisputed evidence of human occupation in the Americas. Apparently, early peoples were seafarers (or at least seafood lovers): among the artifacts found there were 10 different species of seaweed.
The earliest peoples were nomadic hunter-gatherers who lived in small groups. Agriculture likely developed around 5000 BC with the planting of wild tubers such as manioc and sweet potato under systems of shifting cultivation. About the same time, highland people began to farm seed crops, such as beans, and to domesticate animals, such as the llama. One of South America's greatest foodstuffs is the humble but versatile potato, a root crop domesticated in the Andean highlands. Today, more than 6000 varieties of potato are cultivated there.
Complex societies first developed in the valleys of coastal Peru. Their growth was unsustainable, however – it's thought that the population of some of these valleys grew until all the cultivable land was occupied. The need to expand into neighboring valleys led the inhabitants to organize, innovate and conquer. Not dissimilar from what would happen after the Europeans arrived, conquerors became the rulers and the conquered became their subjects; this helped develop the social and economic hierarchies of these early states and beyond.
These embryonic societies ultimately developed into major civilizations, such as the Wari empire of the Peruvian central highlands, the Tiahuanaco culture of highland Bolivia, the Chimú culture of northern coastal Peru and the Inca empire of Cuzco.
Andean women in Quito, Ecuador | GABRIEL PEREZ / GETTY IMAGES ©
## Inca Empire
According to legend, the Inca civilization was born when Manco Cápac and his sister Mama Ocllo, children of the sun, emerged from Lake Titicaca to establish a civilization in the Cuzco Valley. Whether Manco Cápac was a historical figure is up for debate, but what is certain is that the Inca civilization was established in the area of Cuzco at some point in the 12th century. The reign of the first several incas (kings) was largely unremarkable, and for a couple of centuries the Incas remained a small, regional state.
Expansion took off in the early 15th century, when the ninth king, Inca Yupanqui, defended Cuzco – against incredible odds – from the invading Chanka people to the north. After the victory, he took on the new name of Pachacutec (Transformer of the Earth) and spent the next 25 years bagging much of the Andes. Under his reign, the Incas grew from a regional fiefdom in the Cuzco Valley into a broad empire of about 10 million people known as Tawantinsuyo (Land of Four Quarters). The kingdom covered most of modern Peru, in addition to pieces of Ecuador, Bolivia and Chile. The empire traversed the Andes with more than 8000km of highways and managed to control peoples from 100 separate cultures and 20 different language groups for about a century. This was made more remarkable by the fact that the Incas, as an ethnicity, never numbered more than about 100,000.
Pachacutec allegedly gave Cuzco its layout in the form of a puma and built fabulous stone monuments in honor of Inca victories, including Sacsaywamán, the temple-fortress at Ollantaytambo and possibly Machu Picchu. He also improved the network of roads that connected the empire, further developed terrace agricultural systems and made Quechua the lingua franca.
## Portuguese Arrival
The Portuguese were the first Europeans to set foot on the South American continent. In 1500 a fleet of 12 Portuguese ships carrying nearly 1200 men dropped anchor near what is today Porto Seguro. There they erected a cross and held Mass in the land they baptized Terra da Vera Cruz (Land of the True Cross) before taking to the waves once again. Over the next century, the Portuguese set up coastal colonies in present-day Salvador, Rio de Janeiro and other coastal areas. There they harvested the profitable _pau brasil_ (brazilwood), which gave the country its name.
Over the following centuries a four-front war was waged on the indigenous way of life. It was a cultural war, as well as a physical, territorial and biological one. Many indigenous peoples fell victim to the _bandeirantes_ – groups of roaming raiders who spent the 17th and 18th centuries exploring Brazil's interior, pillaging indigenous settlements as they went. Those who escaped such a fate were struck down by the illnesses which traveled from Europe, against which they had no natural resistance. Others were worked to death on sugar plantations.
## Defeat of the Incas
The vast Inca empire was in trouble even before the Spanish arrived in present-day Peru. Smallpox and other epidemics transmitted by European soldiers were sweeping through the entire American continent. Thousands of indigenous people were killed by the disease – including, in all likelihood, Inca emperor Huayna Cápac, who died in 1525.
Without a clear plan of succession, the emperor's untimely death left a power vacuum. The contest turned into a face-off between two of his many children: the Quito-born Atahualpa, who commanded his father's army in the north, and Huáscar, who was based in Cuzco. The ensuing struggle plunged the empire into a bloody civil war, reducing entire cities to rubble. Atahualpa emerged as the victor in April 1532. But the vicious nature of the conflict left the Incas with a lot of enemies throughout the Andes – which is why some tribes were so willing to cooperate with the Spanish when they arrived just five months later.
While the Portuguese were battling for control over the eastern half of the continent, the Spaniards set their sights on South America's Pacific coast. Following rumors of golden splendor in the interior, Francisco Pizarro led an exploratory journey to the north coast of Peru in 1528. There, near Tumbes, a crew of welcoming indigenous people offered them meat, fruit, fish and corn beer. To their delight, a cursory examination of the city revealed an abundance of silver and gold. The group quickly returned to Spain to court royal support for a bigger expedition.
They returned in September 1532, with a shipload of arms, horses and slaves, as well as a battalion of 168 troops. Tumbes, the rich town Pizarro had visited just four years earlier, had been devastated by epidemics, and the recent Inca civil war. Atahualpa, in the meantime, was in the process of making his way down from Quito to Cuzco to claim his hard-won throne. When the Spanish arrived, he was in the highland settlement of Cajamarca, enjoying the area's mineral baths.
Pizarro quickly deduced that the empire was in a fractious state. He and his men charted a course to Cajamarca and approached Atahualpa with royal greetings and promises of brotherhood. But the well-mannered overtures quickly devolved into a surprise attack that left thousands of Incas dead and Atahualpa a prisoner of war. (Between their horses, their armor and the steel of their blades, the Spanish were practically invincible against fighters armed only with clubs, slings and wicker helmets.)
In an attempt to regain his freedom, Atahualpa offered the Spanish a bounty of gold and silver. Thus began one of the most famous ransoms in history – with the Incas attempting to fill an entire room with the precious stuff in order to placate the unrelenting appetites of the Spanish. But it was never enough. The Spanish held Atahualpa for eight months before executing him with a garrote at the age of 31.
The Inca empire never recovered from this fateful encounter. The arrival of the Spanish brought on a cataclysmic collapse of the indigenous society. It is estimated that the local population – around 10 million when Pizarro arrived – was reduced to 600,000 within a century.
oBest Historical Reads
_1491: New Revelations of the Americas before Columbus_ (Charles C Mann)
_Last Days of the Incas_ (Kim MacQuarrie)
_Open Veins of Latin America_ (Eduardo Galeano)
_Prisoner without a Name, Cell without a Number_ (Jacobo Timerman)
Traveler on an Inca trail, Lake Titicaca, Peru | FABIO LAMANNA / SHUTTERSTOCK ©
## The Dark Era of Slavery
The slave trade practiced by early European traders from the 1500s to 1866 enslaved as many as 12.5 million people – with just around 10.7 million surviving the grueling journey from Africa to the Americas. Only a fraction (around half a million) ended up in North America. The rest were destined for Latin America and the Caribbean with the majority (as many as six million) ending up in Brazil, most of them working on the backbreaking sugarcane plantations. They were torn from a variety of tribes in Angola, Mozambique and Guinea, as well as the Sudan and Congo. Whatever their origins and cultures, their destinations were identical: slave markets such as Salvador's Pelourinho or Belém's Mercado Ver-o-Peso. Elsewhere, smaller numbers of Africans were taken to Peru, Colombia, the Guianas and all along the Caribbean coast.
For those who survived the ordeal of removal and transfer, arrival in the Americas meant only continued suffering. A slave's existence was one of brutality and humiliation. Kind masters were the exception, not the rule, and labor on the plantations was relentless. Slaves were required to work as many as 17 hours each day, before retiring to the squalid _senzala_ (slave quarters), and with as many as 200 slaves packed into each dwelling, hygiene was a concept as remote as the distant coasts of Africa. Dysentery, typhus, yellow fever, malaria, tuberculosis and scurvy were rife, and malnutrition was a fact of life.
Syphilis also plagued a slave population sexually exploited by its masters. Sex slavery was so common that a large mixed-race population soon emerged. Off the plantations there were greater numbers of white men than white women, and many black or indigenous women were used by white men as live-in sex slaves.
Many slaves escaped from their masters to form _quilombos,_ communities of runaway slaves that quickly spread across the countryside. The most famous, the Republic of Palmares, which survived through much of the 17th century, was home to some 20,000 people before it was destroyed by Brazilian federal troops.
Most countries in South America banned slavery between 1816 and 1831, but in Brazil, it wasn't until 1888 that slavery was finally outlawed. Unsurprisingly, this didn't make a huge immediate difference to the welfare of the 800,000 freed slaves, who were largely illiterate and unskilled. Thousands were cast into the streets without any kind of infrastructure to support them. Many died, while others flooded to Brazil's urban centers, desperately in search of jobs.
Critics point out that the end of slavery didn't really bring justice to newly freed people. Gross inequalities continue to plague South America up to the present day. One 2015 study in Brazil showed that people of color earn 59% of what whites earn, and that 132% more Afro-Brazilians than whites are killed by violence each year.
Settlements in the Amazon
New discoveries are reshaping the dominant thinking about pre-Columbian societies. The Amazon, once thought to be a wilderness incapable of supporting large populations, is now viewed as home to mound-building societies with some settlements containing as many as 100,000 inhabitants. At least 12% (and probably more) of the nonflooded Amazon forest is of anthropogenic origin (directly or indirectly altered by humans). Evidence of agriculture in the rainforest exists as far back as 4000 years ago, with as many as 140 different crops grown. Anthropologists have even found proof that early peoples used complex farming techniques to enrich the earth with microorganism-rich _terra preta_ (black soil).
## Independence
By the early 19th century, _criollos_ (creoles, born in South America to Spanish parents) in many Spanish colonies had grown increasingly dissatisfied with their lack of administrative power and the crown's heavy taxes – leading to revolutions all over the continent. Argentine revolutionary José de San Martín led independence campaigns in Argentina and Chile (1818), before sailing up the coast to take Lima in 1821. From the opposite direction came Simón Bolívar, the Venezuelan revolutionary who had been leading independence fights in Venezuela, Colombia and Ecuador.
The two liberators met in Guayaquil, Ecuador, in 1822. At this famous meeting, the apolitical San Martín found himself in conflict with Bolívar, who had strong political ambitions. San Martín considered the installation of a powerful leader, even a monarch, as essential to avoid the disintegration of Peru, while Bolívar insisted on a constitutional republic. In a complicated exchange, which aroused ill feeling in both camps, Bolívar won the day and San Martín returned to the south. In the long run, both were disappointed. The proliferation of _caudillos_ (local warlords) set a deplorable pattern for most of the 19th century.
Ever in its own world, Brazil followed quite a different path to independence. Unlike its neighboring countries, Brazil had a European monarch living within its borders in the early 1800s. Brazil became a temporary sanctuary to the Portuguese royal family, who fled from the advance of Napoleon in Iberia in 1807. The prince regent – and future king, Dom João VI – fell in love with Rio, naming it the capital of the United Kingdom of Portugal, Brazil and the Algarves. His affection for Brazil was so strong that he didn't want to return to Portugal even after Napoleon's defeat at Waterloo in 1815. He finally returned to Europe six years later, leaving his son Pedro as prince regent. When the Portuguese parliament attempted to restore Brazil to its previous status as subservient colony, Dom Pedro rebelled and declared Brazil independent, announcing himself the country's head as Emperor Dom Pedro I. Portugal was too weak to fight its favorite son, so without spilling blood, Brazil attained its independence in 1822.
## Military Dictatorships
The 20th century was a tumultuous period for South America, with political turmoil and economic crises paving the way for the rise of military dictatorships. The social unrest that followed the Great Depression of 1929 provided justification for the army to intervene in countries across the continent. In Argentina, the pro-fascist general José Félix Uriburu seized control during a military coup in 1930, ushering in the so-called Infamous Decade. Likewise the 1930s saw military coups and repressive regimes rise in Peru and Chile. In Brazil, it was the era of the autocratic Getulio Vargas, when rival political parties were banned, the press was muzzled and opponents were imprisoned.
Unfortunately, this was just the beginning, with far more horrifying dictatorships on the horizon. The 1960s and 1970s were an even darker period in South America when military dictatorships ruled in Argentina, Bolivia, Brazil, Chile, Paraguay, Peru, Suriname and Uruguay. Student- and worker-led movements crying out for social justice were met with increasing brutality.
In the late 1960s and '70s in Argentina, antigovernment feeling was rife and street protests often exploded into all-out riots. Armed guerrilla organizations emerged as radical opponents of the military, the oligarchies and US influence in Latin America. In 1976 the army general Jorge Rafael Videla seized power, ushering in a bloody seven-year period known as the Dirty War. Security forces went about the country arresting, torturing and killing anyone on their hit list of suspected leftists. As many as 30,000 people were 'disappeared' – that is, murdered.
In Chile, there was hope for a brighter future when socialist candidate Salvador Allende was elected in 1970. This was soon crushed, however, when Augusto Pinochet led a coup in 1973. Ruling until 1989, he became Latin America's most notorious dictator, with thousands of suspected leftists jailed, tortured and executed, and hundreds of thousands fleeing the country.
Meanwhile in Brazil, military dictators ran the show from 1964 to 1984. Though not as brutal as the Chilean or Argentine regimes, it was still a period when dissent was crushed, political parties were banned and the media was muzzled. Things remained grim throughout South America until the early 1990s, when democracy at last returned to most of the continent.
## The 21st Century
Towards the end of the 1990s and into the 21st century, things took a remarkable turn for the better in South America. A rising middle class, falling poverty rates and strong economies were hallmarks of the early 2000s. As the continent veered to the left, wage disparities fell slightly and social justice seemed to be the hot topic of the day.
Progressives like former Brazilian president Lula helped pave the way, demonstrating that you could both grow an economy and help lift people out of poverty. Machismo took a blow, with the first female presidents in South America helping to break down barriers. Cristina Kirchner of Argentina, Dilma Rousseff of Brazil and Michelle Bachelet of Chile all recently served as presidents of some of South America's largest economies. Leaders from indigenous backgrounds also emerged on the political stage, including Evo Morales of Bolivia and Alejandro Toledo of Peru.
In the last few years, the political landscape has again shifted dramatically. As of 2019, no women were serving as heads of state in any South American country. Right-wing governments seemed everywhere on the rise, and some leaders – like Brazil's firebrand Jair Bolsonaro – even spoke fondly of the military dictatorship of the past.
## Timeline
**14,000–23,000 BC**
Humans traveling from Asia, probably across the Bering land bridge, reach the Americas during great human migration.
**1493**
Inca Huayna Cápac begins his reign, pushing his vast empire north; his untimely death in 1525 leaves the kingdom divided.
**1494**
Spain and Portugal sign the Treaty of Tordesillas, dividing colonized lands in the Americas between them.
**1548**
Wine comes to Chile via missionaries and conquistadores. Jesuit priests cultivated early vineyards of rustic pais grapes.
**1550**
Facing a labor shortage, Portugal turns to the African slave trade; open-air slave markets flourish in the growing colony.
**1807**
Napoleon invades Portugal; the Portuguese prince regent (future Dom João VI) and his entire court of 15,000 flee to Brazil.
**1819**
Simón Bolívar defeats the Spanish army at Boyacá near Bogotá; the Republic of Gran Colombia is founded.
**1830**
Gran Colombia splits into Colombia, Ecuador and Venezuela. Bolívar sends himself into exile; he dies in Santa Marta.
**1834–35**
HMS Beagle explores South America with Charles Darwin on board; the expedition gives Darwin fodder for his theory of evolution.
**1865–70**
Brazil, allied with Uruguay and Argentina, wages the 'War of the Triple Alliance' on Paraguay, which leaves untold thousands dead.
**1879–83**
Chile wages war against Peru and Bolivia over nitrate-rich lands in the Atacama Desert; Bolivia loses its coastline.
**1888**
Slavery is abolished in Brazil, the last country in South America to do so.
**1890s**
Brazil opens its borders. Millions arrive from Italy, Portugal, Spain, Germany and later Japan and other countries.
**1967**
Argentine revolutionary Ernesto 'Che' Guevara is executed by a US-backed military squad in Bolivia.
**1970**
A 7.7-magnitude earthquake in northern Peru kills almost 80,000 people, leaves 140,000 injured and another 500,000 homeless.
**1970**
Salvador Allende, Chile's first democratically elected Marxist president, introduces radical social reform and massive income redistribution.
**1973–89**
Following a military coup, General Augusto Pinochet takes charge of Chile. He dissolves Congress and rules by decree.
**1976–83**
A military junta takes control of Argentina, launching the 'Dirty War.' In eight years an estimated 30,000 people 'disappear.'
**1992**
Thousands of indigenous protesters march in Quito. In ensuing negotiations they are granted title to 2.5 million acres in Amazonia.
**2001**
A vast financial crisis hits Argentina. Interim president Duhalde devalues the peso and defaults on US$140 billion in debt.
**2014**
Brazil hosts the 2014 FIFA World Cup, spending around US$12 billion in preparation for the event.
**2015**
Pope Francis visits Bolivia and begs forgiveness for the grave sins committed against indigenous peoples in the name of God.
**2016**
Rio becomes the first city in South America to host the Summer Olympics. The city spends US$13 billion in preparations.
**2018**
Peruvian President Pedro Pablo Kuczynski resigns from office and is succeeded by Vice President Martín Vizcarra.
# People & Culture
There are many layers to South American culture. Religion is a key component of life on the continent, where Christianity, indigenous beliefs and African religions have all shaped identity. South America is also the birthplace of samba, tango, Andean sounds and countless other regional genres. While religion and music can be a great unifier, there is still a huge gulf between haves and have-nots in this stratified society.
Peruvian woman in the Sacred Valley, Peru | HADYNYAH / GETTY IMAGES ©
## Multiculturalism
South America boasts astonishing diversity. The continent has been shaped by the original indigenous inhabitants, European colonists and Africans brought over as slaves to toil in the plantations and mines. The level of intermixing varies greatly from country to country, which has led to the relative diversity or homogeneity of the population.
Immigration has also added to the complex ethnic tapestry of the continent. For decades, the US was a major destination for migrants from South America. These days, there's much more migration happening internally (ie between South American countries), with Brazil, Argentina and Chile attracting the largest numbers of migrants, largely from neighboring countries. There are also complicated dynamics at work in each of the continent's 13 countries. Ecuador for instance, with a population of just 16 million, has an estimated two million emigrants, living in the US, Italy and Spain; in the latter, they make up the largest contingent of Latin Americans. On the flipside, Ecuador has seen an influx of refugees (in the 1980s and 1990s) fleeing from conflicts in neighboring Colombia, and opportunity-seeking migrants from Peru.The situation, however, remains quite fluid. Since 2008, with improved job opportunities in Ecuador, a growing number of Ecuadorians are choosing to return home.
In the 19th century Brazil and Argentina saw mass immigration from Europe: Spaniards, Italians, Germans and Eastern Europeans were the most frequent immigrants moving to the Americas. Brazil also welcomed immigrants from Japan, Portugal and the Middle East. They worked in a wide variety of fields, from coffee plantations and farming to heavy industry in the continent's growing cities. The influx of new arrivals continued well into the 20th century, with tumult in Europe causing the flight of Jews from Nazi persecution. They were soon followed by Nazis looking to avoid being put on trial for war crimes, in addition to Italians and others escaping their ravaged cities in the postwar period.
Baby llama | MEDIAPRODUCTION / GETTY IMAGES ©
## Lifestyles
No matter where you go in South America, you'll likely encounter a yawning divide between rich and poor. Modern-day South Americans inherited a highly stratified society from the slave-owning European colonists, and this dispiriting chasm persists centuries later between the haves and have-nots of both urban and rural society. At the bottom of the heap are those struggling in low-wage jobs in the city, or scraping out a meager existence in the countryside – many are barely able to put food on the table. Those who live in rural areas are practically invisible to the urban middle and upper classes.
The middle and upper class live in comfortable apartments or houses, with all the trappings of the developed world, including good health care in private clinics, cars, vacations away and easy access to the latest gadgets and trends (though iPhones and laptops are pricier here). Owing to low wages, maids are common, even among the middle class. Crime is of high concern, and those that can afford it live in high-security buildings or gated residential complexes.
The divide is greatest in struggling countries like Bolivia, where nearly half the population lives below the poverty line. There many live without running water, electricity and heat, and the threat of illness looms high over children – with the majority of childhood deaths associated with malnutrition and poverty.
oBest Classic Albums
Volver (Carlos Gardel; 1934)
Contigo Peru (Arturo 'Zambo' Cavero; 2008)
Elis & Tom (Elis Regina & Tom Jobim 1977)
Gracias a La Vida (Violeta Parra; 1966)
Ojos que matan (Julio Jaramillo; 1970)
Flower seller, Quito, Ecuador | PHILIP LEE HARVEY / LONELY PLANET ©
## Religion
### Christianity
The dominant religion in South America is Roman Catholicism, a legacy from the early Spanish and Portuguese colonizers. The number of followers varies by region, with no small degree of complexity from country to country (in Argentina, for instance, 92% of the population call themselves Catholic, though less than 20% practice regularly). On average though, 70% or more of each country's population professes to be Roman Catholic. The ranks are declining from year to year, and many people (particularly in urban areas) merely turn up to church for the basics: baptism, marriage and burial. Nevertheless, the church still has a strong visible presence here. Nearly every city, town or village has a central church or cathedral, and the calendar is loaded with Catholic holidays and celebrations.
Evangelical Christianity, meanwhile, is booming. All across the continent, especially in poor communities where people are most desperate, simple, recently built churches are full of worshippers. The religion has done particularly well here, with converts from Catholicism often citing a more personal relationship with God, as well as receiving more direct guidance in the realm of health, jobs and living a moral life. The firebrand Pentecostal branch attracts many new followers, with its emphasis on divine healing, speaking in tongues and receiving direct messages from God. With the current growth of the Evangelical church, some predict that the majority of South America will be Protestant by 2050.
### Indigenous Beliefs
Among indigenous peoples, allegiance to Catholicism was often a clever veneer adopted to disguise traditional beliefs ostensibly forbidden by the church. In parts of the interior, the Amazon and the Andes, shamanism and animism still flourish. There is also a strong belief in powerful spirits that inhabit the natural world – the sky, mountains, lightning and the wind. Some groups, like the Andean Aymará, practice a syncretic religion that pays equal homage to both deities and Catholic saints. They may attend mass, baptisms and saint's day celebrations, while also paying respect to Pachamama (Mother Earth) come harvest time. The old Inca celebration of Inti Raymi (Festival of the Sun) is celebrated with fervor in some parts of the Andes. It happens on the winter solstice (late June) and commemorates the mythical birth of the Inca.
### African Religions
Enslaved peoples brought over ancient African religions to the Americas, which were adapted over the centuries. The best known and most orthodox is Candomblé, which arrived in Brazil via the Nago, Yoruba and Jeje peoples. It later found root in Bahia, where it is still practiced today. Candomblé denotes a dance in honor of the gods and indeed trancelike dancing is an essential part of the religion. Afro-Brazilian rituals are directed by Candomblé priests, the _pai de santo_ or _mãe de santo_ (literally 'saint's father' or 'saint's mother'), and practiced in a _casa de santo_ or _terreiro_ (house of worship).
The religion centers upon the _orixás_. Like the gods in Greek mythology, each _orixá_ has a unique personality and history. Although _orixás_ are divided into male and female types, there are some that can switch from one sex to the other, such as Logunedé, the son of two male gods, or Oxumaré, who is male for six months of the year and female for the other six months. (Candomblé, not surprisingly, is much more accepting of homosexuality and bisexuality than other religions.)
Candomblé followers believe that every person has a particular deity watching over them, and followers give food or other offerings to their respective _orixá_.
A Visionary Leader
One of the most famous figures from the Amazon is Raoni Metuktire, chief of the Kayapó people. Born in 1930, Chief Raoni was an early advocate for indigenous rights and the preservation of the Amazon, and his hard-fought campaigns have brought worldwide attention to the plight of the rainforest and its inhabitants. Though made more than 40 years ago, the French-Belgian documentary _Raoni_ (1978) sheds more light on his tireless efforts to protect the Amazon that continue up to the present. There's also a surprising cameo by Marlon Brando, a strong supporter of the Kayapó.
## Music
South America has given the world a rich musical heritage. This is the birthplace of tango, samba, bossa nova and haunting Andean folk music, but this is just the beginning of the dizzying soundtrack, of wide-ranging rhythms with roots in Europe, Africa and indigenous pre-Columbian villages.
### Tango
Music plays a key part in festivities across the continent, and it also takes center stage when it comes to nightlife in many cities. The tango is deeply linked to Buenos Aires (though the music also has ties to lesser-known Montevideo in Uruguay). It emerged from the country's bordellos in the late 19th century, though it didn't become mainstream until Carlos Gardel helped popularize the songs in the 1920s and 1930s. Although Gardel was born in France, he was brought by his destitute single mother to Buenos Aires when he was three years old. In his youth he entertained neighbors with his rapturous singing, then went on to establish a successful performing career. He single-handedly helped bring tango out of the tenement and onto the world stage. He died tragically in a plane crash at the height of his career and was mourned around the world.
Another seminal figure in the tango world was Astor Piazzolla, who moved the genre from the dance halls into the concert halls. Tango nuevo, as it was called when it emerged in the 1950s, was given a newfound respect, with its blend of jazz and classical elements and new forms of melodic structures. Piazzolla also paved the way for the tango fusion, which emerged in the 1970s and continues to this day with _tango electrónico_ groups such as Gotan Project, Bajofondo Tango Club and Tanghetto.
### Samba & Bossa Nova
The birth of modern Brazilian music essentially began with the birth of samba, first heard in the early 20th century in a Rio neighborhood near present-day Praça Onze. Here, Bahian immigrants formed a tightly knit community in which traditional African customs thrived – music, dance and the Candomblé religion. Such an atmosphere nurtured the likes of Pixinguinha, one of samba's founding fathers, as well as Donga, one of the composers of 'Pelo Telefone,' the first recorded samba song (1917) and an enormous success at the then-fledgling Carnaval.
Samba continued to evolve in the homes and _botequims_ (neighborhood bars) around Rio. The 1930s are known as the golden age of samba. Sophisticated lyricists such as Dorival Caymmi and Noel Rosa wrote popular songs featuring sentimental lyrics and an emphasis on melody (rather than rhythm), foreshadowing the later advent of cool bossa nova. The 1930s were also the golden age of samba songwriting for Carnaval.
In the 1950s came bossa nova (literally, 'new wave'), sparking a new era of Brazilian music. Bossa nova's founders – songwriter and composer Antônio Carlos (Tom) Jobim and guitarist João Gilberto, in association with the lyricist-poet Vinícius de Moraes – slowed down and altered the basic samba rhythm to create a more intimate, harmonic style. Bossa nova was also associated with the new class of university-educated Brazilians. Its lyrics reflected the optimistic mood of the middle class in the 1950s, and by the following decade it had become a huge international success.
### Andean Sounds
The breathy, mournful songs played by groups across the western half of the continent (from Chile up to Venezuela) are all part of the legacy of Andean folk music. Its roots date back to pre-Inca times when music was largely played during religious ceremonies. It was viewed as a sacred art with connections to the divine world, and it paid homage to the spirits that were believed to inhabit the natural world.
Musical styles vary from region to region (with four-, five-, six- or seven-note scales), but the instruments are often quite similar. Panpipes are a staple: usually made of bamboo, these instruments consist of a single or double row of hollow tubes, and come in a bewildering variety of sizes. These are often accompanied by a smaller flutelike _quena,_ a bass drum and a stringed instrument (an influence adopted from Europe), such as the 10-string _charango,_ which is similar to a mandolin. Prior to the Spaniards, wind and percussion were the dominant sounds – fitting for a region of fiery volcanoes and bone-chilling gales that blow across the highlands.
# Outdoor Activities
South America has countless activities for adventure seekers. There's fabulous trekking throughout the continent (particularly in the Andes), whitewater rafting on rushing mountain rivers, and legendary surfing on both the Pacific and the Atlantic coasts. Wildlife-watching here is among the best on earth, whether you're traveling through the Amazon, the Pantanal, the Galapagos or the cloud forests of Peru.
Traveler exploring Machu Picchu, Peru | GLEB AITOV / SHUTTERSTOCK ©
## Hiking & Trekking
The opportunities for hiking are practically limitless. Stunning scenery is a guarantee wherever you go, with snow-covered peaks, cloud forests and verdant lowland jungle setting the stage for hiking and wildlife-watching. If you head to the mountains, be prepared to spend a few days acclimating to the dizzying altitudes – or face a heavy-duty bout of altitude sickness.
The Andean countries are famous for their old Inca roads, which are ready-made for scenic excursions. The four-day tramp along the Inca Trail to Machu Picchu is, of course, the classic, but there are alternative routes around Cuzco, including a six-day trek around the venerated Ausangate (6372m). This takes you over 5000m passes, through huge herds of alpacas and past tiny hamlets unchanged in centuries. In the Peruvian Andes, trekking is most rewarding during the dry season (May to September). Avoid the wet season (December to March), when rain makes some areas impassable.
In Peru, Cuzco and Huaraz have outfitters that can provide equipment, guides and even _arrieros_ (mule drivers). If you prefer to trek ultralight, you might want to purchase your own gear, especially a sleeping bag, as old-generation rental items tend to be heavy. Whether you'll need a guide depends on where you trek. Certain areas of Peru, such as along the Inca Trail, require guides; in other places, like the Cordillera Blanca, you'll find many trekking routes that are wonderfully DIY. Equip yourself with topographic maps for major routes in the nearest major gateway towns.
Chile's sublime Torres del Paine is one of the continent's most beloved hiking destinations, graced by glaciers, gemstone lakes and the world-famous granite spires. The park has good public access, _refugios_ (huts) and campsites that allow for multiday treks. To combat overcrowding, new regulations require reservations for all camping and lodgings on the 'W' and Paine Circuit hikes. For awe-inspiring isolation, Tierra del Fuego's Dientes de Navarino hiking circuit is also stunning but harder to access.
Chile's Lakes District abounds with trails and tantalizing terrain. Across the border Argentina's storm-pounded but spectacular Fitz Roy Range (located in the northern section of Parque Nacional Los Glaciares) is a superb setting for trekking. It's blessed with excellent trail infrastructure and accessibility.
Hiking in Brazil is best done during the cooler months of April to October. During the summer, the tropical sun heats the rock to oven temperatures and turns the jungles into steamy saunas. If you plan to hike in the Amazon, aim to come when the water levels are low (roughly August to December); at other times the forest is flooded and virtually all your activities will be by canoe.
oBest Hiking Destinations
The Inca Trail, Peru
Torres del Paine, Chile
Cordillera Blanca, Peru
Fitz Roy Range, Argentina
Hiker in Parque Nacional Torres del Paine | DUDAREV MIKHAIL / SHUTTERSTOCK ©
Responsible Trekking
Exercise caution with campfires on the windy Andean slopes and the Patagonian steppe.
Cook on a camp stove (not an open fire) and dispose of butane cartridges responsibly.
Carry out all rubbish.
If there's no toilet, the best practice is to carry out waste. If you decide to bury it, dig a small hole 15cm deep and at least 100m from any watercourses. Cover with soil and a rock. Pack out toilet paper (take it with you in a ziplock bag).
Wash with biodegradable soap at least 50m away from any watercourses.
Do not feed wildlife.
Stick to existing trails. Avoid cutting corners; it contributes to erosion.
Trails can pass through private property. Ask permission before entering, and leave all livestock gates as you found them.
### Hiking Safety
Many trails in South America are poorly marked or not marked at all. Trekkers should always inquire ahead, as navigation may be a major component of hiking and many travelers come unprepared. When it comes to packing, quality mountain gear is a must. Even in summer the Andes can have extreme temperatures – summer sleeping bags and nonwaterproof gear will not work!
### What to Bring
Trekking poles are recommended, as some hikes (particularly the Inca Trail) feature a cartilage-crunching number of downhill steps. Other items that will come in handy: first-aid kit, sunscreen, sandals for camp, a down jacket for cold nights, a waterproof jacket, a warm hat and gloves, sun hat, travel towel, broken-in hiking boots, warm trekking socks, thermal underwear top and bottom, a fleece, water bottle or hydration pack, insect repellent, long pants and sunglasses. Make sure that the weight of your pack is comfortable and that you have enough juice (portable battery packs) for cameras and other electronics.
## Mountaineering
On the continent with one of the world's greatest mountain ranges, climbing opportunities are almost unlimited. Ecuador's volcanoes and the high peaks of Peru's Cordillera Blanca all offer outstanding mountaineering opportunities. Despite its relatively low elevation, Argentina's Fitz Roy Range – home to Cerro Torre, one of the world's most challenging peaks – chalks in as a major climbing destination.
In Chile, a charm bracelet of lower volcanic cones rises through the Lakes District and Torres del Paine. Popular climbs here include Volcán Osorno, which has summit ice caves.
## Rock Climbing
Argentina's Parque Nacional Los Glaciares, home to Cerro Torre and Cerro Fitz Roy, is one of the world's most important rock-climbing destinations. Cerro Torre is considered one of the five toughest climbs on the planet. The nearby town of El Chaltén is a climber's haven, and several shops offer lessons and rent equipment. If you don't have the time or talent for climbs of Cerro Torre's magnitude, there are plenty of other options.
## Mountain Biking & Cycling
From a leisurely ride around enchanting lowland scenery to bombing down smouldering volcanoes, South America has some exciting destinations for mountain bikers. The Andes are blessed with some of the most dramatic mountain-biking terrain in the world, and offer relatively easy access to mountain ranges, magnificent lakes, precolonial ruins and trails, and myriad eco-zones connected by an extensive network of footpaths and jeep roads. Mountain bikes are widely available for hire, though quality varies considerably. Make a thorough inspection of key components (brakes, tires, gears) before committing. For longer multiday trips, it's better to bring your own bike.
There is no shortage of incredible terrain in Peru. Single-track trails ranging from easy to expert await mountain bikers outside Huaraz. If you're experienced, there are incredible mountain-biking possibilities around the Sacred Valley and downhill trips to the Amazon jungle, all accessible from Cuzco.
One of **Bolivia** 's most famous mountain-biking experiences is the thrilling 3600m trip down the World's Most Dangerous Road from La Cumbre to Coroico. Outfitters in La Paz arrange excursions.
A favorite mountain-biking destination in **Chile** 's north is San Pedro de Atacama. Fabulous trips in the Lakes District access pristine areas with limited public transportation. The bike lane around Lago Llanquihue is very popular, while Río Puelo Valley and Río Cochamó Valley (also in the Lakes District) are fabulous destinations. In the south, the Carretera Austral remains one of the most popular biking destinations in South America. From December to March the road is packed with cyclists.
At outdoor hot spots in **Argentina** you can rent a mountain bike for a day of independent pedaling or for guided mountain-bike rides. Patagonia is a popular and mythical destination, with its desolate, beautiful landscapes and wide-open skies. However, be ready for fierce, multidirectional winds and rough gravel roads. Take four-season gear, even in summer, when long days and relatively warm weather make for the best touring. The classic road down here is RN 40, but cycling is tough because of the winds and lack of water; most cyclists alternate sections with Chile's Carretera Austral.
In recent years, Buenos Aires has become a more bike-friendly destination, with an expanding system of dedicated bike lanes, along with a free bike-share program.
It's hard to beat the adrenaline-charged downhills on the flanks of Cotopaxi in **Ecuador**. From Baños, you can travel 'La Ruta de las Cascadas' (Highway of the Waterfalls), a 61km (mostly) downhill ride to Puyo, with some refreshing dips in waterfalls along the way.
## Horseback Riding
Saddling up and following in the path of Chile's _huasos_ (cowboys) is a fun and easy way to experience the wilderness. Chilean horses are compact and sturdy, with amazing skill for fording rivers and climbing Andean steps. Now more than ever, multiday horseback-riding trips explore cool circuits, sometimes crossing the Andes to Argentina, on terrain that would be inaccessible otherwise. Except in the far north, opportunities can be found just about everywhere.
With strong initiatives for community-based rural tourism in the south, guided horseback riding and trekking with packhorses is a great way to discover remote areas. Rural guides charge affordable rates, provide family lodging in their own homes and offer invaluable cultural insight.
Adventure outfitters offer multilingual guides and a more elaborate range of services. Most places offer first-time riders preliminary lessons before taking to the trails. Favorites for single- or multiday horse treks are San Pedro de Atacama and around Torres del Paine.
Ecuador also has some fine horseback-riding opportunities, particularly on the slopes of Cotopaxi, a few hours south of Quito.
## Wildlife-Watching
The number of creatures great and small reaches epic proportions in the land of cloud forests, Andean mountains and Amazon rainforest. Whether you're an avid birdwatcher or just want to see monkeys in the wild, South America is hard to top. Its biodiversity is staggering.
Brazil has superb places for seeing wildlife. You can look for toucans, sloths, river dolphins and various monkey species at a jungle lodge in the Amazon. Odds are higher that you'll see even more species in the Pantanal, an extensive wetlands area that can also be accessed from Bolivia or Paraguay.
Ecuador is a major birdwatching hotspot. The Galápagos Islands have their own feathery appeal, owing to their 28 endemic species, which have evolved in extraordinary ways. Isla Santa Cruz boasts the highest bird count overall, and it's a good place to begin to find the 13 species of Darwin's finches. Various large species are easily seen around Puerto Ayora harbor, including blue-footed boobies, magnificent frigate birds and lava herons.
## River Rafting
You'll find churning white water all over the continent. The settings are spectacular: splashing through deep canyons or pounding down the forest-lined banks of a Class IV rapid.
The wealth of scenic rivers, lakes, fjords and inlets in southern Chile makes it a dream destination. Chile's rivers, raging through narrow canyons from the Andes, are world class. Northern Patagonia's Futaleufú River offers memorable Class IV and V runs. Less technical runs include the beautiful Petrohué, near Puerto Varas. Near Santiago, the Cajón del Maipo offers a gentle but enjoyable run.
In Peru, Cuzco is the launch point for the greatest variety of river-running options. Choices range from a few hours of mild rafting on the Urubamba to adrenaline-pumping rides on the Santa Teresa to several days on the Apurímac, technically the source of the Amazon. A river-running trip on the Tambopata, available from June through October, tumbles down the eastern slopes of the Andes, culminating in a couple of days of floating in unspoiled rainforest.
## Skiing & Snowboarding
Powder junkies rejoice: world-class resorts in the Chilean and Argentine Andes, South America's most important downhill ski areas, offer myriad possibilities for skiing, snowboarding and even heliskiing (the season is roughly June to September). Don't expect too many bargains; resorts are priced to match their quality. 'First descents' of Chilean Patagonia's numerous mountains is a growing (but limited) trend.
Most resorts in Chile are within an hour's drive of Santiago, including a wide variety of runs at family-oriented La Parva, all-levels El Colorado, and Valle Nevado, with a lot of terrain and renowned heliskiing. Legendary Portillo, the site of several downhill speed records and the summer training base for many of the Northern Hemisphere's top skiers, is northeast of Santiago near the Argentine border crossing to Mendoza.
## Surfing
Brazil is South America's best-known surfing destination, with great breaks near Rio and in the southeast, and sprinkled all along the coast.
With breaks lining the long Pacific coast, Chile nurtures some serious surf culture, most active in middle and northern Chile. Keep in mind, you'll need a wetsuit. The coastal Ruta 1 is lined with waves.
Ecuador's best breaks are off Isla San Cristóbal in the Galápagos.
## Hang-Gliding & Paragliding
Rio de Janeiro is probably one of the world's most scenic places to go hang-gliding, offering memorable tandem flights over tropical rainforest with views of the beach and island-filled ocean in the distance. Paragliding _(parapente)_ can also be arranged in Rio.
Survival Guide
#### Directory A–Z
#### Accessible Travel
#### Accommodations
#### Customs Regulations
#### Electricity
#### Discount Cards
#### Emergency & Important Numbers
#### Food
#### Health
#### Insurance
#### Internet Access
#### Legal Matters
#### LGBT Travelers
#### Maps
#### Money
#### Opening Hours
#### Photography
#### Post
#### Safe Travel
#### Telephone
#### Toilets
#### Tourist Information
#### Visas
#### Women Travelers
#### TRANSPORT
#### Getting There & Away
#### Getting Around
# Directory A–Z
## MAccessible Travel
In general, South America is not well set up for travelers with disabilities, but the more modernized Southern Cone countries are more accommodating – notably Chile, Argentina and the bigger cities of Brazil. For city travel, Santiago is the most accessible city in South America. Things are slowly improving in some places, particularly in Ecuador, which now has a president who uses a wheelchair.
Unfortunately, cheap local lodgings probably won't be well equipped to deal with physically challenged travelers; air travel will be more feasible than most local buses (although this isn't impossible); and well-developed tourist attractions will be more accessible than off-the-beaten-track destinations. Start your research here:
**Lonely Planet** (<http://lptravel.to/AccessibleTravel>) Free Accessible Travel guides.
**Emerging Horizons** (www.emerginghorizons.com) Features well-written articles and regular columns full of handy advice.
**Go Wheel the World** (<http://gowheeltheworld.com>) Chile-based outfit that leads excellent adventure tours. Check the website for inspiring documentaries about traveling off the beaten path.
**Mobility International** (www.miusa.org) This US-based outfit advises travelers with disabilities and runs educational-exchange programs overseas.
**Royal Association for Disability and Rehabilitation** (www.disabilityrightsuk.org) Good resource for travelers.
**Society for Accessible Travel & Hospitality** (www.sath.org) Good, general travel information; based in the USA.
## iAccommodations
South America has plenty of variety, with options for all budgets. Be sure to book well in advance when visiting during big festivals (particularly Carnaval time in Brazil) and in popular resort areas during high season.
### Costs
Costs vary from country to country, with Andean countries (especially Bolivia) being the cheapest (from around US$10 per night) and Chile, Brazil and Argentina the costliest (upwards of US$30).
### Camping
Camping is an obvious choice in parks and reserves and is a useful budget option in pricier countries such as Chile. Bring all your own gear. While camping gear is available in large cities and in trekking and activities hubs, it's expensive and choices are usually minimal. Camping gear can be rented in areas with substantial camping and trekking action (eg the Lakes District and Huaraz), but quality is sometimes dubious.
An alternative to tent camping is staying in _refugios_ (simple structures within parks and reserves), where a basic bunk and kitchen access are usually provided. For climbers, most summit attempts involve staying in a _refugio._
Book Your Stay
For more accommodation reviews by Lonely Planet authors, check out <http://hotels.lonelyplanet.com/southamerica>. You'll find independent reviews, as well as recommendations on the best places to stay. Best of all, you can book online.
### Hostels
_Albergues_ (hostels) have become increasingly popular throughout South America and, as throughout the world, are great places to socialize with other travelers. Across South America, there are numerous private hostels as well as scores of official _albergue juvenil_ (youth hostel), where you can get a small discount if you're a card-carrying member of Hostelling International–American Youth Hostel (HI-USA).
### **Hotels & Guesthouses**
When it comes to hotels, both terminology and criteria vary. The costliest in the genre are _hoteles_ (hotels) proper. A step down in price are _hostales_ (small hotels or guesthouses). The cheapest places are _hospedajes,_ _casas de huéspedes,_ _residenciales,_ _alojamientos_ and _pensiones._ A room in these places includes a bed with (hopefully) clean sheets and a blanket, maybe a table and chair and sometimes a fan. Showers and toilets may be shared; there may not be hot water. Cleanliness varies widely, but many places are remarkably tidy. In some areas, especially southern Chile, the cheapest places may be _casas familiares,_ family houses whose hospitality makes them excellent value.
In Brazil, Argentina and some other places, prices often include breakfast, the quality of which is usually directly related to the room price.
Hot-water supplies are often erratic, or may be available only at certain hours of the day. It's something to ask about (and consider paying extra for), especially in the highlands and far south, where it gets cold.
When showering, beware the electric showerhead, an innocent-looking unit that heats cold water with an electric element. Don't touch the shower head or anything metal when the water is on or you may get shocked – never strong enough to throw you across the room, but hardly pleasant.
## ECustoms Regulations
Customs vary slightly from country to country, but you can generally bring in personal belongings, camera gear, laptops, handheld devices and other travel-related gear. All countries prohibit the export (just as home countries prohibit the import) of archaeological items and goods made from rare or endangered animals. Avoid carrying plants, seeds, fruits and fresh meat products across borders.
## RElectricity
Electricity is not standard across South America. Voltage ranges from 100V to 240V, with the most common plug types being flat-pronged American style and rounded European style.
## aDiscount Cards
The HI-USA membership card can be useful in some places (particularly in Brazil, where there are dozens of HI-affiliated lodging options.
An International Student Identity Card (ISIC) can provide discounted admission to archaeological sites and museums. It may also entitle you to reductions on bus, train and air tickets. In less developed countries, student discounts are rare, although high-ticket items such as the entrance to Machu Picchu (discounted nearly 50% for ISIC holders under 26) may be reduced. In some countries, such as Argentina, almost any form of university identification will suffice where discounts are offered.
## KEmergency & Important Numbers
There is no continent-wide emergency number.
**Argentina** | %107 ambulance, %100 fire, %101 police
---|---
**Bolivia** | %123 ambulance, fire & police
**Brazil** |
%192 ambulance, %193 fire, %190 police
**Chile** |
%131 ambulance, %132 fire, %133 police
**Colombia** |
%123 ambulance, fire & police
**Ecuador** |
%131 ambulance, %102 fire, %101 police
**Peru** |
%106 ambulance, %116 fire, %105 police
## kFood
Reservations aren't necessary for casual eateries, but are recommended for midrange and top-end restaurants. Reserve several weeks in advance for the best places.
**Cafes** Snacking and sipping with less formality amid a bar or coffeeshop vibe.
**Cevicherías** In Peru and Ecuador you'll find casual lunch spots serving up fresh fish marinated in lime juice, with many variations on the theme.
**Grill houses** Known as _parrillas_ in Argentina and _churrascarias_ in Brazil, these places specialize in barbecue meat.
**Pay-by-Weight** You'll find extensive lunch buffets in some places (particularly in Brazil), where you select what you want and then pay per 100 grams.
**Restaurants** Ranging from fine dining to basic, these sometimes serve a discounted set menu known as _menú del día_ at lunch.
## FHealth
Prevention is the key to staying healthy while in South America. Travelers who receive the recommended vaccines and follow common-sense precautions usually go away with nothing more than a little diarrhea.
### Before You Go
#### Recommended Vaccinations
Since most vaccines don't produce immunity until at least two weeks after they're given, visit a physician four to eight weeks before departure. Ask your doctor for an International Certificate of Vaccination (otherwise known as the yellow booklet), which will list all the vaccinations you've received.
The only required vaccine is yellow fever, and that's only if you're arriving from a yellow-fever-infected country in Africa or the Americas. However, a number of vaccines are recommended.
#### Health Insurance
If your health insurance doesn't cover you for medical expenses abroad, consider getting extra insurance. Find out in advance if your insurance plan will make payments directly to providers or reimburse you later for overseas health expenditures. (In many countries doctors expect payment in cash.)
#### Websites
There is a wealth of travel health advice on the internet. The World Health Organization (www.who.int/ith) maintains up-to-date info on disease outbreaks. Another resource of general interest is MD Travel Health (www.mdtravelhealth.com), which provides complete travel health recommendations for every country in the world.
It's usually a good idea to consult your government's travel health website before departure, if one is available:
**Australia** (www.smartraveller.gov.au)
**Canada** (www.travelhealth.gc.ca)
**UK** (www.fco.gov.uk)
**USA** (wwwnc.cdc.gov/travel)
### In South America
Good medical care may be more difficult to find in smaller cities and impossible to locate in rural areas. Many doctors and hospitals expect payment in cash, regardless of whether you have travel health insurance. If you develop a life-threatening medical problem, you'll probably want to be evacuated to a country with state-of-the-art medical care. Since this may cost tens of thousands of dollars, be sure you have insurance to cover this before you depart. You can find a list of medical-evacuation and travel-insurance companies on the US State Department website (<http://travel.state.gov>).
#### Infectious Diseases
##### Dengue
Dengue fever is a viral infection found throughout South America. Dengue is transmitted by Aedes mosquitoes, which bite preferentially during the daytime and are usually found close to human habitations, often indoors. Dengue is especially common in densely populated, urban environments.
Dengue usually causes flu-like symptoms, including fever, muscle aches, joint pains, headaches, nausea and vomiting, often followed by a rash. The body aches may be quite uncomfortable, but most cases resolve uneventfully in a few days.
There is no treatment for dengue fever except to take analgesics such as acetaminophen/paracetamol (Tylenol) and drink plenty of fluids. Severe cases may require hospitalization for intravenous fluids and supportive care. There is no vaccine. The cornerstone of prevention is protection against insects.
Keep an eye out for outbreaks in areas where you plan to visit. A good website on the latest information is the CDC (wwwnc.cdc.gov/travel).
##### Hepatitis A
Hepatitis A is the second most common travel-related infection (after traveler's diarrhea). It's a viral infection of the liver that's usually acquired by ingestion of contaminated water, food or ice, though it may also be acquired by direct contact with infected persons. Symptoms may include fever, malaise, jaundice, nausea, vomiting and abdominal pain. Most cases resolve themselves without complications, though hepatitis A occasionally causes severe liver damage.
The vaccine for hepatitis A is extremely safe and highly effective. If you get a booster six to 12 months later, it lasts for at least 10 years. You really should get it before you go to any developing nation.
##### Malaria
Malaria occurs in every South American country except Chile and Uruguay. It's transmitted by mosquito bites, usually between dusk and dawn. The main symptom is high spiking fevers, which may be accompanied by chills, sweats, headache, body aches, weakness, vomiting or diarrhea. Severe cases may involve the central nervous system and lead to seizures, confusion, coma and death.
There is a choice of three malaria pills, all of which work about equally well. Protecting yourself against mosquito bites is just as important as taking malaria pills, since none of the pills are 100% effective.
If you develop a fever after returning home, see a physician, as malaria symptoms may not occur for months.
##### Typhoid
Typhoid fever is caused by ingestion of food or water contaminated by a species of salmonella known as _Salmonella typhi_. Fever occurs in virtually all cases. Other symptoms may include headache, malaise, muscle aches, dizziness, loss of appetite, nausea and abdominal pain. Either diarrhea or constipation may occur. Possible complications include intestinal perforation, intestinal bleeding, confusion, delirium or (rarely) coma.
Unless you expect to take all your meals in major hotels and restaurants, the typhoid vaccine is a good idea.
##### Yellow Fever
Yellow fever is a life-threatening viral infection transmitted by mosquitoes in forested areas. The illness begins with flu-like symptoms, which may include fever, chills, headache, muscle aches, backache, loss of appetite, nausea and vomiting. These symptoms usually subside in a few days, but one person in six enters a second, toxic phase characterized by recurrent fever, vomiting, listlessness, jaundice, kidney failure and hemorrhage, leading to death in up to half of the cases. There is no treatment except for supportive care.
Yellow-fever vaccine can be given only in approved yellow-fever vaccination centers, which provide validated International Certificates of Vaccination (yellow booklets). The vaccine should be given at least 10 days before any potential exposure to yellow fever and remains effective for approximately 10 years.
##### Zika
Zika virus is primarily transmitted by infected mosquitoes, typically active from dawn to dusk. It can be transmitted from a pregnant woman to her fetus. Human transmission can also occur through unprotected sex, and on occasion through saliva and urine. Symptoms include mild fever, headache, muscle and joint pain, nausea, vomiting and general malaise. Symptoms may present three to 12 days after being bitten.
The best prevention is to wear long sleeves, repellent with 20% to 30% DEET and avoid being outdoors at dawn and dusk when mosquitoes are most common. High-altitude destinations are not considered a risk.
#### Altitude Sickness
Altitude sickness may develop in those who ascend rapidly to altitudes greater than 2500m. Being physically fit offers no protection. Those who have experienced altitude sickness in the past are prone to future episodes. The risk increases with faster ascents, higher altitudes and greater exertion. Symptoms may include headaches, nausea, vomiting, dizziness, malaise, insomnia and loss of appetite. Severe cases may be complicated by fluid in the lungs (high-altitude pulmonary edema) or swelling of the brain (high-altitude cerebral edema).
When traveling to high altitudes, it's also important to avoid overexertion, eat light meals and abstain from alcohol.
If your symptoms are more than mild or don't resolve promptly, see a doctor. Altitude sickness should be taken seriously; it can be life-threatening when severe.
#### Cold Exposure & Hypothermia
Cold exposure may be a significant problem in the Andes, particularly at night. Be sure to dress warmly, stay dry, keep active, consume plenty of food and water, get enough rest, and avoid alcohol, caffeine and tobacco.
Hypothermia occurs when the body loses heat faster than it can produce it and the core temperature of the body falls. If you're trekking at high altitudes or simply taking a long bus trip over mountains, particularly at night, be prepared. In the Andes, you should always be prepared for cold, wet or windy conditions even if it's just for a few hours. It is best to dress in layers. A hat is also important.
#### Traveler's Diarrhea
To prevent diarrhea, avoid tap water unless it has been boiled, filtered or chemically disinfected (with iodine tablets); only eat fresh fruits or vegetables if cooked or peeled; be wary of dairy products that might contain unpasteurized milk; and be highly selective when eating food from markets and street vendors.
If you develop diarrhea, be sure to drink plenty of fluids, preferably an oral rehydration solution containing salt and sugar. Gastrolyte works well for this. A few loose stools don't require treatment but you may want to take antibiotics if you start having more than three watery bowel movements within 24 hours, and it's accompanied by at least one other symptom – fever, cramps, nausea, vomiting or generally feeling unwell. Effective antibiotics include Norfloxacin, Ciprofloxacin or Azithromycin – all will kill the bacteria quickly. Note that an antidiarrheal agent (such as loperamide) is just a 'stopper' and doesn't get to the cause of the problem. Don't take loperamide if you have a fever or blood in your stools. Seek medical attention quickly if you don't respond to an appropriate antibiotic.
#### Tap Water
Tap water is generally not safe to drink. Vigorous boiling for one minute is the most effective means of water purification. At altitudes greater than 2000m, boil for three minutes.
Other methods of treating water include using a handheld ultraviolet light purifier (such as a SteriPEN), iodine and water filters.
**Argentina** Generally safe to drink, but best to check in rural areas.
**Bolivia** Not safe to drink.
**Brazil** Generally safe to drink in major cities; not safe in rural areas.
**Chile** Tap water is generally safe to drink from Santiago down to Patagonia, but generally unwise in the Atacama Desert.
**Colombia** Not safe to drink, except in Bogotá and Cartagena.
**Ecuador** Not safe to drink.
**Peru** Not safe to drink.
Recommended Vaccinations
**Vaccine** | **Recommended for** | **Dosage** | **Side effects**
---|---|---|---
Chickenpox | Travelers who've never had chickenpox | 2 doses one month apart | Fever; mild case of chickenpox
Hepatitis A | All travelers | One dose before trip; booster 6-12 months later | Soreness at injection site; headaches; body aches
Hepatitis B | Long-term travelers in close contact with the local population | 3 doses over 6-month period | Soreness at injection site; low-grade fever
Measles | Travelers born after 1956 who've had only one measles vaccination | 1 dose | Fever; rash; joint pains; allergic reactions
Rabies | Travelers who may have contact with animals and may not have access to medical care | 3 doses over 3- to 4-week period | Soreness at injection site; headaches; body aches
Tetanus-diphtheria | Travelers who haven't had booster within 10 years | 1 dose lasts 10 years | Soreness at injection site
Typhoid | All travelers | 4 oral capsules, 1 taken every other day | Abdominal pain; nausea; rash
Yellow fever | Travelers to jungle areas at altitudes below 2300m | 1 dose lasts 10 years | Headaches; body aches; severe reactions are rare
## aInsurance
A travel-insurance policy covering theft, loss, accidents and illness is highly recommended. Many policies include a card with toll-free numbers for 24-hour assistance, and it's good practice to carry it with you. Note that some policies compensate travelers for misrouted or lost luggage. Baggage insurance is worth its price in peace of mind. Also check that the coverage includes worst-case scenarios: ambulances, evacuations or an emergency flight home. Some policies specifically exclude 'dangerous activities,' such as scuba diving, motorcycling or even trekking. If such activities are on your agenda, avoid this sort of policy.
There are a wide variety of policies available and your travel agent will be able to make recommendations. The policies handled by student-travel organizations usually offer good value. If a policy offers lower and higher medical-expense options, the low-expenses policy should be OK for South America – medical costs are not nearly as high here as elsewhere in the world.
If you have baggage insurance and need to make a claim, the insurance company may demand a receipt as proof that you bought the stuff in the first place. Make a list of stolen items and their value. At the police station, you complete a _denuncia_ (statement), a copy of which is given to you for your insurance claim.
Worldwide travel insurance is available at www.lonelyplanet.com/bookings. You can buy, claim and extend online anytime – even if you're already on the road.
## IInternet Access
Wi-fi access is widely available, with many hostels, cafes and guesthouses offering free wi-fi. In contrast, internet cafes are a rarity.
## HLegal Matters
In city police stations, an English-speaking interpreter is rare. In most cases you'll either have to speak the local language or provide an interpreter. Some cities have a tourist police service, which can be more helpful.
If you are robbed, photocopies (even better, certified copies) of original passports, visas and air tickets and careful records of credit-card numbers and traveler's checks will prove invaluable during replacement procedures. Replacement passport applications are usually referred to your home country, so it helps to leave a copy of your passport details with someone back home.
## tLGBT Travelers
Buenos Aires, Rio de Janeiro, São Paulo and Santiago are the most gay-friendly cities, though gay couples are openly out only in certain neighborhoods. Elsewhere on the continent, where public displays of affection by same-sex couples may get negative reactions, do as the locals do – be discreet to avoid problems.
Despite a growing number of publications and websites devoted to gay travel, few have specific advice on South America. One exception is Purple Roofs (www.purpleroofs.com), an excellent guide to gay-friendly accommodations throughout South America.
## YMaps
International Travel Maps & Books (www.itmb.com) produces a range of excellent maps of Central and South America. For the whole continent, it has a reliable reference map at a 1:4,000,000 scale and a commemorative edition of its classic 1:5,000,000 map. The maps are huge for road use, but they're helpful for pretrip planning. More detailed ITMB maps are available for the Amazon Basin and every country in South America. All are available on the ITMB website.
Maps of the South American continent as a whole are widely available; check any well-stocked map or travel bookstore.
## aMoney
It's convenient to have a small wad of US dollars tucked away (in US$20 denominations and less; US$100 bills are difficult to exchange). US currency is by far the easiest to exchange throughout South America. Of course, unlike traveler's checks, nobody will give you a refund for lost or stolen cash. When you're about to cross from one country to another, it's handy to change some cash. Trying to exchange worn notes can be a hassle, so procure crisp bills before setting out.
In some countries, especially in rural areas, _cambio_ (change) can be particularly hard to come by. Businesses even occasionally refuse to sell you something if they can't or don't want to change your note. So break down those larger bills whenever you have the opportunity, such as at busy restaurants, banks and larger businesses.
### ATMs
ATMs are available in most cities and large towns, and are almost always the most convenient, reliable and economical way of getting cash. The rate of exchange is usually as good as any bank or legal money changer. Many ATMs are connected to the Cirrus or Plus network, but many countries prefer one over the other. If your ATM card gets swallowed by a machine, generally the only thing you can do is call your bank and cancel the card. Although such events are rare, it's well worth having an extra ATM card (to a different account), should something go wrong.
If possible, sign up with a bank that doesn't charge a fee for out-of-network ATM withdrawals. Also, find a bank that offers a low exchange-rate fee (1% to 2%). Before hitting the road, call your bank, informing them of your travel plans – that way the bank won't put a hold on foreign withdrawals while you're on the road.
Many ATMs will accept a personal identification number (PIN) of only four digits; find out whether this applies to the specific countries you're traveling to before heading off.
### Bargaining
Bargaining is accepted and expected when contracting long-term accommodations and when shopping for craft goods in markets. Haggling is a near sport in the Andean countries, with patience, humor and respect serving as the ground rules of the game. Bargaining is much less common in the Cono Sur (Southern Cone; a collective term for Argentina, Chile, Uruguay and parts of Brazil and Paraguay).
### Credit Cards
Visa and MasterCard are accepted at most large stores, travel agencies and better hotels and restaurants. Credit-card purchases sometimes attract an extra _recargo_ (surcharge) on the price (from 2% to 10%), but they are usually billed to your account at favorable exchange rates. Some banks issue cash advances on major credit cards. The most widely accepted card is Visa, followed by MasterCard (those with UK Access should insist on its affiliation with MasterCard). American Express is accepted at fewer places.
### Exchanging Money
Traveler's checks and foreign cash can be changed at _casas de cambio_ (currency-exchange offices) or banks. Rates are usually similar, but _casas de cambio_ are quicker, less bureaucratic and open longer hours.
It is preferable to bring money in US dollars, although banks and _casas de cambio_ in capital cities will change euros, pounds sterling, Japanese yen and other major currencies. Changing these currencies in smaller towns and on the street is next to impossible.
### Traveler's Checks
Traveler's checks are not nearly as convenient as ATM cards, and you may have difficulty cashing them – even at banks. High commissions (from 3% to upwards of 10%) also make them an unattractive option. If you do take traveler's checks, American Express is the most widely accepted brand, while Visa, Thomas Cook and Citibank are the next best options. To facilitate replacement in case of theft, keep a record of check numbers and the original bill of sale in a safe place. Even with proper records, replacement can be a tedious, time-intensive process.
### Tipping
**Restaurants** In some countries (such as Brazil and Chile), a 10% service charge is typically included.
**Tours** When booking tours (such as to the Galápagos or the Amazon), it's customary to tip your guide – from a few dollars per day to 15%, depending on service.
**Taxis** Not expected, though you can round up the bill.
**Bars** Not expected.
### Fraud
Unfortunately, ATM-card cloning is a big worry in Brazil and your account can be drained of thousands of dollars before you even realize it.
## JOpening Hours
On Sunday, nearly everything is closed. In the Andean countries, businesses tend to close earlier.
**Banks** Monday to Friday (for money changing).
**Businesses** 8am or 9am–noon and 2pm–8pm or 9pm Monday to Friday. Shorter hours on Saturday.
## APhotography
Consumer electronics are readily available throughout South America, but taxes can kick prices through the roof. You'll probably find better deals at home. _Lonely Planet's Guide to Travel Photography_ is full of helpful tips for photography while on the road.
## CPost
International postal rates can be quite expensive. Generally, important mail and parcels should be sent by registered or certified service; otherwise, they may go missing. Sending parcels can be awkward: often an _aduana_ (customs) officer must inspect the contents before a postal clerk can accept them, so wait to seal your package until after it has been checked. Most post offices have a parcels window, usually signed _encomiendas_ (parcels). The place for posting overseas parcels is sometimes different from the main post office.
UPS, FedEx, DHL and other private postal services are available in some countries, but are prohibitively expensive.
Etiquette
**Greetings** These are important. In Spanish-speaking countries, greet people you encounter with _buenos días_ (good morning), _buenas tardes_ (good afternoon) or _buenas noches_ (good evening). Use _bom dia,_ _boa tarde_ and _boa noite_ in Brazil.
**Shops** Always greet people when entering and exiting a shop.
**Cheek kissing** When meeting people socially, give _besos_ (kisses) on the cheek (both cheeks for Brazilians). Men shake hands.
**Shorts** Dress for the occasion; for example, usually only tourists and athletes wear shorts in Buenos Aires.
**Photographs** Ask before photographing people, particularly in indigenous communities – payment may be requested.
## LSafe Travel
To avoid becoming a victim, use common sense and take general precautions throughout South America:
o Carry only the minimum cash needed when out for the day.
o Dress down, leave the jewelry at home and don't walk around flashing iPhones, iPads and other expensive electronics.
o Be alert and walk purposefully. Criminals home in on dopey, hesitant, disoriented-looking individuals.
o Use ATMs inside buildings. Before doing so, be very aware of your surroundings.
o Check windows and doors of your room for security, and don't leave anything valuable lying around.
### Theft
Theft can be a problem, but remember that fellow travelers can also be accomplished crooks, so where there's a backpacker scene, there may also be thievery. Here are some common-sense suggestions to limit your liability:
o Even if you're just running down the hall, never leave your hotel door unlocked.
o Always conceal your money belt and its contents, preferably beneath your clothing.
o Keep your spending money separate from the big stuff (credit cards, tickets etc).
o Be aware of the risk of bag slashing and the theft of your contents on buses. Keep close watch on your belongings – the bag isn't safe under your seat, above your head or between your legs (it's better on your lap). Be mindful in crowded markets or terminals where thefts are more likely to occur.
o When exploring cities, consider ditching the daypack and carrying what you need in a plastic bag to deter potential thieves.
### Tours
There are loads of great adrenaline activities on offer, from rafting to mountain biking, but do your research on an agency before joining a tour. Travelers have lost their lives owing to poorly maintained equipment and reckless, ill-prepared guides. It's never wise to choose an operator based on cost alone.
### Druggings
Lonely Planet has received correspondence from travelers who were unwittingly drugged and robbed after accepting food from a stranger.
Be very careful in bars as there are occasional reports of folks being unwittingly drugged then raped or robbed. Always keep a close eye on your drink and be cautious when meeting new friends.
## KTelephone
### Mobile Phones
Cell (mobile) phone numbers in South America often have different area codes that are different to fixed-line numbers, even if the cell-phone owner resides in the same city. Calling a cell-phone number is always more expensive (sometimes exorbitantly so) than calling a fixed line.
If you plan to travel with a smartphone, you may want to purchase an international plan or local SIM to minimize (what could be) enormous costs. Remember it's possible to call internationally for free or very cheaply using Skype or other Voice over Internet Protocol (VoIP) systems.
### Phone Cards
Aside from Skype, the cheapest way to make an international call is by using a phone card, the type you purchase at a kiosk or corner store. These allow you to call North America or Europe for as little as US5¢ per minute with a good card. The caveat is that you need a private phone line or a permissive telephone kiosk operator to use them.
## GToilets
There are two toilet rules for South America: always carry your own toilet paper and don't ever throw anything into the toilet bowl. Except in the most developed places, South American sewer systems can't handle toilet paper, so all paper products must be discarded in the wastebasket. Another general rule is to use public bathrooms whenever you can, as you never know when your next opportunity will be. Folks posted outside bathrooms proffering swaths of paper require payment.
## CTourist Information
Every country in South America has government-run tourist offices, but their quality and breadth of coverage vary.
## EVisas
Some travelers – including those from the USA – may require visas to enter several countries, including Bolivia and Brazil. Arrange these well in advance of your departure. Some countries don't generally require visas but may require a reciprocity fee (such as the $117 fee Chile charges to Australian travelers), paid upon arrival. If no visa is required, a tourist card is issued upon arrival. See individual countries for more details. Suriname charges U$35 for a tourist card, available upon arrival if flying in; if coming overland, get this from a Surinamese embassy before heading to the border.
Carry a handful of passport-sized photos for visa applications. Hold onto any entry-exit cards you are given. There can be serious fines and complications if you lose them!
If you need a visa for a country and arrive at a land border without one, be prepared to backtrack to the nearest town with a consulate to get one. Airlines won't normally let you board a plane for a country to which you don't have the necessary visa. Also, a visa in itself does not guarantee entry: you may still be turned back at the border if you don't have 'sufficient funds' or an onward or return ticket.
## OWomen Travelers
At one time or another, solo women travelers will find themselves the object of curiosity – sometimes well intentioned, sometimes not. Avoidance is an easy, effective self-defense strategy. In the Andean region, particularly in smaller towns and rural areas, modest dress and conduct are the norm, while in Brazil and the more liberal Cono Sur, standards are more relaxed, especially in beach areas.
_Machista_ (macho) attitudes, stressing masculine pride and virility, are fairly widespread among South American men (although less so in indigenous communities). They are often expressed by boasting and in exaggerated attention toward women. Snappy put-down lines or other caustic comebacks to unwanted advances may make the man feel threatened, and he may respond aggressively. Most women find it easier to invent a husband.
There have been isolated cases of South American men raping women travelers. Women trekking or taking tours in remote or isolated areas should be especially cautious. Some cases have involved guides assaulting tour group members, so it's worth double-checking the identity and reputation of any guide or tour operator. Also be aware that women (and men) have been drugged, in bars and elsewhere, using drinks, cigarettes or pills. Police may not be very helpful in rape cases – if a local woman is raped, her family usually seeks revenge instead of calling the police. Tourist police may be more sympathetic, but it's possibly better to see a doctor and contact your embassy before reporting a rape to police.
Tampons are generally difficult to find in smaller towns, so stock up in cities or bring a supply from home.
# Transport
## (Getting There & Away
Flights, cars and tours can be booked online at lonelyplanet.com/bookings.
### Air
North American, European and Australian airlines offer regular South American connections.
**Argentina** The main airports are **Aeropuerto Internacional Ministro Pistarini** (Ezeiza; %011-5480-6111; www.aa2000.com.ar) and **Aeroparque Jorge Newbery** ( %011-5480-6111; www.aa2000.com.ar; Av Rafael Obligado; g33, 45), both in Buenos Aires. There are several other international airports around Argentina; find info online at Aeropuertos Argentina 2000 (www.aa2000.com.ar). Aerolíneas Argentinas (www.aerolineas.com.ar) is the national carrier.
**Bolivia** The principal international airport is La Paz' **El Alto International Airport** (LPB; Héroes Km 7, El Alto). The national airline is the state-owned Boliviana de Aviación (www.boa.bo), which has international flights to Madrid, Barcelona and Miami.
**Brazil** The most popular international gateways are **Galeão International Airport** (Aeroporto Internacional Antônio Carlos Jobim; %21 3004-6050; www.riogaleao.com; Domestic Arrival Hall, Av Vinte de Janeiro) in Rio de Janeiro and São Paulo's **GRU Airport** (Aeroporto Guarulhos; %11 2445-2945; www.gru.com.br; Rod Hélio Smidt s/n). **Salvador** (SSA; %71 3204-1010; Praça Gago Coutinho s/n, São Cristóvão) receives a few direct scheduled flights from Europe. Though headquartered in Chile, LATAM (www.latam.com) is Brazil's largest international carrier.
**Chile** Santiago's **Aeropuerto Internacional Arturo Merino Benítez** (Santiago International Airport, SCL; %2-2690-1796; www.nuevopudahuel.cl) is the country's main gateway. LATAM (www.latam.com) is the chief international airline serving Chile.
**Colombia** **Aeropuerto Internacional El Dorado** ( %1-266-2000; www.eldorado.aero; Av El Dorado) in Bogotá is the main gateway. Avianca (www.avianca.com) is the national carrier.
**Ecuador** **Quito** ( %02-395-4200; www.aeropuertoquito.aero) and **Guayaquil** (GYE; %04-216-9000; www.tagsa.aero; Av de las Américas) airports are both international hubs. TAME (www.tame.com.ec) is the national carrier but Avianca (www.avianca.com), based in Quito, is close behind.
**Peru** Lima's **Aeropuerto Internacional Jorge Chávez** ( %01-517-3500, schedules 01-511-6055; www.lima-airport.com; Callao) is the country's major hub. LATAM (www.latam.com) has the most flights domestically and internationally.
### Land
From North America, you can journey overland only as far south as Panama. There is no road connection onward to Colombia: the Carretera Panamericana (Pan-American Hwy) ends in the vast wilderness of the Darién Province, in southeast Panama. This roadless area between Central and South America is called the Darién Gap. In the past it has been difficult, but possible, to trek across the gap with the help of local guides, but since around 1998 it has been prohibitively dangerous, especially on the Colombian side. The region is overrun with smugglers and is positively unsafe.
Climate Change & Travel
Every form of transport that relies on carbon-based fuel generates CO2, the main cause of human-induced climate change. Modern travel is dependent on aeroplanes, which might use less fuel per kilometre per person than most cars but travel much greater distances. The altitude at which aircraft emit gases (including CO2) and particles also contributes to their climate change impact. Many websites offer 'carbon calculators' that allow people to estimate the carbon emissions generated by their journey and, for those who wish to do so, to offset the impact of the greenhouse gases emitted with contributions to portfolios of climate-friendly initiatives throughout the world. Lonely Planet offsets the carbon footprint of all staff and author travel.
### Sea
One of the most popular modes of travel between South and Central America is by booking passage on one of the foreign sailboats that travel between Cartagena (Colombia) and the San Blás islands, with some boats continuing to Colón (Panama). The typical passage takes about five days and costs between US$450 and US$650. A good source of information regarding schedules and available berths is at **Blue Sailing** ( %310-704-0425, 300-829-2030; www.bluesailing.net; Calle San Andrés No 30-47; 5-day trips US$450-650; h9am-5pm Mon-Sat) in Cartagena. Do some serious research before joining any tour; there are many unsavory operators out there and a few boats have even sunk.
A less expensive way to reach Panama from Colombia is via small boat from Capurgana to Puerto Obaldia (COP$30,000 for the 30-minute trip), from where you can take a domestic flight to Panama City (US$115, flying Tuesdays, Thursdays and Sundays) or continue up through the San Blás islands.
Officially, both Panama and Colombia require an onward or return ticket as a condition of entry. This may not be enforced in Colombia, but it's wise to get one anyway, or have lots of money and a plausible itinerary. Panama requires a visa or tourist card, an onward ticket and sufficient funds, and has been known to turn back arrivals who don't meet these requirements.
There are occasional reports of pirate attacks off the coast of South America, most of which occur in the Caribbean region.
## PGetting Around
### Air
There is an extensive network of domestic flights, with refreshingly low price tags, especially in the Andean countries (Bolivia, Ecuador and Peru). After 18-hour bus rides across mountainous terrain on atrocious roads, you may decide to take the occasional flight.
There are drawbacks to flying, however. Airports are often far from city centers, and public buses don't run all the time, so you may end up spending a bit on taxis (it's usually easier to find a cheap taxi _to_ an airport than _from_ one). Airport taxes also add to the cost of tickets; they are always higher for international departures. If safety concerns you, check out the 'Fatal Events by Airline' feature at www.airsafe.com.
Avoid scheduling a domestic flight with a close connection for an international flight or vice versa. Reconfirm all flights 48 hours before departure and allow ample extra time at the airport.
#### Air Passes
A few South America air passes exist and can save you a bit of money, provided you can deal with a fixed itinerary. These mileage-based passes allow travelers to fly between cities in a limited set of countries. The restrictions vary, but flights must be completed within a period ranging from 30 days to 12 months. You'll pay higher rates (or be ineligible) if you arrive in South America on a carrier other than the one sponsoring the air pass.
**Gol South America Airpass** (www.voegol.com.br) Includes Brazil, Argentina, Bolivia, Chile, Paraguay and Uruguay.
**One World Alliance Visit South America Airpass** (www.oneworld.com) Includes Argentina, Bolivia, Brazil, Chile, Colombia, Ecuador, Paraguay, Peru, Uruguay and Venezuela.
**LATAM South American Airpass** One of the most extensive networks around the continent; covers some 124 different destinations in Argentina, Bolivia, Brazil, Chile, Colombia, Ecuador, Paraguay, Peru, Uruguay and Venezuela.
### Bicycle
Cycling South America is a challenging yet highly rewarding alternative to public transport. While better roads in Argentina and Chile make the Cono Sur (Southern Cone; a collective term for Argentina, Chile, Uruguay and parts of Brazil and Paraguay) countries especially attractive, the entire continent is manageable by bike, or – more precisely – by mountain bike. Touring bikes are suitable for paved roads, but only a _todo terreno_ (mountain bike) allows you to tackle the spectacular back roads (and often main roads!) of the Andes.
There are no multicountry bike lanes or designated routes. Mountain bikers have cycled the length of the Andes. As for road rules, forget it – except for the logical rule of riding with traffic on the right-hand side of the road, there are none. Hunt down good maps that show side roads, as you'll have the enviable ability to get off the beaten track at will.
Bring your own bicycle since locally manufactured ones are less dependable and imported bikes are outrageously expensive. Bicycle mechanics are common even in small towns, but will almost invariably lack the parts you'll need. Before setting out, learn bicycle mechanics and purchase spares for the pieces most likely to fail. A basic road kit will include extra spokes and a spoke wrench, a tire patch kit, a chain punch, inner tubes, spare cables and a cycling-specific multitool. Some folks box up spare tires, leave them with a family member back home and have them shipped to South America when they need them.
Drawbacks to cycling include the weather (fierce rains, blasting winds), high altitude in the Andes, poor roads and reckless drivers – the biggest hazard for riders. Safety equipment such as reflectors, mirrors and a helmet are highly recommended. Security is another issue: always take your panniers with you and lock your bike (or pay someone to watch it) while you sightsee and bring your bike into your hotel room overnight.
### Boat
From cruises through the mystical fjords of Chilean Patagonia and riverboat chugs up the Amazon to outboard canoe travel in the coastal mangroves of Ecuador, South America offers ample opportunity to travel by boat. Safety is generally not an issue, especially for the established ferry and cruise operators in Chile and Argentina. There have been a couple of problems with tourist boats in the Galápagos (including a few that have sunk over the years), so do some research before committing to a cruise.
### Bus
In general, bus transport is well developed throughout the continent. Note that road conditions, bus quality and driver professionalism vary widely. Much depends on the season: vast deserts of red dust in the dry season become oceans of mud in the rainy season. In Argentina, Ecuador, and coastal and southern Brazil, roads are generally better. Chile and much of Argentina have some of the best-maintained roads and most comfortable and reliable bus services in South America.
Most major cities and towns have a _terminal de autobuses_ or _terminal de ómnibus_ (bus terminal); in Brazil, it's called a _rodoviária,_ and in Ecuador it's a _terminal terrestre._ Terminals are often on the outskirts of town and you'll need a local bus or taxi to reach it. The biggest and best terminals have restaurants, shops, showers and other services, and the surrounding area is often a good (but frequently ugly) place to look for cheap sleeps and eats. Village 'terminals' in rural areas often amount to dirt lots flanked by dilapidated metal hulks called 'buses' and men hawking various destinations to passersby; listen for your town of choice.
Some cities have several terminals, each serving a different route. Sometimes each bus company has its own terminal, which is particularly inconvenient. This is most common in Colombia, Ecuador and Peru, especially in smaller towns.
### Car & Motorcycle
Driving around South America can be mentally taxing and at times risky, but a car allows you to explore out-of-the-way places – especially parks – that are totally inaccessible via public transport. In places like Patagonia and other parts of Chile and Argentina, a short-term rental car can be well worth the expense.
There are some hurdles to driving. First off, it's a good idea to have an International Driving Permit to supplement your license from home. Vehicle security can be a problem anywhere in South America. Avoid leaving valuables in your car, and always lock it. Drive carefully. Throughout South America, if you are in an accident that injures or kills another person, you can be jailed until the case is settled, regardless of culpability.
#### Driver's License
If you're planning to drive anywhere, obtain an International Driving Permit or Inter-American Driving Permit (Uruguay theoretically recognizes only the latter). For about US$10 to US$15, any motoring organization will issue one, provided you have a current driver's license.
#### Insurance
Home auto-insurance policies generally do not cover you while driving abroad. Fender benders are generally dealt with on the spot, without involving the police or insurance agents. When you rent, be certain your contract includes _seguro_ (insurance).
#### Rental
Major international rental agencies such as Hertz, Avis and Budget have offices in South American capitals, major cities and at major airports. Local agencies, however, often have better rates. To rent a car, you must be at least 25 years old and have a valid driver's license from home and a credit card. Some agencies rent to those under 25 years, but charge an added fee. If your itinerary calls for crossing borders, know that some rental agencies restrict or forbid this; ask before renting.
Rates can fluctuate wildly (ranging from US$40 to US$80 per day).
#### Road Conditions & Road Rules
South Americans drive on the right-hand side of the road. Road rules are frequently ignored and seldom enforced; conditions can be hazardous; and many drivers, especially in Argentina and Brazil, are reckless and even willfully dangerous. Driving at night is riskier than during the day due to lower visibility and the preponderance of tired and/or intoxicated nighttime drivers sharing the road.
Road signs can be confusing, misleading or nonexistent – a good sense of humor and patience are key attributes. Honking your horn on blind curves is a simple, effective safety measure; the vehicle coming uphill on a narrow road usually has the right of way. If you're cruising along and see a tree branch or rock in the middle of the road, slow down: this means there's a breakdown, rock slide or some other trouble up ahead. Speed bumps can pop up anywhere, most often smack in the center of town, but sometimes inexplicably in the middle of a highway.
### Local Transportation
Local and city bus systems tend to be thorough and reliable throughout South America. Although in many countries you can flag a bus anywhere on its route, you're best off finding the official bus stop. Still, if you can't find the stop, don't hesitate to throw your arm up to stop a bus you know is going in your direction. Never hesitate to ask a bus driver which is the right bus to take; most of them are very generous in directing you to the right bus.
As in major cities throughout the world, pickpockets are a problem on crowded buses and subways. Avoid crowded public transport when you're loaded down with luggage.
Taxis in most big cities (but definitely not all) have meters. When a taxi has a meter, make sure the driver uses it. When it doesn't, always agree on a fare _before_ you get in the cab.
### Train
Trains have slowly faded from the South American landscape, but several spectacular routes still operate. Ecuador has invested heavily in rehabilitating its old lines.
For great scenery with a touch of old-fashioned railway nostalgia, try the **Puno–Juliaca–Cuzco** route in Peru. This train runs for group bookings in high season. Departures are unpredictable, but when it does run, it's open to nongroup passengers.
# Behind the Scenes
### Send Us Your Feedback
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## Acknowledgements
Climate map data adapted from Peel MC, Finlayson BL & McMahon TA (2007) 'Updated World Map of the Köppen-Geiger Climate Classification', _Hydrology and Earth System Sciences_ , 11, 163–344.
Cover image: Iguazú Falls, Brazil; Giordano Cipriani/4Corners ©
Illustrations Machu Picchu by Michael Weldon.
## This Book
This 1st edition of Lonely Planet's _Best of South America_ guidebook was curated by Regis St Louis and researched and written by Regis, Isabel Albiston, Robert Balkovich, Alex Egerton, Anthony Ham, Mark Johanson, Brian Kluepfel, Tom Masters, Carolyn McCarthy, MaSovaida Morgan, Kevin Raub, Brendan Sainsbury, Andy Symington, Phillip Tang, and Luke Waterson. This guidebook was produced by the following:
**Destination Editor** Bailey Freeman
**Senior Product Editors** Martine Power, Saralinda Turner
**Regional Senior Cartographer** Corey Hutchison
**Product Editor** Jenna Myers
**Book Designer** Meri Blazevski
**Assisting Editors** Sarah Bailey, Michelle Bennett, Nigel Chin, Jacqueline Danam, Andrea Dobbin, Emma Gibbs, Carly Hall, Victoria Harrison, Gabrielle Innes, Alison Morris, Lauren O'Connell, Charlotte Orr, Monique Perrin, Christopher Pitts, Maja Vatrić
**Cover Researcher** Naomi Parker
**Thanks to** **Shona Gray, Angela Tinson**
**eBook thanks to** Julie J. Dodkins, Craig Kilburn, Wayne Murphy, John Taufa and Juan Winata.
### Our Story
A beat-up old car, a few dollars in the pocket and a sense of adventure. In 1972 that's all Tony and Maureen Wheeler needed for the trip of a lifetime – across Europe and Asia overland to Australia. It took several months, and at the end – broke but inspired – they sat at their kitchen table writing and stapling together their first travel guide, _Across Asia on the Cheap_. Within a week they'd sold 1500 copies. Lonely Planet was born.
Today, Lonely Planet has offices in Franklin, London, Melbourne, Oakland, Dublin, Beijing and Delhi, with more than 600 staff and writers. We share Tony's belief that 'a great guidebook should do three things: inform, educate and amuse'.
# Our Writers
###### Regis St Louis
Regis grew up in a small town in the American Midwest – the kind of place that fuels big dreams of travel – and he developed an early fascination with foreign dialects and world cultures. He spent his formative years learning Russian and a handful of Romance languages, which served him well on journeys across much of the globe. Regis has contributed to more than 50 Lonely Planet titles, covering destinations across six continents. His travels have taken him from the mountains of Kamchatka to remote island villages in Melanesia, and to many grand urban landscapes. When not on the road, he lives in New Orleans.
###### Isabel Albiston
After six years working for the _Daily Telegraph_ in London, Isabel left to spend more time on the road. A job as writer for a magazine in Sydney, Australia was followed by a four-month overland trip across Asia and five years living and working in Buenos Aires, Argentina. Isabel started writing for Lonely Planet in 2014 and has contributed to 12 guidebooks. She's currently based in Ireland.
###### Robert Balkovich
Robert was born and raised in Oregon, but has called New York City home for almost a decade. When he was a child and other families were going to theme parks and grandma's house he went to Mexico City and toured Eastern Europe by train. He's now a writer and travel enthusiast seeking experiences that are ever so slightly out of the ordinary to report back on.
###### Alex Egerton
A news journalist by trade, Alex has worked for magazines, newspapers and media outlets on five continents. Having had his fill of musty newsrooms and the insatiable corporate appetite for superficial news, Alex decided to leap into travel writing in order to escape the mundane. He spends most of his time on the road checking under mattresses, sampling suspicious street food and chatting with locals as part of the research process for travel articles and guidebooks. A keen adventurer, Alex has hiked through remote jungles in Colombia, explored isolated tributaries of the mighty Mekong and taken part in the first kayak descent of a number of remote waterways in Nicaragua. When not on the road, you'll find him at home amongst the colonial splendor of Popayán in southern Colombia.
###### Anthony Ham
Anthony is a freelance writer and photographer who specialises in Spain, East and Southern Africa, the Arctic and the Middle East. When he's not writing for Lonely Planet, Anthony writes about and photographs Spain, Africa and the Middle East for newspapers and magazines in Australia, the UK and US.
###### Mark Johanson
Mark Johanson grew up in Virginia and has called five different countries home over the last decade while circling the globe reporting for British newspapers ( _the_ _Guardian_ ), American magazines ( _Men's Journal_ ) and global media outlets (CNN, BBC). When not on the road, you'll find him gazing at the Andes from his current home in Santiago, Chile.
###### Brian Kluepfel
Brian lived in three states and seven different residences by the time he was nine years old. From then, he just kept moving, making stops in Berkeley, Bolivia, the Bronx and the 'burbs further down the line. His journalistic work across the Americas has ranged from the Copa America soccer tournament in Paraguay to an accordion festival in Quebec. Brian has covered far-flung destinations for Lonely Planet such as Venezuela, Costa Rica, Belize, Guatemala, Bolivia and Ecuador.
###### Tom Masters
Dreaming since he could walk of going to the most obscure places on earth, Tom has always had a taste for the unknown. This has led to a writing career that has taken him all over the world, including North Korea, the Arctic, Congo and Siberia. Despite a childhood spent in the English countryside, as an adult Tom has always called London, Paris and Berlin home.
###### Carolyn McCarthy
Carolyn McCarthy specializes in travel, culture and adventure in the Americas. She has written for _National Geographic_ , _Outside_ , _BBC Magazine_ , _Sierra Magazine_ , _Boston Globe_ and other publications. A former Fulbright fellow and Banff Mountain Grant recipient, she has documented life in the most remote corners of Latin America. Carolyn has contributed to 40 guidebooks and anthologies for Lonely Planet, including Colorado, USA, Argentina, Chile, Trekking in the Patagonian Andes, Panama, Peru and USA National Parks guides.
###### MaSovaida Morgan
MaSovaida is a travel writer and multimedia storyteller whose wanderlust has taken her to more than 40 countries and all seven continents. Previously, she was Lonely Planet's Destination Editor for South America and Antarctica for four years and worked as an editor for newspapers and NGOs in the Middle East and United Kingdom.
###### Kevin Raub
Atlanta native Kevin Raub started his career as a music journalist in New York, working for _Men's Journal_ and _Rolling Stone_ magazines. He ditched the rock 'n' roll lifestyle for travel writing and has written over 70 Lonely Planet guides, focused mainly on Brazil, Chile, Colombia, USA, India, the Caribbean and Portugal. Raub also contributes to a variety of travel magazines in both the USA and UK. Along the way, the self-confessed hophead is in constant search of wildly high IBUs in local beers.
###### Brendan Sainsbury
Born and raised in the UK in a town that never merits a mention in any guidebook (Andover, Hampshire), Brendan spent the holidays of his youth caravanning in the English Lake District and didn't leave Blighty until he was nineteen. Making up for lost time, he's since squeezed 70 countries into a sometimes precarious existence as a writer and professional vagabond. His rocking chair memories will probably include staging a performance of 'A Comedy of Errors' at a school in war-torn Angola, running 150 miles across the Sahara Desert in the Marathon des Sables, and hitchhiking from Cape Town to Kilimanjaro with an early, dog-eared copy of LP's _Africa on a Shoestring_.
###### Andy Symington
Andy has written or worked on over a hundred books and other updates for Lonely Planet (especially in Europe and Latin America) and other publishing companies, and has published articles on numerous subjects for a variety of newspapers, magazines and websites. He part-owns and operates a rock bar, has written a novel and is currently working on several fiction and non-fiction writing projects. Andy, from Australia, moved to Northern Spain many years ago. When he's not off with a backpack in some far-flung corner of the world, he can probably be found watching the tragically poor local football side or tasting local wines after a long walk in the nearby mountains.
###### Phillip Tang
Phillip grew up on a typically Australian diet of _pho_ and fish'n'chips before moving to Mexico City. A degree in Chinese- and Latin-American cultures launched him into travel and then writing about it for Lonely Planet's _Canada_ , _China_ , _Japan_ , _Korea_ , _Mexico_ , _Peru_ and _Vietnam_ guides.
###### Luke Waterson
Raised in the remote Somerset countryside in Southwest England, Luke quickly became addicted to exploring out-of-the-way places. Completing a Creative Writing degree at the University of East Anglia, he shouldered his backpack and vowed to see as much of the world as possible. Luke specialises in writing on South America (he writes for the _Peru_ and _Ecuador_ Lonely Planet guides). His other areas of expertise are the Caribbean (LP's _Cuba_ and _Puerto Rico_ guides), Scandinavia and Eastern Europe. That's alongside contributing to travel reference books such as _Food Lover's Guide to the World_ , _Best in Travel_ and _National Parks of Europe_.
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November 2019
ISBN 9781788686716
All rights reserved. No part of this publication may be copied, stored in a retrieval system, or transmitted in any form by any means, electronic, mechanical, recording or otherwise, except brief extracts for the purpose of review, and no part of this publication may be sold or hired, without the written permission of the publisher. Lonely Planet and the Lonely Planet logo are trademarks of Lonely Planet and are registered in the US Patent and Trademark Office and in other countries. Lonely Planet does not allow its name or logo to be appropriated by commercial establishments, such as retailers, restaurants or hotels. Please let us know of any misuses: lonelyplanet.com/ip.
Although the authors and Lonely Planet have taken all reasonable care in preparing this book, we make no warranty about the accuracy or completeness of its content and, to the maximum extent permitted, disclaim all liability arising from its use.
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Also, tagging this post to #atozchallenge.
Also, I don't know whether I have been completely able to do justice to today's prompt. But this is what I chose to write, which is straight from the heart.
This is so sweet, what better way to say thanks to your readers. Love it!
I love this!! Using the prompt as self promotion!
That's a nice ode for your friends 🙂 And I like the 2nd Burma-Shave poem 🙂 And about the hen and fish, well, it does pay to advertize.
Thanks Parthima 🙂 yes it does pays to advertise.
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Yeah, but you have to cheat on trainee ...
Postby Ben K » Mon Aug 25, 2008 9:49 pm
Got to level 14 last night. Phoenix is my friend. Between that and an energy centre it makes things so much easier. :]
Energy becomes a problem in almost every level on Insane - Caldera was one of the few where I didn't have to go far to rearm. I haven't actually run out yet though, and I don't expect anyone who chooses to play on Insane is going to be less capable than I am, so perhaps it's about right.
Terapinners are rare in Gauss bots in that they're actually formidable. Not as much as the Piper, but they aren't as easy to kill either. Pipers generally need one shot and a bit of range. When range is available. You have a habit of putting robots behind every blind corner, Wolf.
Postby darkflamewolf » Mon Aug 25, 2008 10:01 pm
true, I'll admit that is my fault for doing that for robot positioning. But if you think about it, wouldn't robots know the structural integrity of the mines and better suit themselves in positions of fortification based on their knowledge of the mine layout? At least that's what smart AI should be doing.
DH, hope you are going to input the new mantid HXMs soon and see if that makes any difference at all in facing the Mantids.
And for robot positioning, I believe we need a few blind corners every now and then, force players to think smart and be cautious rather than ramming into unknown rooms like 90% of the community does. That's likely to get you killed. I have never used secondaries as much as I have in this set. This set is Secondary weapon HEAVY. Good thing we supply a bunch.
Postby Sirius » Mon Aug 25, 2008 10:35 pm
Ben K wrote: Caldera was one of the few where I didn't have to go far to rearm.
I'm nice like that.
And yes, re: the Bolas - I'm not sure a smart missile robot being a sitting duck is a bad thing.
yes, the Bolas may not be much of a factor in early levels, esp. since I didn't use them much until the last 7-8 levels. But I remember them giving me major headaches in the final stages of the mission. So don't hold your breath on them just yet!
Postby karx11erx » Mon Aug 25, 2008 11:12 pm
darkflamewolf wrote: true, I'll admit that is my fault for doing that for robot positioning. But if you think about it, wouldn't robots know the structural integrity of the mines and better suit themselves in positions of fortification based on their knowledge of the mine layout? At least that's what smart AI should be doing.
You want smart AI in Descent 2? Be careful, or you'll get what you're asking for, and it might not be what you wanted ...
I could add that. I think AI is the most fascinating computer related subject, and that is something I haven't yet delved into ...
How about some swarm AI for small, fast bots? They would coordinate their attacks, retreat to form up and attack together, some trying to get around you in your flanks and back ...
Some cannonfodder tactics? Have some smaller bots take the fire meant for more dangerous ones around, keeping them alive and pounding you?
Reinforcement tactics? Whoops, 60 % of nearby forces destroyed - let's get away and find some buddies, before we come back!
Detours? Whoopsy, the player opened a second route for us ... lets sneak up at him through half of the mine and see the silly expression in his face before we blast him to smithereens.
And why can't bots open doors when they're so smart?
You can have that. Don't provoke me.
Ben K wrote: You have a habit of putting robots behind every blind corner, Wolf.
True, and it gets kind of repetitive and tedious. Grant Hallman, quite a few levels of whom I have collected in my level spotlight, had a different approach: He had a knack for luring the player into traps, and he was really great in building them - they were very hard to predict. Absolutely scary, and entertaining. He got me every time. I recommend (re) playing his missions. You won't regret it. Find them in my level spotlight's G.E.L.D. section.
Sirius wrote: And yes, re: the Bolas - I'm not sure a smart missile robot being a sitting duck is a bad thing.
Well ... then there should at least be packs of two or three in level 7. As you have a lot of room to maneuver there, they are a piece of cake for super lasers. Didn't even need missiles. And they really didn't even seem to move.
Last edited by karx11erx on Tue Aug 26, 2008 12:15 am, edited 1 time in total.
but you forget, its still early in the mission at level 7, trust me, there is plenty of time for the Bolas to get tougher and plenty more spots where you DON'T have room to maneuver.
Postby karx11erx » Tue Aug 26, 2008 12:14 am
darkflamewolf wrote: but you forget, its still early in the mission at level 7, trust me, there is plenty of time for the Bolas to get tougher and plenty more spots where you DON'T have room to maneuver.
Sure, I was just talking about it in spacious ... ummm ... spaces. I don't know whether you're already done with testing in two weeks when I am back from my vacation. But so or so will I see how I'll fare with them in later missions. The large areas of level 7 could use a Bolas and a Mantid more or two though.
Postby Ben K » Tue Aug 26, 2008 12:17 am
OK, so the most dangerous level so far: Secret level 2. I remember the old days when robots in secret levels were just target practice...
It's like fighting a normal level on Insane with no save/reload and no extra lives. You can, I suppose, complete it before destroying the boss, but with THAT boss, I wasn't about to risk it.
Suddenly the Bolas was dangerous. Very, very dangerous. Mantids had the habit of sneaking up from miles away, but smart missiles with blind corners... thankfully they weren't player smarts because I copped a few of them.
The bonus at the end suddenly seemed like a really nice touch. 6 extra lives for managing to survive to the end. Hehe.
That puts me up to level 16 so far.
Postby Sirius » Tue Aug 26, 2008 12:33 am
Umm. Are RLE-compressed POGs not supported by D2X-XL? I just checked a tweaked version of level 4 in there and all the custom textures either looked odd or didn't load at all. Strange since other versions don't seem to mind...
Postby darkflamewolf » Tue Aug 26, 2008 1:35 am
Darkhorse, the secret levels, as a whole, are actually harder than most of the standard levels. They are a step above in difficulty than the levels you probably enter them from. I did this on purpose, I missed the days of Descent 1, where the secret levels were indeed harder than the current levels you were working through. But yes, the final prize of many extra lives, omega cannon, fusion cannon, guided missiles and two earthshakers was more than enough for your trouble. Not to mention the other shakers and megas you find throughout the level.
Postby Alter-Fox » Tue Aug 26, 2008 2:12 am
I've always wondered why no one puts bosses on secret levels. I think having a boss robot that can only be faced on a secret level would really add to replay value, not to mention the difficulty of that particular secret level.
Postby Sirius » Tue Aug 26, 2008 2:14 am
Black Rabite comes to mind...
level 10 boss uncompletable - new robot coding makes him insanely fast, insane 'bob and weavy' and he shoots around the arena like a bullet bill. He dodges and weaves like crazy around even insta-hit Gauss. Most other robots are showing similar traits unless they are set to 'still.' 1.13.80 of D2-XL has made this mission impossible.
Postby Ben K » Tue Aug 26, 2008 6:02 am
While I still remember, do you want to keep the Guidebots in Pumo's levels? No-one else has added them.
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From the desk of Greg Goldfinch, Owner of Goldfinch Bros., Inc.
I've always said I can't be a hippie because I can't grow my hair that long. Yet Earth Day 2013 is a special one for our company.
Today, The Bullitt Center in Seattle cuts the ribbon on the first commercial office building to pursue the Living Building Challenge. It has truly been an honor to be on the team developing the most energy efficient commercial office building ever constructed.
Based in Everett, Washington, Goldfinch Bros., Inc. (GBI) is a full-service glass and glazing company with over 100 years of experience serving Puget Sound communities for all residential and commercial needs. We were first contacted about the Bullitt Center back in 2011 by the Schuchart Corporation, general contractor for the project. To meet the Living Building challenge, all heavy materials for the Center need to be sourced from a radius of 300 miles.
To achieve the ambitious goals of meeting Living Building Challenge and Net Zero Energy standards, it was essential that the Bullitt Center team give particular attention to the building envelope design and construction, including installing super energy-efficient windows. Yet the manufacturer of one of the best windows, Schuco, was in Germany.
That's where Goldfinch stepped in. We worked with Schuco and now locally manufacture and distribute the windows. The Bullitt Center could then meet both the needs of a highly efficient product to maximize its energy use, and source the material locally to meet the radius challenge as well. Goldfinch is now the sole West Coast manufacturer of these windows which remain in high-demand as both building retrofit and new construction firms look to save energy.
The Bullitt Center can now boast net-zeros in energy, carbon, waste, water and toxics. Yet the Center's construction has provided a net gain for companies like ours. And that's not only in dollars but in demand: the knowledge gained through this highly collaborative design and construction process is invaluable as the demand increases for more sustainable buildings of all types.
The entire Goldfinch team is proud to have partnered with the Bullitt Foundation, Schuchart Construction, Schuco, Point32, the Miller Hull Partnership, and all others on the project. The Bullitt Center represents both a milestone for collaboration and a model to drive innovation. Goldfinch Brothers is already working as a glazing subcontractor for two additional Living Building Challenge projects, Bertschi School and Stone 34, and continues to see a growing demand for higher standards of energy efficient envelope design. 100+ year old companies can and should lead in building the clean energy economy and our eyes remain open to new opportunities for our business, our community and solving our climate crisis.
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{"url":"https:\/\/sfb1044.uni-mainz.de\/m2-meson-dynamics\/","text":"# M2: Meson dynamics\n\n### Goals\n\nThe strong-interaction part of the Standard Model is described by an SU(3) gauge theory - Quantum Chromodynamics (QCD) - in terms of quarks and gluons as the fundamental dynamical degrees of freedom. However, experimentally only colour-neutral combinations, namely, mesons and baryons, are observed as the asymptotic states of the theory. Unraveling the structure and dynamics of mesons is still one of the fascinating challenges of the strong interactions. In the low-energy regime, the interaction of pseudoscalar mesons (\u03c0, K, \u03b7) among themselves, with vector mesons, photons, and baryons is constrained by spontaneous chiral symmetry breaking in QCD in combination with explicit symmetry breaking by the quark masses and their electric charges. On the other hand, the study of \"exotic\" heavy states in charmonium and charm mesons allows one to address structure and dynamics beyond the interpretation as conventional quark-antiquark systems. In this project we will, in particular, focus on:\n\n\u2022 Hadronic \u03b7, \u03c9 and \u03b7'\u00a0decays\nThe neutral decays \u03b7'\u00a0\u2192\u00a0\u03c00\u03c00\u03b7\u00a0 and \u03b7'\u00a0\u2192\u00a0\u03c00\u03c00\u03c00\u00a0will be measured with the Crystal Ball detector using\u00a0\u03b7'\u00a0mesons photoproduced at the new endpoint tagging facility at MAMI. These decays will provide strong experimental constraints on the influence of quark-mass differences and rescattering effects in the decay amplitudes. We will also study $\\omega$ decays to final states including real or virtual photons.\n\u2022 Anomalous processes in chiral effective field theory including vector mesons\nAt leading order in chiral effective field theory, anomalous processes such as \u03b7 \u2192 \u03b3\u03b3, \u03b7 \u2192\u00a0\u03c0+\u03c0-\u03b3 are driven by the Wess-Zumino-Witten effective action. We will study the influence of vector-meson degrees of freedom on higher-order corrections. To that end, we will combine a Dirac constraint analysis with a renormalizability analysis to constrain the possible vector-meson interactions. We will investigate various vector-meson approaches with respect to their VP\u03b3 interactions. We plan to develop a power-counting scheme which allows for the implementation of unstable particles (complex-mass scheme).\n\u2022 Light hadron dynamics in charmonium decays\nA simultaneous amplitude analysis of the decays J\/\u03c8 and\u00a0\u03c8'\u00a0\u2192\u00a0\u03c0+\u03c0-\u03c00\u00a0will provide insights into the source of the large and unexpected differences in their branching fractions and decay dynamics and allows the study of rescattering effects.\n\u2022 \"Exotic\" charmonium states\nA long standing question in hadron spectroscopy is experimental evidence for mesons which fall outside of the conventional quark-antiquark system. The recent discoveries of charmonium-like states which cannot be explained by potential models and which often have exotic quantum numbers (`XYZ states') have been interpreted as hybrid mesons, meson molecules (loosely bound states), and four-quark states. Combined fits of the large data samples available at BESIII as well as searches for isospin singlet states will help to shed light on these questions.\n\u2022 \"Exotic\" charm states\nWe plan to search for new states containing a charm quark in the recoil spectra of charmed mesons with BESIII data in order to investigate whether there are states analogous to the XYZ particles. The first surprising discovery was the Ds0(2317)+\u00a0(seen in 2003 by BABAR), which is lighter and more narrow than expected from a quark model calculation. LHCb has recently also observed higher excited states with open charm.","date":"2017-06-26 13:42:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6879266500473022, \"perplexity\": 1393.3052525196408}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-26\/segments\/1498128320763.95\/warc\/CC-MAIN-20170626133830-20170626153830-00644.warc.gz\"}"}
| null | null |
Айкана (Aikanã, также в англоязычной литературе используются следующие названия для данного языка: Masaká, Kasupá, Mundé, Huari, Corumbiara, Tubarão, Corumbiara/Kolumbiara, Huari/Uari/Wari) — находящийся под угрозой исчезновения изолированный индейский язык, на котором говорит народ айкана, проживающий в штате Рондония (к западу от города Вильена, около шоссе Куяба-Порто — Велью) в Бразилии. Живут вместе с носителями языка кваза. Айкана имеет диалекты масака (массака) и тубанао. Также носители тубанао и масака используют португальский язык.
По состоянию на 2007 год насчитывается около 200 говорящих на айкана.
Фонология
Гласные
Согласные
Ссылки
Айканан на Ethnologue
Примечания
Языки Бразилии
Неклассифицированные языки
Индейские языки
|
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Produced by David Edwards, Demian Katz and the Online
Distributed Proofreading Team at http://www.pgdp.net (Images
courtesy of the Digital Library@Villanova University
(http://digital.library.villanova.edu/))
MOTOR STORIES
THRILLING
ADVENTURE
MOTOR
FICTION
No. 29
SEPT. 11, 1909
FIVE
CENTS
MOTOR MATT'S
MAKE UP
OR PLAYING
A NEW ROLE
_BY
THE AUTHOR
OF
"MOTOR MATT"_
_Street & Smith
Publishers
New York_
[Illustration: _"Maskee!" cried the astounded Hindoo as Motor Matt
leaped at him_]
MOTOR STORIES
THRILLING ADVENTURE MOTOR FICTION
_Issued Weekly. By subscription $2.50 per year. Copyright, 1909, by_
STREET & SMITH, _79-89 Seventh Avenue, New York, N. Y._
=No. 29.= NEW YORK, September 11, 1909. =Price Five Cents.=
MOTOR MATT'S MAKE-UP;
OR,
PLAYING A NEW RÔLE.
By the author of "MOTOR MATT."
CONTENTS
CHAPTER I. HIGH JINKS IN THE SIDE SHOW.
CHAPTER II. THE "BARKER" SHOWS HIS TEETH.
CHAPTER III. THE MAN FROM WASHINGTON.
CHAPTER IV. A CLUE IN HINDOOSTANEE.
CHAPTER V. SOMETHING WRONG.
CHAPTER VI. A BLUNDER IN THE RIGHT DIRECTION.
CHAPTER VII. THE HOUSE WITH THE GREEN SHUTTERS.
CHAPTER VIII. THE PILE OF SOOT.
CHAPTER IX. MATT MEETS AN OLD ACQUAINTANCE.
CHAPTER X. RESCUE!
CHAPTER XI. BILL WILY REPENTS.
CHAPTER XII. MATT LAYS HIS PLANS.
CHAPTER XIII. MOTOR CAR AND AEROPLANE.
CHAPTER XIV. THE OAK OPENING.
CHAPTER XV. AEROPLANE WINS!
CHAPTER XVI. CONCLUSION.
A BRAVE DEED.
A LOCOMOTIVE HERO.
GEESE DROWN A SQUIRREL.
CHARACTERS THAT APPEAR IN THIS STORY.
=Matt King=, otherwise Motor Matt.
=Joe McGlory=, a young cowboy who proves himself a lad of worth and
character, and whose eccentricities are all on the humorous side. A
good chum to tie to--a point Motor Matt is quick to perceive.
=Carl Pretzel=, an old chum who flags Motor Matt and more trouble
than he can manage, at about the same time. In the rôle of detective,
he makes many blunders, wise and otherwise, finding success only to
wonder how he did it.
=Ping=, the Chinese boy.
=Ben Ali=, the Hindoo hypnotist and elephant trainer, who executes a
master-stroke in the matter of his niece, Margaret Manners, and finds
that a letter in Hindoostanee can sometimes prove a boomerang.
=Dhondaram and Aurung Zeeb=, two Hindoos who have appeared before as
confederates of the crafty Ben Ali, and who now show themselves for
the last time in their villainous part, and vanish--one into prison
and the other into parts unknown.
=Margaret Manners=, the niece of the rascally Ben Ali and a ward of
the British nation temporarily. In her particular case, justice is
slow in righting a grievous wrong--and would have been slower but for
Motor Matt and his aëroplane.
=Reginald Pierce Twomley=, who represents the British ambassador,
wears a monocle, and who, in a passage at arms with Dhondaram, proves
himself a man in McGlory's eyes and a near-pard.
=Boss Burton=, manager and proprietor of the "Big Consolidated," who,
in his usual manner, forms hasty conclusions, discovers his errors,
and shows no sign of repentance.
=The Bearded Lady, the Armless Wonder, the Elastic Skin Man, the Zulu
chief and the Ossified Man=, all freaks in the side-show tent, who
appear briefly but brilliantly in the light of a Roman candle.
CHAPTER I.
HIGH JINKS IN THE SIDE SHOW.
"Hello, dere, Viskers!" grinned Carl Pretzel, reaching up to grab the
hairy paw of the Zulu chief.
"Howdy, Dutch!" answered the chief, with a nasal twang that suggested
New England. "By Jocks, I ain't seen yeou in quite a spell. How's
tricks, huh?"
"Dricks iss fine, I bed you. Say, sheef, dis iss mein leedle shink
bard, Ping Pong. He iss der pest efer--oxcept me. Shake hants, Ping,
mit a Zulu sheef vat vas porn near Pangor, Maine."
"Tickled tew death," said the chief effusively, taking the yellow palm
of a small Chinaman who pushed himself closer to the platform.
The scene was the side-show tent of the "Big Consolidated," Boss
Burton's "Tented Aggregation of the World's Marvels." The show had
raised its "tops" at Reid's Lake, near the city of Grand Rapids. A high
wind had prevented Motor Matt from giving his outdoor exhibition of
aëroplane flying, and the disappointed crowds were besieging the side
show, eager to beguile the time until the doors for the big show were
open.
With the exception of Carl and Ping, no outsiders had yet entered the
side-show tent. Carl, having once played the banjo for the Zulu chief
while he was dancing on broken glass in his bare feet, was a privileged
character. He had walked into the tent without so much as a "by your
leave," and he had escorted Ping without any adverse comment by the man
on the door.
The freaks and wonders of the side show were all on their platforms and
ready to be viewed. The Ossified Man had been dusted off for the last
time, the Bearded Lady had just arranged her beard most becomingly, the
Elastic Skin Man was giving a few warming-up snaps to his rubberoid
epidermis, the Educated Pig was being put through a preliminary stunt
by the gentlemanly exhibitor, and the Armless Wonder was sticking a
copy of the Stars and Stripes in the base of a wooden pyramid--using
his toes.
The Armless Wonder occupied the same platform as the Zulu chief. His
specialty was to stand on his head on the wooden pyramid, hold a Roman
candle with one foot, light it with the other, and shoot vari-
balls through a hole in the tent roof. In front of the Wonder,
neatly piled on the little stage, were half a dozen long paper tubes
containing the fire balls.
"How you was, Dutch?" inquired the Wonder, doubling up in his chair and
drawing a bandanna handkerchief over his perspiring face with his foot.
"_Ganz goot_," laughed Carl, carelessly picking up one of the Roman
candles. "I vill make you acguainted, oof you blease, mit mein leedle
shink bard."
"Shake!" cried the Wonder heartily, offering his right foot. "It does
me proud to meet up with a friend of Pretzel's."
"Allee same happy days," remarked Ping, releasing the foot and backing
away.
"Yeou tew kids aire chums, huh?" put in the Zulu chief, leaning down to
arrange the row of photographs in front of him.
"Surest t'ing vat you know," answered Carl.
"Dutchy boy heap fine," declared Ping. "We both one-piecee pards."
"That's the talk!" exclaimed the Armless Wonder. "Too much weather for
the flyin' machine to-day, huh? Motor Matt was afeared to go up, I
reckon, Dutch?"
"Afraidt?" protested Carl. "Modor Matt vasn't afraidt oof anyt'ing.
He couldn't haf shtaid ofer der show grounds, und dot's der reason he
dit'nt go oop. Der vind vould haf plowed him galley-vest, und den some."
"I see. These here aëroplanes are hard things to handle, and----Holy
smoke! Drop it! Put it out!"
Carl, as has already been stated, had picked up one of the Roman
candles. While talking with the Armless Wonder, he leaned back against
a tent pole and clasped his hands--the candle in one of them--behind
him.
Ping had stepped back. The Roman candle, held fuse end outward, looked
most inviting. Digging a match out of his kimono, Ping scratched it on
the pole and applied the flame unseen to the fuse.
While the Armless Wonder was talking, Carl heard a long-drawn-out hiss,
a smell of smoke came to his nostrils, and a Niagara of sparks floated
around him. Naturally he was startled, and it flashed over him that
something was wrong with the Roman candle. Bringing the candle around
in front of him for examination, he had it leveled at the Wonder the
very instant the first fire ball was due. The ball was not behind
schedule. Rushing from the end of the tube, it caught the Wonder in the
breast, and he turned a back somersault off the platform.
Bewildered by the mysterious cause of the situation, Carl swerved the
candle in order to get a look through the smoke and sparks at the place
where the Wonder had been seated.
A roar came from the Zulu chief. A ball of flaming red had slapped
against his shoulder, and he jumped for the next platform on the right.
Landing on the edge, his weight overturned the structure. There was a
scream from the Bearded Lady and a whoop from the Elastic Skin Man, and
the next moment they landed in a tangled heap on top of the Zulu chief.
"Put it out!" the Armless Wonder continued to yell.
"Point it up or down!" bellowed the gentlemanly trainer of the Educated
Pig.
"Ged some vater!" howled Carl, running back and forth and waving the
candle; "ged a pucket oof vater und I vill drown der t'ing in it!"
The Dutch boy didn't know what to do. If he dropped the candle he
might get hit with some of the balls himself, and if he turned it
straight upward he might set fire to the top of the tent. While he was
running up and down, trying frantically to think of some way out of the
trouble, of course the fire stick was continuing to unload.
Whizz--slap!
A wad of yellow fire hit the Pig, which squealed and bolted. The
gentlemanly attendant tried to head off the Porcine Marvel, but it ran
between his outspread feet and knocked him off the stand. A rain of
lettered blocks followed.
The frantic Pig bunted into Ping, tripped him, and hurled him against
Carl. Both boys went down, and Carl rolled over and over, discharging
red, white, and blue balls as he revolved.
Up to that moment the Ossified Man had escaped. But now his turn had
come. He was said to have been turning to stone for thirty years, and
was supposed to be so brittle that he had to be handled with extreme
care.
The first ball that struck him, however, caused him to jump off his
board slab with a yell. From the way he rushed to get out of the tent,
it was pretty certain that he was as wiry and pliable as the average.
The Educated Pig, to an accompaniment of yells, howls, and screams,
and with the lurid glare of the popping balls lighting the smoky
interior of the tent, ran on blindly, overturned the stage set aside
for the Zulu chief and the Armless Wonder, showered broken glass over
everybody, and then tore through the tent wall and out into the open.
Naturally, this Bedlam, suddenly turned loose in the tent, had excited
the wonder and curiosity of the ticket seller, the "barker," and the
man at the door.
As the man at the door looked in, the last of the balls struck him
below the belt, and he collapsed in the arms of the "barker," who was
crowding in behind him.
The last of the balls! That hollow, pasteboard tube seemed to have been
a perfect mine of shooting stars. It had disgorged itself of a dozen.
Carl had not counted them--he was too busy with other matters--but it
seemed to him as though the tube had been fully an hour getting rid of
its contents.
A madder assortment of freaks it would have been harder to find than
wrangled and protested, there in the side-show tent, while they rubbed
their bruises and shook the kinks out of themselves.
"It was one of the Armless Wonder's Roman candles," came in sepulchral
tones from the Ossified Man as he climbed back to his slab.
"I'll quit the show, and give two weeks' notice this minute," piped the
Bearded Lady as she picked her way through the scattered glass, "if
they don't cut out these fireworks. My goodness! You might just as well
be killed outright as scart to death. Wha'ju jump onto our stage for?"
and she glared at the chief, who was gently massaging his burned spot.
"By Jocks," answered the chief, "I didn't care where I jumped s'long's
I got away from the fireworks."
"It was the Dutchman done it," flared the Wonder.
"He's a freak," rumbled the Ossified Man. "Kick him out."
"I don'd peen a freak," said Carl angrily, throwing the burned-out tube
at the O. M. "Oof I vas, den here iss vere I should shday."
"Did you set that Roman candle to goin'?" demanded the "barker"
fiercely.
"I don'd set him to going, py chimineddy! I hat him in my handt, und he
vent off mit himseluf. Dot's all aboudt it."
"This ain't no place for them kind o' jokes," cried the Elastic Skin
Man. "He's played hob with this outfit: Give him a h'ist!"
The ticket seller, the "barker," and the man on the door all three
fell upon Carl. Between them they had the Dutch boy turning cartwheels
through the entrance.
Ping, the cause of all the trouble, slipped away quietly under the
canvas wall--but not until he had picked up something white from the
earthen floor of the tent. The object lay close to where Carl had lain,
and Ping conceived the idea that it belonged to the Dutch boy and that
it was his duty to recover it and return it to the owner.
CHAPTER II.
THE "BARKER" SHOWS HIS TEETH.
When Carl finally rounded up his wits he found himself sitting under
the lee of the "animal top," leaning against one of the guy ropes. The
wind was blowing half a gale, and the big tents swayed and tugged at
their fastenings. There was only one idea just then in the Dutch boy's
mind, and that was this:
"How dit dot Roman gandle go off mit itseluf? I remember taking him in
my handt und holting him pehindt me, und den--whizz, bang! Ach, how der
shparks dit fly! Dere vas fordy-'lefen palls in der gandle, und I hit a
freak mit efery pall. Donnervetter, vat a hot time!"
At this point Ping came rounding the curved canvas wall, head to the
wind, blouse and wide trousers flapping, and pulling himself along by
means of the guy ropes.
"Hello, Clal!" he called, mooring himself to a tent stake.
"Hello yourseluf once!" answered Carl, drawing one powder-blackened
hand up and down his trousers leg. "How you like der pooty firevorks?"
"By Klismus!" grinned the Chinaman, "him velly fine. Fleaks no likee."
"How dit der gandle go off mit itseluf? Tell me dose."
Ping's grin faded from his yellow face, and he grew solemn and serious.
"No savvy, Clal. Him devil joss stick, awri'. Whoosh!"
A sudden suspicion darted through Carl's brain as he stared at Ping.
The Chinese boy was altogether too serious.
"Py shiminy grickets!" whooped Carl, "vas it you dot douched him off
ven der gandle vas my pack pehindt und I don'd see? Dit you make all
der drouples? Oof I vas sure oof dot, den I vould eat you oop like some
ham santviches."
Ping gave a yell of protest.
"We allee same fliends, huh?" he demanded. "Why my makee tlouble fo'
fliend?"
"Vell, I don'd know for vy, aber such chokes iss nod vat I like. Oof I
findt oudt dot you lit der gandle, den I vill ged efen for dot. You bed
my life, I pay efery debt vat I owe."
Ping looked serious. Then, glad that he was able to change the subject,
he remarked:
"You losee one piecee papel in tent, Clal?"
"I don't got one piecee paper, shink. How could I lose somet'ing vat I
don't got?"
"My findee him same place you makee tumble. Look."
Ping drew the folded sheet from his blouse. Carl stretched out his hand.
"I vill take a look at dot," said he.
When opened flat, the sheet contained writing, but it was not writing
that Carl could read.
"Vedder it iss a ledder or nod," mused Carl, "I don'd know. Vat I see
on dis paper looks schust like hen dracks. It don'd vas English, und it
don'd vas German. Iss it shink wriding, Ping?"
Ping dropped to his knees and examined the sheet of paper upside down
and sideways.
"My no savvy," he answered. "Him not China writing. Some fleak lettee
dlop--him fleak writing. Him no gottee sense."
Carl wrinkled his brows ominously.
"I tell you somet'ing," said he. "Dere iss more to dis alretty as we
know, Ping. I peen a tedectif. Meppy you vill make a tedectif, too.
Subbose we findt oudt vat der ledder iss aboudt?"
"Plaps we no makee find out."
"Dot's vere der tedectif part comes in."
"Plaps we no gottee sense enough, Clal."
"_Ach, du lieber!_" grunted Carl. "Ditn't I findt dot Margaret Manners
vat vas draveling mit der show? Ditn't I get dot Ben Ali Hindoo feller
on der run? Ditn't I vin fife tousant tollars?"
"You no gettee fi' thousan' dol'."
"I vill get dot. It has to come from Inchia, und Inchia iss more as ten
tousant miles from vere I am. It takes time to get money from Inchia. I
was a shmard feller to do all dot. Meppy I gif you some lessons und you
vill be as shmard as vat I am."
"Plaps."
"You vant to choin in mit me, hey?"
"Awri'. No savvy pidgin, Clal. What we do?"
Before Carl could answer, the "barker" for the side show came running
around the tent wall. Carl grabbed the letter out of Ping's hand and
thrust it into his pocket.
"What yuh got there?" demanded the "barker," coming to a halt and
glaring at Carl.
"You don'd got some pitzness to know," was the Dutch boy's calm reply.
The "barker's" name was Bill Wily, but, on account of his shady
character, he was generally known as Wily Bill.
"I lost a letter durin' that shake-up in the tent," said Wily Bill,
truculently, "an' it looked to me as though that sheet yuh just tucked
away in your jeans was the one. Hand it over."
"Don'd get gay mit yourseluf," warned Carl, rising to his feet.
"Where'd yuh git that paper?"
"Dot's for me to know. Oof you get pitzness any blace else, don'd let
us keep you a minid. Moof on. I don'd like you none too vell, anyhow."
"You'll give me that paper," declared Wily Bill angrily, "or I'll twist
that Dutch neck o' yours."
"Meppy you vill," answered Carl, "aber I don'd tink. Here it iss
different as it vas in der show. You don'd got der freaks und der
odders to helup."
"I'll find Burton," fumed Wily Bill, "and I'll tell him yuh've stole
that there paper off me."
"Den you vill be telling Purton vat ain'd so."
The "barker" took a step forward.
"Yuh goin' to give me that?" he shouted.
"Say," answered Carl, with a happy thought, "you tell me vat iss in der
ledder, den oof it agrees mit vat iss dere you prove he belong mit you,
und I gif him oop. Oddervise, nod. Hey?"
"Oh, you fall off the earth!" growled Wily Bill. "I don't have to tell
what's in the letter in order to prove it's mine, see? Fork over."
Carl had thought he might get Wily Bill to translate the "hen tracks,"
but the "barker" either could not or would not.
"You und me don'd agree on dot," said Carl stoutly. "You tell me vat
iss in der ledder, oder you don'd get him. Dot's all aboudt it."
"Look here," and Wily Bill made a threatening gesture with his clinched
fist, "pass that over or I'll push yer face inter yer back hair. Now,
then. Cough up or take the consequences."
"I dradder fighdt as eat some meals!" whooped Carl. "Come on vonce,
oof dot's der game. Hit me in der eye! Dot geds my madt oop kevicker
as anyt'ing, und I fighdt pedder der madder vat I ged. Eider eye, it
_machts nichts aus_. Blease!"
With a savage exclamation, Wily Bill threw himself forward and lunged
with the full force of his right. Carl ducked sideways. The fist missed
him, and the impetus of the blow hurled Wily Bill over the guy rope.
Boss Burton, the proprietor of the show, seeing the clash from a
distance, was hurrying up to take a part in proceedings. He arrived
just in time to collide with the tumbling form of the "barker."
It was with difficulty that Burton retained his footing. The breath was
knocked out of him, and as he tottered and gasped he glared at Wily
Bill.
"Dere iss Poss Purton," chuckled Carl. "Schust tell him vat you vant
und see vat he say."
"What're you roughing things up like this for, Wily?" demanded the
showman. "You know very well I don't allow any fighting on the show
grounds."
"That Dutchman," answered the "barker," getting his temper a little in
hand, "has got a letter belongin' to me. I want it, an' he won't give
it up."
"Is that so, Carl?" asked Burton, whirling on the Dutch boy.
"I don'd know vedder or nod it iss so," replied Carl. "I got a ledder,
und he say it pelongs by him. Aber he von't say vat iss in der ledder,
so how could I know?"
"Isn't the envelope addressed?"
"Dere iss no enfellup."
"Isn't there a name on the letter?"
"Dere iss no name anyvere."
"It's from a pal o' mine, Burton," explained Wily Bill, "and I dropped
it out of my kick in the tent. This Dutch lobster and that chink turned
on a row in the side show. The Dutchman got one of the Armless Wonder's
Roman candles, and while he held it behind him the chink touched a
match to it, and we had all kinds of fireworks for a----"
"Donner und blitzen!" yelled Carl, facing Ping and shaking his fist.
"Den it _vas_ you, hey? I von't be no tedectif mit you! You vas no bard
to blay sooch a choke! I vill ged efen, yah, so hellup me! Oof you----"
"That will do," cut in Boss Burton sternly. "We'll settle this letter
business before we do anything else. Where did you get the thing, Carl?"
"Dot false-alarm chink gif him by me," answered Carl, watching angrily
while Ping allowed the wind to waft him out of sight around the
side-show tent.
"Where did he get it?"
"He picked him oop from vere I lay on der groundt. Dot's vat he say,
aber my confidences in him vas padly shook."
"Give it to me."
There was no dodging such an order from the proprietor of the show, and
the folded sheet was handed over.
Burton looked at the letter. While he was doing so, Wily Bill made a
desperate grab for it. The showman was too quick for the "barker," and
jerked the sheet out of reach.
"That's your game, is it?" growled Burton. "Go back to your job, Wily.
Come to me after the show, and we'll talk this over. I don't like the
way you're acting in this matter, and if you know when you're well off,
you'll put your foot on the soft pedal and keep it there. Not a word!
Clear out!"
With a black scowl, and a look at Carl that boded him no good, Wily
Bill turned on his heel and made his way back to the side show.
CHAPTER III.
THE MAN FROM WASHINGTON.
"Sufferin' hurricanes, what a blow!" remarked Joe McGlory. "What good's
a flying machine, pard, when a spell of weather puts it down and out?
The _Comet's_ a back number in a hatful of wind."
"Hatful!" repeated Motor Matt. "If this breeze isn't doing fifty miles
an hour I'm no hand at guessing."
The two motor boys were in their old rendezvous, the calliope tent,
sitting on a couple of overturned buckets and listening to the roar
and boom of bellying canvas, the flutter and snap of banners, and the
whistle of violently disturbed air around the tent poles.
The big card played by Burton was the aëroplane flights, two of
which were given every day, before the afternoon and the evening
performance--wind and weather permitting. Since the motor boys'
engagement with Burton, Matt had not failed to take the aëroplane aloft
on an average of more than two days a week. This violent wind made
the morning flight at Reid's Lake one of the "off" days. There was a
chance, however, that the wind would go down with the sun, and that it
would be possible to do a little flying before the evening show.
It was Saturday, and the "Big Consolidated" was to remain at Reid's
Lake over Sunday and give two performances Monday. On Monday,
therefore, it was quite possible the _Comet_ would be able to carry out
her part of the circus programme.
"Up in North Dakota," observed Joe McGlory, "where it blows like sin
when it _does_ blow, you've capered around in the sky in the face of a
breeze every bit as strong as this, Matt."
"There it was different," answered the young motorist. "I didn't have
to manipulate the machine over the show grounds, and there were not
thousands of people directly underneath to suffer if the aëroplane
didn't come down in the place from which it started. I don't want any
more accidents like the one we had at Jackson."
"Where a snake short-circuited the engine, and you had all kinds of
hair-raising experiences," breathed McGlory. "Speak to me about that!
By gorry, I wouldn't even look on while you pulled off another such
performance, pard, for a million in yellow boys!"
Before the king of the motor boys could make any reply, Landers, the
man who had charge of the calliope, showed himself in the tent door.
Behind him trailed a smooth-faced man of forty, in a cap and gray
tweeds.
"That's Motor Matt," said Landers, pointing to the young motorist.
"This gentleman wants a word with you, Matt," he added, "and I
volunteered to show him where you could be found."
Landers ducked away again, and the stranger pushed into the tent.
"Fancy!" he exclaimed, staring at Matt, then at McGlory, and then
letting his eyes wander around the tent. "So this is Motor Matt. Ah, by
Jove!"
McGlory picked up a bucket, emptied the water out of it, and turned it
upside down.
"Sit down, pilgrim," said the cowboy, "and make yourself comfortable."
The other pulled up his trousers at the knees and deposited himself
carefully on the bucket. He laughed a little, lifted a round piece of
glass from his coat and tucked it into his right eye, and then took
another look at Matt and McGlory.
"Only fancy!" he murmured.
"If you want to join the show," said McGlory, with a wink at Matt,
"you'll have to see Burton."
"Join the show?" returned the other. "Why, I don't want to join the
blooming circus. I'm just looking for Motor Matt, don't you know."
"You're not looking for him, neighbor, but at him. It's your move."
"Deuced odd, that. My move. In other words, I'm to tell my business,
eh? It's private, very. I want to talk with Motor Matt alone."
McGlory started to get up, but Matt stopped him with a gesture.
"This is my chum, Joe McGlory," said he. "I have no secrets from him.
Fire away, sir."
"Aw," drawled the other. "Well, if that's the way of it, then here
goes."
Drawing a morocco case from his pocket, the stranger extracted a card
and handed it to Matt.
"Reginald Pierce Twomley," ran the legend on the card; then, down in
the lower left-hand corner were the words: "Attaché British Embassy,
Washington."
Matt passed the card to McGlory.
"Glad to see you, Mr. Twomley," said Matt. "What can we do for you?"
Reginald Pierce Twomley lighted a cigarette. It was a pretty cigarette,
with a gilt monogram on one side. He offered the case to the boys, but
they respectfully declined.
"Aw, let us approach our business with method," said Mr. Twomley. "I
have come from Washington--aw--on very important business. Allow me to
prove my right to act as agent for his excellency the Ambassador by
recapitulating a few facts with which you must be familiar.
"At one time, my dear sir, there was with this circus a Hindoo mahout
who called himself Ben Ali. That was not his real name, but it will
serve. With Ben Ali was a young lady who was called Haidee. Ben
Ali was a rotter--the worst case of thug that ever came out of the
Bombay presidency--and he had a powerful rajah for a brother. Ben Ali
took care of the rajah's elephant herd. The rajah's sister married
one Lionel Manners. Manners died, his wife perished by the infernal
practice of _suttee_--even now secretly practised in spite of the
English government--and Ben Ali left India with Manners' only daughter,
Margaret. The girl known as Haidee was in reality Margaret Manners. Am
I correct?"
Matt nodded.
"Ben Ali was an adept in the hypnotic line," proceeded Twomley, looking
thoughtfully into the smoke of his cigarette, "and Miss Manners was
in this country and with the show against her will. Her uncle, the
rascally Ben Ali, kept her under his evil influence, and was gradually
causing her to forget even her own identity. The mahout bore a grudge
against his powerful brother, the rajah, and he had stolen the girl in
a spirit of revenge. Eventually, he hoped to force the rajah to pay
many rupees for Miss Manners before Ben Ali released her. But this is
beside the mark. I don't care a hap'orth about that part of it. The
point that concerns the British Ambassador, Sir Roger Morse-Edwards, is
this:
"You and your friends, Motor Matt, discovered who Haidee really was.
You rescued her from the evil spell of the mahout, and she was left in
Lafayette, Indiana, in charge of a worthy English lady, pending advices
from her uncle, the rajah, in India. We have received advices, not from
the rajah, but direct from our foreign office. I was sent forthwith
to Lafayette to get Miss Manners, take her to New York, and, with a
suitable maid as companion, send her by first steamer to Liverpool, and
so to London."
"Good!" exclaimed Matt, with visible satisfaction. "Miss Manners is
a very fine girl, and I suppose her future will make up for the many
hardships she has undergone while in this country."
"Exactly," answered Twomley, "if we could find her. But we can't. She
has disappeared."
"Disappeared?" gasped Matt.
"That is the way of it. I went to this English lady in Lafayette,
and she received me with astonishment. Several days before a man,
professing to be from the ambassador, had called and taken Miss
Manners away. We are done, done as brown as a kipper, and a telegram
to Washington brought an answer requesting me to hunt up this show and
have a talk with you."
Motor Matt was astounded. And so was McGlory.
"Have you any idea who the man was that called on the English woman in
Lafayette and took Miss Manners away?"
"No. The Lafayette police are looking for him."
"Have you any idea that Ben Ali is mixed up in the affair?"
"I have, Motor Matt, and a very clear idea. I was ten years in India,
and learned the natives there, and their ways. It was for that, I
fancy, that Sir Roger asked me to come for Miss Manners. While I was
about taking the train at Lafayette, yesterday, I received another
message from the ambassador. That message informed me that a telegram
had been received from Ben Ali, informing Sir Roger that he again
had the girl in his possession, and that she would be delivered to
any agent Sir Roger might send after her on payment of ten thousand
pounds."
"Fifty thousand dollars!" exclaimed Matt. Then he whistled.
"Old Ben Ali is out for the stuff," muttered McGlory grimly.
"He's a crafty beggar!" commented Twomley. "I left all the telegrams
with the police, and Sir Roger is taking the whole matter up with the
United States state department. The Secret Service of the government
will presently be at work on this case, for it is of international
importance. Can you give any information, Motor Matt, that will help us
find Ben Ali, or Miss Manners?"
Matt shook his head.
"Why doesn't the ambassador agree to send some one to meet Ben Ali?
Then the rascal could be caught."
"He's too clever to let himself be caught. He----"
Just here Boss Burton strode into the tent, followed by Carl.
"Shut up about that, Carl," the showman was growling. "You haven't any
right to that letter, and I'm going to keep it."
"I'm in der tedectif pitzness," returned Carl, "und I need dot ledder,
py shinks, to helup unrafel der case. Modor Matt," and Carl appealed to
his pard, "make Purton gif me der ledder."
"What letter?" demanded Matt.
"I'll tell you what we'll do," said Burton to Carl; "we'll leave the
letter with Matt. If Wily can prove it's his, then Matt can turn the
thing over to him."
Burton handed a folded sheet to Matt. The latter, entirely in the dark,
opened the sheet and laid it on his knee.
"What sort of writing is this?" he asked.
"That's too many for me. It isn't Chinese--Carl said Ping told him
that--and it isn't Dutch. Of course, it's not English. And who it
belongs to, or where it came from, or what's the good of it, is more
than I know. But it appears to have caused a lot of bother."
"It's Hindoostanee," spoke up Twomley, staring at the open sheet. "I
can read the language. If you wish, I'll translate it."
Then, for the first time, Burton and Carl turned on the Englishman and
took his measure.
CHAPTER IV.
A CLUE IN HINDOOSTANEE.
"Who are you, my friend?" inquired Burton bluntly.
"A friend of Motor Matt," replied Twomley easily. "He'll vouch for me,
I fancy."
"Mr. Twomley, attaché of the British Legation at Washington, Burton,"
said Matt. "Mr. Burton," Matt added to the Englishman, "is the
proprietor of the show. The other lad is Carl Pretzel, who is also
a chum of mine. We can talk over this matter before them. Carl had
everything to do with the finding of Margaret Manners, back there at
Lafayette."
"Aw," drawled Twomley, screwing his monocle in his eye, and regarding
the Dutch boy, "he's the claimant for that thousand pounds reward, I
dare say."
Tremors of excitement ran galloping through Carl.
"Haf you prought der money?" he fluttered. "Vas you looking for me to
pay ofer dot rewart?"
"I am sorry to say that I haven't brought the money. That matter is
still in abeyance."
"Vat iss dot?" asked the puzzled Carl. "I don'd _verstch_ dot vort
apeyance."
"He means the matter is still pending, Carl," put in Matt. "In other
words, you haven't got the money yet."
"I know dot, aber vill I ged it? Dot's vat gifs me some vorries."
"The rajah's a regular topper," said Twomley. "He'd never miss a
thousand pounds, and I fancy he'll do the right thing."
"Mooch opliged," breathed Carl, in deep satisfaction. "It vas a
habbiness to know dot I ged him some dime."
"Now, if you wish," went on Twomley, stretching out his hand for the
letter.
"Just a moment, Mr. Twomley," said Matt. "We don't know much about this
letter, and I'd like to find out where and how Carl got it, and what
the dispute is about."
The Dutch boy launched into an explanation, beginning with the Roman
candle and ending at the place where Burton refused to turn the letter
over to Wily Bill. Carl touched but lightly on the culpability of Ping
in the matter of the Roman candle. In this he was wise. Motor Matt's
orders were to the effect that there should be no bickering between the
Dutch boy and the Chinese lad. They had been at swords' points for a
long while and had only recently developed a friendly feeling for each
other.
"I always sized up that Wily Bill for a false alarm," remarked McGlory.
"Can he read that Hindoostanee lingo? I'll bet my spurs he can't! If
that's the case, what's he doing with the letter?"
"He must have wanted it a whole lot," said Matt, "or he wouldn't have
made such a fight to get it. Perhaps the letter itself will be a clue.
Tell us what's in it, Mr. Twomley," and Matt passed the letter to the
Englishman.
The latter studied the sheet with absorbed attention. Finally he sprang
up.
"By Jove!" he exploded.
"What's the matter?" inquired Matt.
"This is luck! Just fancy such a clue coming into our hands at this
very moment when it is most needed. Aw, it's--aw--incredible."
"You might give us a chance to pass judgment on that, Mr. Twomley,"
returned Burton. "Maybe it's not so incredible as you seem to think."
"It was written by Ben Ali," said the attaché.
"_That_ tinhorn!" exclaimed McGlory. "I thought we'd cut him out of our
herd altogether. Beats creation how he keeps bobbing up."
"Who's it for?" spoke up Matt. "Has Bill Wily any right to it?"
"The name of Wily doesn't appear anywhere in the writing," answered
Twomley. "In fact, the letter's addressed to a fellow named Dhondaram."
Here was another hot shot. Both McGlory and Matt were brought excitedly
to their feet.
"Dhondaram!" growled Burton, with an expressive glance at the king of
the motor boys. "I thought we'd heard the last of that villain."
"Who was he?" demanded Twomley.
"A Hindoo----"
"So I gather from the name."
"He blew into the show grounds with a cobra and a home-made flute, when
we were at Jackson, and I gave him Ben Ali's place as driver of our
man-killin' elephant, Rajah. Oh, he did a lot of things, Dhondaram did.
We captured him, but he got loose and dropped off the train between
stations."
"Aw, Ben Ali didn't know that," reflected Twomley. "Ben Ali must have
thought he was still with the show, and sent this letter to him."
"What does the letter say?" asked Matt, with some impatience.
"It asked Dhondaram to finish his work as soon as possible and to join
Ben Ali, with the money, in short order."
A silence followed, and during the silence the motor boys exchanged
wondering looks.
"What was Dhondaram's work?" queried Twomley.
"Nothing more or less than putting Pard Matt out of the running,"
replied McGlory. "Ben Ali's on the warpath against Matt, because of
what he did in Lafayette, and Dhondaram tried hard to wipe my pard off
the slate."
"Ben Ali speaks of money," went on Twomley. "What does that mean?"
Burton muttered wrathfully.
"I'll bet a thousand," said he, "that refers to the proceeds of
the afternoon performance in Jackson, which the ticket man and
this Dhondaram tried to get away with. Ben Ali put up the job with
Dhondaram, and the ticket man was helping them out."
"Matters must have been lively all around in Jackson," observed
Twomley. "Dhondaram didn't get the money?"
"Not so you could notice," answered McGlory. "Pard Matt jumped in and
plugged that little game."
"Ben Ali," reasoned the king of the motor boys, "has probably been
thinking of recapturing Miss Manners for some time. All he had
Dhondaram try to do, in Jackson, was to help on his villainous schemes.
But Dhondaram failed. Probably Ben Ali is needing some money pretty
badly, about now. What is the date of that letter, Mr. Twomley?"
"There is no date."
"Then there's no telling how long Bill Wily has carried it in his
pocket?"
The attaché shook his head.
"He must have got it after we left Jackson, pard," interposed McGlory.
"If he had got it before, he'd have passed it on to Dhondaram."
"How he got it at all is a mystery," mused the young motorist. "He has
probably seen and talked with Ben Ali."
"Before the show got to Jackson, then," continued the cowboy, who was
doing a little sharp thinking. "If he had talked with Ben Ali after the
doings in Jackson, he'd have told the old skinner how Dhondaram fell
down."
"There's a clue here, but it's not so promising as it might be," came
disappointedly from the Englishman.
Matt walked toward the tent door.
"Our best clue," said he decisively, "is Bill Wily. We'd better go to
the side show and have a talk with him."
"Bring him here, Matt," suggested Burton. "We can talk with him in this
place to better advantage than in the side-show tent. I'll go with you
and make sure he comes. The rest of you wait," and the showman started
from the calliope tent after Matt.
Inquiry of the man on the door at the side show developed the fact that
Bill Wily had started for town. He had been gone about five minutes,
Matt and Burton were informed, and had left the show grounds for the
street-car track.
"He's making a getaway!" averred Burton.
"That's the way it looks," agreed Matt. "We've got to stop him, if we
can."
Without loss of time the king of the motor boys and the showman hustled
for the place where the street-car track made a loop, just beyond a big
concert garden. They were hoping to catch Wily before he could board a
car.
But in this they were disappointed. A car was moving off in the
direction of town, and all their frantic yells and gestures were
powerless to secure the attention of the conductor.
"It'll be fifteen minutes before there's another car," panted Burton,
"and by that time the 'barker' will be--the deuce only knows where.
It's a cinch, Matt, that he's scared, and is running away. If there was
an automobile handy, we could overhaul the car." Burton looked in every
direction. "But, of course," he added, "whenever you want a chug-wagon
there's none in sight."
A familiar humming drew Motor Matt's attention. Looking in the
direction of the sound, he saw a motor-cycle spinning along the road
from the direction of Grand Rapids. A young fellow of nineteen or
twenty was in the saddle.
"There's something that will do--if we can borrow it," said Matt, and
jumped into the road and waved his hands.
The motorcycle came to a stop.
"Are you flagging me?" asked the driver of the machine.
"Yes," said Matt hurriedly. "I want to overhaul the street car that
just left here. There's a man aboard that we've got to catch. Will you
let me take your motorcycle?"
"Well, I guess not!" was the reply. "The last time I loaned this
machine I was two days getting it back into shape again."
"I'll give you twenty dollars for the use of it, young man," put in
Burton eagerly.
"No inducement," was the answer.
"There's hard luck for you, Motor Matt," grunted Burton.
The young fellow had been on the point of starting away, but he
suddenly paused and turned to Matt.
"Are you Matt King," he asked, "the fellow they call Motor Matt?"
"Yes," was the reply.
"Doing an aëroplane stunt with the show?"
"Yes."
"Well, take the machine. It won't cost you a cent, either. I work in
a motor-car factory in the Rapids, and we've heard a good deal about
you there. I'm tickled to death to be able to help you out. Bring the
machine back here when you're done with it, and you'll find me waiting."
"Such is fame!" laughed Burton.
With a hasty word of thanks, Matt headed the machine the other way and
got into the saddle.
One turn of the pedal and the motor took up its cycle. Half a minute
later the king of the motor boys was out of sight down the road.
CHAPTER V.
SOMETHING WRONG.
McGlory, Carl, and Twomley waited in the calliope tent until their
patience was exhausted.
"Py shiminy," fluttered Carl, "I bed you somet'ing for nodding dot Vily
Pill don'd vas by der site show yet."
"I reckon you've dropped a bean on the right number," agreed the
cowboy. "What's our next jump, your highness?"
The question was put to the Englishman.
"Aw, I say," said the latter, in remonstrance, "I'm not that, don't
you know. I'm not of the peerage. An uncle and three cousins, all
distressingly healthy, stand between me and an earldom."
"I want to know!" murmured McGlory, in mock surprise. "Why, I didn't
think any one this side a lord could wear one of those little window
panes in the right eye."
"You jest," said Twomley, with a faint smile. "Fancy!"
"Well, anyhow, what are we going to do? Sit here and wait, or hit the
trail ourselves and find out what's doing?"
"Hit the trail?" echoed Twomley, lifting his brows. "Deuced odd, that.
Why should we hit it, and what shall we hit it with?"
"Vat a ignorance!" murmured Carl.
"We'll hit it with our feet, excellency," went on McGlory.
He had a hearty contempt for the monocle, and took it out on the wearer.
"I don't know whether I rise to that," returned Twomley, "but if it
means to go forth and look into the cause of our friends' delay in
returning with Wily Bill, then, it's ay, ay, with a will."
"Come on, then, and we'll vamose."
McGlory led the way to the side-show tent, and Twomley and Carl
followed him closely.
The crowds had long since entered the big tents, and the performance
in the "circus top" was in full blast. With the beginning of the "big
show" there was no business left for the annex, and the ticket seller
was withdrawn under the lee of a canvas wall, hobnobbing with the man
on the door. These two informed McGlory, Twomley, and Carl that Wily
Bill had left for town on the street car, and that Motor Matt and
Burton had started for the car line in the hope of overhauling him. But
that had been all of half an hour before.
The three searchers immediately departed for the car-line loop. There
they found Burton and a young fellow kicking their heels impatiently
and keeping their eyes down the track.
"Where's Matt?" asked McGlory.
"Ask us something easy," replied Burton. "Wily has hiked for town. When
we got here the car he was on was too far down the track to stop. This
young man"--the showman indicated his companion--"came along on a motor
cycle. Matt borrowed the machine with the intention of overtaking
the car and bringing Wily back, but neither has shown up yet. Must be
something wrong."
"Vell, I bed you!" said Carl anxiously. "On some modor cycles
Mile-a-minid Matt alvays geds vere he iss going pefore he shtarts.
Somet'ing has gone crossvays alretty, und dot's no tream."
"I'm doing a century to-day," remarked the motor cycle owner, "and this
is cutting into my time."
"Don't fret about your wheel, neighbor," spoke up McGlory. "You'll get
it back, all right."
"I'm not fretting. Motor Matt's welcome to a dozen of the gasoline
bikes if I had 'em. But I'd like to be moving on."
Burton looked at his watch.
"Matt's been gone thirty-five minutes," he announced.
"If he was running all the time," observed the lad from the motor-car
works, "he could be thirty-five miles from here."
"Perhaps," ventured Twomley, "he has mucked the play, somehow."
"Mucked the play!" exclaimed the exasperated McGlory. "That's not his
style, your lordship."
"We'll wait twenty-five minutes longer," announced Burton. "If Matt
isn't back by then, this young man and I will start along the car track
in my runabout and we'll see what we can find."
"Dake me along," clamored Carl. "I vas afraidt somet'ing iss wrong mit
Matt."
"If there are any extra passengers in the runabout," said McGlory
resolutely, "I'm the one."
"My word!" muttered Twomley. "I hope everything's all serene, I do,
indeed. I'm a juggins at waiting when there's so much excitement going
on."
"Juggins is good," grunted McGlory. "You can retire somewhere, Mr.
Twomley, and hold onto your nerves while the rest of us hunt up the
'barker.' You'll not shine much till we find Wily Bill, anyhow."
"You're an odd stick," answered Twomley, whose good nature was not a
thing to be ruffled.
He was sharp enough to see that the cowboy had a pique at him, and he
had sufficient good sense to take it calmly.
"Py shinks," said Carl, after ten more weary minutes had passed, "Matt
has hat time to do some centuries himseluf, und I can't guess it oudt
for vy he don'd get pack. Oof you don'd dake me in der runaboudt, den,
so helup me, I vill valk. Anydink is pedder to shtand as uncerdainties."
Carl constantly watched the road that paralleled the car track. And so,
for the most part, did the Englishman.
"My word, but it is trying!" murmured Twomley. "If we could only see a
bit of dust, then we'd know Motor Matt was coming, and my relief would
be profound."
"Dust! _Ach, himmelblitzen!_ Vy, Matt vill go so fast on dot machine
der dust vill be a mile pehindt und you don'd see dot."
"Here's something," came from McGlory. "Speak to me about it, will
you? Where's Ping? Little Slant-eyes is always around when anything is
doing, but I haven't seen him since he finished watering the calliope."
Carl knew why Ping wasn't around. Ping was afraid Carl would do
something to him to play even for the Roman-candle business. Oh, yes,
that was an easy one for Carl to guess. There was secret satisfaction
for the Dutch boy in the reflection. And he gloated over it and kept it
to himself.
"Time's up," announced Burton, snapping his watch, "and here's where
I go for the runabout. My thoroughbred is hitched to the buggy, so be
ready to go with me," he added to the owner of the motor cycle.
"I'm not worrying about the wheel, understand," said the lad, "but
about the century I'm to turn. I'm making it right in the teeth of this
wind."
Inside of five minutes Burton came with the runabout, his Kentucky
thoroughbred stamping off the ground at a record pace.
The runabout seat was narrow, and Burton and the lad from the motor-car
factory filled it comfortably. But they took McGlory on their knees and
whipped away, leaving Twomley and Carl gazing after them disconsolately.
Hardly were the runabout and its passengers out of sight when a car
rounded the loop and deposited its passengers on the platform.
"Led's ged on der car, Misder Dumley," suggested Carl. "Ve vill vatch
der road as ve go, und oof ve see somet'ing ve vill trop off. I peen a
tedectif feller, und oof dere iss any clues dey von't ged avay from me."
"Go you!" answered Twomley heartily.
Any sort of action was a relief for his impatience, and he and Carl
scrambled aboard the car.
Meanwhile the pedigreed Kentucky cob was pounding off the distance.
In the horse's performance the proud showman lost sight of the main
business in hand--temporarily.
"See that knee action!" he exulted. "Did either of you ever see a
prettier bit of traveling? We're doing a mile in two-thirty!"
"Bother the horse!" growled McGlory. "Keep your eyes on the road for
clues."
"Clues! I'll bet money the 'barker' wouldn't get off the car. How could
Matt make him? He couldn't, of course. Nothing short of a cop and a
warrant could make Wily Bill leave the car if he was set for reaching
Grand Rapids. I might have known that, if I had stopped to think. We'll
have to keep right on into town--and, then, like as not, we won't find
either Matt or Wily. Now----"
"Whoa!" cried McGlory. "You're shy a few, Burton. Here's where we stop."
"What's up?" returned Burton, reining in his spirited roadster.
"Look there!"
McGlory pointed to the left-hand side of the road. Close to a steep
bank, against a clump of bushes, stood the motor cycle.
"Jupiter!" exclaimed Burton.
"Great Scott!" cried the owner of the machine.
McGlory tumbled clear of the runabout and started toward the bushes.
He had not taken half a dozen steps, however, before he came to a dead
stop.
A form fluttered out of the bushes and approached him excitedly.
"Ping!" gasped the cowboy. "Speak to me about this! Where'd you come
from, Ping? And where's Pard Matt?"
The Chinese boy's feelings apparently defied expression. He tried to
speak, but his lips moved soundlessly. Hopping up and down in his
sandals, he waved his arms and pointed--not toward Grand Rapids, but
off across a piece of rough woodland.
CHAPTER VI.
A BLUNDER IN THE RIGHT DIRECTION.
Ping had felt certain that his move in touching off the Roman candle
had not been seen. It was a disagreeable surprise to him, therefore,
when Bill Wily told Carl just who was responsible for the fireworks.
Ping and Carl were trying hard to be pards. Their hearts were not in
the attempt, for deep in the spirit of each one slumbered a latent
animosity against the other. But they had to try to fraternize. Motor
Matt had issued an edict to the effect that, if they did not become
pards, he and McGlory would cut them out of the motor boys' combination.
So the lads did their utmost to appear friendly. They wandered around
together, and whenever Matt or McGlory was in sight they locked arms
and addressed each other in terms of endearment. When they were away
from Matt and McGlory they still kept up the pretense, but in a manner
that was more subdued.
Ping could not resist the temptation to touch a match to the Roman
candle. He had not expected to cause such a disturbance, and the fact
that chaos had reigned in the side show, and that his culpability had
become known, filled him with apprehension.
Carl would tell Matt, and Matt would sidetrack his Chinese pard. Ping
worried, and had no desire to see Matt, or any one else. The show
was to be at Reid's Lake for three days, and there was no Sunday
performance. Ping, therefore, could flock by himself until Monday
afternoon.
Ping's work consisted of watering the steam calliope, and in helping
the aëroplane take its running start for the flights. Owing to the
wind, there would be no morning flight, and--very likely, as he argued
to himself--no afternoon ascension, either. And Ping knew Motor Matt
would not work on Sunday.
Taken all in all, this was a most propitious time for Ping to absent
himself from the show grounds. With the idea that he would go into
Grand Rapids and hunt up some of his countrymen, he left the grounds
and made his way around the concert garden to the car-line loop.
Here his nerve began to fail him, and he allowed two or three cars
to come and go without getting aboard. Finally he bolstered up his
tottering resolution and climbed into one of the cars.
Looking through the open window, after he had taken his seat, he saw
Wily Bill swing up by the hand rails.
Ping was asking himself what this could mean when the car pulled out.
A little worried, he knew not for what reason, he got up from his seat
and walked to the forward platform, thinking it well to keep out of
Bill Wily's sight.
Suddenly he became aware of something. A voice, from far behind, was
shouting for the car to stop. The passengers, thrusting their heads
from the windows, were looking back, and some of them were talking
excitedly.
Ping, hanging out from the lower step, turned his gaze rearward, and
what he saw caused his heart to thump wildly against his ribs.
One of the little two-wheeled devil wagons was rushing along the road
that paralleled the track, coming like a limited choo-choo train, and
Motor Matt was in the saddle!
Ping had but one thought. The Dutch boy had told Matt about the Roman
candle, and Matt was chasing the street car in order to remove his
Chinese pard, read the riot act to him, and cast him adrift.
What a turn Ping had! He crouched down on the step, and the clatter of
the gong, as the conductor gave the motorman the bell from the rear
platform, sent a shiver of dread through his nerves.
Rather than face Matt and be cut out of the motor boys' combination,
Ping would have done almost anything. The only thing that suggested
itself at that moment was to jump and run. His original intention to
lie low until the Roman-candle incident blew over grew stronger in his
mind.
The car was beginning to slow down, but it was still proceeding at a
lively gait when Ping threw himself straight out from the lower step.
The Chinese boy did not know the proper way to alight from a swiftly
moving trolley car, and the result of his leap can be imagined.
The passengers who were looking out from that side of the car had a
vision of a small Chinaman in the air, pigtail flying. The next instant
the Chinaman touched ground, but found it moving too fast for a secure
foothold. Ping bounded into the air again, his slouch hat going in one
direction, his sandals in another, and he himself describing what is
technically known as a parabola. The Le Bons--the best "kinkers" in the
Big Consolidated--could not have twisted themselves into more fantastic
shapes than did Ping during that stunt of ground-and-lofty tumbling. He
landed on the ground like a frog taking to the water from the top of
a toadstool, and he wound up his performance by throwing a number of
choice cartwheels and then sitting up in the dust and looking around in
considerable mental perturbation.
About the first thing he saw and was able to realize was that another
besides himself had made a jump from the car. The other was Wily Bill,
and he must have dropped from the rear platform a little before Ping
dropped from the platform forward.
Wily Bill, however, must have known how to jump from a swiftly moving
car and yet keep his balance, for he was on his feet and making a dash
for a brushy bank at the roadside.
Motor Matt had swerved his motor cycle and was making in the "barker's"
direction, calling loudly the while for him to stop.
The light that dawned on Ping, just then, was a good balm for his
bruises.
Matt was not chasing him, after all, but had been hot on the trail of
Wily Bill!
While Ping sat there in the dust, hat and sandals gone, his clothes
torn and awry, and himself more or less disorganized, he saw Wily Bill
scramble up the steep bank and vanish among the bushes on the top of
it. Possibly thirty seconds later, Matt sprang from the motor cycle,
leaped up the ascent like an antelope, and likewise vanished.
"By Klismus!" murmured Ping, rubbing his knees. "Velly funny pidgin! My
no savvy. One piecee queer biz, you bettee. Wow! China boy all blokee
up! Motol Matt no wanchee pullee pin on China boy. Hoop-a-la!"
Between his physical pain on account of his bruises and his rejoicing
over the discovery that Matt had not been following him, Ping failed
to observe that the street car had stopped and backed up to the place
nearest the spot where he was crooning to himself and rubbing his
bruised limbs. It was not until the conductor and the motorman faced
him that Ping realized that he was the object of their consideration.
"Didju fall off?" asked the conductor.
"No makee fall," answered Ping, cocking up his almond eyes, "makee
jump."
"Blamed wonder yu didn't break yer neck!" growled the motorman. "Chinks
don't know nothin' anyhow."
"Hurt?" asked the conductor, animated by a laudable desire to avoid a
damage suit in behalf of the company.
"Heap sore," chattered Ping, "no bleakee bone. Hoop-a-la!" he
jubilated, a wide grin cutting his yellow face in half. "Woosh!" he
added, as the grin faded and a look of pain took its place.
"Well, I'm stumped!" muttered the conductor. "Is he crazy, or what?" he
added, looking at the motorman.
"Pass it up," snapped the motorman. "Chinks is only half baked, best
you can say for 'em. Let's snake 'im aboard and go on. We've lost
enough time."
One got on either side of Ping and lifted him to his feet. They would
have dragged him to the car had he not resisted.
"Leavee 'lone!" he shouted, squirming.
"Oh, snakes!" ground out the exasperated motorman. "Ain't you fer the
Rapids?"
"No wanchee go Glan' Lapids!" declared Ping. "Why my makee jump my
wanchee go Glan' Lapids?"
"That's so," said the conductor. "What did he jump from the car for if
he wanted to go on with us? We'll leave him, Jim. I thought, when I saw
him hit the ground, we'd have to take him to the hospital, but he seems
to be all right."
Jim, with an angry exclamation, let go of Ping and hustled back to
his place at the front end of the car. The conductor mounted the rear
platform, and the starting bell jingled.
As the passengers looked back, they saw the Chinese boy attempt a war
dance in his stocking feet, then suddenly cease and reach down to clasp
his right shin.
"He's got out o' some lunatic asylum," thought the conductor. "Well,
it's none o' my funeral," he added, and went into the car and began
collecting fares.
Ping, when the car was out of sight, limped around collecting his
scattered wardrobe. While he was about it, he was wondering, in his
feeble way, why Motor Matt was chasing Bill Wily.
Probably, he reasoned, Wily had cut up so rough with Carl that Matt had
thought best to pursue the man and call him to account.
Ping was not in very good condition to take part in the chase, but if
he could manage it, and proved of some assistance to Motor Matt, such
a move would go far toward making his peace with the king of the motor
boys.
"My makee tly," groaned Ping, limping to the place where the motor
cycle had been left.
With infinite patience he crawled up the steep <DW72>. One of his legs
felt as though it didn't belong to him--it seemed more like a cork leg
than anything else, and was numb from ankle to thigh. But, somehow, he
managed to get up the bank with it. Pausing there, he called aloud for
Motor Matt. His voice echoed weirdly in the scant timber of the rocky
ground in front of him, and the shout brought no response.
"My findee Motol Matt," declared the Chinese lad to himself, as he
limped into the timber. "My ketchee Motol Matt, mebby ketchee Wily
Bill. Woosh! Hoop-a-la!"
CHAPTER VII.
THE HOUSE WITH THE GREEN SHUTTERS.
While making his slow and painful way among the scrub oaks that grew
out of the stony earth, Ping was looking in all directions for Matt and
Wily. He was listening, too, with all his ears. But he could neither
see nor hear anything of the two for whom he was searching.
"My findee!" he said, with dogged determination. "Motol Matt no chasee
China boy, him chasee Wily Bill," and again he exulted.
Action was perhaps the best tonic he could have had. As he swung
onward, the leg which did not seem to belong to him began to remind
him, in no uncertain manner, that it was really his, and that he was
responsible for its condition.
A slow pain made itself manifest, running up the member like a streak
of lightning and giving Ping a "gone" feeling in the pit of his
stomach. But he was "game." Shutting his teeth on more than one groan,
he kept resolutely on through the bleak timber, looking and listening.
Finally he came out on a rough crossroad, which he followed. Five
minutes of wabbling along this road brought him to the end of it--and
across the end squatted a dingy white house with green shutters.
The shutters were closed, and the house had the appearance of being
deserted.
Here, Ping felt, was the end of his trail. He was on the wrong track,
and the question that pressed upon him was what he should do next.
Withdrawing to a clump of bushes, he sat down and gave the matter
extended thought.
Who lived in the house? And was there any one at home? If there was any
one in the place, would they talk with him and tell him whether they
had seen Matt or the side-show man?
Ping, unlike Carl, made no boasts of being a "tedectif." He could
blunder around and, maybe, stumble upon something worth while, but it
would be purely a hit-and-miss performance.
Yes, he decided, he had better go to the house and see whether there
was anybody there.
Barely had he made up his mind when, with amazing suddenness, Bill Wily
rushed around the corner of the house, jammed a key into the door, and
disappeared.
He did not close the door behind him, being, as it seemed, in too much
of a hurry to attend to such trifling matters.
While Ping was still wrenched with this startling exhibition, an even
more astounding spectacle was wafted his way.
Motor Matt followed Wily around the house corner, paused an instant in
front of the open door, then was swallowed up in the dark interior.
Ping had not called out, for amazement had held him speechless.
The Chinese boy had blundered in leaping from the street car, but, as
it had chanced, that had been a blunder in the right direction. All the
heathen gods of luck had been ranged on his side, too, when he followed
the crossroad and went into communion with himself in the clump of
bushes facing the green-shuttered house.
In about two minutes, Ping figured, Matt would have Bill Wily by the
heels. So it followed, if Ping was to have any part in the capture, he
would have to hurry.
In the excitement of the moment he forgot his bruises, emerged from the
undergrowth, and made his way rapidly toward the house.
At the open door he stopped, thrust his head into the hallway, and used
his ears.
The silence was intense, and not the faintest sound was to be heard.
There was something weirdly mysterious about this. With Matt and Wily
both in the house, and each more or less hostile toward the other,
there should have been a good deal of noise.
A qualm raced through Ping's nerves.
There was something ominous about mysteries, and he had made it a rule
to fight shy of ominous things. He did not consider them at all good
for a Chinaman's health, or his peace of mind.
And a Melican house, too, deserted and with closed shutters, offered
dangers not lightly to be reckoned with.
But Ping, as yet, was Motor Matt's pard; and whereever Motor Matt led
the way, then Ping would be more of a hired man than a pard if he did
not follow. Shutting his teeth hard, and breathing only when necessary,
the Chinese boy crossed the threshold of the house with the green
shutters.
He was in a narrow hall that extended through the house from front to
rear. A stairway led to the second floor, and two doors opened off to
left and right.
Throttling his fears, Ping moved toward the door on the right, his
sandals scuffling over the uncarpeted floor. There was no furniture in
the house, and the floor was bare.
The swish of the sandals sent vague fears cantering through the little
Celestial, and he curled up his toes in order to wedge the soles of his
footgear closer to the bottoms of his feet.
The room he entered was dark. With a trembling hand he groped in his
blouse for matches. Had he lost his matches in taking that header from
the street car? His fears in that respect were short-lived, for he
quickly found half a dozen of the small fire-sticks.
Scratching one, he held it up and peered around. The room was
empty--bare as a last year's bird's nest. Going back into the hall, he
examined a room on the opposite side. That one also was empty, and over
all the emptiness arose a musty odor as of a building long untenanted.
Two more rooms remained to be examined on the first floor.
One of these was the kitchen, and a quantity of soot had drifted down
and lay in a heap on the floor. Ping kept away from the soot, and was
glad afterward that he had done so. Across the hall was the last of
the four rooms comprising the lower part of the house--dark, deserted,
and musty as were the other three.
Failure to encounter danger of any visible sort had heartened Ping
wonderfully.
"My makee go up stlails," he thought. "Mebby my ketchee something
top-side."
He moved softly, but the stairs creaked and rasped under his sandals in
spite of his wariness.
There were four rooms upstairs, just as there were below, and in none
of the dark chambers did he discover any trace of Motor Matt or of Wily
Bill.
Ping was "stumped." The longer he thought of the mystery the more
terrified he became.
He believed in demons. Ben Ali, he knew, was possessed of them, for he
had heard how the Hindoo, with his eyes alone, had put people to sleep
and made them do strange things while they dreamed.
Ping, naturally, had no idea that Ben Ali was in any way concerned with
Matt's pursuit of Wily Bill, but the Chinaman's mind reverted to Ben
Ali, and Aurung Zeeb, and Dhondaram, three Hindoos, all of whom, at
various times, had formed a part of the Big Consolidated.
Had he dared, Ping would have shouted Matt's name at the top of his
voice. But he was afraid. A dragon, spouting fire from its red mouth,
and with a hundred claw-armed feet, might materialize and attack him,
did he dare awake the echoes of that sombre house.
Turning swiftly away from the last room, Ping got astride the
banisters, slid to the bottom of the stairs, and ducked through the
front door.
The bright sunshine was never pleasanter to him than at that moment. He
gulped down a few draughts of pure outside air and started off toward
the bushes, bent upon a little solitary reflection.
By a sudden thought, he whirled abruptly, softly drew the door shut,
turned the key in the lock, and then slipped the key into his pocket.
He had locked the door on the mysteries, and he hoped the fiends of
darkness would respect the barrier until he could think of some way to
exorcise them.
Once more in his original place among the bushes, Ping watched the
house warily and tried to approach the problem in a reasonable way.
But it was not a question of reason. His investigation had developed
facts that defied every logical process.
What had become of Motor Matt?
This was the point that disturbed the Chinese boy most. If he could
find Motor Matt, he would be content to leave the question of Wily's
whereabouts out of the count.
Abruptly Ping had an idea. Perhaps Wily had rushed out of a rear door,
and Matt had followed him? During his investigations, Ping had tried no
doors or windows.
Getting to his feet, he made a circle around the house. There was one
door in the rear, and only one. Cautiously he approached and tried the
knob. The door was locked.
As for the windows, every one was tightly closed in with the green
shutters.
These discoveries left Ping in a daze. After several minutes of
bewilderment, he finally made up his mind to return to the show
grounds, find McGlory, and acquaint him with the situation. McGlory
would know what to do!
Then, there was the two-wheeled devil wagon Motor Matt had left at the
foot of the bank, by the roadside. A hazy idea of riding the machine
back to the show grounds passed through the Chinaman's mind.
To regain the road by the street-car track took time, but the distance
was covered much more rapidly than Ping had covered it coming the other
way.
Strange to relate, the Chinese boy's bruises caused him little concern.
All his aches and pains were lost in the details of the inexplicable
situation connected with the deserted house.
While he was in the brush, at the foot of the bank, eying the motor
cycle a bit dubiously, he heard a patter of hoofs, a grind of wheels,
and a sound of voices.
Looking up, he saw Burton's runabout at a stop. Burton was in the
buggy, and so was a young fellow Ping had never seen before--and
McGlory. The cowboy was just scrambling out of the vehicle and starting
in the direction of the motor cycle.
The sight of reinforcements caused all Ping's wonder, and doubt, and
apprehension to revive with redoubled force. He attempted to shout, but
no words escaped his lips. Rushing forth to meet McGlory, he waved his
arms and pointed in the direction of the house with the green shutters.
CHAPTER VIII.
THE PILE OF SOOT.
Ping was not many minutes recovering the use of his tongue. McGlory
grabbed him and shook his powers of speech back into their normal
condition.
"Where's Motor Matt?" cried McGlory.
"My no savvy!"
"How did you happen to be here?"
"Stleet cal."
"What're you making a run from the show grounds for without saying a
word to Matt?"
That was a point which Ping did not care to reveal. He was not above
being careless with the truth in a pinch, having been raised that way.
But, while he might resort to a little harmless fiction with McGlory,
he would have cut his tongue out before he would have fibbed to Motor
Matt.
"Makee see Wily Bill ketchee cal," Ping explained; "my ketchee same
cal. Follow Wily Bill. Wily Bill jump from cal. My jump, too. Tumble
all ovel load. Wily Bill lun fo' top-side bank. Motol Matt chasee.
Motol Matt leavee gas hlorsee by bank. My follow, no findee."
Out of this pigeon English McGlory captured a few germs of sense.
"What the nation was he following Wily for?" demanded Burton. "How did
he know we wanted Wily?"
Ping was still equal to the emergency.
"Dutchy boy havee low with Wily Bill," he explained.
"That's right," went on Burton; "you _were_ around during the row. I'd
forgotten that. That may have been enough to put you on Wily's trail,
although I can't figure it out exactly. But you followed him, and then
you followed Matt when he ran after Wily. They went up the bank and
into the woods, you say?"
"Allee same."
"Then where did they go?" demanded McGlory.
"Makee tlacks fo' house with green blinds."
"They made tracks for a house with green blinds? Now we're getting at
it. Where's this house?"
"Othel side woods. My findee, you savvy; makee sit down, do heap big
think. Bymby, 'long come Wily Bill, unlock do', go in house. Plenty
soon, 'long come Motol Matt, go in house, too." Ping became oppressed
with the awe aroused by the event next to be described, and his voice
sank into a husky whisper. "My makee tlacks inside, hunt evel place, no
can find. House allee same empty. Motol Matt disappeal, vanish, makee
go up in smoke. Woosh! My plenty 'flaid."
"What's he givin' us?" snorted Burton. "He's talking through his hat,
seems like, to me."
"He's run into something that he can't cumtux," returned McGlory. "It's
plain enough, though, that a house with green shutters is at the end of
our trail. Ping can take us there, and it will be up to us to do the
rest."
"Say, young feller!" cried Burton, standing up in the runabout and
addressing the lad from the motor-car works.
The latter was pulling his motor cycle out of the bushes and making
ready to forge away on the rest of his "century" run.
"Well?" returned the youth, one leg over the saddle and ready to pedal
off.
"Load that machine into the runabout and drive this rig back to the
show grounds for me, will you?" requested Burton. "I'm hungry to see
this game through, and I can't leave the horse hitched in the road."
"Couldn't get the motor cycle into the buggy," was the answer. "Anyhow,
I guess I've helped you about as much as you could reasonably expect."
"There's twenty coming to you," went on Burton. "Take the rig back and
I'll make it thirty."
"There's nothing coming to me. I told Motor Matt he could use the
machine, and welcome. Now that he's done with it, I'll go on with my
run."
The motor began to pop, and presently settled into a steady hum. A
minute later the motor cycle and its rider were out of sight.
Just then, when it looked as though Burton was to be permanently
retired from the rest of the pursuit, a street car from the lake
rattled to a halt, and Carl and Twomley dropped from the steps.
"Here's the Englishman," muttered McGlory, without much enthusiasm.
"And Carl!" added Burton. "He'll take the rig back for me, and the rest
of us will start for the house with the green shutters."
"Vat's to pay?" clamored Carl, running toward McGlory and Ping.
Ping's confidence in Carl, like Carl's confidence in Ping, was badly
"shook." The Chinese boy backed away.
"Here, Carl," cried Burton. "Jump into the runabout and take it back to
the grounds for me. I've got business with McGlory."
"Meppy I don'd got some pitzness mit McGlory, same as you," demurred
Carl. "Vere iss Modor Matt?"
"There's no time to palaver, Carl," interposed McGlory. "Take the rig
back."
When Matt was away, McGlory was the boss. Carl could not very well
disobey such a pointblank order. Much against his will, he climbed into
the runabout.
"My word!" cried Twomley. "You seem to have discovered a clue of some
sort. Who's the Chinaman?"
"Never mind that, now," returned Barton. "Come with us, Twomley, and
we'll tell you as we go along."
"Lead off, Ping," ordered McGlory.
Carl, very much out of temper, shook his fist at Burton, and then at
Ping. Following this, he turned the rig the other way and rode moodily
back toward the show grounds.
Ping, meanwhile, had climbed the bank, and was leading the party of
investigators through the woods in the direction of the crossroad. As
they went along, Burton was telling Twomley what Ping had discovered.
The information given by the Chinaman was lacking in many important
points, but its very incompleteness added to the tensity of the
situation.
When they came to the end of the crossroad, Ping halted and indicated
the house with the green shutters.
"You say," remarked McGlory, giving the house a swift sizing, "that
Wily Bill ran into the house?"
"All same," answered Ping.
"And that Pard Matt trailed after him?"
"All same."
"Then you went in, looked around, and couldn't see anything of either
of them?"
"My no findee." Ping shivered. "When my makee come out, my lockee do'."
He dug up the key and handed it to McGlory.
"Well," declared McGlory, "if Motor Matt and Wily Bill went in there,
and didn't come out again, we'll find them."
"If the Chinaman didn't find them," struck in Twomley, "they must have
come out."
"We'll soon know what's what," and the cowboy made his way to the door,
thrust the key into the lock, and pushed the door ajar.
The same dark, funereal silence that had greeted Ping stared McGlory,
Burton, and Twomley in the face.
"My no findee," chattered Ping, drawing back; "you no findee."
McGlory pressed into the hall.
"I'll take the rooms on the left," said he, "and the rest of you take
the ones on the right. Do your bushwhacking, and then, if you don't
find anything, meet me at the foot of the stairs for a look overhead."
Nothing was found. The back door was securely bolted on the inside, and
all the windows and blinds of the various lower windows firmly fastened.
The situation upstairs was exactly the same. Puzzled and bewildered,
the party returned to the lower hall.
"If Ping's giving it to us straight," said McGlory, "neither Matt nor
Wily got out of here. They couldn't have gone through the rear door or
any of the windows, without leaving them open. And they couldn't have
left by the front door because it was locked, and Ping had the key."
"They might have slipped out while Ping was nosing around upstairs,"
suggested Burton.
"They'd have made some noise," objected the cowboy. "Matt didn't have
any call to keep quiet, and Ping would surely have heard him. Let's go
back to the rear rooms again."
Burton and Twomley had examined the kitchen. McGlory now looked that
room over for himself.
He was no more than two minutes in picking up a clue. The lighted match
which he held close to the floor showed footprints outlined in black.
He traced them to the pile of soot under the chimney.
"Here's where we find something!" he cried. "Open those shutters, you
fellows! We want light while we run out this trail of soot."
Twomley and Burton unfastened the windows and pushed back the blinds on
their screeching hinges. The sunlight, drifting into the room, brought
out the trail with weird distinctness.
"Maybe the Chinaman blundered into the soot and left the trail,"
hazarded Burton.
"My no makee tlail," declared Ping. "No touchee soot."
"There's only one of the chink, anyhow, pards," said McGlory, "and at
least two pairs of feet walked through that pile of black stuff. One
man wore shoes, and the other wore slippers. The slippers left marks
a good deal like Ping's sandals, but the marks are too big for Ping.
We'll find out a few things now, I reckon."
With eyes bent sharply on the floor, the cowboy crossed the kitchen
into the hall, and then moved along the hall to a spot under the stairs.
The stairs were not enclosed, but sprang directly from the hall floor.
In the angle formed by the flight and the floor the sooty trail
vanished.
"Now what?" queried Burton. "It looks like we were up in the air as
much as ever."
Without replying, McGlory drew his knife from his pocket, opened it,
and went down on his knees.
CHAPTER IX.
MATT MEETS AN OLD ACQUAINTANCE.
Matt's pursuit of the street car reminded him of his old motor-cycle
days in Arizona. The familiar hum of the twin cylinders between his
knees carried his mind back to his ill-fated gasoline marvel, the
_Comet_, in honor of which he had named the aëroplane he was using with
the show.
The borrowed motor cycle had all the improvements, and the way it could
run warmed the cockles of Matt's heart. In less than a minute after
leaving Burton and the machine's owner, the king of the motor boys was
shooting along the road like a bullet out of a gun.
He was pursuing an electric car that ran at a high rate of speed, but
the motor cycle must have been going five feet to the car's one. Before
Matt fairly realized it he was within sight of the car.
When he was close enough to be heard he began to call to the conductor
to stop. The passengers heard him, as the row of heads thrust out from
each side of the car conclusively proved; and the conductor also heard
him, for he appeared on the rear platform.
Matt could see the conductor reaching for the bell rope. At the same
time, Wily Bill rushed out on the back platform, took in the situation
at a startled glance, and then dropped dexterously from the car at the
track side.
Matt was so wrapped up in what Wily Bill was doing that Ping's leap
from the front platform escaped him entirely.
Wily Bill scurried for the side of the road, and Matt shut off the
power and glided after him.
"Hold up there, Wily!" cried Matt.
The "barker" paid no attention, but plunged up the bank and darted off
into the timber.
By that time Motor Matt's blood was up. He knew that a great deal
depended on the capture of Wily. If the "barker" could be made to tell
when and how he had received that note in Hindoostanee, a clue to
the whereabouts of Ben Ali and the missing Margaret Manners would be
secured.
Appreciating fully the exigencies of the case, Matt sprang from the
wheel and leaped up the bank. From the top of the rise he could see
nothing of Wily, but a crashing of the undergrowth told him plainly in
which direction the man had gone. He was but an instant in taking after
him.
Wily's actions were those of a guilty man; in fact, they inferred a
deeper guilt than the mere possession of a note in Hindoostanee would
indicate.
This, naturally, made the fellow's capture all the more important.
For a quarter of a mile, Matt judged, Wily led him a chase through the
woods. The "barker" had lost a little of his lead, but was keeping up
his fierce pace with a good deal of vigor. Then, suddenly, he began to
double. Matt would run on, looking and listening, only to find that
there was no thrashing brush ahead. When he stopped, the sounds made
by the fleeing fugitive had changed their direction, and the young
motorist had to whirl and take another course.
For some time this variation of the game of hare and hounds continued,
Matt drawing steadily nearer and nearer.
At last Matt caught his first glimpse of Wily, since he had fled over
the bank from the street car, at the rear of a house whose windows were
closed with green shutters.
Wily stood out against the house wall, his form sharply defined, just
as Matt rushed from a fringe of hazels. The "barker" cast a look over
his shoulder, gave vent to a panting exclamation, and darted around the
end of the house.
When Matt reached the front of the structure, Wily had vanished. The
key to his disappearance was furnished by the wide-swinging front door,
key still in the lock. Besides, Wily had not had time to go around the
other side of the house, or to get into the woods again, so Matt knew
he must have entered the building.
With scarcely a moment's hesitation, the king of the motor boys
followed the fugitive.
Coming in out of the bright sunshine, the darkness of the shut-in hall
was intense. As Matt ran on past one of the doors leading to a room on
the right a sinewy, turbaned form leaped out and a fist shot through
the gloom, landing on the back of Matt's head with tremendous force.
Matt staggered, regained his balance, and whirled around. His brain
was reeling, but, looking toward the light that entered at the open
door, he saw that the man who had struck him was not Wily, as he had
imagined, but a Hindoo--none other than his old acquaintance, Dhondaram.
Flinging out his arms, he leaped at the Hindoo. Then it was that Wily
completed the work that Dhondaram had begun. Another blow from behind,
savagely given with all the "barker's" strength, caused Matt to sink to
his knees and then straighten out unconscious on the bare floor.
"You saw what was goin' on?" asked Wily breathlessly.
"Even so, sahib," answered the other, in a low tone.
"I'm in luck to find you here. Wasn't intendin' to blow in at this
place till night--but any port in a storm. Pick him up and let's get
away somewhere."
"The kitchen, sahib."
Between them, the unconscious king of the motor boys was lifted and
carried into the kitchen.
"Hang it!" growled Wily, floundering through the soot pile; "this
won't do. There may be more after me. There's another place, under the
stairs. Sharp's the word, now. Carry him there."
Matt was not bereft of his senses for long. There was too much steel
and whalebone in his athletic body to keep him steeped in oblivion for
any great length of time.
The first thing he saw, when his eyes slowly opened, was a candle
planted in the earth.
He was lying, hands and feet bound and a cloth over his mouth, in a
sort of pit. Above him were the stringers and boards of a floor.
A few moments passed while he was picking up the thread of events.
While he was piecing details together, he heard a light footfall on the
floor overhead, advancing and retreating. Later there came the creaking
of boards as of some one climbing a flight of stairs.
Wily and Dhondaram, silent and motionless as statues, knelt in the
earth, the fluttering gleam of the candle over them, and were listening
to the footfalls with bated breath.
From the manner of these two Matt understood forthwith that the person
in the upper part of the house must be one whom his captors feared.
Had it not been for the cloth that smothered his lips, Matt would have
shouted at the top of his voice and so have informed a possible friend
where he was.
Inasmuch as he could neither move nor make an audible sound, the
prisoner lay quiet.
There was no cellar under this house with the green shutters, only
a scooped-out place in the earth where possibly potatoes and other
vegetables had been kept.
Presently the footsteps once more descended the stairs and could be
heard leaving the house. Wily turned to Dhondaram with a deep breath of
relief.
"That was a close call," he muttered. "If we'd been a second later
gettin' down here----"
He bit off his words quickly. The door had slammed and the grating of a
key could be heard.
"_Maskee!_" rumbled Dhondaram. "The door has been closed and locked,
sahib. You left the key in the door."
"I was in too big a hurry to do anythin' else. As it was, Motor Matt
came within one of layin' hands on me. See if he's got his wits back."
On hands and knees the Hindoo crept to Matt's side and peered into his
face. Matt kept his eyes closed.
"Not yet, sahib," answered Dhondaram. "It is well. He shall not waken
in this world. The goddess Kali----"
Dhondaram did not finish the sentence. He had referred to the malign
Hindoo deity invoked by thugs, and it may be he thought the talk
unsuited to American ears. Lifting himself on his knees, he drew from
the breast of his jacket a glittering blade.
The next moment Wily Bill had caught his arm.
"Chuck it!" he growled sternly.
The Hindoo turned his glittering eyes on the "barker."
"Sahib, you do not understood," said he, in a hissing voice.
"I understood you're intendin' to use the knife," answered Wily Bill,
"an' I won't have it. What d'you take me for? They don't hang people
in this State, but I don't intend to pass the rest o' my days in the
'pen.' Put that knife back where you took it from."
"It is my duty to do this thing," flared the Hindoo.
"Go on!"
"Ben Ali saved my life in my own country, and I joined the show of
Burra Burton because he told me. I tried to remove Motor Matt because
he told me. That will pay my debt to Ben Ali. I failed in my work while
I was with the show, but now----"
"You're goin' to fail here, too. I've got a tender regard for my
liberty, an' that's why I was runnin' away from the show grounds. There
was a fracas turned on in the side-show tent, an' I got mixed up in
it. Durin' the row I lost a letter that came to me by mail--a letter
that contained somethin' for you. Ben Ali, in my letter, said where he
wanted to meet you. I don't know what he said in your letter, as that
was in Hindoostanee."
Dhondaram's eyes glowed expectantly, and he held out his hand.
"The writing, sahib."
"I haven't got it. Didn't I just tell you it was lost? That's what made
me bolt from the grounds. One of Motor Matt's friends got the thing,
and when I tried to get it, Burton took possession of it. If that
letter's ever translated, I'll bet it contains stuff that would make
the show too hot to hold me. I got away while there was time--but there
wasn't any too much time, at that. If----"
Dhondaram drew back.
"Motor Matt, sahib," muttered Dhondaram, "he's listening to your talk."
The prisoner had opened his eyes, and the keen glance of the Hindoo had
detected it. Both Dhondaram and Wily turned their gaze on Matt.
CHAPTER X.
RESCUE!
Motor Matt understood full well the gravity of his situation. Never
until that moment had he known the cause of the murderous Dhondaram's
hostility to him, but now it appeared that he was merely seeking to
cancel a debt which he owed Ben Ali.
Bill Wily's regard for his own welfare was all that stood between Motor
Matt and the knife of the misguided Hindoo.
"Give me that knife, Dhondaram," ordered Wily.
"I will keep the knife, sahib," replied the other.
"Keep it, then, and be hanged to you," answered Wily angrily, "but
you'll settle with me if you try any knife tricks on the prisoner. I
guess you rise to that, all right enough. Take off the gag. I want to
talk with Motor Matt."
Dhondaram bent down and removed the cloth.
"I'm a 'barker,'" went on Wily, still addressing the Hindoo and making
brief display of a revolver, "but here's somethin' that bites as well
as barks. Put away that knife."
Silently the Hindoo returned the knife to his jacket and sank back on
his heels.
"What was you chasin' me for, Motor Matt?" asked Wily.
"Why were you running away from me?" Matt countered.
"That's my business. You answer my question. I guess you'd better treat
me white, 'cause it's me that keeps the Hindoo from doin' a little
knife work on you."
"Burton wanted you to tell him something about that letter," Matt
answered, making up his mind that a little of the truth would not be
out of place.
"Oh, ho!" muttered Wily. "Does he think I can read Hindoostanee?"
"No. What he wanted to know was where you got the letter. The Hindoos
who have been connected with the show haven't turned out very
well--they are all fugitives from the law, even Dhondaram."
Not a ripple crossed the placid brown face of the Hindoo; only his
glittering eyes revealed the feeling that slumbered in the depths of
his soul.
"I guessed there'd be a stir about that letter," went on Wily, "an'
that's the reason I made up my mind to pull out. I'd had to explain,
an' no matter what I'd said I'd have been fired, anyway. I used to live
in Grand Rapids, and the home town was a good place for me to cut loose
from the show, see?"
"Why are you treating me like this?" asked Matt quietly.
"Couldn't help it. Them kid pards o' yours was the cause o' the hull
bloomin' twist-up!" Wily Bill swore savagely under his breath. "I'd
like to take the kinks out o' that Dutchman. He's too much on the
buttinsky order. You chased after me, hung on, an' wouldn't let go.
What else could I do but make myself safe?"
"You didn't have to have Dhondaram knock me down."
"It wasn't him did that. He tried, but I had to finish the job. But I
was treatin' you well, at that. I could have dropped down back of a
clump o' bushes, there in the timber, and picked you off with this."
Wily touched his hip pocket. "But I didn't. That ain't my style. I'd
rather have you like this an' come to a little agreement with you. As
for Dhondaram, I hadn't an idea he was in the house. I'd given him a
key, an' I knew he might be here, but I wasn't expectin' him so soon.
Mebby it was lucky for me that he was around."
"So that's it, eh?" commented Matt sarcastically. "You've been meeting
Dhondaram, and helping him, when you knew he had been a prisoner of
Burton's and had escaped from the show train between Jackson and
Kalamazoo. If a person helps a fugitive of the law to escape, he is
guilty of a crime and can be punished for it."
"There you hit it! But I was ducking out--and you wouldn't let me duck.
I'm going to leave, in spite of you and Burton. That's the worst I've
done--talkin' with Dhondaram and carryin' Hindoostanee letters. But
I'll not be jugged for that, or----"
A hiss of warning came from Dhondaram. At the same moment he leaned
down and replaced the cloth over Matt's lips.
Distant voices were heard, then the sound of a key rattling in a lock.
"The fellow that was here before has brought some others," whispered
Wily. "Hang the luck! I wish we had got out o' here while we had the
chance. Now, then, we're in for it an' no mistake."
"Listen, sahib!" frowned the Hindoo.
The voices that had been heard outside the house were now talking in
the hall. It was impossible to distinguish words, but Matt's heart
leaped as he recognized McGlory's voice and Burton's.
They were looking for him!
"They cannot find us down here, sahib," murmured the Hindoo, his voice
soft and purring as that of a tiger cat. "They will go as the first one
went, then we can leave."
This was Wily's hope. Breathlessly he listened to the sounds above.
The footsteps and the voices faded away into the upper regions of the
building.
"Now," muttered Wily, "we might be able to dodge through the front
door. They're all upstairs."
Dhondaram shook his head.
"The door in the floor, sahib, cannot be found," he whispered
reassuringly. "The _feringhis_ will not discover us. Be patient."
Presently Matt heard his friends returning to the lower floor, heard
them enter the kitchen, heard the sound of lifted windows and opening
blinds, marked the slow and steady advance from the kitchen into the
hall, and along the hall to a point under the stairs.
By then, even Dhondaram had begun to take alarm.
"They're at the trap!" gasped Wily Bill.
"Is there no way out of this hole, sahib?" demanded Dhondaram through
his teeth.
"Only by the way we came in. I lived in this house and I know all about
it."
Dhondaram smashed the flat of his hand down over the light of the
candle. The Stygian blackness that reigned showed plainly the rim of
daylight under the lifting door.
"The revolver!" hissed Dhondaram. "Shoot, sahib!"
"No, I tell you!" answered Wily. "I'll have none o' that, or----"
With a savage snarl, Dhondaram hurled himself on Wily Bill in a furious
effort to secure the revolver and fight off the approaching rescuers.
The trapdoor had been thrown entirely back, and daylight was flooding
the pit. The sounds of the struggle between the Hindoo and Wily Bill
reached the ears of those above.
"Here they are!" cried the voice of McGlory, and instantly he leaped
downward.
With a blow of his fist the Hindoo staggered the cowboy, leaped upward,
and gained the floor.
"Dhondaram!" yelled Burton, who was just preparing to follow McGlory
down under the floor.
The word was hardly out of his lips before the showman was compelled to
drop back to avoid a sweeping blow of the knife in the Hindoo's hand.
McGlory was looking for Matt, and paid little attention to the Hindoo.
He found his pard with his groping hands, for his eyes were blinded by
the sudden change from day to the darkness of the pit.
"Bully for you, pard!" exclaimed McGlory. "Lashed hand and foot, or I'm
a Piegan! Speak to me about this, will you? And gagged, too. Sufferin'
blazes, but you've had a time! There, how's that?"
The cowboy pulled away the cloth.
"Wily's here," were Matt's first words. "He and the Hindoo had a fight,
and----"
"Bother Wily! It's you I'm after," and, with his open knife, McGlory
slashed at the cords. "Now we can look after Wily."
Leaving that part of the work to his chum, Matt leaped upward and
climbed over the edge of the floor. Burton was running toward one of
the front rooms.
"Where's the Hindoo?" cried Matt.
"The Englishman tagged him in here, after heading him off at the door,"
panted Burton. "I always knew that thug was a killer, and if I hadn't
been quick he'd have knifed me."
A smash of glass came from the front room and two of the blinds were
smashed open. The light afforded by this gave Matt and Burton a view of
a desperate struggle in which the attaché of the British Legation was
proving himself a whole man, in every sense of the word.
Unarmed, and with every disregard for his personal danger, Twomley had
set upon the Hindoo. Dhondaram's knife had ripped Twomley's coat and
brought a stain of red, but the Englishman had both hands around the
Hindoo's throat, and they were flinging here and there around the room.
The smash of glass and the crash of the blinds had been caused by
Dhondaram falling heavily against one of the windows. Then suddenly,
before either Matt or Burton could go to his aid, Twomley hurled his
antagonist from him with terrific force. The Hindoo fell sprawling
against the wall, and dropped stunned to the floor. His knife slipped
from his hand, and Burton kicked it aside while he and Matt threw
themselves upon the supine figure.
"Take his turban," said Matt, "and bind his hands with it."
The turban was merely a long strip of twisted cloth, and there were two
or three yards of it--enough for both his wrists and ankles.
Barely was the tying finished when McGlory drove Wily into the room
with his own six-shooter.
"Talk about this, friends," laughed McGlory. "Wily Bill fights with
the Hindoo, and has the tuck about all taken out of him. I snatch his
revolver, and then we come out from under the floor, Wily in the lead
and acting real peaceable. You've caught Dhondaram, too. Everything's
lovely, eh?"
"All serene," answered the Englishman.
He had removed his coat and was binding his handkerchief about his arm.
"Twomley captured Dhondaram, Joe," said Matt, "and did it alone."
"Getting stabbed for his pains," added Burton.
"A scratch," was Twomley's cool response. "How could you expect me
to do a thing like that without getting a nick or two? A pretty show
altogether. And it might have been a good deal worse."
CHAPTER XI.
BILL WILY REPENTS.
McGlory motioned Wily to take a seat on the floor, near Dhondaram, and
then turned toward Twomley.
"So you put the kibosh on our brown friend all by yourself, did you?"
he asked.
"It wasn't much," was the diffident answer. "I know these Hindoos
somewhat."
"You're the clear quill," said McGlory, "and I've got a different
estimate of you. What do you think?" he added to Burton. "They had my
pard down in the spud cellar, covered with ropes and gagged."
"Nice how-d'ye-do!" growled Burton. "What sort of a way is that to act,
Bill Wily?" and he flashed a look of anger and contempt at the "barker."
"I've made a holy show of myself," mumbled Bill Wily. "That comes of
gettin' confidential with these here chocolate- crooks. They're
no good."
"What do you think of yourself, hey?"
"Not much, Burton, an' that's a fact. I'm down and out, and just
because I wanted to shake your show an' not have any trouble. What a
lot of excitement over nothin' at all!"
"Fancy that!" remarked Twomley, mildly surprised. "I guess the man
doesn't know the true state of affairs."
"He'll know everything before we're done with him," snapped Burton.
"You're not goin' to bear down too hard on me, are you, Burton?"
pleaded Wily.
"Why shouldn't I?"
"What've I done?"
"I can't tell that till I hear what happened to Motor Matt. If these
disgraceful proceedings get out, it will be a black eye for the show."
Boss Burton was a queer fish. He had always a high regard for carrying
out every promise he made in his show "paper," and was also solicitous
about the good name of the Big Consolidated; at the same time, he had
done a number of things which gave Matt a poor opinion of his character.
Matt, taking advantage of the opening afforded him, told what had
happened after he had left Burton on the motor cycle. The rough
treatment he had received brought scowls to the faces of McGlory and
Burton.
"That Hindoo might have knifed you, and all on account of Wily there!"
breathed the showman.
"But he didn't," returned Matt, "and that was on account of Wily, too.
Keep that in mind, Burton."
"Your head, pard," said the cowboy solicitously. "You've had a couple
of good hard raps, and I'll bet that block of yours feels as big as a
barrel."
"I'm like Twomley," smiled Matt, "and couldn't expect to come through
such a tussle without a few marks. But it's nothing serious. Another
thing, Burton," he added, turning to the showman, "just recollect that,
if Wily wanted to, he could have used that thing Joe has in his hand.
But he wouldn't, and he fought with Dhondaram rather than let him use
it."
"Wily hadn't the nerve," commented Burton. "He's in the parlor class
when it comes to strong-arm work. He's more of a shell worker and a
confidence man."
"Don't be rough, Burton," begged Wily Bill.
"What've you got to say for yourself?"
"I'm blamed sorry things turned out like they did. That's all."
"Just how sorry are you? Sorry enough to make a clean breast of
everything?"
"That depends on what'll happen to me. You let the ticket man off when
he and Dhondaram tried to loot the Jackson proceeds. I didn't do half
as much as him."
"Tell me what you've done, and then I'll tell you what I'm goin' to
do," said Burton.
"I knew Ben Ali pretty well when he was with the show," returned Wily,
"but he didn't put it up with _me_ to help steal the ticket-wagon
money. I'm not makin' such a terrible sight as spieler for that
side-show outfit, and when I get a letter in Kalamazoo, inclosin'
another in Hindoostanee and askin' me to deliver same, what am I goin'
to do? That letter contained a money order for ten dollars."
"And it was from Ben Ali?" asked Motor Matt.
Wily nodded.
"We got into Kalamazoo about three in the morning," proceeded Wily
Bill, "and when I dropped off the train, Dhondaram stepped out from
between a couple o' box cars----"
"It was the night we left Jackson that we had Dhondaram lashed and
lying in the aisle of the sleeper on section two of the show train,"
interrupted Burton. "He got loose and skipped. I fired a shot at him,
but he jumped off the train. How could he have done that and then shown
up in Kalamazoo the morning we got there?"
The showman was trying to pick flaws in Wily's narrative, but the
"barker" was equal to the emergency.
"For the reason, Burton, that he didn't jump off the train. Dhondaram
rode the platform, and now and then he dodged down on the bumpers when
the train men came too close. As I say, he met me as I dropped off, and
we had a bit of a chin together."
"Why didn't you grab him," demanded Burton, "and turn him over to me?"
"That's where I was lame, I expect, but you forget I was a friend of
Ben Ali's, and Dhondaram was also a friend. That made a sort of hitch
between us. Then, too, Dhondaram told me he was expecting word from
Ben Ali in my care. I hadn't received any word, and I told him so.
Dhondaram said that I would get a letter, sooner or later, and that
he'd like to meet me somewhere near Grand Rapids. That's when I told
him about this house and gave him one of my keys to it."
"What have you got to do with this house?" queried Burton.
"I happen to own it," was the surprising answer. "It ain't worth much,
an' it's been condemned by a railroad that intends runnin' a line of
rails and ties right over the place where it stands. For that reason
it's closed up. I'm to get twelve hundred dollars for the property any
day now. Why," and Wily Bill looked around, "when I was a kid I used to
live here. When the folks died I rented the house an' took to roamin'
around. It was a good place to meet Dhondaram and give him a letter if
there was any come from Ben Ali. I wasn't expectin', though, to call
here before night. The letter from Ben Ali reached me in Kalamazoo in
the afternoon, at a time when Dhondaram must have been travelin' north."
"What did you do with your part of the letter?"
Wily's profession of repentance seemed to be sincere, and Burton and
Matt were doing their utmost to find out everything he was able to
tell. Dhondaram, sitting on the floor with his back against the wall,
glared at Wily fixedly while he talked. The savage menace of the
Hindoo's look, however, seemed to make not the slightest impression on
the "barker."
"I tore up my part o' the letter, Burton," replied Wily. "Didn't think
it best to carry it around. If I'd torn up Dhondaram's part, too, I
guess I'd have been a whole lot better off."
"I guess you would," agreed the showman dryly. "What had Ben Ali to say
to you?"
"He told me where he wanted Dhondaram to meet him. You see, Ben Ali's
been busy, an' hasn't been payin' much attention to what's been goin'
on in the show."
"By Jove," put in Twomley, "I should say he had been busy."
"Ben Ali didn't know Dhondaram had cooked his goose, so far as the show
was concerned, in Jackson, the same day he joined on."
"Where did Ben Ali send his letter from?" inquired Matt.
"Lafayette."
"And where does he want to meet Dhondaram?"
"Five miles west of the Rapids, on the wagon road to Elgin. There's an
openin' in the woods, somewhere there, and Ben Ali wants Dhondaram to
join him at the place to-morrow morning. I don't know what's up, but I
guess it's somethin' mighty important for the Hindoos."
"Does Ben Ali know about this house of yours?"
"Not a thing. I never told him. I guess I was foolish to jump off the
car and run over here, but the ruction in the side show and the loss
o' that Hindoostanee letter sure got me on the run. I thought mebby,
if I couldn't dodge Motor Matt in the woods, I could get him somewhere
and have a talk with him that would let me out. But things didn't come
out as I wanted. I couldn't shake him in the timber, so I rushed for
the house. Dhondaram was here, ahead o' schedule, an' he complicated
matters a-considerable."
"Do you think," asked Matt, "that we could go to that place on the
Elgin road and meet Ben Ali instead of letting Dhondaram do it?"
Twomley started, for he instantly caught Matt's idea. Dhondaram
likewise showed much concern, and undoubtedly he surmised what was at
the back of the young motorist's head.
"I don't think you could," replied Wily. "Ben Ali ain't nobody's fool,
and he'll have the road watched to see that only the right party comes.
If the wrong party comes, then Ben Ali, more'n likely, 'll fade out of
the oak openin'. You can't get there any way by road without Ben Ali
findin' out just who's after him. That's my notion."
"Suppose we should come in on him from both sides at once?" suggested
Burton.
"Then he'd slide out between you. Oh, he's a slippery proposition, that
boy!"
Twomley nodded affirmatively.
"He speaks the truth," he averred. "A man who can do what Ben Ali has
done is a rogue of the first water."
"There's a way to get at him," said Matt confidently. "Here, in a
thickly populated country, that scoundrel can't have things his own
way."
"He's takin' chances," put in Wily, "but that's his stock in
trade--takin' chances an' throwin' in a little hypnotism now an' then.
Why he's so particular about meetin' Dhondaram is what gets me."
"He needs money," said Burton sarcastically, "and he has to run a few
risks to get it."
"I've got a plan," said Matt, starting toward the door.
"What is it?" asked Burton and McGlory.
Matt turned around in the doorway and cast a suggestive glance at Wily
and Dhondaram.
"I'll not go into it now," said he, "but it all depends on the
truthfulness of Bill Wily. If Wily has given us a straight story,
then the plan will work. If it does, then I shall insist that Wily be
allowed to go free, without any punishment for what he has done. If
the plan doesn't succeed, and Ben Ali is not out on the Elgin road
to-morrow morning, I think Wily can be put through for the work he has
done here in this old house."
"I'm willin' to leave it that way," said Wily, "providin' you're
careful how you come onto Ben Ali, so as not to scare him away, an'
providin' Boss Burton gives me his word to back up Motor Matt's."
"I'm in on the deal," declared Burton.
"Both Wily and Dhondaram will have to be left here under suitable guard
until after the plan is executed," continued Matt.
"Count me in as one of the guard," spoke up Twomley, lighting a
cigarette, "but send over some food and something to sit on. And," he
finished, pointing to the weapon in the cowboy's hand, "Mr. McGlory
might lend me that."
"McGlory will stay and help you with your guard duty," said Matt. "I'll
have to hurry off now. I suppose Ping and Carl are at the show grounds
and are looking after the aëroplane?"
"Ping!" exclaimed McGlory, looking around. "Why, where the nation is
he? He was the one who brought us here, and I haven't thought of him
until this minute. But Carl's at the grounds, Matt. Anyhow, one of the
canvasmen is on duty at the aëroplane's berth."
"Don't fret about the machine," reassured Burton. "I'm going right back
to the grounds and I'll look after it personally."
"Just a minute, gents," called Wily. "How did you fellows know we were
under the floor."
"You walked in the soot," laughed Burton derisively. "McGlory can tell
you all about that."
Thereupon he and Motor Matt left the room. They passed the trap in the
hall floor, and Matt observed that it was flush with the boards and
difficult to locate for any one who did not know it was there.
"I guess the trouble I had here, Burton," remarked Matt, as he and the
showman passed through the front door, "will turn out to be a pretty
good thing, after all."
"Not for Ben Ali," returned Burton, "if he is caught and turned over to
Twomley."
"I was thinking of Margaret Manners," said Matt.
CHAPTER XII.
MATT LAYS HIS PLANS.
On the way through the woods and back to the road by the car track,
Motor Matt was extremely thoughtful.
By Ben Ali's cleverness in getting some white man to represent the
agent of the British ambassador, the Hindoo had succeeded in luring his
niece from the home of the English woman in whose care the girl had
been left.
Once this was accomplished, it was easy to guess how the artful Hindoo
had proceeded. Miss Manners had been a hypnotic subject for so long
that it was useless for her to attempt to fight against the black
magic of her rascally uncle. He had but to catch her eye and snap his
fingers, and the girl would be utterly in his power.
To fight such a man as Ben Ali called for ways and means at once bold
and wary. He was not to be easily snared.
"You're as mum as an oyster," grunted Burton, as they neared the road.
"I've spoken to you half a dozen times, and you didn't seem to hear me.
Come back to earth now, and tell me what's on your mind?"
"I'll tell you later, Burton," laughed Matt. "I've got a hard problem
to solve, and I don't want to say anything about it until it's all
worked out."
"From what you said at that house with the green shutters, I take it
you're not going back to the show with me?"
"No."
"Be back there in time to take the aëroplane aloft at six-thirty? The
wind's down, and you can pull off the trick."
"There'll be no aëroplane flight this afternoon, Burton. I have more
important matters to attend to."
Burton began to bristle.
"By Jerry," he cried, "what am I giving you your salary for? We've
missed one ascension to-day, and the people will be wild if we don't
have one this afternoon."
"Then," answered Matt, "tell them that we'll give an aëroplane
performance for the whole of Grand Rapids to-morrow. That ought to
satisfy them, and I know you'll make a lot of capital out of it."
Burton stopped stock-still and stared.
"You're crazy?" he bluntly inquired. "To-morrow's Sunday, and I've
never yet been able to get you to make an ascension on Sunday.
Backsliding, eh?"
"For this one time," said Matt. "I'm not doing this for the benefit of
your show, Burton, but because, as I size the matter up now, there's
nothing else to be done."
"Whew!" whistled the showman, "you're about the biggest conundrum, now
and then, that I ever tackled. When'll you get back to the grounds?"
"This evening, some time."
"Hunt for me the minute you get there, and let me know what's up."
They found Ping waiting for them in the road. He was a
disconsolate-looking Chinaman, and ran up to Matt the moment he slipped
down the steep bank.
"You heap mad with Ping, huh?" the Chinese boy chattered. "You know him
makee shoot Loman candle, play plenty hob with side show? Woosh! My
velly bad China boy."
Matt laughed. That laugh caused Ping to brighten.
"I'll have to forgive you this once, Ping," said Matt. "A whole lot of
good has resulted from that flare-up in the side-show tent. But I don't
like practical jokes--you know that. Get on the car and go back to the
grounds with Burton. As for the Roman-candle business, we'll talk about
that later."
"You no pullee pin on China boy?" faltered Ping.
"No. You make your peace with Carl, that's all."
"Hoop-a-la!" said Ping, and limped aboard an electric car that Burton
had flagged.
Matt caught a car going the other way, and, as soon as he reached
Monroe Street, hurried to the nearest automobile garage, bent upon
making the most of the daylight that remained.
He hired a car and a driver who knew the city. It was a small roadster,
and Matt had the driver take him beyond the city limits and out for
five miles on the Elgin road.
They passed through a small oak opening, which looked as though it
might be the place where Ben Ali was to meet his crony, Dhondaram.
"This will be far enough," said Matt. "Now, turn around and take us
back to town."
The king of the motor boys gave careful attention to all the landmarks,
going both ways. Returning, dusk had begun to fall, and his survey
could not be as comprehensive as the one made on the outward trip.
However, he was abundantly satisfied with the information he had
acquired.
When they reached the garage, Matt bargained with the proprietor for a
powerful touring car, with the same driver who had already been with
him, to be at the show grounds at Reid's Lake at eight o'clock the
following morning.
After that, he dropped in at a restaurant and had a good meal, then
boarded a car for the lake, and rode back to the grounds with a crowd
of people who were going to the evening performance of the show.
He had a good deal of amusement listening to the disappointed
expressions of the people regarding the failure of Burton to have any
aëroplane flights. Mixed up in the talk were a number of complimentary
references to Motor Matt and his chums. These, so far as they applied
to himself, the king of the motor boys tried not to hear. But,
nevertheless, they caused a glow of satisfaction to mount to his face.
It was certainly pleasant to know how his efforts in the line of duty
had struck a popular chord.
That wild half-hour in the air, over Jackson, when Matt found his
batteries short-circuited by a coiling cobra, had been exploited
through the press. These, while arousing the popular admiration, only
made the general disappointment more keen because of the failure of the
Saturday flights at Reid's Lake.
When Matt got off the car at the lake, he made his way to the
brilliantly lighted show grounds, and repaired immediately to the
calliope tent.
Burton was there, smoking a cigar and nervously walking back and forth
in front of the canvas-covered calliope.
"The people are pulling me all to pieces, Matt," he cried the moment
the king of the motor boys entered the tent. "They're saying we could
just as well have had a flight to-night, that I'm not living up to my
promises, and all that. By Jerry, it hurts!"
"Let it be announced in the circus tent," said Matt, "that there'll
be a flight to-morrow morning at nine o'clock--not for exhibition
purposes, as Motor Matt doesn't give a performance on Sunday--and that
all who wish to can see it."
"Good!" declared Burton. "I guess that'll catch them. But what are you
making the flight for, if not to please the people?"
"For the purpose of backcapping Ben Ali, capturing him, and finding
out where he has taken Margaret Manners."
Burton whirled around and gave Matt a steady look.
"What have you got up your sleeve?" he demanded curtly. "Are you going
to try that, all alone, in the _Comet_?"
"Not all alone. You, and Twomley, and Joe are going to help. Send
Harris and another trusty man over to that house with the green
shutters, will you, and have them relieve the Englishman and McGlory. I
want them here to talk with them."
Harris was Burton's brother-in-law, and a thoroughly reliable man in
every respect.
"I've already sent them supper, a lantern, and a couple of chairs,"
said Burton, "but it seems to me all foolishness to hold the prisoners
in the house. Why not send 'em to jail, where they belong?"
"Because Wily may not belong in jail, and because, if Dhondaram is
taken there to-night, Ben Ali might hear of it and not present himself
in that oak opening on the Elgin road to-morrow."
"Can't you tell me what you're going to do?"
"Not till Twomley and Joe get here."
With that, Matt dropped down on a cot, at one side of the tent, and
tried to get a little rest. He was used to the band, and to the many
other sounds that characterized a show just preceding a performance,
and these did not bother him; but his head! that had suddenly begun to
remind him that it had been badly treated during the afternoon.
CHAPTER XIII.
MOTOR CAR AND AEROPLANE.
It was about nine o'clock in the evening when Matt was awakened by the
arrival of McGlory and Twomley. Burton, curious and eager, came into
the calliope tent with them.
"I'll tell you what my plan is," said Matt, sitting up on the edge of
the cot, "and then you can all go to bed and get a good night's rest.
Ben Ali is a crafty scoundrel, and it is necessary for us to capture
him in order to find out what he has done with Miss Manners."
"That's the point," approved Twomley. "If we can't get hold of Ben Ali,
the Secret Service men will have a bally time locating the girl."
"I'm inclined to think that Bill Wily told nothing but the truth,"
proceeded Matt.
"You never can tell about Wily," struck in Burton. "It's because he's
so shifty and unreliable that they call him Wily Bill. I wouldn't bank
too much on what he says."
"It's neck or nothing with him," suggested Twomley. "He has everything
to lose by not telling the truth, and I believe the fellow appreciates
that fact."
"You can gamble a blue stack he does!" declared McGlory. "Did you see
the look Dhondaram gave him while he was handing us that long palaver?
If the Hindoo ever gets foot-loose, I wouldn't stand in Wily's shoes
for a bushel of pesos."
"To my mind," said Matt, "the fact that Dhondaram was in that house
proves the truth of Wily's story. Well, true or false, my whole plan
is built up on what the 'barker' told us. We're to assume that Ben Ali
will be in that oak opening, five miles from Grand Rapids on the Elgin
road."
"Who knows whether there's an opening there or not?" asked Burton.
When the showman once lost confidence in a man, he put no trust in
anything the man might do or say.
"The opening is there," said Matt. "I went out in an automobile and saw
it for myself."
"Ah! So that's what you passed up the afternoon flight for, eh?"
"Partly," answered Matt. "Now, let us suppose that Ben Ali is in that
opening to-morrow, waiting for Dhondaram to arrive with money which
Ben Ali thinks he has stolen. Quite likely the Hindoo will have some
one with him--perhaps the old ticket man whom you discharged, Burton,
and perhaps Aurung Zeeb. This ticket man has played the part of the
agent representing the British ambassador in turning that trick in
Lafayette----"
"Sufferin' traitors!" chanted McGlory. "I've a hunch, pard, your
finger's on the right button."
"So," pursued Matt, "it is fair to assume that Ben Ali has some one
to watch the Elgin road in the vicinity of the oak opening. If he is
warned that any suspicious persons are approaching, the Hindoo will
slide away snakelike and dodge pursuit."
Twomley nodded.
"You're a fair daisy, Motor Matt, in placing the situation squarely in
front of us. By Jove, it looks like a hard nut to crack."
"Matt will crack it," averred McGlory. "Listen, now, to how he proposes
to do it."
"How are you going about it?" inquired Burton impatiently. "I've had
this on my mind ever since you and I left the house with the green
shutters, and I can't tell how nervous you make me hanging fire about
it. Seems like a mighty simple thing to go out in the woods, meet a
fellow where he intends to be, and nab him."
"Not so deuced simple as you suppose, Mr. Burton," returned Twomley,
"when you consider the character of the man, and his ability to make
passes, look at you, and give you your ticket to the Land of Nod."
"We're going to work out this problem by motor car and aëroplane," said
Matt.
"Aëroplane!" exclaimed McGlory. "That means you and me, pard."
"The motor car for you, Joe," smiled Matt. "You and Twomley, and Burton
will go along the Elgin road in that."
"What's the good?" demurred Burton. "You all seem to think it a cinch
that the car will be seen, and that Ben Ali will get out of the way."
"You'll lag behind, you and your car," continued Matt, "and you'll let
me and the aëroplane move ahead. I'll keep over the road as well as I
can, and you can see me. When I sight our quarry I'll descend; then you
can put on all speed and come up."
"The aëroplane will be a dead give-away!" asserted Burton. "Ben Ali and
his outposts will see that as quick, or quicker, than they will the
automobile."
"Suppose Ben Ali sees only one man on the machine, and thinks that the
man is Dhondaram?" asked Matt. "Would he run, then?"
There was a silence, a startled silence, while the words of the young
motorist were being pondered by his listeners.
"How'll Ben Ali think Dhondaram is running the _Comet_, pard?" queried
McGlory.
"Because the man on the aëroplane will not look very much like Motor
Matt, and _will_ look a little like a Hindoo."
"You're going to make up for the part?"
"It won't be much of a make-up. A white robe over my ordinary clothes
will do."
"But your face----"
"In the air and at a distance, my face won't tell against the
deception. When the _Comet_ has landed in the opening, then it will
be Ben Ali and me for it--with an automobile full of reinforcements
rushing to the scene."
"It sounds good," said McGlory thoughtfully.
"Here's something," observed Twomley, who had a clear head and a quick
brain. "Ben Ali can think for himself. Won't he think it queer that
Dhondaram is navigating the flying machine? Dhondaram, I make no doubt,
is highly gifted, but will Ben Ali credit him with skill enough to
operate the aëroplane?"
"He may not," admitted Matt; "still, if Ben Ali sees the machine, and
a man in it who looks like Dhondaram, even if Ben Ali doubts he'll
hold his ground in order to make sure. Ben Ali won't run from one man.
Besides, he's expecting Dhondaram. That's a weighty point."
"I believe it will work," said Twomley. "At any rate, it will hold Ben
Ali in the opening until the automobile has a chance to come close.
Then the scoundrel is ours, no matter what he tries to do. By Jove, I
like the idea!"
"Another thing," spoke up McGlory. "If Ben Ali smells a rat and tries
to make a run, Matt can keep over him and follow him."
"Hardly that, Joe," returned Matt. "The woods are pretty thick along
the Elgin road, and you know how big the top of a tree looks when
you're gazing down on it. Besides, if there's any wind, the _Comet_ is
going to be a fair-sized handful to take care of."
"There you are," said Burton. "How do you know the opening is big
enough for you to come down in? It won't do," and something akin to
panic took hold of the showman, "to damage the aëroplane."
"Oh, go off somewhere, Burton, and wring out your wet blanket," growled
McGlory. "You're tryin' to throw it over everything."
"We've got to get a look at this business from every angle," said
Burton doggedly.
"Well, be easy about the oak opening," came from Matt. "It's large
enough to alight in and to start from. If there's only a little wind,
there'll be no danger."
The Englishman reached over and took Matt's hand.
"Allow me," said he, with a solemn handshake. "Win or lose, my bucko,
you have my admiration."
Matt flushed.
"Why," said he, "this is all talk, as yet, Twomley."
"It's the sort of talk, my lad, that precedes notable achievements.
Nine-tenths of all the great work that's done owes more to the head
than to the hands. What about the automobile?"
"That will be here at eight o'clock in the morning."
"You even thought of that! I suppose I'll have to be catching a car for
town."
Twomley got up and flung away the remains of a cigarette.
"You'd better stay here," suggested Matt. "There's an extra cot behind
the calliope, and I'm sure Burton will give you your breakfast in the
morning."
Twomley cast a glance around him. The odor from the animal tent,
of which the calliope house was only a lean-to, was strong and
disquieting. A lantern, tied to one of the tent poles, shed a murky
light over the litter of buckets and ropes that strewed the tent floor.
Matt had made ready for bed by kicking off his shoes and removing his
coat and hat. It was all very primitive. In Washington Twomley looked
as though he might have been of a fastidious nature. But, whatever he
was at Washington, he was "game" at Reid's Lake.
"Go you," said he briefly. "Just where is that cot, my dear sir?"
McGlory dragged it out for him and opened it up.
"I'll pull it away from the wall of the animal top," said the cowboy.
"Rajah, the bad elephant, is just on the other side of that piece of
canvas, and he has the habit of snooping around in here with his trunk."
"I don't fancy Rajah will bother me," and Twomley shucked out of his
low patent leathers.
"I could almost make a pard out of you," remarked McGlory.
"Nice work you've mapped out for Sunday," was Burton's sly fling as he
paused at the door on his way out. "Motor Matt, who refused to make
flights on Sunday for me for an extra hundred a week, lays out to pull
off a go like this! Well, I'm surprised."
"Fate is no respecter of the calendar, Burton," Matt replied, with
some show of feeling. "I'll work all day to-morrow if I can accomplish
anything for Margaret Manners."
"Shake again," said the attaché.
CHAPTER XIV.
THE OAK OPENING.
Reid's Lake was a popular resort, and a large crowd rendezvoused there
on Sundays and holidays. The coming of the crowd, however, had shifted
to the beginning of the day, so that the start of the aëroplane might
be witnessed.
Owing to Burton's enterprise, an "extra" of one of the evening dailies
was on the Grand Rapids streets at nine in the evening, announcing, in
large type, that Boss Burton, regretting the disappointment caused the
Grand Rapids people because of the failure of the aëroplane ascensions
on the first day of the show, was glad to announce that the king of the
motor boys would take his famous machine aloft on the following morning
at nine o'clock.
This was one of the little things Burton could do, on occasion, which
jarred on Matt's nerves. He made it appear in the news columns as
though Matt was making the ascension because Burton had so willed it,
and as though the showman had willed it because of the disappointment
which had been caused the Great Rapids people on the first day of the
show.
When Matt discovered this, it was too late to remedy it. He had the
satisfaction, however, of telling Burton just what he thought.
Extra cars were put on the run between town and the lake to accommodate
the crowds. And the people came not only in the street cars, but also
in carriages, wagons, and automobiles.
Carl and Ping had slept under the lower wings of the _Comet_, as was
their usual custom when the weather was at all propitious, and to
the casual observer it would have looked as though the Roman-candle
incident had been entirely forgotten.
Matt was early at the machine, looking it over carefully and making
sure that everything was in readiness. The _Comet_, he found, had never
been in better trim for work than she was that morning.
Then, too, such a day for aëroplane flying could not have been
surpassed. There was not enough wind stirring to flutter the banners on
the tent tops.
It was necessary for McGlory, Twomley, and Burton to get away somewhat
in advance of Matt, and to take up a position beyond the outskirts of
the city on the Elgin road. At sharp eight-forty-five the motor car
got away.
McGlory was usually in charge of the start during the aëroplane
flights, but now Matt placed Carl in command. The importance of the
position filled Carl with glory, and was correspondingly depressing to
Ping, who really knew more about the aëroplane than Carl could have
learned in a hundred years.
Carl and Ping were assisted by half a dozen stout canvasmen.
Before Matt took his seat, to the wonder of the crowd pressing against
the guard ropes, he shook out a white robe and arranged it about him in
such a manner as to leave his arms perfectly clear, but covering every
part of his clothing.
After that he stepped on the footboard and dropped down in front of the
motor.
The canvasmen, divided by Carl into two groups of three each, were
placed behind the wings.
"All ready, Carl!" called Matt.
"Retty it iss!" shouted Carl.
The motor started merrily, the bicycle wheels began to turn, and the
canvasmen to push.
Slowly the _Comet_ gathered headway. Faster and faster it went, leaving
the canvasmen behind; then, like a great bird, it soared into the air,
followed by wild cheering.
A vagrant puff of wind struck the planes, just over the concert garden,
and only quick work on the part of the intrepid young motorist averted
a disaster. Gathering headway under the impetus of the thrashing
propeller, the aëroplane darted upward into the blue and began reaching
out toward the city.
Matt, while manipulating the aëroplane, had little time for sights and
scenes below him. He was obliged to keep every faculty riveted on his
work. Now and again, however, as he took his bearings and laid his
course, he glimpsed the staring people in the roadways and on rooftops.
Some of these spectators had opera glasses and binoculars.
Over the flat roofs of the city he whirled, cheered almost continuously.
The motor had never worked better. Everything depended on the motor.
If the power had happened to fail, Matt could have glided harmlessly
down the airy <DW72> to earth--providing the city afforded him a good
clear space in which to alight. A street zigzagged with telegraph, and
telephone, and electric light wires was not such a place.
Passing the close-packed buildings of the business section, Matt gained
the residence districts, and held on in a straight line for the Elgin
road. He watched his landmarks, and, while they looked differently to
him from aloft than they did from the ground, he knew he was going
right when he saw the waiting automobile.
McGlory was standing up and waving his hat.
Throwing full speed into the propeller, Matt set the automobile a
fifty-mile pace. At such a speed only a few minutes were necessary to
carry the flying machine close to the oak opening where Ben Ali was to
be in waiting for Dhondaram.
Peering forward and downward, Matt guided and manoeuvred the _Comet_ by
sense of touch alone, watching eagerly the while for the great gap in
the woods.
Finally he saw it, and what he glimpsed in the centre of the cleared
space--etched into his brain as by the instantaneous operation of a
photographic lens--was startling, to say the least.
The irregular circle of the opening was crossed through its centre
by the hard, level road. Off to one side of the road were the dying
embers of a fire, and near the fire lay a bundle, on which a young
woman was sitting, her head bowed dejectedly. A turbaned figure stood
at a distance from the girl--the figure covered with a red robe and
its brown, staring face uplifted. This was Ben Ali. And the girl--who
was she? Was it possible, _could_ it be possible, that the girl was
Margaret Manners? A wild hope leaped in Motor Matt's breast.
Ben Ali leaned on a club, leaned and watched with never a move
toward running away. Probably he was speculating as to whether his
confederate, Dhondaram, had learned to operate the air craft.
Matt gave Ben Ali scant time to come to a conclusion. Quick work was
now in order, and the _Comet_ ducked downward and slid through the air
with slowing motor. Guided by a true, steady hand, the wheels brushed
the roadway, then began to turn as the weight of the machine rested
more heavily upon them. A short run of a dozen feet brought the _Comet_
to a stop.
Ben Ali had not stirred from the place where Matt had first seen him
standing.
Gathering the white robe about him, Motor Matt stepped hurriedly to the
ground and ran toward Ben Ali.
The Hindoo, staring serpent-like, recoiled, his red robe falling away
slightly as his hands raised the club.
"Ben Ali," cried the king of the motor boys, "I have caught you at
another of your tricks. Did you think I was Dhondaram? Dhondaram is a
prisoner, and you will soon join him in jail."
There followed a tense moment, during which Ben Ali's eyes glowed and
scintillated with their marvelous powers, and his hands tightened on
the bludgeon.
It was not a time to delay matters, and the young motorist made ready
for desperate work against the arrival of the automobile.
"_Maskee!_" cried the astounded Hindoo, as Motor Matt leaped at him.
Ben Ali's amazement appeared to hold him paralyzed for the moment. It
was not until Matt had caught the club that he aroused himself and
began vigorous resistance.
Every instant Matt expected the automobile to come whirling to the spot
with his friends.
He had the club, but Ben Ali, with a tigrish spring, seized him about
the throat and clung to him like a leech, and all the while Ben Ali's
eyes were rolling about in a way that was horrible to behold.
Matt dropped the club to catch at the Hindoo's straining arms. He felt
a wave of weakness sweep through him, while the flashing eyes continued
to exercise their baneful spell.
Was he being hypnotized in spite of himself? He had read that this was
impossible, and that no man could be put in a state of hypnosis against
his will. Yet what did that strange weakness mean?
A tremor ran through Matt's body. He tried to call aloud, but his lips
framed voiceless words. By degrees he felt himself growing weaker and
weaker, yielding more and more to the spell of the baneful orbs that
sought his undoing.
Then, when it seemed as though he was about to come entirely under Ben
Ali's power, there fell a blow--sudden, quick, and accompanied by a
wild, feminine cry.
Ben Ali's tense fingers relaxed their grip, his form slumped forward,
and Matt stood staring at the girl.
She was Margaret Manners, there was not the least doubt of that. In
order to save him, the girl had seized the bludgeon, had approached her
uncle from behind, and struck him down.
The girl's face was wild with grief, but there was a burning resolution
in the eyes.
"I had to!" she cried hysterically. "I had to do that in order to save
you. It was the spell, the spell of the eyes! He would have made you
his victim, Motor Matt, just as certainly as he has worked his will
with me! Oh, let us get away from here! Quick!" In a frenzy of fear
she cast aside the club and seized his arm with both hands. "There are
others--Aurung Zeeb is one. They are armed, and they will soon be here."
Matt dashed a hand across his forehead, as though to free his brain
from some frightful dream.
"There are others, you say?" he gasped.
"Yes, yes," she answered distractedly.
"Where?"
"Watching the road! They---- Ah, too late, too late!"
Matt whirled and looked across the oak opening. From the side lying
nearest the town came a running figure. It was Aurung Zeeb.
Where was the automobile? Matt could not hear it, and there was now no
time to wait.
The girl had dropped to her knees and thrown her hands over her face.
"Come!" he called, bending down and catching her by the arm. "We can
get away from here. Be brave, and trust to me!"
The girl started up, and he ran with her toward the aëroplane. As they
drew near the machine, Matt saw another Hindoo coming into the opening
along the other road.
CHAPTER XV.
AEROPLANE WINS!
Matt supposed that the automobile must have broken down somewhere on
the road. His friends had not arrived in time to help him, so he was
thrown upon his own resources.
While he and Miss Manners were racing toward the aëroplane, Matt was
measuring his chances. The appearance of the second Hindoo, on the
other side of the opening, complicated the dangers of the situation.
If these Hindoos were armed, as the girl had declared, then the case
was indeed desperate. In making its start, however, the _Comet_ would
be running away from Aurung Zeeb, and straight toward the other Hindoo.
This second man would have to leave the road or be run down; and if the
start was made quickly enough, the _Comet_ could get away from Aurung
Zeeb.
"Sit there," cried Matt, lifting the girl to a seat on the lower plane.
"Hold on," he added, starting the motor, "and don't move."
The girl's small fingers twined convulsively into the hand-holds. Matt
dropped into his own seat and turned the power into the bicycle wheels.
Slowly they took the push, the great wings lurching and swaying as the
aëroplane moved.
Would it be possible for the machine, unaided by a crew of men behind
the wings, to take to the air before the trees on the opposite side of
the opening interfered?
This was a momentous, nay, a vital, question, and could only be solved
by actual trial.
Out of the tails of his eyes Matt saw Ben Ali rising groggily to his
feet. He flung up his arms and shouted.
Crack!
From behind came a bullet, ripping through the canvas of the upper
plane, but, fortunately, doing no damage to the machinery. Aurung Zeeb
was doing the firing.
And this same Aurung Zeeb had failed Ben Ali once in a dangerous pinch.
This had caused a rupture of the friendly relations between the two
men, but their differences had evidently been patched up. Now Aurung
Zeeb was doing his utmost to help Ben Ali--and, perhaps, to land
himself in the same trouble in which Dhondaram had been entrapped.
Another bullet was fired, but Aurung Zeeb must have been shooting as he
ran, for his aim was poor.
Faster and faster raced the aëroplane, and Matt kept measuring the
distance between the machine and the trees on the farther side of
the opening. The Hindoo, in the road ahead, was running out of the
aëroplane's path like a frightened hare.
By then, Ben Ali had joined in the chase, but the speed of the _Comet_
was too great for the pursuers.
They were close to the edge of the timber, very close, when Matt felt
the wings beginning to lift. A dozen feet farther and they were in the
air.
In a flash the power was switched from the wheels to the propeller. The
aëroplane dropped a little before it yielded to the thrashing blades of
the screw; then it picked up the lost headway and arose.
The upward tilt was frightful, but necessary if a wreck in the treetops
was to be avoided.
Never a word had come from Margaret Manners. White as a ghost, she held
to her place, swaying her body to preserve a poise against the tilt and
pitch of the huge framework.
The wheels brushed against the outer ends of the tree limbs, but the
machine continued to glide into the air, walking upward as though
climbing the rounds of a ladder.
If the motor had failed from any cause, there could have been no
harmless gliding back to earth. A sheer drop downward would have been
the result.
But the motor performed its work, and the trees presently hid the
Hindoos and screened the _Comet_ from any further attack.
Then, and not till then, did the king of the motor boys draw a full
breath.
"Are you holding on, Miss Manners?" asked Matt.
"Yes," was the reply in a stifled voice.
"You're not afraid?"
"No."
"Bravo! We'll soon be back at the show grounds. You have seen the last
of Ben Ali."
High above the trees Matt brought the _Comet_ to an even keel, then
laid out in a straightaway flight toward the lake. This time he did not
follow the Elgin road, but struck across country the nearest way home.
That was not the first time Margaret Manners had had a ride in the
aëroplane. Some time before, when, under the name of Haidee, she had
traveled with the Big Consolidated, she had ridden on a trapeze swung
below the machine. It was against Matt's will, and only a trick of
Burton's had made it possible for the girl to make the ascension. At
the time she was under hypnotic influence, and could not realize what
she was doing. So, it followed, this was really the first ride she had
ever taken in the aëroplane while mistress of her own faculties and
able to understand her situation.
She behaved admirably, and did not even cry out when the wings tilted
sideways, or ducked forward with the seeming intention of hurling her
and Matt to the earth.
There was no talk between the two. In silence Matt attended to his
work, drove the _Comet_ at speed over the show grounds, circled, and
came down in the roped-off space set apart for the machine.
The crowds were still lingering, waiting for the aëroplane to return.
Cheering began as soon as the _Comet_ was in sight, and was kept up
until she was safely on the ground in the position from which she had
originally started.
Carl and Ping were waiting, too, and the eyes of both boys were big
with astonishment when they saw and recognized Margaret Manners.
"Vell, py shiminy grickets!" exclaimed Carl.
The girl smiled at him wanly as Matt helped her from her seat.
"You and Ping take care of the machine, Carl," cautioned Matt, as he
led Miss Manners to the guard ropes and parted a course for her through
the jostling mob.
"Hurrah for Motor Matt!" shouted some one. "He goes out alone and comes
back with a passenger!"
A laugh followed the cheer.
"What's the price for a trip on the _Comet_?" called some one else.
"Where does your air-ship line run?"
"Give me a ticket to San Francisco!"
Matt met the joking good-naturedly and piloted Miss Manners to the
calliope tent. The girl was tired and worn out.
"You'd better get a little rest, Miss Manners," Matt suggested. "What
you have passed through this morning would have shaken nerves much
stronger than yours."
"I don't want to rest," she answered; "I want to talk. You have saved
me again, Motor Matt, but what is the use of it all if I can't leave
this country and go to England, or back to India? Ben Ali will find me
again."
"You are through with him," said Matt, "just as I told you. A man has
come from the British legation in Washington to get you and send you
away by the first boat leaving New York."
"The man who came to Mrs. Chadwick's in Lafayette said the same thing,"
answered the girl wearily. "It seems as though there is no escaping Ben
Ali."
"Has he hypnotized you many times since he took you from Mrs.
Chadwick's?" asked Matt anxiously.
"Only once. I gave up hope, and went with him without trying to resist.
He said he intended to send me back to India, but not until the rajah
had paid him a lot of rupees."
"He treated you well?"
"He always treated me well--in his way--but the horror of going into
a trance and saying and doing things I know nothing about is more
terrible than ever to me. It was the fear of a trance that made me
promise not to make Uncle Ben any trouble."
"Who was the man who impersonated the agent of the British ambassador?"
"I had never seen him before."
"I thought that perhaps he might have been the man who sold tickets in
the ticket wagon for Burton--the one who was with the show when you and
Ben Ali were traveling with us."
She shook her head.
"I should have known that man if it had been he."
"Where did the man take you?"
"On the train somewhere. I thought we were going to Washington until we
got off the train at a little station and met Uncle Ben. It was then he
threw me into a trance, and when he awoke me we were at a little house
near the place where we went this morning to wait for Dhondaram. Aurung
Zeeb was at the house, and so was the other Hindoo--a man I had never
seen before. You are sure," the girl asked tremblingly, "that this
other agent of the British ambassador is really the person he pretends
to be?"
The girl's lack of confidence was pitiable. She had suffered so much
that Matt could readily understand her feelings.
"I am positive, Miss Manners," he answered gently. "You must rest now.
I will have Mrs. Harris come and stay with you for a while."
The girl did not object, and Matt had soon found Mrs. Harris and sent
her to the calliope tent.
Two hours later, while Matt was lounging around the front of the animal
tent, a tired party consisting of Burton, Twomley, and McGlory arrived
from the direction of the street-car line.
"You Matt!" cried McGlory. "Why didn't you wait and give us a chance?"
"If I'd waited much longer," answered Matt, "there wouldn't have been a
chance for anybody. Did you see me coming back from the oak opening?"
"Did we?" echoed Twomley, putting his monocle in his eyes. "By Jove, I
should say we did. Fancy! You up aloft, sailing as nice as you please
with Miss Manners beside you, and Burton, McGlory, and me tramping
along the road."
"What was the matter?" asked Matt.
"Matter?" fumed Burton. "What's the matter when you set out in an
automobile and don't arrive where you're going? The motor bucked, three
miles out of Grand Rapids, and you sailed right along and never paid
any attention to us. McGlory, Twomley, and I started to walk the rest
of the distance, when we saw the machine couldn't be fixed up for an
hour or so, and before we'd gone a mile you sailed off in the direction
of the show grounds--and never looked our way! Oh, blazes! I'm done
with automobiles."
CHAPTER XVI.
CONCLUSION.
Motor Matt's regret was keen over the failure to catch Ben Ali, Aurung
Zeeb, and the unknown Hindoo. It was one of those cases, however, where
it was best to be satisfied with the work accomplished, and to forget
the failure whereby three miscreants escaped the consequence of their
evil deeds.
And it was possible that Ben Ali was not long to enjoy his freedom,
for Twomley asserted that all the powers of the United States Secret
Service would be bent toward accomplishing his ultimate capture.
When it came to dealing legally with Dhondaram, a serious question
arose. If the Hindoo was to be punished severely, it would be necessary
to take him to Jackson, where the worst of his crimes had been
committed. This would require the presence of complaining witnesses,
of which Burton was one. For a man traveling from place to place
constantly, as was Burton, such a move could not be made without great
sacrifices.
It was deemed better, therefore, to have Dhondaram brought to book for
the lesser crime committed in the house of the green shutters. "Assault
with murderous intent" was the charge, and a light sentence followed.
Bill Wily, agreeably to promises given him, was released. Whether
he profited by his experience or not, Motor Matt never afterward
discovered. Such a lesson as he had had, however, should have been
enough for any man. For a little matter of ten dollars, he had entered
blindly into the schemes of Ben Ali--and Ben Ali's schemes left their
mark on every person who had anything to do with them.
Twomley was a delighted Englishman, if there ever was one. He had
fulfilled the mission with which he had been intrusted by Sir Roger,
and he had done so after discovering that his errand to Lafayette, so
far as securing Miss Manners was concerned, was useless.
A Roman candle in the side-show tent had lent itself to the
perpetration of a practical joke; and out of that joke had come the
clue which had made possible the second rescue of Margaret Manners.
Carl was very much pleased to learn that so much good had developed
from a row in the freak tent, but whether or not he forgave Ping for
setting off the Roman candle is open to question.
Carl had declared that he would "play even" with Ping for the candle
episode, and those who knew Carl best believed that he would prove as
good as his word.
Monday morning Twomley and Miss Manners took a train for New York, but
not until both the attaché and the girl had expressed to Matt and the
motor boys their appreciation of all that they had done.
It was somewhat indelicate of Carl, perhaps, to mention the matter of
his five thousand dollars before Miss Manners, but he was beginning
to worry about the money. As he expressed it, "Der longer vat der time
iss, der more vat I don'd seem to ged dot rewart. I peen sefendeen
years olt, und meppy I don'd lif more as sixdy years from now."
Twomley assured Carl that he would do whatever he possibly could to
hurry the money along. And with this promise Carl had to be satisfied.
With the turning over of Dhondaram to the police, the liberating of
Bill Wily, and the departure of Twomley and Miss Manners, a series of
thrilling incidents connected with Motor Matt's show career came to a
close.
And Motor Matt's show experiences were likewise drawing near an end.
Just how close this end was he did not dream that Monday morning when
he and McGlory accompanied the attaché and his charge to the train.
When the two boys got back to the show grounds, however, Boss Burton
had a telegram for Matt.
Burton was frankly worried about that telegram. Some other showman,
he felt sure, was offering Matt a bigger salary for his aëroplane
performances.
"Don't you forget for a minute," said Burton, watching keenly as Matt
opened the telegram, "that you're hooked up with me on a contract for
the season. You can't break that contract, you know."
"There were conditions, Burton," said Matt.
"The only condition I remember was something about the government
buying the aëroplane--which is all a dream. The government has bought
one of the machines, and that's enough. It takes a Motor Matt to run
one of those cranky Traquair air ships. It'll be a long while before
Uncle Sam buys another."
Matt read the message through, gave a whoop of delight, and passed the
yellow slip on to McGlory.
Then McGlory jubilated.
"What's to pay?" demanded Burton.
"Uncle Sam has done the trick!" crowed the cowboy. "He takes the
_Comet_ at the same price he paid for the _June Bug_--fifteen thousand
spot--machine to be crated and shipped immediately, if not sooner.
Whoop-ya! That settles the aëroplane business for King & McGlory. The
next game we get into will be something, I reckon, that I can take a
hand in, and not leave Pard Matt to do all the work."
Burton's face grew gloomy.
"Let me look at that message," he requested.
Matt handed it to him, and he read it over two or three times, then
dropped it savagely, and ground it under his heel.
"You don't _have_ to sell," said he angrily. "You can turn that offer
down if you want to."
"No, I can't," Matt answered. "The sale was virtually made up in North
Dakota weeks ago. Besides, I'm not the only one interested in the
deal."
"Who else besides McGlory?"
"Why, Mrs. Traquair, the widow of Harry Traquair, who invented the
extension wings and a few other things that have made the aëroplane
a success. Half of the fifteen thousand the government pays for the
machine goes to Mrs. Traquair."
"Oh, blazes!" growled Burton. "Don't tell the woman anything about it.
Send word back to the war department you don't want to sell; then I'll
make a new contract with you for a thousand a week. In seven or eight
weeks you boys will receive all your share of what the government pays
for the _Comet_."
Matt listened to the showman gravely.
"You don't mean what you say, Burton," said he. "If you think for
a minute that I'd play crooked with Mrs. Traquair, or with the
government, then you've got pretty far off your track. It's in our
contract that, if the government wants the machine, the contract
terminates. Here's where the motor boys' engagement with the Big
Consolidated comes to a close."
"You'll make a couple of flights to-day, won't you?" asked Burton,
swallowing his disappointment.
"Yes, I'll do that much for you," Matt answered, "and then, bright and
early to-morrow morning, we begin crating the machine for shipment."
"Blamed if I don't sort of hate to see the machine go," murmured
McGlory. "Many a hair-raising old trip you've had in the _Comet_, pard,
with me below lookin' up at you and almost kicking the bucket with
heart failure! Mainy a thriller the machine has given us, and--well, I
reckon it's done some good, too."
"That's the best part of it, Joe," said the king of the motor boys.
THE END.
THE NEXT NUMBER (30) WILL CONTAIN
Motor Matt's Mandarin
OR,
Turning a Trick for Tsan Ti.
On the Mountainside--The Yellow Cord--The Glass Balls--The Paper
Clue--Putting Two-and-two Together--A Smash--Nip and Tuck--Tsan Ti
Vanishes Again--Tricked Once More--The Diamond Merchant--The Old
Sugar Camp--A Tight Corner--The Glass Spheres--A Master Rogue--The
Eye of Buddha--The Broken Hoodoo.
MOTOR STORIES
THRILLING ADVENTURE MOTOR FICTION
NEW YORK, September 11, 1909.
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79-89 Seventh Avenue, New York City.
A BRAVE DEED.
The mining town of Capelton was alive with excitement. A
long-looked-for event was about to take place. Mr. Hilton, the owner of
the mines and more than half the village, was to give a ball in honor
of his son's twenty-first birthday, and also to celebrate the return of
his only daughter from the Parisian school to which she had been sent
when but ten years old.
Carl Hilton was an only son, and because of his parents' indulgence had
become selfish and tyrannical. His father idolized him, and was blind
to his faults. He was to become a partner in the mines on attaining his
majority. As Mr. Hilton had been out of health for more than a year,
Carl had attended to most of the business, and he had so tyrannized
over the miners that they one and all hated him; but they loved and
respected his father, and for his sake bore in silence the abuse of the
son.
To this birthday ball all the miners and their families had been
invited, and the rumors of the great beauty of Nina Hilton only added
to the excitement and anticipation.
I will not weary the reader by a description of the affair, and no
event of interest occurred until supper was announced. It fell to the
lot of Fred Chase, one of the foremen in the mines, to escort the
beautiful Nina, and so deeply did they become engaged in conversation
that it was some minutes before Fred noticed that Carl sat directly
opposite, and was watching them closely. With an effort the young man
concealed his annoyance, and continued his attentions to Nina.
"I intend to visit the mines to-morrow," said the girl, in tones loud
enough to be heard by her brother. "I want to descend the new shaft."
"I shall be very happy to conduct you through the mines, but you must
not descend the new shaft, for it is not safe. I have warned your
brother that the roof of the mine is in danger of falling, but he only
laughs at me, and I fear some terrible accident will be the result of
his neglect."
"You are a fool, Fred Chase! The shaft is safe enough; if you talk like
this, the men will all be afraid of it, and refuse to work. I shall
take Nina there myself to-morrow," said Carl angrily.
The young man's face flushed, but he controlled himself, and answered
coldly:
"I spoke the truth; the shaft is not safe, and unless more timber is
put in to support the roof, you will soon have proof that I am right. I
only hope that no lives will be lost."
"Pooh! You are a coward. I will show you to-morrow how little faith I
put in your words."
The eyes of all present were drawn to the two by Carl's excited tone,
and Fred's reply was plainly heard.
"Call me a coward, if you will, but time will prove the truth of my
assertion. Neglect for twenty-four hours to order more timber to be
placed in the new shaft for the support of its roof, and you alone will
be responsible for what follows."
Carl did not answer, but glanced angrily at Fred, who, after a minute's
pause, turned to Nina again, and changed the subject of conversation.
The following morning Carl started for the new shaft alone. Nina
refused to accompany him, and begged him to delay his visit until the
roof was made secure.
"Nonsense, sis! It is safe enough. That fool, Fred Chase, wanted to
impress you."
Carl believed what he said. He had not visited the shaft for several
weeks, and had not seen the timbers bend beneath the weight of earth
above them. He reached the shaft just as half a dozen miners came
from it, and in answer to his inquiries, was told that Fred Chase and
another man had remained behind to finish filling the last car with ore.
"I am going down," he said, and in a few minutes was lowered to the
bottom of the shaft. In the distance he could see the lights of the
two miners. He advanced toward them. By the light of his own lantern
he saw that some of the beams were bent; all seemed weighted to their
utmost capacity, and he could not but own to himself that Fred Chase
was right. He involuntarily shuddered as, in passing one large post, a
slight crackling sound was heard; but it was not repeated, and he went
on, determined to again make light of the matter.
"You see, I am not afraid of your shaft," he said sneeringly, as he
reached the spot where the two men were standing with the now loaded
car beside them.
"Only cowards need boast of their bravery," said Fred sternly.
"I am going on a short distance to look at the ore; you may wait for
me at the foot of the shaft, and we will all be drawn up at once,"
continued Carl.
He strolled on, while Fred and his companion returned, as directed, to
the entrance. They had barely reached it when they heard a loud report
behind; a cry of fear mingled with the noise of falling rocks; then all
was still.
With pallid faces the men looked at each other, for each knew what
had happened. The roof had fallen, and Carl Hilton was either crushed
beneath the rocks or imprisoned in the opening beyond.
Only an instant did they stand motionless. Then Fred grasped the rope
and gave the signal to be hoisted to the top.
They told their sad story, and a messenger was dispatched to Mr.
Hilton's residence. Soon the entrance to the shaft was a scene of wild
excitement. The stricken relatives of the buried man had reached the
spot as soon as possible. The father offered large rewards to any who
would attempt the rescue of his son; but not a man would volunteer.
Mr. Hilton doubled and trebled his reward, but to no avail; to his
entreaties were added the frantic pleading of the mother and Nina's
distressed sobs.
Fred had stood silent, with his eyes bent on the ground, until the old
man, in sheer despair, cried out:
"I will give half of my fortune, and it is a large one, to the men who
will help me reach my boy!"
Fred came forward with a look of resolve on his face. "Mr. Hilton, not
for your entire fortune would I enter that mine to save your son; but
for humanity's sake, I will do my best to rescue him."
A cheer from the miners greeted these brave words. With a wave of his
hand, Fred commanded silence, and running his eye over the crowd, said
slowly:
"I must have three trusty men to help me. Who will go?"
For an instant no one responded; then Charles Gray, Fred's chosen
companion, stepped to his side.
"I will go, Fred," he said quietly.
Two more men quickly followed the example of their brave leader, and,
armed with spades, bars, ropes, and a bottle of brandy, they were
lowered into the shaft.
Then followed a time of anxious suspense to the waiting crowd, who
could only pray for the safety and success of the rescuing party.
The first act of the workers was to place extra beams, a few of which
were lowered down the shaft for the purpose, as near as they could
to the fallen roof, to help bear any strain that might be resting on
those already there. In a few minutes they realized their wisdom, for a
cracking sound was heard which caused them to retreat toward the shaft;
but it was not repeated, and they returned to their work. At the end
of three hours of cautious digging they came to the car which Fred and
his companion had stayed behind to fill, and they stopped for a few
moments' rest.
"He cannot be far from here, for we had barely reached the shaft when
the roof fell. Hark! What was that?"
Fred stopped suddenly to listen.
"It was a groan! He is alive! Let us get to work, for he must be quite
near," said Charlie Gray excitedly.
With new zeal they worked on, and in half an hour they had reached an
opening caused by two large rocks, which had fallen together in such a
manner as to leave a space between them. In that space lay Carl, with
one arm doubled under him, and one foot pinioned by a large stone. The
poor fellow was terribly bruised and cut, but conscious. Very gently he
was lifted by the men and borne to the foot of the shaft. The signal
was given, and they were carefully drawn to the top, and when they laid
Carl on the ground a shout went up from the miners that echoed loudly
over the hills.
"God bless you, Fred, and your brave companions!" said Mr. Hilton
huskily, as he grasped the young man by the hand. "From my heart I
thank you."
"No thanks are due. I could not bear to see a fellow creature die
without trying to save him."
The crowd soon dispersed, and Carl was conveyed to his home. After many
weeks of suffering he recovered; but the crushed foot was useless--he
was a <DW36> for life.
As soon as he was able to do so, Carl sent for Fred.
"Forgive me, Fred," he said frankly. "I was wrong not to heed your
advice, but my punishment has been great. Forget the past, and allow me
to thank you for saving my life."
Fred could not refuse the apology thus offered, and the two became fast
friends.
About a year afterward Mr. Hilton bestowed his daughter's hand upon the
brave young man who had saved his son's life, and on his wedding day
Fred became one of the owners of the mines. He is now a wealthy and
prosperous man, and, with his beautiful wife, is almost worshiped by
the miners.
A LOCOMOTIVE HERO.
Well, boys, if you wish it, I'll tell you the story. When I was a youth
of eighteen, and lived with my parents, I had a boyish ambition to
become an engineer, although I had been educated for loftier pursuits.
During my college vacation, I constantly lounged about the station,
making friends with the officials, and especially with an engineer
named Silas Markley. I became much attached to this man, although he
was forty years of age and by no means a sociable fellow.
He was my ideal of a brave, skillful, thoroughbred engineer, and I
looked up to him as something of a hero. He was not a married man, but
lived alone with his old mother. I was a frequent visitor at their
house, and I think they both took quite a fancy to me in their quiet,
undemonstrative way.
When this Markley's fireman left him, I induced him to let me take his
place during the remainder of my vacation. He hesitated for some time
before he consented to humor my boyish whim; but he finally yielded,
and I was in great glee.
The fact was that, in my idleness and the overworked state of my brain,
I craved for the excitement, and, besides, I had such longing dreams
of the fiery ride through the hills, mounted literally on the iron
horse. So I became an expert fireman, and liked it exceedingly; for the
excitement more than compensated for the rough work I was required to
do.
But there came a time when I got my fill of excitement. Mrs. Markley
one day formed a plan which seemed to give her a good deal of
happiness. It was her son's birthday, and she wanted to go down to
Philadelphia in the train without letting him know anything about it,
and there purchase a present for him. She took me into her confidence,
and asked me to assist her. I arranged the preliminaries, got her into
the train without being noticed by Markley, who, of course, was busy
with his engine.
The old lady was in high glee over the bit of innocent deception she
was practicing on her son. She enjoined me again not to tell Silas, and
then I left her and took my place.
It was a midsummer day, and the weather was delightful.
The train was one which stopped at the principal stations on the route.
On this occasion, as there were two specials on the line, it was run by
telegraph--that is, the engineer has simply to obey the instructions
which he receives at each station, so that he is put as a machine in
the hands of one controller, who directs all trains from a central
point, and thus has the whole line under his eye. If the engineer does
not obey to the least tittle his orders, it is destruction to the whole.
Well, we started without mishap, and up to time, and easily reached the
first station in the time allotted to us. As we stopped there, a boy
ran alongside with the telegram, which he handed to the engineer. The
next moment I heard a smothered exclamation from Markley.
"Go back," he said to the boy; "tell Williams to have the message
repeated; there's a mistake."
The boy dashed off; in a few minutes he came flying back.
"Had it repeated," he panted. "Williams is storming at you; says
there's no mistake, and you'd best get on."
He thrust the second message up as he spoke.
Markley read it, and stood hesitating for half a minute.
There was dismay and utter perplexity in the expression of his face as
he looked at the telegram and the long train behind him. His lips moved
as if he were calculating chances, and his eyes suddenly quailed as if
he saw death at the end of the calculation. I was watching him with
considerable curiosity. I ventured to ask him what was the matter, and
what he was going to do.
"I'm going to obey," he said curtly.
The engine gave a long shriek of horror that made me start as if it
were Markley's own voice. The next instant we slipped out of the
station and dashed through low-lying farms at a speed which seemed
dangerous to me.
"Put in more coal," said Markley.
I shoveled in more, but took time.
"We are going very fast, Markley."
He did not answer. His eyes were fixed on the steam gauge, his lips
close shut.
"More coal," he said.
I threw it in. The fields and houses began to fly past half-seen.
We were nearing Dufreme, the next station. Markley's eyes went from
the gauge to the face of the timepiece and back. He moved like an
automaton. There was little more meaning in his face.
"More!" he said, without turning his eye.
"Markley, do you know you are going at the rate of sixty miles an hour?"
"Coal!"
I was alarmed at the stern, cold rigidity of the man. His pallor was
becoming frightful. I threw in the coal. At least we must stop at
Dufreme. That was the next halt. The little town was approaching. As
the first house came into view the engine sent its shrieks of warning;
it grew louder--still louder.
We dashed over the switches, up to the station, where a group of
passengers waited, and passed it without the halt of an instant,
catching a glimpse of the appalled faces and the waiting crowd. Then we
were in the fields again. The speed now became literally breathless,
the furnace glared red hot. The heat, the velocity, the terrible
nervous strain of the man beside me seemed to weight the air. I found
myself drawing long, stertorous breaths, like one drowning.
I heaped in the coal at intervals as he bade me. I did it because I
was oppressed by an odd sense of duty which I never had in my ordinary
brainwork. Since then I have understood how it is that dull, ignorant
men, without a spark of enthusiasm, show such heroism as soldiers,
firemen, and captains of wrecked vessels.
It is this overpowering sense of routine duty. It's a finer thing than
sheer bravery, in my idea. However, I began to think that Markley was
mad--laboring under some frenzy from drink, though I had never seen him
touch liquor.
He did not move hand or foot, except in the mechanical control of
his engine, his eyes going from the gauge to the timepiece with a
steadiness that was more terrible and threatening than any gleam
of insanity would have been. Once he glared back at the long train
sweeping after the engine with a headlong speed that rocked it from
side to side.
One could imagine he saw a hundred men and women in the cars, talking,
reading, smoking, unconscious that their lives were all in the hold of
one man, whom I now suspected to be mad. I knew by his look that he
remembered that their lives were in his hand. He glanced at the clock.
"Twenty miles," he muttered. "Throw on more coal, Jack; the fire is
going out."
I did it. Yes, I did it. There was something in the face of that man I
could not resist. Then I climbed forward and shook him roughly by the
shoulder.
"Markley," I shouted, "you are running this train into the jaws of
death!"
"I know it," he replied quietly.
"Your mother is on board."
"Heavens!"
He staggered to his feet. But even then he did not remove his eyes from
the gauge.
"Make up the fire," he commanded, and pushed in the throttle valve.
"I will not."
"Make up the fire, Jack," very quietly.
"I will not. You may kill yourself and your mother, but you shall not
murder me!"
He looked at me. His kindly gray eyes glared like those of a wild
beast, but he controlled himself in a moment.
"I could throw you off this engine, and make short work of you," he
said. "But, look here, do you see the station yonder?"
I saw a faint streak in the sky about five miles ahead.
"I was told to reach that station by six o'clock," he continued. "The
express train meeting us is due now. I ought to have laid by for it at
Defreme. I was told to come on. The track is a single one. Unless I
make the siding at the station in three minutes, we shall meet it in
yonder hollow."
"Somebody's blunder?" I said.
"Yes, I think so."
I said nothing. I threw on coal. If I had had petroleum, I should have
thrown it on; but I never was calmer in my life. When death actually
stares a man in the face, it often frightens him into the most perfect
composure. Markley pushed the valve still farther. The engine began
to give a strange panting sound. Far off to the south I could see the
dense black smoke of a train. I looked at Markley inquiringly. He
nodded. It was the express. I stooped to the fire.
"No more," he said.
I looked across the clear summer sky at the gray smoke of the peaceful
little village, and beyond that at a black line coming closer, closer,
across the sky. Then I turned to the watch. In one minute more--well,
I confess I sat down and buried my face in my hands. I don't think
I tried to pray. I had a confused thought of mangled, dying men and
women--mothers and their babies.
There was a terrible shriek from the engine, against which I leaned,
another in my face. A hot, hissing tempest swept past me. I looked up.
We were on the siding, and the express had gone by. It grazed our end
car in passing. In a sort of delirious joy, I sprang up and shouted to
Markley. He did not speak. He sat there immovable and cold as a stone.
I went to the train and brought his mother to him, and, when he opened
his eyes and took the old lady's hand in his, I turned hastily away.
Yes, gentlemen, I have been in many a railway accident, but I have
always considered that the closest shave I ever had.
What was the blunder?
I don't know; Markley made light of it ever afterward, and kept it a
secret; but no man on the line stood so high in the confidence of the
company after that as he. By his coolness and nerve he had saved a
hundred lives.
GEESE DROWN A SQUIRREL.
Jack, a big gray squirrel, who, with his mate, Jill, inhabited the
island in the duck inclosure in the Bronx Park Zoo, New York City,
sacrificed his life to his love of high living. It was this way:
Jack and Jill long ago discovered that by crossing over the
ten-foot-wide stream of water which separates the island from the
mainland on all sides they could reach a trough filled with corn,
which was replenished daily, for the ducks and geese, which rightfully
inhabit the pond and island. A wire fence dividing the inclosure used
by the mallard ducks from that enjoyed by the Canada geese offered a
means of communication between the island and the corn trough, and
Jack and Jill long ago became expert in running along the top of this
ticklish pathway.
Daily the two squirrels made pilgrimages to the corn trough, eaten to
repletion, and then returned to the island. The ducks and the geese
always swam close to the fence, flapping their wings and uttering
hoarse cries of rage, but were never able to catch the nimble
squirrels. Little by little, however, Jack lost his native agility as
he partook of more and more of the rich food, and when he started back
from a particularly heavy feast he waddled slowly along the top of the
fence instead of hopping nimbly along as had been his wont.
One of the mallards saw him and realized that he was too heavy and too
well fed to move hurriedly. The duck sounded a cry which brought all
of its mates, and they attacked Jack viciously. The squirrel tried to
hurry, but at last was pushed off the fence and fell into the pond.
In an instant he was surrounded by big Canada geese. Persons on shore
saw him fight desperately for life, but finally he was forced under
water. The geese churned the pond into a foam, and when they swam
majestically away there was nothing to be seen of Jack.
Jill, who ran back and forth on the shore of the island while Jack was
fighting for his life, retired to a tree after the tragedy, and has not
been seen since. Keepers think that she will not try to reach the corn
trough any more.
LATEST ISSUES
MOTOR STORIES
The latest and best five-cent weekly. We won't say how interesting it
is. See for yourself. =High art covers. Thirty-two big pages.
Price, 5 cents.=
16--Motor Matt's Quest; or, Three Chums in Strange Waters.
17--Motor Matt's Close Call; or, The Snare of Don Carlos.
18--Motor Matt in Brazil; or, Under the Amazon.
19--Motor Matt's Defiance; or, Around the Horn.
20--Motor Matt Makes Good; or, Another Victory for the Motor Boys.
21--Motor Matt's Launch; or, A Friend in Need.
22--Motor Matt's Enemies; or, A Struggle for the Right.
23--Motor Matt's Prize; or, The Pluck That Wins.
24--Motor Matt on the Wing; or, Flying for Fame and Fortune.
25--Motor Matt's Reverse; or, Caught in a Losing Game.
26--Motor Matt's "Make or Break"; or, Advancing the Spark of
Friendship.
27--Motor Matt's Engagement; or, On the Road With a Show.
28--Motor Matt's "Short Circuit"; or, The Mahout's Vow.
29--Motor Matt's Make-up; or, Playing a New Rôle.
TIP TOP WEEKLY
The most popular publication for boys. The adventures of Frank and Dick
Merriwell can be had only in this weekly. =High art covers.
Thirty-two pages. Price, 5 cents.=
687--Dick Merriwell's Colors; or, All For the Blue.
688--Dick Merriwell, Driver; or, The Race for the Daremore Cup.
689--Dick Merriwell on the Deep; or, The Cruise of the _Yale_.
690--Dick Merriwell in the North Woods; or, The Timber Thieves of the
Floodwood.
691--Dick Merriwell's Dandies; or, A Surprise for the Cowboy Nine.
692--Dick Merriwell's "Skyscooter"; or, Professor Pagan and the
"Princess."
693--Dick Merriwell in the Elk Mountains; or, The Search for "Dead
Injun" Mine.
694--Dick Merriwell in Utah; or, The Road to "Promised Land."
695--Dick Merriwell's Bluff; or, The Boy Who Ran Away.
696--Dick Merriwell in the Saddle; or, The Bunch from the Bar--Z.
697--Dick Merriwell's Ranch Friends; or, Sport on the Range.
698--Frank Merriwell at Phantom Lake; or, The Mystery of the Mad
Doctor.
699--Frank Merriwell's Hold-back; or, The Boys of Bristol.
700--Frank Merriwell's Lively Lads; or, The Rival Campers.
NICK CARTER WEEKLY
The best detective stories on earth. Nick Carter's exploits are read
the world over. =High art covers. Thirty-two big pages. Price,
5 cents.=
649--The Triangle Clue; or, Nick Carter's Greenwich Village Case.
650--The Hollingsworth Puzzle; or, Nick Carter Three Times Baffled.
651--The Affair of the Missing Bonds; or, Nick Carter in the Harness.
652--The Green Box Clue; or, Nick Carter's Good Friend.
653--The Taxicab Mystery; or, Nick Carter Closes a Deal.
654--The Mystery of a Hotel Room; or, Nick Carter's Best Work.
655--The Tragedy of the Well; or, Nick Carter Under Suspicion.
656--The Black Hand; or, Chick Carter's Well-laid Plot.
657--The Black Hand Nemesis; or, Chick Carter and the Mysterious
Woman.
658--A Masterly Trick; or, Chick and the Beautiful Italian.
659--A Dangerous Man; or, Nick Carter and the Famous Castor Case.
660--Castor the Poisoner; or, Nick Carter Wins a Man.
661--The Castor Riddle; or, Nick Carter's Search for a Hidden Fortune.
662--A Tragedy of the Bowery; or, Nick Carter and Ida at Coney Island.
663--Four Scraps of Paper; or, Nick Carter's Coney Island Search.
_For sale by all newsdealers, or will be sent to any address on receipt
of price, 5 cents per copy, in money or postage stamps, by_
STREET & SMITH, Publishers, 79-89 Seventh Avenue, New York
=IF YOU WANT ANY BACK NUMBERS= of our Weeklies and cannot procure them
from your newsdealer, they can be obtained from this office direct.
Fill out the following Order Blank and send it to us with the price
of the Weeklies you want and we will send them to you by return mail.
=POSTAGE STAMPS TAKEN THE SAME AS MONEY.=
________________________ _190_
_STREET & SMITH, 79-89 Seventh Avenue, New York City._
_Dear Sirs: Enclosed please find_ ___________________________
_cents for which send me_:
TIP TOP WEEKLY, Nos. ________________________________
NICK CARTER WEEKLY, " ________________________________
DIAMOND DICK WEEKLY, " ________________________________
BUFFALO BILL STORIES, " ________________________________
BRAVE AND BOLD WEEKLY, " ________________________________
MOTOR STORIES, " ________________________________
_Name_ ________________ _Street_ ________________
_City_ ________________ _State_ ________________
A GREAT SUCCESS!!
MOTOR STORIES
Every boy who reads one of the splendid adventures of Motor Matt, which
are making their appearance in this weekly, is at once surprised and
delighted. Surprised at the generous quantity of reading matter that we
are giving for five cents; delighted with the fascinating interest of
the stories, second only to those published in the Tip Top Weekly.
Matt has positive mechanical genius, and while his adventures are
unusual, they are, however, drawn so true to life that the reader can
clearly see how it is possible for the ordinary boy to experience them.
_HERE ARE THE TITLES NOW READY AND THOSE TO BE PUBLISHED_:
1--Motor Matt; or, The King of the Wheel.
2--Motor Matt's Daring; or, True to His Friends.
3--Motor Matt's Century Run; or, The Governor's Courier.
4--Motor Matt's Race; or, The Last Flight of the "Comet."
5--Motor Matt's Mystery; or, Foiling a Secret Plot.
6--Motor Matt's Red Flier; or, On the High Gear.
7--Motor Matt's Clue; or, The Phantom Auto.
8--Motor Matt's Triumph; or, Three Speeds Forward.
9--Motor Matt's Air Ship; or, The Rival Inventors.
10--Motor Matt's Hard Luck; or, The Balloon House Plot.
11--Motor Matt's Daring Rescue; or, The Strange Case of Helen Brady.
12--Motor Matt's Peril; or, Cast Away in the Bahamas.
13--Motor Matt's Queer Find; or, The Secret of the Iron Chest.
14--Motor Matt's Promise; or, The Wreck of the "Hawk."
15--Motor Matt's Submarine; or, The Strange Cruise of the "Grampus."
16--Motor Matt's Quest; or, Three Chums in Strange Waters.
17--Motor Matt's Close Call; or, The Snare of Don Carlos.
18--Motor Matt in Brazil; or, Under the Amazon.
19--Motor Matt's Defiance; or, Around the Horn.
20--Motor Matt Makes Good; or, Another Victory for the Motor Boys.
21--Motor Matt's Launch; or, A Friend in Need.
22--Motor Matt's Enemies; or, A Struggle for the Right.
23--Motor Matt's Prize; or, The Pluck that Wins.
24--Motor Matt on the Wing; or, Flying for Fame and Fortune.
25--Motor Matt's Reverse; or, Caught in a Losing Game.
26--Motor Matt's "Make or Break"; or, Advancing the Spark of
Friendship.
27--Motor Matt's Engagement; or, On the Road With a Show.
28--Motor Matt's "Short Circuit"; or, The Mahout's Vow.
To be Published on September 6th.
29--Motor Matt's Make-up; or, Playing a New Role.
To be Published on September 13th.
30--Motor Matt's Mandarin; or, Turning a Trick for Tsan Ti.
To be Published on September 20th.
31--Motor Matt's Mariner; or, Filling the Bill for Bunce.
To be Published on September 27th.
32--Motor Matt's Double-trouble; or, The Last of the Hoodoo.
PRICE, FIVE CENTS
At all newsdealers, or sent, postpaid, by the publishers upon receipt
of the price.
STREET & SMITH, _Publishers_, NEW YORK
Transcriber's Notes:
Added table of contents.
Retained some inconsistent hyphenation; in many cases, words are
hyphenated when used as adjectives but unhyphenated when used as nouns.
Italics are represented with _underscores_, bold with =equal signs=.
Front and rear covers, accent is missing from "Role" in original;
retained inconsistency.
Page 1, corrected ? to ! after "Howdy, Dutch!"
Page 3, corrected "shimiiy" to "shiminy" in "Py shiminy grickets!"
Page 4, corrected "Wiley" to "Wily" in "Go back to your job, Wily."
Page 6, removed stray single quote after "going to keep it."
Page 8, added missing quote before "I'm the one."
Page 9, corrected typo "minues" in "Inside of five minutes."
Page 11, removed unnecessary quote before "Yes, he decided." (Adding a
quote after "Yes" would also have been an option; however, this series
usually does not quote thoughts).
Page 16, changed "doin 'a" to "doin' a."
Page 18, changed "go" to "got" in "What have you got to do with this
house?"
Page 19, changed "he" to "the" in "the loss o' that Hindoostanee."
Page 21, changed "foolishnes" to "foolishness."
Page 22, changed "fair to asume" to "fair to assume." Changed "every"
to "ever" in "on my mind ever since."
Page 24, expanded oe ligature to "oe" for this text edition. Ligature
retained in HTML version.
End of Project Gutenberg's Motor Matt's Make Up, by Stanley R. Matthews
***
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 5,487
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Cincinnati Kid – amerykański film psychologiczny z 1965 roku na podstawie powieści Richarda Jessupa.
Główne role
Steve McQueen – Cincinnati Kid
Edward G. Robinson – Lancey Howard
Ann-Margret – Melba
Karl Malden – Shooter
Tuesday Weld – Christian
Joan Blondell – Lady Fingers
Rip Torn – Slade
Jack Weston – Pig
Cab Calloway – Yeller
Jeff Corey – Hoban
Theodore Marcuse – Felix
Milton Selzer – Sokal
Karl Swenson – Pan Rudd
Émile Genest – Cajun
Ron Soble – Danny
Irene Tedrow – Pani Rudd
Midge Ware – Pani Slade
Fabuła
Nowy Orlean, lata 30. W mieście dochodzi do spotkania najlepszych pokerzystów w USA. Jednym z nich jest Cincinnati Kid, który jest niepokonanym mistrzem w mieście. Przybywa też Lancey Howard – pokerowy mistrz kraju. Kiedy rozgrywa pojedynek z biznesmenem Saldem, ogołaca go ze wszystkiego. Kid chce konfrontacji z Howardem, dzięki Shooterowi do niej dochodzi. Ale Cincinnati Kid nie wie, że wynik jest ustawiony, gdyż Howard przekupił krupiera!
Nagrody i nominacje
Złote Globy 1965
Najlepsza aktorka drugoplanowa – Joan Blondell (nominacja)
Linki zewnętrzne
Plakat
Amerykańskie filmy psychologiczne
Amerykańskie adaptacje filmowe
Filmy w reżyserii Normana Jewisona
Filmy wytwórni Metro-Goldwyn-Mayer
Amerykańskie filmy z 1965 roku
Filmy o tematyce hazardowej
|
{
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| 5,638
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import * as fs from 'fs';
import * as path from 'path';
import * as os from 'os';
import * as assert from 'assert';
import * as vscode from 'vscode';
import { formatImports } from '../../format-imports';
interface Case {
input: string;
output: string;
}
const cases: Case[] = [{
input:
`'use strict';
import { foo } from "./b";
import { bar } from './a';
import * as b from '../b';
import baz from '../../c';
import { writeFileSync } from 'fs';
import {
one,
two,
three
} from './four';
require('./style.scss');
import { join as pathJoin } from 'path';
export const lorem = 'ipsum';
export const dolor = 'sit';
`,
output:
`'use strict';
import { writeFileSync } from 'fs';
import { join as pathJoin } from 'path';
import baz from '../../c';
import * as b from '../b';
import { bar } from './a';
import { foo } from './b';
import {
one,
two,
three
} from './four';
require('./style.scss');
export const lorem = 'ipsum';
export const dolor = 'sit';
`
}];
function runCase(input: string) {
let testFilePath = path.join(os.tmpdir(), 'format-imports-' + (Math.random() * 100000) + '.ts');
fs.writeFileSync(testFilePath, input);
return vscode.workspace.openTextDocument(testFilePath).then((document) => {
return vscode.window.showTextDocument(document).then((editor) => {
formatImports();
return new Promise((resolve) => {
setTimeout(resolve, 200);
}).then(() => {
return editor.document.getText();
});
});
});
}
suite('formatImports', () => {
test('formatImports', () => {
return cases.reduce((p, c) => {
return p.then(() => {
return runCase(c.input).then(output => {
assert.equal(output, c.output);
});
})
}, Promise.resolve());
});
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,499
|
Home/Others/In Beijing, Trump declines to hit President Xi Jinping on commerce: 'I don't blame China'
In Beijing, Trump declines to hit President Xi Jinping on commerce: 'I don't blame China'
U.S. President Donald Trump and U.S. first woman Melania go to the Forbidden City on Wednesday with Chinese President Xi Jinping and China's first woman Peng Liyuan in Beijing. (Jonathan Ernst/Reuters)
BEIJING — President Trump lavished reward on Chinese chief Xi Jinping right here Thursday, touting "great chemistry" between them whereas refusing to criticize his counterpart for the commerce imbalance that Trump railed in opposition to throughout his marketing campaign.
Speaking at a joint look with Xi in entrance of enterprise leaders, Trump stated the U.S. commerce relationship with China is "a very one-sided and unfair one." But, he rapidly added: "I don't blame China. Who can blame a country that is able to take advantage of another country for the benefit of its citizens. I give China great credit."
During the marketing campaign, Trump threatened to label China a "currency manipulator" — though financial badysts have stated Beijing has not artificially inflated the renminbi for years. In his remarks right here, Trump reiterated that the United States should "change its policies," however he supplied no particulars about actions his administration will pursue.
"We've gotten so far behind on trade with China and frankly many other countries," Trump stated forward of a bilateral badembly with Xi, earlier than saying he had "great respect" for Xi for "representing China."
Trump blamed previous administrations "for having allowed it to get so far out of kilter. We'll make it fair, and it'll be tremendous for both of us. My feeling toward you is incredibly warm. We have great chemistry. I think we'll do tremendous things, China and the U.S."
Their high-stakes, two-day summit is being carefully watched for indicators of how the leaders of the world's two greatest economies will be capable of cooperate on points from North Korea to commerce to cyber safety amid mounting challenges within the Asia-Pacific. Trump is hoping to win concessions from Xi, however the Chinese chief is in a powerful place after having consolidated energy at a Communist Party congress final month.
The two nations introduced memorandums of understanding to extend commerce by $253 billion, which the leaders stated was an indication of better cooperation.
In distinction, Xi appeared reserved and spoke in fastidiously scripted language about "win-win" cooperation and a "new starting point" for the bilateral relationship — language Beijing has employed in a bid to get the United States to agree to permit China to function in its "sphere of influence" in Asia with out meddling. Xi didn't discuss in private phrases about Trump, even because the U.S. chief praised him repeatedly.
The United States and China had clashed on points from cyber safety to commerce within the closing years of the Obama administration, although that they had struck a landmark local weather deal throughout Obama's 2014 go to to Beijing that served as a prelude to the Paris local weather accord.
Trump has introduced intentions to withdraw the United States from that settlement, however Xi has pledged to make China a pacesetter on lowering carbon emissions.
Xi has responded to Trump's push to extend stress on North Korea over its nuclear program, however U.S. officers have stated they hope Beijing will do extra to chop off commerce and monetary flows with North Korea.
Xi vowed to work collectively within the "spirit of mutual respect and mutual benefit."
On Trump's first full day in China — the third cease on a five-country, 12-day journey by Asia — Xi greeted Trump on the Great Hall of the People, a show that included three horn gamers in crimson uniforms, a army band and ceremonial cannon hearth.
Trump, who has recommended he wish to stage a army parade in Washington over the July four weekend, appeared impressed. He referred to as the parade "magnificent" and stated the world was watching.
"I already had people calling from all parts of the world," he added. "They were watching. Nothing you can see is so beautiful."
The Chinese have described Trump's journey to the nation as a "state visit plus" and to date have lavished him with particular remedy. He arrived Wednesday afternoon for a sundown tour of the Forbidden City, the ornate Chinese imperial palace stretching from the Ming to Qing dynasties, earlier than taking in a efficiency of the Peking Opera.
Trump effusively thanked Xi for internet hosting him and first woman Melania Trump at a dinner after his arrival an evening earlier, saying meal was scheduled to final lower than half an hour as a result of Trump was drained after an extended day of touring from Seoul. Instead, Trump stated, it went on for greater than two hours as they loved every others' firm.
"I enjoyed every minute of it," Trump stated. At one other level, Trump informed Xi: "You are a very special man."
The elaborate stagecraft and flattery belies the true points — and actual tensions — between the 2 nations. Trump arrived in China whereas being dogged with political issues again dwelling and going through the bottom approval numbers of his presidency.
On Tuesday night time, the president obtained extra dangerous information, within the type of a Democratic rout within the off-year elections throughout the nation. Ed Gillespie, the Republican candidate for governor in Virginia whom Trump endorsed, misplaced to Lt. Gov. Ralph Northam, who was propelled to victory within the state by extremely motivated Democrats who turned out to vote.
Trump is anticipated to press the Chinese to place extra diplomatic and monetary stress on North Korea to stop its nuclear and ballistic missile applications, and has referred to as out China by identify a number of instances already on the journey, to ask them to do extra.
Trade might be one other subject of debate. The president, whose high commerce advisers embody Robert E. Lighthizer and Peter Navarro — writer of the guide "Death by China" — has lengthy railed in opposition to the commerce imbalance with China, which he argues is harming the American employee, and is probably going to make use of this go to to stress Xi for extra favorable phrases.
But regardless of the pageantry surrounding the go to and an eagerness on the a part of China to reset their relationship with the United States, Xi — who's now arguably his nation's strongest chief since Mao Zedong — and the Chinese additionally really feel emboldened to demand concessions from the United States
"One of the things I find interesting is that they also seem quite confident, though, that if they're not able to push the reset button on the relationship, they're ready to turn in the other direction as well," stated Christopher Okay. Johnson, senior adviser in China research on the Center for Strategic and International Studies, briefing reporters earlier than the journey. "So if they're not able to turn it around, I think we could see sort of a snarkier-looking position from the Chinese on the bilateral relationship after the summit's over."
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 971
|
:mod:`diffprivlib.accountant`
=============================
.. automodule:: diffprivlib.accountant
Base class
-----------------------------
.. autoclass:: BudgetAccountant
:members:
:inherited-members:
:exclude-members: slack,spent_budget,epsilon,delta
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,286
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Weberbauerocereus churinensis är en kaktusväxtart som beskrevs av Friedrich Ritter. Weberbauerocereus churinensis ingår i släktet Weberbauerocereus och familjen kaktusväxter. Inga underarter finns listade i Catalogue of Life.
Källor
Kaktusväxter
churinensis
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,334
|
Late last week I posted a screenshot of the November 2018 sales statistics for Metro Denver. While the state economists today suggested 2019 should be a positive year for Colorado's economy concerning job and wage growth across all sectors with a mild slowing; the housing market may be advising differently.
Our housing market did not go into a free-fall unlike Las Vegas and Phoenix.
We have been in a 5+-year expansion concerning housing prices.
Wages are not keeping up with housing costs in Metro Denver.
New construction did not keep up with demand over the last 5 years.
In general housing costs in San Francisco and Seattle are more expensive then Denver HOWEVER their average incomes are higher and by geography their ability to expand and build outward is limited.
While housing prices in metro Denver were on what seemed like an exponential trajectory I have suggested prior and statistics may be validating we peaked a few months back. While sales prices continue to climb, inventory is increasing, days on market are increasing and eventually prices may begin to adjust downward or keep with inflation and not show oversized gains.
The November 2018 #'s are interesting and showing an impressive gain on a year-to-year basis and while month-over-month does not show a trend I suggest the real estate market is looking outward and showing some hesitation similar to how the stock market projects out 6-12 months.
What will be interesting in to see what November 2019 stats show. My gut is we will see prices either static or lower. Inventory will be higher and days on market will also increase.
This is not necessarily negative, as markets should over time trend back towards normalcy. For too many years we have been in a seller market and it is time to move back to equilibrium of sorts. In high-end neighborhoods there seems to be a glut of expensive homes waiting a buyer or rental signs as owners wait our the market conditions. While there continues to be some blockbuster sales they are more of an anomaly versus weekly updates. Two recent high profiles sales in Cherry Creek North and Belcaro were to out-of-state buyers relocating as part of VF Corp. relocation to Denver.
My concern is for our local and regional population of move up and move down buyers. At present 1sttime homebuyers continue to be challenged in the market and even as prices may be stabilizing; interest rate increases negate the opportunity of lower pricing.
Move-up buyers are being challenged in finding suitable inventory. This is worrisome as families outgrow their first home or desire more space find inventory challenged in central Denver and will migrate to the suburbs/exurbs or worse leave the state. Move-down buyers those who may be downsizing can take advantage of the sellers market HOWEVER again their inventory for replacement is challenged and thus may consider regional relocation or out of state.
While I am not predicting a severe downtown I would not be surprised to see a 5%-10% correct concerning housing prices over the next year across Metro Denver. I believe there are segments i.e. the luxury housing niche i.e. $750K and above that will see more severe adjustments.
Let's just use this blog posting as an opportunity to revisit in one year.
With the run-up in real estate prices in Metro Denver since The Great Recession we are finally witnessing the cooling of the market memorialized in the New York Times a few weeks ago in an article titled: Housing Market Slows, as Rising Prices outpace Wages.
While those sellers who have owned their residences for over 3 years are probably fine with selling and gaining a small profit; over the past few months I have written about a few residential sales which has actually taken a loss via actual recorded sales price and additional losses when factoring in commission and of course inflation which seems to rarely be factored into the transaction.
In the present environment of housing prices adjusting downward, interest rates continuing to increase and signs of instability in the equities markets those taking losses on their residences may become more commonplace depending when they purchased and how motivated they are to sell.
Of note in general a loss on the sale of a home is NOT deductible on one's income tax. In general a loss concerning real estate is only deductible when the property has been used for business or investment purposes. One tip if a loss may be forthcoming consider turning the residence into a rental and then sell; the property is now considered related to investment. Of course one would need to consult with their professional tax advisor or financial planner to ascertain the legality and proper filing but this is an option.
I predict we will start seeing some losses on homes in the Denver Metro area that had been purchased within the last 12-36 months when the market seems unstoppable concerning price appreciation coupled with historically low interest rates. While it may seem counterintuitive when the employment market is at its zenith that housing should be lagging yet that is generally how the market behaves. This is partially due to interest rate impacts, inflation eroding the value of money and other factors. This is not a new phenomenon; happens with every business cycle. This is why longer-term holds on housing usually generates a hedge against inflation but the key is long-term i.e. 7,10, 20 years out. A few examples if I may including my personal residence.
Two years ago (April 2016) I sold my personal residence for $535,000. The net after commission and closing costs was $520,000 (and no I did not pay myself a commission).
I purchased the house in October 1989 for $140,000 or $266,692 in inflation dollars. Thus my actual net gain was $253,308….no I am not complaining. On a monthly basis I made about $800 +/- and when factoring in taxes, maintenance, upkeep…..lets just say I had a house over my head.
Thus not including commissions in real dollars the seller not only took a $60,000 loss from his purchase to the sale in 5 years, when factoring in inflation i.e. $38,566 and commission (6% at $140,000 = $8,400) the seller lost $45,000+ or almost a quarter of the value of his home in that 5 year period and the loss was not deductible.
As mentioned I sold the house in April 2016 for $535,000.
The buyers actually resold the house in June 2017 for $560,000 due to a relocation thus even after commissions and closing costs they did OK. From what I understand the new owners plan to reside long-term and thus are somewhat insulated from the pending adjustments in housing prices I believe will be headwinds in the near future.
The single biggest sale last month (September 2018), at $42 million, was a penthouse covering the entire 77th floor of One57, the vitreous skyscraper in the heart of Manhattan's Billionaires' Row, at 157 West 57th Street. Monthly carrying charges are $15,214. The unnamed European seller took a loss, however, having paid nearly $47.8 million for the unit in May 2015. The 6,240-square-foot apartment has four bedrooms and five and a half baths, not to mention breathtaking views.
While a $6M loss is painful when you consider the apartments were delivered with interiors unfinished, at that price-point you bring in your own designers and architects which can easily add $500,000 to $1M+ in finishes AND the monthly carrying charges i.e. HOA fees ranging increasing from $12,500/month to $15,214 at the time of sale that is over $150,000 annually just in common charges or another $500,000 paid during ownership. Thus all losses are relative; as we say on Wall Street you will never sell at the high and buy on the bottom.
A house is a home and should not necessarily be considered an investment or a hedge against inflation, it is shelter first and foremost.
While some brokers continue to suggest the recent slowdown in sales and significant and immediate price reductions is seasonal (and they may be correct) a few outlets are advising the slowdown in the market may be more serious. An article from The New York Times titled Housing Market Slows as Rising Prices Outpace Wages provided their national and international readership with an interesting overview of Denver which is not flattering. Even during my recent trip to Hong Kong more than one person when realizing I reside in Denver mentioned the article.
Related according to the monthly report from the Denver Metro Association of Realtors: In September (2018), housing inventory continued to move higher, even though it typically decreases this time of year, and home prices dropped nearly 5 percent since its record-peak highs this past May and June. Good for prospective buyers not necessarily welcome news for sellers.
Some of my readers have advised privately that I am a pessimist as I have been advising a downturn or the moving towards a more stable market. I do not consider myself a pessimist; more a realist. With 20+ years as a broker literally been there and gone through that. While I too have been impressed with the most recent expansion post The Great Recession I have been concerned about headwinds in the market from out-migration to increasing interest rates to incomes lagging housing price appreciation.
To place this in prospective, a undivided flat means the 200 sq. ft. Ms. Feng rents is part of another apartment. Their rent in US Dollars is $1,021/month. The average hotel room in the United States is 325 sq. ft. or 125 sq. ft. larger than the living space for this family of 5!
While I have in general been against the concept of slot homes due to its impact on the existing urban fabric of traditionally single-family and duplex neighborhoods; all of a sudden Hong Kong makes such density look palatable even preferable.
My wife and I have been looking for a home (for followers of my blog we sold our primary residence of just shy of 30 years back in April 2017). We have kept our eye on a listing in one of Denver's most desirable and stable (concerning values over the long-term) neighborhoods. The home we expressed interest in is small (similar houses have been expanded), requires updating to present code including electrical, no garage and the basement shows evidence of past and more recent water damage. Coupled with all the above information the most recent index by Beracha, Hardin & Johnson Buy vs. Rent Index suggests we would be better of renting than purchasing at present yet as brokers we too sometimes operate on emotion and we are looking longer-term.
While the index does somewhat influence my decision; being a logical broker I conducted my due diligence concerning comparable properties in the same block on the same side of the street. I went back a few years and extrapolated the comparable's using an inflation calculator to justify our offer.
-This home is in meticulous shape including the architecturally designed addition on the rear with the expanded kitchen, family room with fireplace, 2-car garage and professionally landscaped front, rear and side.
-While I have not seen the inside except from the exterior new lighting, new windows, architect-designed extensions on the rear, garage parking to match. It is a duplex and both sides sold together as one structure. Each 1/2 of the duplex has 3 bedrooms and 2.5 bathrooms, larger than the subject property.
-While used as a pied-a-terre the interior condition is similar. The kitchen was outdated however larger space, has a garage and deep south setback with a lot that is 1,000+ SF larger than subject property.
–The house is very similar to Comp 1 (next door) yet narrower lot and smaller size overall. Excellent design and layout. The rear and upper extension were beautifully designed and executed with functionality i.e. den w/ fireplace, expanded kitchen with breakfast area, 2 car garage made of brick to match the historic urban fabric coupled with a professionally landscaped yard.
Thus concerning the comparable properties using 2018 dollars the prices per square foot above grade range from $420 to $546. While 4 homes do not make a proper statistical average would be $463.50 PSF based on inflation with the $546/PSF sale skewing the average upward do to limited sample size. Of note the Median is $444/PSF.
Many of my peer brokers believe the peak of the market was 6-12 months prior as prices are beginning to slip, inventory is increasing coupled with rising mortgage interest rates.
Based on the $463 PSF average noted the house we made the offer upon should be priced at approximately $625,000 which may even be somewhat aggressive as the comparables are homes that have been extensively renovated or updated and all include alley access garages.
We offered $560 PSF or 20% above the comparable properties identified on a PSF basis.
Our offer was promptly rejected as the seller is asking $625 PSF.
While no fault of the out-of-state seller if /when the residence goes under contract and assuming there is an appraisal there may be a rude awakening. We could have offered full price and use the appraisal and inspection contingencies to eventually close at a lower market oriented price; however that is not our method of operation.
We made a viable offer, provided statistical pricing guidance and was subsequently rejected based on I assume emotion and/or irrational exuberance concerning valuation. I have been incorrect before and the residence may actually sell for asking (of note at present on the market almost two months and one price reduction to date); on this one we like it (we do not love it) however we willing to wait it out or pass altogether as inventory increases and pricing pressures are forecast to be in our (buyers) favor.
It was not so long ago when one purchased a home with the rationale of not only having a roof over's one head but also a vehicle to keep up with and even better beat inflation and have enjoy some added tax deduction benefits.
Assuming a residence can only increase in value (the Great Recession shattered that myth).
Using equity in one's residence as leverage (the House as Personal ATM).
Limitations on the deductibility concerning real estate taxes.
As a broker I completely understand the desire for a home purchase especially when we see markets with low inventory and continued historically low-interest rates. Yet are Millennials setting themselves up for future challenges?
Yes most millennials went through the Great Recession and while experienced may not have been in the workforce or owned a residence. They may not have witnessed the job losses, foreclosures and the evaporation of paper wealth over that period. While the economy has come roaring back (even though I question the longevity of this bull market) as I always advise past performance is not indicative of future returns.
This is why a recent survey from The Bank of the West truly concerns me as follows: "The fact that nearly one in three millennials who already own their homes have dipped into their retirement nest eggs to finance their down payment is alarming. With careful financial planning, millennials can have it all – the dream home today, without compromising their retirement security tomorrow." Ryan Bailey, Head of the Retail Banking Group at Bank of the West.
Basic reality; a mortgage is debt, plan and simple. While a long-term mortgage with a low monthly payment and a fixed interest rate may be attractive and definitely can be a hedge in an inflationary environment, it is still debt.
Yes the mortgage payment may in fact be less than comparable rent (yet did the buyer factor in the down-payment).
While there are tax advantages including mortgage interest and real estate tax deductions, are the benefits truly appreciable concerning one's income? The debt to income ratio can be an eye-opener.
Unlike retirement investing which is usually liquid and easily revised depending on market conditions, a residence is truly illiquid and can incur major costs when trying to sell i.e. commissions, preparation to sell and so forth.
Home ownership can be a foundation for a lifetime. This is not necessarily a positive attribute. What happens if the homeowner decides to entertain an employment opportunity elsewhere? What if the market during that time is a buyer's market? What if market rent would NOT cover the monthly PITI? In such scenarios one may be losing precious investment opportunities while covering the monthly payment coupled with an inflation reduced asset.
Mortgages do provide leverage and equity via one's down-payment HOWEVER during the recession the terms negative equity, short-sales and foreclosures entered the vernacular and unfortunately we all have collective short-memories. Just last week I viewed a home on S. Monaco in the Southmoor neighborhood. While needing some cosmetic updates the home is in good condition and state of repair. Lowest priced home in the area concerning both asking and on a PSF basis. The asking $475,000 yet this is a short-sale with a loan balance of $515,000. Yes in the present sellers market a short-sale!
In addition to all of the above what concerns me locally here in Denver is the type and location of residences millennial's are purchasing. I am seeing a proliferation of townhouse style residences as well as condos and similar attached multi-family construction in all the most desirable neighborhoods i.e. Golden Triangle, LoHi, Highlands, Sloans Lake and others. Concerning affordable, think again, many are $500K+ some pushing 7 figures. Yet I am seeing younger buyers purchasing with the assumption that 1) housing will continue to appreciate, 2) they plan to live in or potentially rent if they move or lifestyle change and 3) using monies allocated for retirement and/or using family capital to assist in purchase with the belief that inflation coupled with low mortgage loan rates is a winning combination.
While these new homes are beautiful and contemporary and perfect for the single or young DINK (dual-income no kids) couple; lifestyles change. Are these buyers considering children in the future? Are the local schools the caliber they desire for their offspring? Is there a risk of a glut in the area when the market adjusts course? How deep is the rental market for their unit style? Will rent cover their PITI?
Years 4-8: Sold West Washington Park Home. Purchased in Stapleton as one child heading to elementary school and another on the way.
Year 10: Sold out of Stapleton, purchased in Littleton, house triple the size of Denver and large lot, literally 1/2 the cost of anything within 8 miles of downtown, more attractive school system yet more challenging commute (both work in downtown) however easy access to light-rail and Santa Fe Drive.
Year 10-15: Forecast – Will stay in Littleton until youngest goes off to college.
Year 16: Forecast – Sell Littleton home, move to Cherry Creek North.
I am a firm believe one's first home can be a great foundation for future success from lifestyle to investing. However I also feel one's first home should not be over-extended i.e. live within one's means, consider allocating some housing expenditures to the equities market to take advantage of compound interest and if planning so change jobs, careers, locations be realistic as if changes are happening in 3-5 years the potential loss of equity concerning one's home can happen. Ask all the buyers in 2006 which sold between 2008 and 2013…..
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My Favorite Magic Books
5 Books | by NickNickleby
The New York Times bestselling novel about a young man practicing magic in the real world, now an original series on SYFY"The Magicians is to Harry Potter as a shot of Irish whiskey is to a glass of weak tea. . . . Hogwarts was never like this." —George R.R. Martin "Sad, hilarious, beautiful, and essential to anyone who cares about modern fantasy." —Joe Hill "A very knowing and wonderful take on the wizard school genre." —John Green "The Magicians may just be the most subversive, gripping and enchanting fantasy novel I've read this century." —Cory Doctorow"This gripping novel draws on the conventions of contemporary and classic fantasy novels in order to upend them . . . an unexpectedly moving coming-of-age story." —The New Yorker"The best urban fantasy in years." —A.V. ClubQuentin Coldwater is brilliant but miserable. A high school math genius, he's secretly fascinated with a series of children's fantasy novels set in a magical land called Fillory, and real life is disappointing by comparison. When Quentin is unexpectedly admitted to an elite, secret college of magic, it looks like his wildest dreams have come true. But his newfound powers lead him down a rabbit hole of hedonism and disillusionment, and ultimately to the dark secret behind the story of Fillory. The land of his childhood fantasies turns out to be much darker and more dangerous than he ever could have imagined. . . .The prequel to the New York Times bestselling book The Magician King and the #1 bestseller The Magician's Land, The Magicians is one of the most daring and inventive works of literary fantasy in years. No one who has escaped into the worlds of Narnia and Harry Potter should miss this breathtaking return to the landscape of the imagination.
#1 New York Times bestselling author!Booklist Editors' Choice 2015 - Youth! Named a "Best Book of 2015" by Time Magazine, School Library Journal, Barnes & Noble, NPR, PopSugar, The Millions, and The News & Observer!Simon Snow is the worst Chosen One who's ever been chosen.That's what his roommate, Baz, says. And Baz might be evil and a vampire and a complete git, but he's probably right.Half the time, Simon can't even make his wand work, and the other half, he starts something on fire. His mentor's avoiding him, his girlfriend broke up with him, and there's a magic-eating monster running around, wearing Simon's face. Baz would be having a field day with all this, if he were here -- it's their last year at the Watford School of Magicks, and Simon's infuriating nemesis didn't even bother to show up.Carry On is a ghost story, a love story and a mystery. It has just as much kissing and talking as you'd expect from a Rainbow Rowell story - but far, far more monsters.
Harry Potter - The Complete Collection 1 - 7
Harry Potter - The Complete Collection 1 - 7 J. K. Rowling
Filled with mystery and an intriguingly rich magic system, Tracy Deonn's YA contemporary fantasy Legendborn offers the dark allure of City of Bones with a modern-day twist on a classic legend and a lot of Southern Black Girl Magic.After her mother dies in an accident, sixteen-year-old Bree Matthews wants nothing to do with her family memories or childhood home. A residential program for bright high schoolers at UNC–Chapel Hill seems like the perfect escape—until Bree witnesses a magical attack her very first night on campus. A flying demon feeding on human energies. A secret society of so called "Legendborn" students that hunt the creatures down. And a mysterious teenage mage who calls himself a "Merlin" and who attempts—and fails—to wipe Bree's memory of everything she saw. The mage's failure unlocks Bree's own unique magic and a buried memory with a hidden connection: the night her mother died, another Merlin was at the hospital. Now that Bree knows there's more to her mother's death than what's on the police report, she'll do whatever it takes to find out the truth, even if that means infiltrating the Legendborn as one of their initiates. She recruits Nick, a self-exiled Legendborn with his own grudge against the group, and their reluctant partnership pulls them deeper into the society's secrets—and closer to each other. But when the Legendborn reveal themselves as the descendants of King Arthur's knights and explain that a magical war is coming, Bree has to decide how far she'll go for the truth and whether she should use her magic to take the society down—or join the fight.
From #1 New York Times bestselling author Brandon Sanderson, the Mistborn series is a heist story of political intrigue and magical, martial-arts action.For a thousand years the ash fell and no flowers bloomed. For a thousand years the Skaa slaved in misery and lived in fear. For a thousand years the Lord Ruler, the "Sliver of Infinity," reigned with absolute power and ultimate terror, divinely invincible. Then, when hope was so long lost that not even its memory remained, a terribly scarred, heart-broken half-Skaa rediscovered it in the depths of the Lord Ruler's most hellish prison. Kelsier "snapped" and found in himself the powers of a Mistborn. A brilliant thief and natural leader, he turned his talents to the ultimate caper, with the Lord Ruler himself as the mark.Kelsier recruited the underworld's elite, the smartest and most trustworthy allomancers, each of whom shares one of his many powers, and all of whom relish a high-stakes challenge. Only then does he reveal his ultimate dream, not just the greatest heist in history, but the downfall of the divine despot.But even with the best criminal crew ever assembled, Kel's plan looks more like the ultimate long shot, until luck brings a ragged girl named Vin into his life. Like him, she's a half-Skaa orphan, but she's lived a much harsher life. Vin has learned to expect betrayal from everyone she meets, and gotten it. She will have to learn to trust, if Kel is to help her master powers of which she never dreamed.This saga dares to ask a simple question: What if the hero of prophecy fails?Other Tor books by Brandon SandersonThe CosmereThe Stormlight ArchiveThe Way of KingsWords of RadianceEdgedancer (Novella)Oathbringer The Mistborn trilogyMistborn: The Final EmpireThe Well of AscensionThe Hero of AgesMistborn: The Wax and Wayne seriesAlloy of LawShadows of SelfBands of MourningCollectionArcanum UnboundedOther Cosmere novelsElantrisWarbreakerThe Alcatraz vs. the Evil Librarians seriesAlcatraz vs. the Evil LibrariansThe Scrivener's BonesThe Knights of CrystalliaThe Shattered LensThe Dark TalentThe Rithmatist seriesThe RithmatistOther books by Brandon SandersonThe ReckonersSteelheartFirefightCalamityAt the Publisher's request, this title is being sold without Digital Rights Management Software (DRM) applied.
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Results show that 450 people expressed their views on budget priorities online or via paper copies; the survey was also available to print off and in Easy Read format. Respondents were approximately 53% male and 47% female, 90% over 35, and 14% with a disability or health problem.
People were asked from the main services provided by Devon County Council what they thought were the priorities for their community.
People were asked from the main services provided by Devon County Council whether they thought spending should be protected, or reduced.
When asked about their community respondents indicated that they felt that they fairly strongly belonged to their local community, with 59% thinking communities could play a bigger role in the running of local services. Suggestions that Devon County Council could help communities with seed funding, training, and signposting were selected by around half of the respondents. 46% indicated they would like to get involved, or more involved, in their local community. 21% said no or don't know when asked whether they would be prepared to be more involved in helping their local community. 86% said they currently or would consider helping family or neighbours who are ill or disabled, 76% to help improve safety and security in their neighbourhood, and 66% volunteer to help run local services.
When asked whether they would pay more council tax to protect services, 74% of respondents selected 'yes', whilst 52% said they were satisfied in the way Devon County Council has managed the tough choices faced.
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Q: OWIN Authentication in MVC 5 Loops while User.IsAuthenticated = false I'm trying to implement authentication and authorization in an MVC 5.1 app. The authentication takes place via Facebook that is custom implemented. (I can post that code if needed.) Once FB authenticates and sends back the code and the Authenticate method of the auth service is called to sign the user into the application. There is no auth code in the application itself (thus not using Identity or other membership services).
public async Task<ActionResult> Connect(string code)
{
if (code == null)
{
return RedirectToAction("Index", "Home");
}
else
{
// get access token
var accessToken = await nApplication.FacebookClient.AccessTokenAsync(code);
// get user info from facebook
var meResult = await nApplication.FacebookClient.MeResultAsync(accessToken);
nApplication.NRepository.SaveChanges();
nAuthorization.Authenticate(member);
return RedirectToAction("Index");
}
}
nAuthorization.Authenticate(member); creates a list of claims and executes OWIN SignIn,
claims.Add(new Claim(ClaimTypes.Name, member.Name));
claims.Add(new Claim(ClaimTypes.Role, "Member"));
var claimIdentity = new ClaimsIdentity(claims, DefaultAuthenticationTypes.ApplicationCookie);
owinContext.Authentication.SignIn(new AuthenticationProperties { IsPersistent = true }, claimIdentity);
I'm using the Authorize attribute from Mvc namespace. But at this point /Profile/Authenticate/ which is my Owin LoginPath get's called again and again to redirect the user to FB and return to the Connect method above.
[Authorize(Roles = "Member")]
public async Task<ActionResult> Index(int? id)
I've checked the User property in the controller and it is not authenticated. I could set that to a new ClaimsPrincipal but I'd like the auth code to be independent of the HttpContext. And it doesn't seem to be right solution.
My Startup class contains:
app.UseCookieAuthentication(new CookieAuthenticationOptions
{
LoginPath = new PathString("/Profile/Authenticate/"),
AuthenticationType = DefaultAuthenticationTypes.ApplicationCookie,
CookieSecure = CookieSecureOption.Always,
ReturnUrlParameter = "next"
});
Maybe I am missing something completely fundamental? Any pointers would help, I've looked through articles such as the following but to no avail:
*
*http://brockallen.com/2013/10/24/a-primer-on-owin-cookie-authentication-middleware-for-the-asp-net-developer/
*http://www.khalidabuhakmeh.com/asp-net-mvc-5-authentication-breakdown-part-deux
A: I think this will solve your problem...it worked for me:
Add an empty method in your global.asax.cs file:
protected void Session_Start()
{
}
for some reason, the asp.net session cookie does not get set at the proper time without this. The Thinktecture devs think this might be happening if your webapp uses http and your identity provider uses https but I have not verified that yet.
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Kevin Smith Is Bringing A Mooby's Restaurant Pop-Up to Phoenix in February
First Taste: Stoop Kid Brings Big Apple Bagels to The Churchill
Making Mendl's Courtesan au Chocolat with Country Velador of Super Chunk Sweets and Treats (VIDEO)
Lauren Saria
| DIY |
Lauren Saria | April 8, 2014 | 11:00am
First things first: If you haven't seen Wes Anderson's newest film, The Grand Budapest Hotel, by now, you're really missing out. The director's latest effort is beautiful, funny, quirky, and, best of all, involves food. Of course, it doesn't feature just any edible creation, but rather a pastry that blends simple confections with just the right amount of whimsy to be the type of fictional dessert that fits in perfectly with Anderson's one-of-a-kind world.
We've already showed you an adorable video recipe for making Mendl's courtesan au chocolat, but when we heard local pastry chef Country Velador would be making the confections to sell at her Scottsdale store we knew we had to get in the kitchen for a firsthand look at the baking process.
See also: Bake Like Wes Anderson's Mendl's Bakery: A Tutorial
For those who have not seen the movie, or maybe weren't paying close enough attention, the Courtesan au Chocolat is a tiered-dessert made of three different sizes of chocolate cream-filled pastries. Each of the puffs, which are staggered in size, is coated in a different color glaze and then decorated with blue and white icings.
Velador, a pretty big Wes Anderson nerd fan, started with the fictional Mendl's recipe that's been floating around the internet based on the video tutorial. It begins:
Mendl's Courtesan au Chocolat
The exact recipe for the Courtesan au chocolat has never been published or publicly disclosed as per the conditions of Herr Mendl's will. However, the following has been collated and adapted from several "pirate" sources in the Nebelsbad archives (including a 1963 recipe from the kitchen of the Grand Budapest Hotel using powdered eggs that was printed in the Lutz Daily Fact).
The first step is to make a choux pastry. For this portion of the recipe Velador uses the same proportions as are provided in the recipe:
1 cup plain flour
1 cup fresh water
1/4 lb (1 stick) butter
4 eggs beaten in a bowl
A larger pinch of sugar
To make the pastry she combines the water, butter, and salt and brought to a boil. (Velador uses a convection burner, but home bakers can use a regular stove top.) Once the mixture has reached a boil, remove it quickly from the heat and mix it with sifted flour. Velador then returns the dough to the heat and cooks until it forms one lump.
You then remove the dough and allow it to cool before adding the eggs. Doing so prevents the eggs from cooking, so Velador says it's important to let it cool at least to the touch. She also recommends adding each egg one at a time with a wooden spoon or plastic spatula to be sure the dough doesn't become too runny to pipe. The correct number of eggs will differ depending on elevation and humidity, but ideally your dough should be about the same texture as cookie dough, soft enough to pipe but not so runny it doesn't keep its shape.
Once the dough is done Velador covers a tray in parchment and puts the dough into a pastry bag. You can also use a spoon if you don't have a bag. You'll need to pipe the same number of small, medium, and large pastry balls -- about the size of a tablespoon, teaspoon, and hazelnut, respectively. The smallest balls should be put on a separate sheet since they'll likely take less time to bake.
Velador recommends baking all three for 10 minutes at 400 degrees to "give them a nice rise" before lowering the temperature to about 350 degrees and baking for another 25 minutes. You'll want to keep an eye on the pastries since the bake time will also be affected by elevation and humidity.
Once the pastries are done remove them from the oven and cut small holes in the bottom to allow the steam to escape. Then let cool.
The second step will be to make the chocolate filling. Velador used the recipe below, taken from the original but with a few minor adjustments. When completed the filling should be about the same consistency as a thick pudding.
1/2 cups whole milk
3 oz semi-sweet chocolate
.5 oz cocoa powder
.4 oz flour (optional)
.4 oz cornstarch
Velador recommends heating the milk and chocolate over a double boiler to avoid getting lumps in the filling. Then premix the cornstarch, sugar, and cocoa prior to adding them to the three egg yolks. Velador says you can remove the flour from the recipe altogether and just use more cornstarch instead, which may also help cut down on lumps. Once the dry ingredients and eggs are mixed, slowly mix in half the hot chocolate then add the second half while the filling sits over a low very heat. Continue stirring until the filling comes to a bubble and it should thicken into a custard. Then remove from the heat, strain, and let chill.
Once the pastry and filling has cooled, pipe the filling to the cut holes in the bottom of the puffs. Velador chose not to fill the smallest pastries.
Then she makes the sugar icing by combining powdered sugar and milk. Velador does this by eye until the icing becomes the right consistency to work as a glaze. To be completely authentic to the movie, you'll want three colors: pale pink, light green, and very light lavender. Be sure to also save a small amount of white icing. Velador also used blue and white buttercreams, which can be made by whipping together butter and powdered sugar.
She then dips the tops of the large pastries in the pink icing, the medium in the green, and the smallest in the lavender and decorates the tops of each with a swirl pattern of white icing. Velador then stacks the three pastries from largest to smallest using a small amount of blue buttercream in between each pastry. She finishes the decorating with a row of blue buttercream dots in between each pastry, a ruffle of white butter cream at the base, and a small piece of chocolate on the top -- though the original recipe calls for a cocoa bean.
If all that work sounds like too much for you (it was for us!) you'll be able to purchase courtesan au choclat at Super Chunk Sweets and Treats next Friday through Sunday, April 11-13. The shop is open from 11 a.m. to 6 p.m. Velador says she plans to sell them for $6 each.
For more information visit the Super Chunk Sweets and Treats website or call (602) 736-2383.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,215
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Watch: Rod Rosenstein says 'it's up to the attorney general' to decide to prosecute a sitting president |
Watch: Rod Rosenstein says 'it's up to the attorney general' to decide to prosecute a sitting president
Image via screengrab.
Brandon Gage
January 15, 2023Bank
On Sunday's edition of Meet the Press, former Deputy Attorney General Rod Rosenstein – whose oversight of the United States Department of Justice's probe into ex-President Donald Trump's ties to Russia sparked massive public controversy –weighed in on the handfuls of classified documents that were found in President Joe Biden's home and offices that he used during his tenure as vice president. The materials were discovered by Biden's attorneys and were quickly handed over to the National Archives last year.
This is notably different than the trove of top-secret texts that Trump intentionally took from the White House to Mar-a-Lago at the end of his term, which he later falsely claimed belonged to him. Biden has made no such declaration of ownership and denies knowing about them at all – even though some were found in his personal garage in Delaware.
Attorney General Merrick Garland appointed special counsels to investigate each of the cases. Republicans have accused Democrats of hypocrisy about Trump, whom they maintain did nothing wrong. But NBC Moderator Chuck Todd pointed out that this has created in a predicament for the GOP about what is and is not criminal behavior, based on the evidence at hand.
READ MORE: Donald Trump accuses the 'fake news media' of downplaying the Joe 'Bidden' documents story
"There's two sort of unique defenses in each of these cases that I'm curious for your take on. One is Donald Trump's claim that he could declassify anything he wanted. Now, he's not made a legal claim that he did that. And the second is Joe Biden's currently president. So is it currently illegal for him to have classified documents in his possession, even if it's at his home next to his Corvette?" Todd asked Rosenstein.
"Right? Well, that's one of the questions the special counsel will have to look into. And in addition, you know the legal status of the vice president, whether or not he has authority to make any declassification decisions. But the key question, the threshold question here is going to be, was President Biden aware of those classified documents," Rosenstein replied.
"And how does that get proven? You think he's going to have to sit for an interview?" Todd pressed Rosenstein.
"Well, that would be a logical step. If I were conducting this investigation, I want to go right to the source and ask the president directly whether or not he was aware of those documents," said Rosenstein.
READ MORE: Chris Wallace blasts Republicans for double standard on Trump and Biden classified docs ordeal
Todd then touched upon what the procedure would be if evidence emerges that Biden broke the law.
"A sitting president can't be indicted, according to the Justice Department. So let's say something is found. What would happen?" he wondered.
"Chuck, I think what should happen in the special counsel investigation is that the special counsel should evaluate the facts and the law, look at Department of justice policies, and make a recommendation to the attorney general about whether or not prosecution is warranted," Rosenstein opined. "Then it's up to the attorney general to make the decision whether to apply that DOJ policy."
\u201cEXCLUSIVE: The key question for special counsel Robert Hur is whether President Biden was aware of the classified documents, says fmr. Deputy AG Rod Rosenstein.\n\n"I would want to go right to the source and ask the president directly whether he was aware of those documents."\u201d
— Meet the Press (@Meet the Press) 1673796864
READ MORE: Joe Biden's document discovery does not cancel out Donald Trump's troubles
GOP strategist pardoned by Donald Trump convicted of illegally funneling him Russian cash ›
Trump plotted to trade Mar-a-Lago files for 'sensitive documents' about his 2016 campaign Russia ties: report ›
What if Donald Trump's conspiracy was even bigger than we thought? ›
Rod Rosenstein, Face Of Russia Probe, Steps Down As Deputy ... ›
Trump: Rosenstein belongs in jail because 'he never should have ... ›
Rod Rosenstein's letter recommending Comey be fired - BBC News ›
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,045
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package org.h2s.jw.soql.annotation;
import java.lang.annotation.ElementType;
import java.lang.annotation.Retention;
import java.lang.annotation.RetentionPolicy;
import java.lang.annotation.Target;
@Retention(RetentionPolicy.RUNTIME)
@Target(ElementType.TYPE)
public @interface SfObject {
String name() default "";
}
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,889
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{"url":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/using-a-formula-to-solve-a-real-world-application\/","text":"Using a Formula to Solve a Real-World Application\n\nMany applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem\u2019s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the area of a rectangular region, $A=LW$; the perimeter of a rectangle, $P=2L+2W$; and the volume of a rectangular solid, $V=LWH$. When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.\n\nExample 3: Solving an Application Using a Formula\n\nIt takes Andrew 30 min to drive to work in the morning. He drives home using the same route, but it takes 10 min longer, and he averages 10 mi\/h less than in the morning. How far does Andrew drive to work?\n\nSolution\n\nThis is a distance problem, so we can use the formula $d=rt$, where distance equals rate multiplied by time. Note that when rate is given in mi\/h, time must be expressed in hours. Consistent units of measurement are key to obtaining a correct solution.\n\nFirst, we identify the known and unknown quantities. Andrew\u2019s morning drive to work takes 30 min, or $\\frac{1}{2}$ h at rate $r$. His drive home takes 40 min, or $\\frac{2}{3}$ h, and his speed averages 10 mi\/h less than the morning drive. Both trips cover distance $d$. A table, such as the one below, is often helpful for keeping track of information in these types of problems.\n\n$d$ $r$ $t$\nTo Work $d$ $r$ $\\frac{1}{2}$\nTo Home $d$ $r - 10$ $\\frac{2}{3}$\n\nWrite two equations, one for each trip.\n\n$\\begin{array}{ll}d=r\\left(\\frac{1}{2}\\right)\\hfill & \\text{To work}\\hfill \\\\ d=\\left(r - 10\\right)\\left(\\frac{2}{3}\\right)\\hfill & \\text{To home}\\hfill \\end{array}$\n\nAs both equations equal the same distance, we set them equal to each other and solve for r.\n\n$\\begin{array}{l}r\\left(\\frac{1}{2}\\right)\\hfill&=\\left(r - 10\\right)\\left(\\frac{2}{3}\\right)\\hfill \\\\ \\frac{1}{2}r\\hfill&=\\frac{2}{3}r-\\frac{20}{3}\\hfill \\\\ \\frac{1}{2}r-\\frac{2}{3}r\\hfill&=-\\frac{20}{3}\\hfill \\\\ -\\frac{1}{6}r\\hfill&=-\\frac{20}{3}\\hfill \\\\ r\\hfill&=-\\frac{20}{3}\\left(-6\\right)\\hfill \\\\ r\\hfill&=40\\hfill \\end{array}$\n\nWe have solved for the rate of speed to work, 40 mph. Substituting 40 into the rate on the return trip yields 30 mi\/h. Now we can answer the question. Substitute the rate back into either equation and solve for d.\n\n$\\begin{array}{l}d\\hfill&=40\\left(\\frac{1}{2}\\right)\\hfill \\\\ \\hfill&=20\\hfill \\end{array}$\n\nThe distance between home and work is 20 mi.\n\nAnalysis of the Solution\n\nNote that we could have cleared the fractions in the equation by multiplying both sides of the equation by the LCD to solve for $r$.\n\n$\\begin{array}{l}r\\left(\\frac{1}{2}\\right)\\hfill&=\\left(r - 10\\right)\\left(\\frac{2}{3}\\right)\\hfill \\\\ 6\\times r\\left(\\frac{1}{2}\\right)\\hfill& =6\\times \\left(r - 10\\right)\\left(\\frac{2}{3}\\right)\\hfill \\\\ 3r\\hfill& =4\\left(r - 10\\right)\\hfill \\\\ 3r\\hfill& =4r - 40\\hfill \\\\ -r\\hfill& =-40\\hfill \\\\ r\\hfill& =40\\hfill \\end{array}$\n\nTry It 3\n\nOn Saturday morning, it took Jennifer 3.6 h to drive to her mother\u2019s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer 4 h to return home. Her speed was 5 mi\/h slower on Sunday than on Saturday. What was her speed on Sunday?\n\nSolution\n\nExample 4: Solving a Perimeter Problem\n\nThe perimeter of a rectangular outdoor patio is $54$ ft. The length is $3$ ft greater than the width. What are the dimensions of the patio?\n\nSolution\n\nThe perimeter formula is standard: $P=2L+2W$. We have two unknown quantities, length and width. However, we can write the length in terms of the width as $L=W+3$. Substitute the perimeter value and the expression for length into the formula. It is often helpful to make a sketch and label the sides.\n\nFigure 3\n\nNow we can solve for the width and then calculate the length.\n\n$\\begin{array}{l}P=2L+2W\\hfill \\\\ 54=2\\left(W+3\\right)+2W\\hfill \\\\ 54=2W+6+2W\\hfill \\\\ 54=4W+6\\hfill \\\\ 48=4W\\hfill \\\\ 12=W\\hfill \\\\ \\left(12+3\\right)=L\\hfill \\\\ 15=L\\hfill \\end{array}$\n\nThe dimensions are $L=15$ ft and $W=12$ ft.\n\nTry It 4\n\nFind the dimensions of a rectangle given that the perimeter is $110$ cm and the length is 1 cm more than twice the width.\n\nSolution\n\nExample 5: Solving an Area Problem\n\nThe perimeter of a tablet of graph paper is 48 in2. The length is $6$ in. more than the width. Find the area of the graph paper.\n\nSolution\n\nThe standard formula for area is $A=LW$; however, we will solve the problem using the perimeter formula. The reason we use the perimeter formula is because we know enough information about the perimeter that the formula will allow us to solve for one of the unknowns. As both perimeter and area use length and width as dimensions, they are often used together to solve a problem such as this one.\n\nWe know that the length is 6 in. more than the width, so we can write length as $L=W+6$. Substitute the value of the perimeter and the expression for length into the perimeter formula and find the length.\n\n$\\begin{array}{l}P\\hfill&=2L+2W\\hfill \\\\ 48\\hfill&=2\\left(W+6\\right)+2W\\hfill \\\\ 48\\hfill&=2W+12+2W\\hfill \\\\ 48\\hfill&=4W+12\\hfill \\\\ 36\\hfill&=4W\\hfill \\\\ 9\\hfill&=W\\hfill \\\\ \\left(9+6\\right)\\hfill&=L\\hfill \\\\ 15\\hfill&=L\\hfill \\end{array}$\n\nNow, we find the area given the dimensions of $L=15$ in. and $W=9$ in.\n\n$\\begin{array}{l}A\\hfill&=LW\\hfill \\\\ A\\hfill&=15\\left(9\\right)\\hfill \\\\ \\hfill&=135\\text{ in}^{2}\\hfill \\end{array}$\n\nThe area is $135$ in2.\n\nTry It 5\n\nA game room has a perimeter of 70 ft. The length is five more than twice the width. How many ft2 of new carpeting should be ordered?\n\nSolution\n\nExample 6: Solving a Volume Problem\n\nFind the dimensions of a shipping box given that the length is twice the width, the height is $8$ inches, and the volume is 1,600 in.3.\n\nSolution\n\nThe formula for the volume of a box is given as $V=LWH$, the product of length, width, and height. We are given that $L=2W$, and $H=8$. The volume is $1,600$ cubic inches.\n\n$\\begin{array}{l}V=LWH\\hfill \\\\ 1,600=\\left(2W\\right)W\\left(8\\right)\\hfill \\\\ 1,600=16{W}^{2}\\hfill \\\\ 100={W}^{2}\\hfill \\\\ 10=W\\hfill \\end{array}$\n\nThe dimensions are $L=20$ in., $W=10$ in., and $H=8$ in.\n\nAnalysis of the Solution\n\nNote that the square root of ${W}^{2}$ would result in a positive and a negative value. However, because we are describing width, we can use only the positive result.","date":"2020-02-26 13:44:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7597653269767761, \"perplexity\": 425.00320791548523}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875146342.41\/warc\/CC-MAIN-20200226115522-20200226145522-00327.warc.gz\"}"}
| null | null |
Did you know you can also save money by lowering your taxes at the same time? There are major medical insurance plans that allow you to do both; offering you lower monthly premiums AND helping you lower your taxable income! Commonly called Health Savings Accounts or HSA's, they are used to pay for future medical expenses. You can deduct your allowable contributions on your Federal Income Tax Return, even if you do not itemize! Distributions may be tax-free if you pay for qualified medical expenses. Many local banks can show you how easy it is to open one of their HSA's, and provide you with more details. To learn more about HSA guidelines, see IRS Publication 969, or click on this IRS web site link: http://search.irs.gov/web/query.html?col=allirs&charset=utf-8&qp=&qs=-Wct%3A%22Internal+Revenue+Manual%22&qc=&qm=&rf=&oq=&qt=hsa+guidelines . Enter the search words "HSA guidelines" at the upper right hand corner of the page.
You will also need to choose an "HDHP", or High Deductible Health Plan to work with your HSA. You can compare the benefits of different "HSA-Qualified High Deductible Health Plans" to determine which one best suits your needs and budget. A licensed independent health insurance agent can help you select a HDHP from a variety of plans that are available through many different insurance companies. Remember to keep your existing health insurance coverage in force until your new HDHP has been issued.
Maintaining a healthy lifestyle plays a big role in keeping your medical costs down. It is wise to avoid excessive use of alcohol and tobacco since those habits can lead to many health problems over time. Regular exercise can reduce body fat, lower blood pressure, and help you to release stress. We have all heard about the importance of eating a balanced diet, taking nutritional supplements, drinking purified water, getting enough sleep, and brushing our teeth. (Did you know that poor oral hygiene can lead to cardiovascular disease, stroke, and diabetes and more?) Remember, "Your good health is your most valuable asset"!
This entry was posted in health Insurance and tagged affordable, Austin, hdhp, health, high deductible health plan, HSA, inexpensive, insurance, plans, Texas. Bookmark the permalink.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,518
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Home / Articles / Summer Research Fellow Spotlight: Alayna Mackiewicz
Summer Research Fellow Spotlight: Alayna Mackiewicz
By Kim Spurr
Alayna Mackiewicz
Alayna Mackiewicz, a Ph.D. student in the department of biology in UNC's College of Arts & Sciences, is the 2021 recipient of The Gordon W. and Janice L. Plumblee Summer Research Fellowship, which allowed her to travel to Woods Hole, Massachusetts, to conduct research at a renowned research lab in biological sciences.
The Summer Research Fellowship, supported by generous donors and by The Graduate School, funded her travel and her research in sensory biology to learn more about how some marine animals use earth's magnetic field for navigation and migration. As Mackiewicz explained, humans use five senses, such as smell and taste. But some marine animals, such as sea turtles and fish, have additional senses, including a magnetic sense—known in the scientific community for several decades.
"The field has made some progress, but we still don't know for certain how animals do it, and there is still a lot to learn," she said. "If we can learn how animals are using earth's magnetic field to figure out where they are in the world — there are huge implications for navigation." Mackiewicz said better understanding the earth's magnetic field and how marine animals use their magnetic sense could have real-world applications, including at the Department of Defense and helping marine animal conservation efforts.
"A lot of people are interested in it in terms of an additional way of navigation," Mackiewicz said. "It has implications for non-GPS satellite ways to navigate."
At the Marine Biological Laboratory, Mackiewicz is working alongside scientists to explore the neurobiology of the magnetic sense. Specifically, the research team is using the oyster toadfish to learn more about where in the brain detection and processing of magnetic information occurs.
Mackiewicz, from Minnesota, said spending time outdoors, including at Lake Superior as an undergraduate student at the University of Minnesota Duluth, influenced her decision to pursue marine biology. At Carolina, Mackiewicz said the research conducted in the Lohmann Lab drew her to the program and that the Summer Research Fellowship allows for hands-on training that she can bring back to the lab.
"I knew how great the program was, and when I was applying for graduate schools, it was on my radar," she said. "The fellowship allows me the freedom to pursue and make progress in my dissertation research and alleviate financial stress."
The Summer Research Fellowships provide summer support to doctoral students so they may focus exclusively on their dissertation research.
By Elizabeth Poindexter, The Graduate School
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,578
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package com.opengamma.analytics.financial.interestrate.payments.derivative;
import static org.testng.AssertJUnit.assertEquals;
import static org.testng.AssertJUnit.assertFalse;
import org.testng.annotations.Test;
import org.threeten.bp.Period;
import org.threeten.bp.ZonedDateTime;
import com.mcleodmoores.date.CalendarAdapter;
import com.mcleodmoores.date.WeekendWorkingDayCalendar;
import com.mcleodmoores.date.WorkingDayCalendar;
import com.opengamma.analytics.financial.instrument.index.IndexON;
import com.opengamma.analytics.financial.instrument.index.IndexONMaster;
import com.opengamma.analytics.financial.instrument.payment.CouponONSpreadSimplifiedDefinition;
import com.opengamma.analytics.financial.schedule.ScheduleCalculator;
import com.opengamma.analytics.util.time.TimeCalculator;
import com.opengamma.financial.convention.businessday.BusinessDayConvention;
import com.opengamma.financial.convention.businessday.BusinessDayConventions;
import com.opengamma.util.money.Currency;
import com.opengamma.util.test.TestGroup;
import com.opengamma.util.time.DateUtils;
/**
* Tests related to the coupon overnight-indexed with spread derivative.
*/
@Test(groups = TestGroup.UNIT)
public class CouponONSpreadTest {
private static final int EUR_SETTLEMENT_DAYS = 2;
private static final BusinessDayConvention EUR_BUSINESS_DAY = BusinessDayConventions.MODIFIED_FOLLOWING;
private static final boolean EUR_IS_EOM = true;
private static final IndexON EONIA = IndexONMaster.getInstance().getIndex("EONIA");
private static final IndexON FEDFUND = IndexONMaster.getInstance().getIndex("FED FUND");
private static final WorkingDayCalendar CALENDAR = WeekendWorkingDayCalendar.SATURDAY_SUNDAY;
private static final Currency EUR = EONIA.getCurrency();
// Coupon EONIA 3m
private static final ZonedDateTime TRADE_DATE = DateUtils.getUTCDate(2011, 9, 7);
private static final ZonedDateTime SPOT_DATE = ScheduleCalculator.getAdjustedDate(TRADE_DATE, EUR_SETTLEMENT_DAYS,
CalendarAdapter.of(CALENDAR));
private static final Period EUR_CPN_TENOR = Period.ofMonths(3);
private static final ZonedDateTime START_ACCRUAL_DATE = SPOT_DATE;
private static final ZonedDateTime END_ACCRUAL_DATE = ScheduleCalculator.getAdjustedDate(START_ACCRUAL_DATE, EUR_CPN_TENOR,
EUR_BUSINESS_DAY, CalendarAdapter.of(CALENDAR), EUR_IS_EOM);
private static ZonedDateTime LAST_FIXING_DATE = ScheduleCalculator.getAdjustedDate(END_ACCRUAL_DATE, -1, CalendarAdapter.of(CALENDAR)); // Overnight
static {
LAST_FIXING_DATE = ScheduleCalculator.getAdjustedDate(LAST_FIXING_DATE, EONIA.getPublicationLag(), CalendarAdapter.of(CALENDAR)); // Lag
}
private static final ZonedDateTime PAYMENT_DATE = ScheduleCalculator.getAdjustedDate(LAST_FIXING_DATE, EUR_SETTLEMENT_DAYS,
CalendarAdapter.of(CALENDAR));
private static final double PAYMENT_ACCRUAL_FACTOR = EONIA.getDayCount().getDayCountFraction(START_ACCRUAL_DATE, END_ACCRUAL_DATE);
private static final double NOTIONAL = 100000000;
private static final double SPREAD = 0.0010;
private static final double SPREAD_AMOUNT = SPREAD * NOTIONAL * PAYMENT_ACCRUAL_FACTOR;
private static final double FIXING_YEAR_FRACTION = EONIA.getDayCount().getDayCountFraction(START_ACCRUAL_DATE, END_ACCRUAL_DATE);
private static final CouponONSpreadSimplifiedDefinition EONIA_COUPON_DEFINITION = new CouponONSpreadSimplifiedDefinition(EUR,
PAYMENT_DATE, START_ACCRUAL_DATE, END_ACCRUAL_DATE,
PAYMENT_ACCRUAL_FACTOR, NOTIONAL, EONIA, START_ACCRUAL_DATE, END_ACCRUAL_DATE, FIXING_YEAR_FRACTION, SPREAD);
private static final ZonedDateTime REFERENCE_DATE_1 = DateUtils.getUTCDate(2011, 9, 7);
private static final double PAYMENT_TIME_1 = TimeCalculator.getTimeBetween(REFERENCE_DATE_1, PAYMENT_DATE);
private static final double START_ACCRUAL_TIME_1 = TimeCalculator.getTimeBetween(REFERENCE_DATE_1, START_ACCRUAL_DATE);
private static final double END_ACCRUAL_TIME_1 = TimeCalculator.getTimeBetween(REFERENCE_DATE_1, END_ACCRUAL_DATE);
private static final CouponONSpread EONIA_COUPON_NOTSTARTED = new CouponONSpread(EUR, PAYMENT_TIME_1, PAYMENT_ACCRUAL_FACTOR, NOTIONAL,
EONIA, START_ACCRUAL_TIME_1,
END_ACCRUAL_TIME_1, FIXING_YEAR_FRACTION, NOTIONAL, SPREAD_AMOUNT);
private static final ZonedDateTime REFERENCE_DATE_2 = DateUtils.getUTCDate(2011, 10, 7);
private static final ZonedDateTime NEXT_FIXING_DATE_2 = ScheduleCalculator.getAdjustedDate(REFERENCE_DATE_2, 1,
CalendarAdapter.of(CALENDAR)); // Overnight
private static final double PAYMENT_TIME_2 = TimeCalculator.getTimeBetween(REFERENCE_DATE_2, PAYMENT_DATE);
private static final double START_FIXING_TIME_2 = TimeCalculator.getTimeBetween(REFERENCE_DATE_2, NEXT_FIXING_DATE_2);
private static final double END_FIXING_TIME_2 = TimeCalculator.getTimeBetween(REFERENCE_DATE_2, END_ACCRUAL_DATE);
private static final double FIXING_YEAR_FRACTION_2 = EONIA.getDayCount().getDayCountFraction(NEXT_FIXING_DATE_2, END_ACCRUAL_DATE);
private static final double NOTIONAL_WITH_ACCRUED = NOTIONAL * (1.0 + 0.01 / 12); // 1% over a month (roughly)
private static final CouponONSpread EONIA_COUPON_STARTED = new CouponONSpread(EUR, PAYMENT_TIME_2, PAYMENT_ACCRUAL_FACTOR, NOTIONAL,
EONIA, START_FIXING_TIME_2, END_FIXING_TIME_2,
FIXING_YEAR_FRACTION_2, NOTIONAL_WITH_ACCRUED, SPREAD_AMOUNT);
/**
*
*/
@Test(expectedExceptions = IllegalArgumentException.class)
public void nullIndex() {
new CouponONSpread(EUR, PAYMENT_TIME_1, PAYMENT_ACCRUAL_FACTOR, NOTIONAL, null, START_ACCRUAL_TIME_1, END_ACCRUAL_TIME_1,
FIXING_YEAR_FRACTION, NOTIONAL, SPREAD);
}
/**
*
*/
@Test
public void getterNotStarted() {
assertEquals("CouponONSpread: getter", EONIA, EONIA_COUPON_NOTSTARTED.getIndex());
assertEquals("CouponONSpread: getter", START_ACCRUAL_TIME_1, EONIA_COUPON_NOTSTARTED.getFixingPeriodStartTime());
assertEquals("CouponONSpread: getter", END_ACCRUAL_TIME_1, EONIA_COUPON_NOTSTARTED.getFixingPeriodEndTime());
assertEquals("CouponONSpread: getter", FIXING_YEAR_FRACTION, EONIA_COUPON_NOTSTARTED.getFixingPeriodAccrualFactor());
assertEquals("CouponONSpread: getter", NOTIONAL, EONIA_COUPON_NOTSTARTED.getNotionalAccrued());
assertEquals("CouponONSpread: getter", SPREAD_AMOUNT, EONIA_COUPON_NOTSTARTED.getSpreadAmount());
}
/**
*
*/
@Test
public void getterStarted() {
assertEquals("CouponONSpread: getter", EONIA, EONIA_COUPON_STARTED.getIndex());
assertEquals("CouponONSpread: getter", START_FIXING_TIME_2, EONIA_COUPON_STARTED.getFixingPeriodStartTime());
assertEquals("CouponONSpread: getter", END_FIXING_TIME_2, EONIA_COUPON_STARTED.getFixingPeriodEndTime());
assertEquals("CouponONSpread: getter", FIXING_YEAR_FRACTION_2, EONIA_COUPON_STARTED.getFixingPeriodAccrualFactor());
assertEquals("CouponONSpread: getter", NOTIONAL_WITH_ACCRUED, EONIA_COUPON_STARTED.getNotionalAccrued());
assertEquals("CouponONSpread: getter", SPREAD_AMOUNT, EONIA_COUPON_STARTED.getSpreadAmount());
}
/**
* Tests the equal and hashCode methods.
*/
@Test
public void equalHash() {
assertEquals("CouponOIS derivative: equal/hash code", EONIA_COUPON_STARTED, EONIA_COUPON_STARTED);
final CouponONSpread other = new CouponONSpread(EUR, PAYMENT_TIME_2, PAYMENT_ACCRUAL_FACTOR, NOTIONAL, EONIA, START_FIXING_TIME_2,
END_FIXING_TIME_2, FIXING_YEAR_FRACTION_2,
NOTIONAL_WITH_ACCRUED, SPREAD_AMOUNT);
assertEquals("CouponOIS derivative: equal/hash code", EONIA_COUPON_STARTED, other);
assertEquals("CouponOIS derivative: equal/hash code", EONIA_COUPON_STARTED.hashCode(), other.hashCode());
CouponONSpread modified;
modified = new CouponONSpread(EUR, PAYMENT_TIME_2, PAYMENT_ACCRUAL_FACTOR, NOTIONAL, FEDFUND, START_FIXING_TIME_2, END_FIXING_TIME_2,
FIXING_YEAR_FRACTION_2,
NOTIONAL_WITH_ACCRUED, SPREAD_AMOUNT);
assertFalse("CouponOIS derivative: equal/hash code", EONIA_COUPON_STARTED.equals(modified));
modified = new CouponONSpread(EUR, PAYMENT_TIME_2, PAYMENT_ACCRUAL_FACTOR, NOTIONAL, EONIA, START_FIXING_TIME_2 + 0.1,
END_FIXING_TIME_2, FIXING_YEAR_FRACTION_2,
NOTIONAL_WITH_ACCRUED, SPREAD_AMOUNT);
assertFalse("CouponOIS derivative: equal/hash code", EONIA_COUPON_STARTED.equals(modified));
modified = new CouponONSpread(EUR, PAYMENT_TIME_2, PAYMENT_ACCRUAL_FACTOR, NOTIONAL, EONIA, START_FIXING_TIME_2,
END_FIXING_TIME_2 + 0.1, FIXING_YEAR_FRACTION_2,
NOTIONAL_WITH_ACCRUED, SPREAD_AMOUNT);
assertFalse("CouponOIS derivative: equal/hash code", EONIA_COUPON_STARTED.equals(modified));
modified = new CouponONSpread(EUR, PAYMENT_TIME_2, PAYMENT_ACCRUAL_FACTOR, NOTIONAL, EONIA, START_FIXING_TIME_2, END_FIXING_TIME_2,
FIXING_YEAR_FRACTION_2 + 0.1,
NOTIONAL_WITH_ACCRUED, SPREAD_AMOUNT);
assertFalse("CouponOIS derivative: equal/hash code", EONIA_COUPON_STARTED.equals(modified));
modified = new CouponONSpread(EUR, PAYMENT_TIME_2, PAYMENT_ACCRUAL_FACTOR, NOTIONAL, EONIA, START_FIXING_TIME_2, END_FIXING_TIME_2,
FIXING_YEAR_FRACTION_2,
NOTIONAL_WITH_ACCRUED + 123.4, SPREAD_AMOUNT);
assertFalse("CouponOIS derivative: equal/hash code", EONIA_COUPON_STARTED.equals(modified));
modified = new CouponONSpread(EUR, PAYMENT_TIME_2, PAYMENT_ACCRUAL_FACTOR, NOTIONAL, EONIA, START_FIXING_TIME_2, END_FIXING_TIME_2,
FIXING_YEAR_FRACTION_2,
NOTIONAL_WITH_ACCRUED, SPREAD_AMOUNT + 12.34);
assertFalse("CouponOIS derivative: equal/hash code", EONIA_COUPON_STARTED.equals(modified));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,087
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Good grief! What have we done to deserve this? National shame, humiliation and surrender. England's non appearance at the match last night was shocking. I'm no football expert, I can't claim to be a football fan, but even I knew that what we watched last night was not "the beautiful game". It wasn't a game: for that you need two teams.
If Capello confesses that he doesn't know what happened what chance have others in diagnosing the sickness that appears to have struck down "our boys".
Some have suggested the weight of expectation of the English fans has chilled the souls and leadened the legs of Gerrard, Lampard and Rooney. Others, now say Capello has it all wrong. How can you build a team if no one knows who'll be playing until 2 hours before the game?
Unkindly, it has been suggested that the reason why the individual players appear to do so well in the domestic league is because they're supported by great foreign players. A team of English players are just not that good.
If we do stumble and are on the plane home next week, we'll be in for an orgy of recriminations, blame and scape - goating. Tribal demarcation lines will be drawn up. Chelsea supporters will blame United players and visa versa. There'll be calls for the FA to find an English manager forgetting the disaster that is English management. But most of all there'll be incomprehension and utter disappointment that once again the country that gave the world the game fails so spectacularly at the highest levels.
It's as if the "Little Englander"'s last refuge is the Premier League with its supporters' myopic belief that it's the best league in the world and its clubs are the most successful and best supported. The obvious question is; if so why does the national team fail so unerringly?
Of course, my despondency and lack of faith could be completely misplaced. As we've been told time and time again, in 1990, the last time we got anywhere near to winning the World Cup, we started with two draws and had to win the third game to get to the knock out stage.
And maybe, just maybe, it is a little unfair that we should invest in our national football team the wherewithal to rejuvenate our national sense of self belief and well-being. Anyone who was there in 1966 will remember the impact of that win against West Germany. It's as if somehow England's victory on the world stage did and would show to others and ourselves what we have lost - a sense of identity.
After the Emergency Budget on Tuesday, we'll need something to lift the spirits. May our endeavour at Rorke's Drift inspire our boys and we can begin to believe in ourselves.
Labels: Budget England football
My Aunt Bertha: The last of Ten
My Fat Greek Wedding
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Disability rates among U.S. residents in their 30s and 40s increased "dramatically" in the last 20 years, in large part because of increased obesity rates, according to a study published in the January/February issue of Health Affairs , USA Today reports (Hellmich, USA Today , 1/9). In the study, Darius Lakdawalla, an economist at RAND, and colleagues analyzed data from the National Health Interview Survey, which collects information from about 36,000 U.S. households each year (McDonough, AP/Dallas Morning News , 1/8). Survey respondents were considered disabled in cases in which they could not take care of their personal needs or had limited ability to perform routine tasks ( USA Today , 1/9). According to the study, between 1984 and 1996, the rate of respondents ages 30 to 39 considered disabled increased by 80%, and the rate among respondents ages 40 to 49 increased by 31% (Winslow, Wall Street Journal , 1/9). The study also found that the rate of respondents ages 18 to 29 and those ages 50 to 59 increased between 1984 and 1996; the rate among respondents ages 60 to 69 decreased by more than 10%, according to the study. The main causes of disability among respondents younger than age 60 were musculoskeletal problems, such as back injuries, and mental illnesses, the study found (Richardson, Los Angeles Times , 1/9). Both problems disproportionately affect obese individuals. According to researchers, the results of the study predict "potentially challenging issues for both employers and policy makers already struggling with skyrocketing health care costs," the Journal reports. Helen Darling, president of the National Business Group on Health, said that the results of the study highlight increased concerns among employers about the effect that overweight employees have on health care costs and productivity. In addition, she said that health insurers also may face problems because they rely on healthy employees to limit premium rate increases ( Wall Street Journal , 1/9).
In related news, a National Council on Disability report scheduled for release Friday warns that a lack of government information on U.S. residents with disabilities has "deprived local officials, schools and businesses of financing they need" to provide them with adequate services, the AP/Philadelphia Inquirer reports. According to the council, accurate information is required because census data are used to allocate federal funds to states, improve civil rights enforcement and predict the number of individuals eligible for Social Security benefits. The report found that the 2000 census failed to account for children younger than age six with disabilities and that the Census Bureau reached two different estimates on the number of U.S. residents with disabilities. The report calls for "drastic changes in the way" that the Census Bureau collects data to ensure adequate funds for disability services, according to the AP/Inquirer ( AP/Philadelphia Inquirer , 1/9).
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{
"redpajama_set_name": "RedPajamaC4"
}
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Surrey Eagles
Valley West Hawks
BC Major Midget League
White Rock Tritons
BC Premier Baseball League
Discover the Peninsula
RCMP officers on the scene of a White Rock home invasion June 7.
Home-invasion victims: How are we not co-operative?
Home-invasion targets dispute police report
Tracy Holmes
Jun. 14, 2011 12:00 p.m.
Victims of a home invasion in South Surrey last week say police weren't being truthful when they told media those targeted were involved in the drug trade and aren't co-operating with the investigation.
"I don't understand what the heck is going on with the cops telling you guys we're not co-operating," said Aaron Semple, 28. "We talked to the police. Everybody in the house gave a full statement to the police.
"How are we not co-operative?"
Semple said he was the only man among two adults and three teenagers in the house June 7 when four armed males broke in at 1:15 p.m. brandishing shotguns and a machete. The intruders forced him to the floor and put a shotgun to his head, then dragged him into another room and kicked him in the face, Semple said.
The attackers caused extensive damage inside the home before fleeing with cash and various belongings, he said.
The day after the incident – which took place at a small white house in the 15200-block of 24 Avenue – police said it was a targeted attack and described the victims as "young people that are known to police to be involved in the drug trade."
"They're not co-operating with the police and they are the primary people that we would need to solve this," a spokesman said.
Semple said Friday that all five victims provided police with their version of events the day of the attack, including descriptions of the assailants and the licence plate of the vehicle they fled in.
While Semple said residents in the house smoke marijuana – including for medicinal reasons – it is not "a drug house," he said.
"Nobody in this house has ever been arrested for drugs," he said. "That doesn't make us involved in the drug trade, just because we smoke it."
One resident, identifying herself as Semple's girlfriend, Michelle, she said she has lived at the house with her son and daughter for four years, and was livid to read what was said.
"The police told you bull—-," the 31-year-old told Peace Arch News Friday. "(The landlord) called me telling me they read what was in the news and that they think it would be best for me to move, and that they would work with me to help me move.
"Thanks to you guys, you have just had my children and myself kicked out of my house."
Michelle said the house was targeted because it was an easy target. "They came in planning to rob a woman and young children."
Semple and Michelle said officers did not follow up with them in the days since the attack, including not returning messages they've left regarding additional information they feel could help the investigation.
Police said Tuesday the incident is still under investigation, and that they stand by their earlier comments.
Surgery shuffle in Surrey a 'charade'
Company fined $140,000 in connection to workplace death
Surrey Board of Trade highlights innovation, policy changes as new U.S. president sworn in
COVID-19, border re-opening among issues affecting city, SBOT says
Explore Peace Arch News
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© 2021, Peace Arch News and Black Press Group Ltd.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Elisenda Barceló Olivé va néixer a Reus fa 39 anys. Actualment viu a Reus. És diplomada en Estadística per la UB i llicenciada en Investigació i Tècniques de Mercat per la UOC. Treballa a la Diputació de Tarragona, actualment a les escoles d'Art i Disseny de Reus i Tarragona. Entre 2015 i 2019 va ser l'alcaldessa d'Arbolí.
Referències
Polítics reusencs
Alumnes de la Universitat de Barcelona
Alumnes de la Universitat Oberta de Catalunya
Polítics catalans del sud contemporanis
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{
"redpajama_set_name": "RedPajamaWikipedia"
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| null | null |
**Begin Reading**
Table of Contents
Newsletters
Copyright Page
All rights reserved. In accordance with the U.S. Copyright Act of 1976, the scanning, uploading, and electronic sharing of any part of this book without the permission of the publisher constitute unlawful piracy and theft of the author's intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained by contacting the publisher at permissions@hbgusa.com. Thank you for your support of the author's rights.
# INTRODUCTION
## Why I Wrote a Book about the World's Most Versatile Ingredient
I was on the phone with author and television personality Alton Brown not long ago, whom I first met while taping season one of _The Next Iron Chef America_. He asked what I was working on.
"I want to write about the egg," I said, "all the things you can do with an egg, and how if you knew everything about cooking with eggs, you would become a better cook in a hundred different ways."
He said, "Yeah, I've always liked to say that the egg is the Rosetta stone of the kitchen."
That is exactly what makes Alton Brown so good on TV—he cuts right to the chase with the perfect metaphor: an ancient stone that helped us decipher a little-known language. Like that Rosetta stone, the egg, far more ancient, unlocks the secret language of the kitchen. Learn the language of the egg—understand completely this amazing and beautiful oblong orb—and you can enter new realms of cooking, rocketing you to stellar heights of culinary achievement.
The greatest of all our foods, the egg combines beauty, elegance, and simplicity, a miracle of natural design and, as food, bounty. Containing all of the nutrients required to create life, eggs give our bodies a powerful combination of proteins, amino acids, fatty acids, antioxidants, minerals, and vitamins, a package unmatched by any other single food.
The egg represents food at an almost primordial level when we eat it raw, consuming the liquidy stuff of life itself, and is capable of being transformed into the most sophisticated of culinary creations. I think of my friend and collaborator Thomas Keller and his truffle custard—egg and cream flavored with white truffle, cooked and served in the shell with a potato chip garnish. It's one of the finest four-star dishes ever created, and it's nothing more than egg, cream, truffle, potato. Genius in its simplicity.
The egg's shell is delicate but sturdy, porous but protective. Inside, more than a dozen different proteins form the white, the albumen, each serving a specific function in the developing creature—some feeding the embryo, some fending off large predators, others disabling harmful microbes. It's evolution at its most elegant and offers the cook a range of culinary acrobatics that give us an airy cake, a crisp meringue, a pillowy soufflé, or a tightly bound seafood terrine.
The yolk, that rich and fatty orb—suspended at either end within the albumen by protein coils called chalazae—is the nutrient center of the egg, accounting for three-quarters of the egg's calories. It also contains iron, thiamin, vitamin A, protein, cholesterol, and lecithin (a badass fat-water-hybrid molecule that gives yolks the ability to emulsify a lot of fat into a small amount of water, for culinary essentials such as the humble mayonnaise and the elegant béarnaise sauce).
A miracle of nature ought to be rare, like a truffle, but the egg is abundant; it ought likewise to be expensive, but instead it is among the most affordable foods in the store, costing just pennies apiece for the mass-produced varieties. At their very finest—from free-range hens fed only organic grain—eggs still cost only thirty or forty cents each.
The egg, as an individual food item (say, a poached egg on whole-wheat toast), is excellent. As an ingredient the egg is also, to the chef, an emblem and test of a cook's skill. More than with any other ingredient, the way an egg is handled by a cook tells a chef 90 percent of what he or she needs to know about a prospective hire. Many chefs ask a young cook to put aside the résumé and make an omelet. That's all that's needed as a tryout, because cooking an omelet well requires skill, knowledge, experience, and finesse.
It's this fact that long ago started me thinking about the importance of the egg to cooking generally. I got down to the business of writing about the egg in _Ruhlman's Twenty_ , devoting an entire chapter to it, beginning thus:
If you could choose to master a single ingredient, no choice would teach you more about cooking than an egg. It is an end in itself; it's a multipurpose ingredient; it's an all-purpose garnish; it's an invaluable tool. The egg teaches your hands finesse and delicacy. It helps your arms develop strength and stamina. It instructs in the way proteins behave in heat and in the powerful ways we can change food mechanically. It's a lever for getting food to behave in great ways. Learn to take the egg to its many differing ends, and you've enlarged your culinary repertoire by a factor of ten.
For this book I wanted to write about all those differing ends, working my way back to the egg. And in thinking about those ends, I began to break down the egg into its many uses. It quickly developed in my head not as ends, or single recipes, but as a unity, a single complex image in which everything is connected. Nature loves simplicity and unity: the hen's egg is an expression of nature's genius.
In the kitchen, the egg is ultimately neither ingredient nor finished dish but rather a singularity with a thousand ends. Scrambled eggs and angel food cake and ice cream and aioli and popovers and _gougères_ and _macarons_ and a gin fizz aren't separate entities, they're all part of the egg continuum, they are all one thing. The egg is a lens through which to view the entire craft of cooking. By working our way through the egg, we become powerful cooks.
Anyone familiar with my work knows my belief in the sanctity of technique. Recipes today are free. The world is awash in recipes. While there's nothing wrong with recipes per se, there's everything wrong with relying solely on recipes if you want to be a better cook. You've got to be very shrewd to disinter an unknown technique from within a recipe. But if you know a single technique, you immediately have hundreds of recipes at hand. That's why cooking schools don't teach recipes, they teach technique.
Recipes are a valuable resource for ideas, and I use them often. Sometimes they offer outright an unfamiliar technique. I love to compare recipes for the same finished product— _Why does this quick bread use both baking powder and baking soda? Why does that one use considerably more egg relative to the flour? What are the resulting differences?_ When I'm writing my own recipes, I often compare various recipes, pick and choose elements from each that I like, filter them through my own particular nature (preferences, biases, practicalities, competencies), to arrive at a recipe that is more or less mine (I don't know that there are any truly unique recipes in traditional cooking). There are many wonderful recipes in this book, so even if you just want to get dinner on the table and have no desire to become a better cook, the preparations in this book will still make you feel like a star. The recipes have also been chosen because they are classic or distinctive examples of a technique made possible by the miraculous egg.
Within the universe of the egg are dozens of techniques. I'd wager that no other single ingredient has as many, not by a long shot. But shouldn't we expect as much from a little package that contains all the stuff of life itself?
# THE EGG FLOWCHART
THE IDEA TO PRESENT THE EGG AS A FLOWCHART CAME NATURALLY, FROM JUST THINKING about it, asking the question, "What can you do with an egg?"
The answer—after the obvious "all kinds of things"—is that it depends. Are you going to cook it in its shell or out? If you're going to leave it in its shell, are you going to cook it hard or soft? If you're going to cook it out of its shell, are you going to cook it whole or separate it? Are you going to cook it at all? Are you going to use it as a tool, to leaven a cake or emulsify a mayonnaise?
You could even make a game of it. Think of a dish that uses eggs—a quiche, a cake, a poached egg, pasta carbonara—and your opponent has twenty guesses. You'd begin with those same questions—Is it cooked in the shell or out? If out, is it whole or separated? If separated, is the white or the yolk used? If used whole, is it cooked as is or blended?
It occurred to me that to show, visually, how versatile the egg is, I'd have to make a flowchart. After thinking about how readily egg preparations would lend themselves to a graphic representation in this format, I sat down at our dining room table with a roll of parchment paper and wrote one out. It was sloppy, the kind of diagram a teacher at a blackboard would improvise. But it worked—the egg really did present itself visually as a flowchart, one that measured about a foot and a half long.
When I was ready to get serious about it, I asked my wife and partner, Donna, to write it out, since she's better at all things visual. I mapped the whole thing out, and she figured out how much space she was going to need to get it all in. The finished diagram required a five-foot length of parchment.
It was a thing of beauty. It mesmerized. I tacked it across a row of bookshelves behind my desk, and when we had a gang over for a New Year's Day brunch, people went into my office and gazed as if it were a museum piece. By twos and threes, guests stood staring at it, pointing things out to each other, and staring some more. A few stood for ten minutes before they could draw themselves away. "Michael," they asked, "what _is_ this?"
The promise of the flowchart demanded to be fulfilled. A removable poster of the flow-chart is included at the back of this book.
# THE WAY I COOK
MY DEAR EDITOR, MICHAEL SAND, SOMETIMES MAKES ME WANT TO PULL MY HAIR OUT. When I went page by page through his edit of the manuscript that would become this book, I saw that he was a comma freak. I kind of toss them willy-nilly onto the page after I've written it and hope they fall into the right places, and so am incredibly grateful for his fastidiousness. He is acutely attuned to confusing or lazy phrasings in a recipe and calls attention to them without making me feel like an idiot. His questions arrive with a delicacy that would please the famously decorous former _New Yorker_ editor William Shawn.
But he continually asked me about butter (salted or unsalted?) and kept writing "large eggs?" in a book _about_ eggs. Why do these most basic questions make me want to pull my hair out? Because _it doesn't matter_.
And because at the same time it _does_ matter. (I could put three commas in that fragment but refuse to.)
And, finally, because it lasers in on the fundamental difficulty of writing about food: Cooking is so infinitely nuanced that to write completely about how to cook any dish would require a manuscript longer than a David Foster Wallace novel and include twice as many footnotes within twice as many endnotes. And then no one would be able to follow it, let alone cook from it—and just as well, because it would _still_ be incomplete. That's how nuanced cooking is.
I'm not a chef but I am a cook; I've written with, and for, a lot of really talented chefs, and there isn't a single one I haven't learned something from—and then incorporated it into my own way of cooking or adapted it to my culinary personality, which is not fussy or detail-oriented, but rather exuberant and life-embracing.
Salt is the most valuable ingredient in the kitchen. Thomas Keller told me this sixteen years ago, when I asked him what he considered the most important thing for a cook to know. After thinking for a little bit he said, "How to salt food. It's the first thing we teach new cooks when they come to work at the French Laundry."
Michael Symon taught me to salt onions the moment they hit the pan (I used to wait till I got at least _some_ liquid in there). He does it to get the seasoning process under way, but when I did it I realized it also pulls out water and gets them cooking faster. That's how my personality fits into what he taught me. And Michael Symon didn't come up with this idea, he got it from another chef early in his own career. Michael Pardus taught me how to salt pasta water ("till it _tastes_ seasoned"), Eric Ripert taught me how to salt fish, Judy Rodgers taught me how to salt meat, and again Keller, who has taught me so much I don't know where he ends and I begin, taught me that it didn't matter what kind of salt I used, but rather that the critical factor was using the _same_ salt every time so that my fingers got used to a specific quantity.
Probably the most important thing I taught myself, when I wanted to convert an older chef's ratios into contemporary ones for a book, was how important weighing your food is as opposed to measuring it by volume. It makes all of cooking so much easier and more consistent. So, if you have a scale, use it, especially for measuring flour and large quantities of salt.
See? I could go on and on about just salt and we haven't even gotten to how to use it in a recipe or how it affects butter (it's added for flavor), and it doesn't have anything to do with eggs (except that it makes them taste better). So Michael Sand's question about salted or unsalted butter requires an essay, but I'll try to boil it down to this: I use salted butter because it's what I've been using since I started cooking in the fourth grade; it's what I'm used to. And since even sweet preparations benefit from salt, salted butter works fine even in pastries. Most chefs use unsalted butter; this is because they want ultimate control over the salt level in their food. I have no issue with this. Indeed, salt levels are especially important in the pastry kitchen. On occasion it will be the butter I choose to use for this reason.
So what do we make of the fact that salted and unsalted butter yield pretty much the same results? We are to take from this a very important lesson: good cooking requires us to pay attention, to think, and to taste our food and evaluate it throughout the cooking process.
Which brings me to the "large egg" question. Large eggs, by definition in the United States, weigh 2 ounces. In his book _On Food and Cooking_ , Harold McGee notes that large eggs weigh 55 grams (about 1 gram less than 2 ounces), with the whites weighing 38 grams and the yolks 17 grams. But the thing is, they don't all weigh exactly that—some "large" eggs weigh a little more, some a little less. Other sources will tell you that they weigh 50 grams and contain 70 calories, whereas McGee says they contain exactly 84 calories. Here's the truth: if you were to crack ten large eggs into a bowl, you'd find that they'd come pretty darn close to 550 grams total, 380 grams of white and 170 grams of yolk. (And if you have to worry about the calorie issue, you need to start eating smarter.)
Moreover, what if you're lucky enough to have a neighbor who raises hens and supplies you with really fresh eggs? They're going to be all different sizes, unless your neighbor has a government inspector living in the backyard to do the sorting. Either weigh them or just use your common sense.
If you want to be completely accurate, you do need to weigh your eggs, and many professional kitchens and bakeries do this. But for most preparations, this is neither practical nor necessary. So the rule is the same for salt, butter, and eggs, those most fundamental ingredients: it doesn't matter what kind you buy, what matters is that you always buy the same kind.
And pay attention.
For health and safety issues, see here.
# ABOUT THE RECIPES IN THIS BOOK
LIKE ALL RECIPES, MINE ARE APPROXIMATIONS ONLY, AND SO REQUIRE THAT YOU PAY attention and adjust as you go. Food will behave differently in an arid Phoenix kitchen than it will in a humid North Carolina kitchen than it will in a mile-in-the-sky Denver kitchen. These recipes have all been tested, and they should work perfectly well no matter your location or equipment, especially since most are very simple anyway and allow for plenty of give.
As for the specifics, unless otherwise stated:
All eggs are large.
All flour is all-purpose unbleached.
All salt is coarse kosher salt.
All butter is salted; if you prefer unsalted butter, just pay attention to the flavor of the recipe you're cooking.
For successful and satisfying cooking, you don't need _lots_ of equipment, but you do need _good_ equipment. You need two good stainless-steel sauté pans (a big one and a little one), one big pot, and one medium saucepan. Occasionally a nonstick pan is useful, but usually it's not. You need a wooden spoon with a flat edge. You need a large, heavy cutting board. Many people hamper themselves by trying to cut up their food on a board the size of a sheet of paper. Give yourself plenty of room.
# NUTRITIONAL DIFFERENCES IN EGGS
There will always be squabbling over food issues, so before you latch on to that 2010 _Time_ magazine article that says that organic eggs are no more healthy for you than factory-farmed eggs, or side with the opposite camp (many organic proponents denounced the article), know what the labels mean (see box on next page), and use your common sense. While there's no definitive way to be sure of the nutritional content of every egg available to you, an independent study by researchers at Pennsylvania State University, published in the journal _Renewable Agriculture and Food Systems_ , found that pastured hens produce more nutritious eggs than factory-farmed eggs. This makes good sense to me.
I will forever urge people to use their common sense first. In an ideal scenario you know the person who sells you the eggs and can ask about how the hens are raised and what they are fed. The Penn State study reported that pastured eggs have three times as much omega-3 fatty acids, twice as much vitamin E, and 40 percent more vitamin A in their yolks than factory-farmed eggs.
Some sources point out that eggs from pastured hens are also free of antibiotic residues and contain no arsenic, which is sometimes added to the feed of factory-farmed chickens to prevent infections and spur growth. Others claim there's no difference.
Given that what goes into our animals affects what comes out, it stands to reason that chickens feeding in a well-tended pasture, or enclosed chickens fed a diet of organic grain, will produce healthy, nutritious eggs. So, if you don't know the source of your eggs, pay attention to what the labels mean but don't worry too much either way, as long as you eat a healthy diet yourself. Even the cheapest of eggs are still nutritious and delicious.
A great alternative to buying eggs is to raise your own chickens, as many people are doing now. The trend is so strong that Williams-Sonoma advertises backyard chicken coops in its catalogs. This is a good option if you like this kind of work. I have one friend who has an egg intolerance; eating them makes her very uncomfortable. But she loves eggs, so she began raising a variety of hen breeds and found that one, Barred Rocks, produces eggs that don't make her ill.
Here are some online resources if you want to look into raising your own layers:
backyardchickens.com
beginningfarmers.org
hobbyfarms.com
mypetchicken.com
By far the biggest problem in home kitchens is the lack of a sharp knife. You need two of them, a big one and a little one. And they need to be sharp. A big knife block with forty-five knives of different sizes is useless if not one is sharp. Please, find a good knife sharpener in your area (preferably a wet-grind service, and not a hardware store that also does lawn mower blades) or buy a good sharpening stone (I swear by the DMT Diamond Whetstone sharpener). And learn how to use a sharpening steel to keep the edge.
After that it's all a matter of practice. And as far as I'm concerned, there's no better practice than that which results in a tasty, healthy meal for your family and your friends.
# EGGS OF A DIFFERENT FEATHER
THIS BOOK IS DEVOTED SOLELY to chicken eggs, produced, packaged, and sold by the billions, a staple of kitchens worldwide, because laying chickens are the easiest and least expensive birds to raise in quantity for egg production. Were geese or ducks or turkeys on par with chickens, their eggs might be what we go to the store to pick up a dozen of. It just so happens that the hen's egg seems to be the perfect size for our consumption; two make a meal, and one makes a portion when it's part of a meal, whether topping a _bibimbap_ or as an integral part of pasta dough.
# WHAT THE EGG LABELS MEAN
While you'll find a wide range of labels on egg cartons, the term "organic" seems to be the primary designation monitored and certified by the United States Department of Agriculture (USDA). Other labels placed on eggs and the way the hens are raised are not regulated by the USDA, although all commercial eggs have stamps noting when they were laid. Here are the definitions of other designations you may find in your grocery store's egg section.
# CAGE-FREE
Hens that are allowed to roost and socialize freely in a room or an open area can be called cage-free; this open area is typically in a barn or poultry house. Cage-free chicken farming is more labor intensive and land intensive for the farmer, which is why these eggs tend to cost more. Being cage-free is regulated, but the USDA's definition is vague, giving more of an indication how the chicken was _not_ raised (that is, confined to a cage).
# FREE-RANGE
Eggs that are laid by hens that have access to the outdoors can be called free-range eggs. These hens have the same benefits as cage-free hens. However, "outdoors" designates only no roof, not necessarily the healthfulness of the environment in which they're raised—it could be on grass, dirt, or a few square feet of concrete.
# PASTURE-RAISED
This is another unregulated term, but it does imply that the hens spend time outside on grass and eat a diet partly, if not entirely, of bugs and plants. This is the type of egg you're likely to find at a farmers' market.
# ORGANIC
The USDA classifies and certifies eggs as organic, but the regulations can differ by state. Organic eggs are raised according to USDA National Organic Program guidelines and are marketed as such on the carton. These hens are allowed free range of their houses, given outdoor access (though time outside is not regulated), and fed an organic diet (meaning that the feed is not treated with pesticides, antibiotics, herbicides, or fertilizers). If the hens do not have access to pasture area, then the growers must provide the chickens with sprouted grains or fresh plants on a daily basis. Organic-egg producers put a USDA emblem on the carton denoting that the eggs are certified organic.
# OMEGA-3 EGGS
These eggs come from hens that were fed a diet of foods rich in omega-3 fatty acids, such as algae, fish oil, and flaxseeds. By adjusting the hen's diet, the level of the omega-3s in the egg increases from 30 milligrams per egg to 100 to 600 milligrams per egg, according to some sources. The USDA does not certify this, but farms can be audited if they make such claims.
# VEGETARIAN
This label denotes that no animal products have been used in the feed given to the hens.
# NATURAL EGGS
This label does not indicate anything about how the chicken was raised; it simply means that no flavoring, brine, or color was added to the egg. The USDA does not regulate this designation.
# NO HORMONES/NO ANTIBIOTICS
These terms indicate that the grower has not used any hormones (prohibited by law) or antibiotics on the chicken itself or included them in its feed. The USDA does not certify this, but farms making such claims can be audited and must be able to document their claims.
# AMERICAN HUMANE CERTIFIED
This label designates that the American Humane Association has deemed that the farm where the hens lay is humane according to its standards. The hens are raised in cage-free barns or warehouses; they are allowed to roost and socialize freely indoors; no antibiotics or hormones are given to them; and their beaks are not cut.
# ANIMAL WELFARE APPROVED
Mainly used by family farms that raise chickens, this term indicates that the eggs come from farms that have been approved by Animal Welfare Approved, a group that strives to identify farms that produce food under the highest standards of animal welfare and environmental consciousness. Eggs with this label come from hens that have access to pasture and shelter, are fed a vegetarian diet, have not been given antibiotics, and have not had their beaks cut.
If ostrich eggs were all that was available, we'd be buying the egg in a different form. Quail eggs are popular and produced on a large scale, but they're so tiny as to be impractical for baking or making a custard—for anything, that is, that doesn't require their specific size. When I was in culinary school, we were rewarded with a quail egg and caviar pizza with Champagne if we'd performed well, an idea borrowed from Jeremiah Tower; the little quail egg fit perfectly on a single slice. They are the perfect hors d'oeuvre size—Thomas Keller serves a poached quail egg as a single bite on a gorgeous spoon, sauced with butter and a colorful garnish. I offer a different take on the quail egg canapé (here); it's decidedly showy but fun to prepare if you love to cook for guests.
Because we live near farms that raise ducks, we sometimes have access to duck eggs. If you do, too, please take advantage of them—they have big, rich yolks (see Poached Duck Egg on Duck Confit Hash, here).
The Whole Foods Market near us used to sell duck and ostrich eggs. Ostrich eggs are huge; if I had access to them, I'd probably scramble one, or make an omelet out of it. It would be a tricky size to fry or poach! But it will respond to heat and our palates pretty much the same way hens' eggs do. In this country we don't customarily raise turkeys to an egg-producing age unless they're specifically raised for making more turkeys.
There are, of course, fish eggs, which can be great to eat, but they, along with reptile eggs (eaten in certain parts of the world), are beyond the scope of this book, which is devoted to the mighty and ubiquitous hen's egg.
The easiest way to cook an egg is to do so before removing it from its shell. It can be considered foolproof, as long as the eggs are roughly the same size and are cooked in water, which works the same no matter the kitchen or the equipment (though eggs may take a little longer at high altitudes). Some chefs recommend taking eggs out of the refrigerator an hour or two before cooking them to reduce the possibility of their cracking while cooking (puncturing the shell with a pin is said to do the same thing), but my own tests of this found little if any difference. Neither practice hurts the egg, but given that these acts have so little impact on the cooked egg, I don't find it practical.
You can also bake an egg in its shell to hard-cook it (20 minutes at 350°F/180°C) for the same result, but ovens differ and the heat is not nearly as uniform and dense as water, so again, baking eggs in their shell is impractical.
Eggs can be cooked in the ash of a fireplace, and there are numerous ancient recipes for this, but the temperature is inconsistent and thus the cooking uneven, often resulting in eggs that are overcooked and nasty tasting, so I don't recommend this except as a novelty method for the curious.
Interesting effects can be had by soaking peeled hard-cooked eggs in beet juice (pickled or not) or by lightly cracking the shells and immersing the eggs in tea to create a "shattered" pattern. Again, all well and good for those who like to play with food (indeed, something to be encouraged), but beyond my interests here, which are the practical matters of cooking eggs in their shell. Eggs can be cooked to varying degrees with a range of delicious, even dramatic results simply by using water in various ways.
#
# **THE HARD-COOKED EGG**
THE HARD-COOKED EGG IS AMONG THE simplest preparations of the egg. I love to eat them while they're still warm. I also love to eat them cold, with salt and pepper and a bite of cheese. They make a good start to the day, a quick fix when there's little time for lunch, and a satisfying canapé. I love deviled eggs, one of the great underused renditions of the egg (see here). And I love that old staple, egg salad. Finely chopped hard-cooked eggs make a fabulous garnish, sometimes called a mimosa (after the yellow flower). And they're also a critical component of the classic vinaigrette known as _sauce gribiche_. Hard-cooked eggs are so versatile, so easy, they're almost easy to overlook. But don't. They're too valuable.
# HOW TO MAKE PERFECT HARD-COOKED EGGS
While it's the simplest and most routine of preparations, hard-cooking an egg can still be done carelessly, overcooked so that the white is rubbery and the yolk gray-green and sulfurous, or undercooked so that the yolk is uneven. On the other hand, when you cook it correctly and slice it open to see the beautifully uniform pastel yellow against the glossy white, it's something to delight in.
There is more than one way to hard-cook an egg, but the easiest and surest method is to use the uniform, dense, gentle heat of water, followed by water's powerful capacity to extract heat (an ice bath)—indeed, the most important part of the preparation is rapid cooling once they're done. The works-every-time method is this: put cold eggs in a pan in one layer, cover them with water by about 1 inch/2.5 centimeters, and put the pan over high heat. When the water comes to a full boil (at least 209°F/98°C), cover the pan, remove it from the heat, and let it sit for 15 minutes. Remove the eggs to an ice bath (half ice, half water) until they're completely chilled, at least 10 minutes but preferably longer, giving the water a gentle stir every now and then to keep the cold circulating.
The result will be a uniformly yellow yolk, which indicates a perfectly cooked egg. If you overcook them or fail to chill them quickly and thoroughly, ferrous sulfide, with a gray-green color, the odor of sulfur, and an off flavor, can form on the surface of the yolk. If you undercook them, they may not look as pretty but they're still delicious. Peel and use the eggs right away, or store them in their shells in a covered container in the refrigerator for up to 2 weeks.
# HARD-COOKED EGGS USING A PRESSURE COOKER
I have a pressure cooker, but I rarely use it. I am, however, frequently on Twitter (@ruhlman), where in the winter of 2012, someone asked me about pressure-cooking eggs. I replied that I had no experience, but asked anyone reading the feed to respond. Laura Pazzaglia (@hippressurecook), an American living in Italy, did. She said that pressure-cooking eggs was a brilliant method. I checked her profile and learned that she writes a blog called hippressurecooking.com. I was so intrigued I asked her to write a guest post for my site on pressure-cooking eggs, and she did.
After reading Laura's post and consulting other sources, I headed to the kitchen for my own highly scientific, exhaustive tests (two dozen eggs, a pressure cooker, and a stopwatch) to see for myself.
She's absolutely right. It's a fabulous way to cook eggs in the shell, primarily because it makes even the freshest eggs easy to peel. Fresh eggs have so little air in them that their shell and membrane tend to stick to the cooked egg white, forcing you to pull divots of white out of the egg as you peel it. A pressure cooker, however, creates a moisture barrier between the shell and the egg white so that the egg easily slips out of its shell. This is especially useful when you're making a lot of eggs, or when it's essential that the exterior of a hard-cooked egg remain pristine and smooth—that is, any time you won't be chopping up the egg for egg salad or for an egg garnish.
To hard-cook eggs using a pressure cooker, it's best to steam them, so you'll need a steamer basket or trivet to keep the eggs above the small amount of water in the pot.
1. Put the eggs in a steamer basket or on a trivet in the pot. Add 1 cup/240 milliliters water and lock the lid, turning the pressure setting to low. (Setting it to high usually results in violently cracked eggs.)
2. Put the pressure cooker over high heat. After the pressure button pops up, the steam will build and begin to whistle out of the valve. As soon as it reaches its maximum pitch, reduce the heat to medium-low and set your timer for 7 minutes.
3. Fill a large bowl with half ice and half water.
4. When 7 minutes have elapsed, remove the pot from the heat and allow it to cool to the point that the pressure button has fallen and you can open the pot. If after 5 minutes, the pressure button remains raised, run cold water over the pot until the pressure button drops.
5. Remove the eggs to the ice bath for at least 10 minutes, stirring the ice water a couple of times during the first minutes of cooling.
6. Peel and use the eggs right away, or store them in their shells in a covered container in the refrigerator for up to 2 weeks.
Times may vary by a minute depending on your cooker and your stove, so pay attention to initial results and adjust accordingly. For more on pressure-cooking eggs to different degrees of doneness, see here.
# Egg Salad Three Ways
EGG SALAD IS ONE OF THE MOST ACCESSIBLE, easy, and delightful egg preparations, though the name confounds me. Why _salad?_ Can't we come up with a better name for chopped eggs bound with mayonnaise? You can put it on lettuce if you want—but be sure it's crunchy lettuce, head lettuce or romaine, which serve as a serving vessel and garnish. Because egg salad is about the softest food you can make, it should always be paired with something crunchy: Toast. Crackers. Celery. Crisp bacon. Try making a lettuce wrap with the tarragon-chive version, or serve pappadams with the curried egg salad.
#### Egg salad rules of thumb:
2 eggs per serving
1 tablespoon mayonnaise per egg
judicious flavoring (herbs, spices, onion, as you wish)
crunch (croutons, celery)
I use a wooden bowl and a mezzaluna, a curved knife with a wooden handle, but a knife and cutting board work fine. If you want a smoother consistency, you can even use a food processor.
Egg salad is especially good when you make your own mayonnaise (Egg Salad with Homemade Lemon-Shallot Mayonnaise, here, is heaven to me). And when you make your own mayonnaise, you can infuse the oil with intense flavors, as in the Curried Egg Salad (here), which also includes the aromatic flavors of garlic and ginger. But it's not essential, so wondrous are the fresh hard-cooked eggs themselves. And sometimes even I don't want to bother with homemade mayo (I always have some Hellmann's on hand for a fried-egg sandwich, here). In that case I make sure to load the eggs with plentiful, flavorful herbs, such as tarragon (my favorite herb with eggs) and the oniony chive.
I love an egg salad sandwich on toasted bread, but egg salad can also make an excellent—and elegant—canapé on crackers or small toasts, an hors d'oeuvre that can be made in advance. A tablespoonful on a water cracker garnished with a sprig of chervil or a small leaf of tarragon is a satisfying and economical bite to serve a lot of guests.
#### EGG SALAD
1. / _A wooden bowl and a rounded knife called a mezzaluna are the perfect tools for quickly and neatly chopping hard-cooked eggs._
2. / _The yolks fall apart easily, so you mainly have to go after the whites._
3. / _Chop the eggs to your taste. I like larger chunks of white, but those pictured here could be chopped even further for more uniformity._
4. / _Add the mayonnaise and the flavoring ingredients—here, chives, tarragon, and minced red onion._
5. / _Continue to chop and mix the ingredients. If you haven't already done so, give the eggs an aggressive dose of salt and several grinds of fresh pepper._
6. / _Egg salad ready to serve, on toast as a sandwich, on toast points as a canapé, or simply as is._
# Egg Salad with Tarragon and Chives
SERVES 4 FOR SANDWICHES OR 12 FOR CANAPÉS
_Tarragon is my favorite herb, both powerful and gentle, assertive yet delicate. It pairs beautifully with eggs. I also love the oniony punch of chives with egg. I make this as a summer lunch, when the herb garden is lush._
3 tablespoons minced red onion
Salt
8 hard-cooked eggs, peeled and coarsely chopped
Freshly ground black pepper
½ cup/120 milliliters Hellmann's mayonnaise (or, better yet, your own, here)
2 tablespoons chopped fresh tarragon
2 tablespoons minced fresh chives
Put the red onion in a small bowl and sprinkle liberally with salt, then cover with water for 5 to 10 minutes.
Put the eggs in a medium bowl. Give them a three- or four-finger dose of salt and a liberal application of freshly ground pepper. Add the mayonnaise. Strain the onion and add it, along with the herbs, and stir with a rubber spatula till all of the ingredients are uniformly combined.
# Egg Salad with Homemade Lemon-Shallot Mayonnaise
SERVES 4 FOR SANDWICHES OR 12 FOR CANAPÉS
_This is a rich delight made with your own homemade mayonnaise. I use a hand blender to make the mayonnaise quickly, which requires less muscle, but a whisk and a bowl work just as well. If the mayonnaise becomes too thick, add a few drops of water to thin it out._
8 hard-cooked eggs, peeled and coarsely chopped
Salt and freshly ground black pepper
2 celery ribs, cut into small dice
½ cup/120 milliliters Lemon-Shallot Mayonnaise (here)
Put the eggs in a medium bowl. Give them a three- or four-finger dose of salt and a liberal application of freshly ground pepper. Add the celery and the mayonnaise and stir with a rubber spatula until all of the ingredients are uniformly combined.
# Curried Egg Salad
SERVES 4 FOR SANDWICHES OR 12 FOR CANAPÉS
_Yes, you're allowed to put 2 teaspoons of raw curry powder into some store-bought mayonnaise and it will be fine, or even into your own mayonnaise, but what's the fun of that? Especially when you can ratchet up the taste by a factor of ten by making your own curried oil with ginger and garlic. In either event, buy a fresh jar of curry powder if you can't remember buying the one currently in your pantry. This mayonnaise can also be used to make excellent deviled eggs (here). I love to serve this in crunchy pappadams, pressed with a ladle into hot oil to give them a bowl shape._
# FOR THE CURRIED MAYONNAISE:
¾ cup/180 milliliters vegetable oil
1 large garlic clove, minced
1 piece ginger (about ½ inch/1 centimeter long), peeled and finely grated
1 tablespoon good curry powder
½ teaspoon ground turmeric (optional)
Pinch of cayenne pepper
2 teaspoons fresh lime juice, plus more to taste
1 teaspoon water
½ teaspoon salt
1 egg yolk
# FOR THE EGG SALAD:
¼ cup/25 grams minced red onion
Salt
8 hard-cooked eggs, peeled and coarsely chopped
2 celery ribs, cut into small dice
First, make the mayonnaise. Combine the oil, garlic, and ginger in a small sauté pan over high heat. When the garlic begins to simmer, reduce the heat to medium and continue cooking until the garlic is cooked (smell it; it should have lost its raw smell). Remove the pan from the heat and add the curry, turmeric (if using), and cayenne. Transfer the mixture to a glass measuring cup to cool.
While the oil is cooling, combine the lime juice, water, and salt in the vessel you'll be mixing your mayonnaise in and allow the salt to dissolve. Stir in the egg yolk. When the oil is cool enough to touch, emulsify it into the water-yolk mixture as you wish (see making mayonnaise, here). Set the mayo aside. (It can be prepared and refrigerated in a covered container for up to 8 hours before mixing the salad.)
When you're ready to make the egg salad, put the red onion in a small bowl and sprinkle liberally with salt, then cover with water for 5 to 10 minutes.
Put the eggs in a medium bowl. Give them a three- or four-finger dose of salt. Strain the onion and add it, along with the celery and ½ cup/120 milliliters of the curried mayonnaise, and stir with a rubber spatula until all of the ingredients are uniformly incorporated.
# Warm Hard-Cooked, Creamy-Yolk Egg with Ham and Cheddar
SERVES 1
_When I was twenty, poor, and a solitary traveler in Amsterdam, I found myself lost and roomless in a seedy part of the city. I worked my way back to the tourist bureau near the train station, which directed me to a cheap B &B. It was dark by the time I found the right tram and the B&B, relieved just to be anywhere safe and dry. When I awoke in my tiny but clean, sun-lit room and went downstairs for break fast, they had warm, hard-cooked eggs and cheese out on the counter. The day was cool and crystal clear. I found the warm eggs an uncommonly comforting and satisfying break fast, especially with the rich, delicious cheese. My whole life I had eaten hard-cooked eggs chilly out of the fridge, and only on those rare occasions when we made them. So I associate warm, hard-cooked eggs and cheese with that long-ago trip, with unexpected safety and a crisp, bright day on friendly, foreign ground, where you discover something new about what had once seemed ordinary._
_I like the yolk to be set in the center, but only just—dark yellow and creamy. Served with a hunk of cheese and a slice of good ham eaten by hand, it's a break fast that lasts all day._
1 egg
1 demi-baguette, sliced in half lengthwise, then cut into the desired size for toast
A gratuitous amount of butter
1 (2-ounce/55-gram) chunk farmhouse cheddar
1 (2-ounce/55-gram) slab country ham
Put the egg in a pot covered with a couple inches of water. Bring the water to a boil over high heat, remove from the heat, and cover for 7 to 9 minutes.
Meanwhile, toast your bread, then butter it (with all other ingredients being dry, don't skimp!), and put it on a plate with the cheese and ham.
Remove the egg from the water and allow it to dry. Clip or peel the top of the egg as you wish. An eggcup and egg spoon are handy but not strictly necessary. Serve with the toast, cheese, and ham.
#
# **THE SOFT-COOKED EGG**
WHEN PREPARING EGGS IN THE SHELL, WE always want the white to be cooked, or at least somewhat opaque, but we don't always want the yolks cooked through. So it's good to have a handle on the range of egg donenesses available to you, from soft-cooked, in which the yolk is completely runny and the white just barely set; to what is called _mollet_ , meaning that the white and part of the yolk are solid but the center of the yolk remains fluid; to just before fully hard-cooked, when the yolk is soft and creamy throughout but still slightly darker in the center than at the edge.
Eggs cooked so soft that the white remains somewhat runny should usually be eaten in the shell. An eggcup is the ideal serving piece, but you can also present a soft-cooked egg on a bed of kosher salt in a ramekin. Of course they're most commonly eaten for breakfast, with toast. But they're good anytime, and if for whatever reason you can't eat solid food, they're excellent, easy nutrition. In his lovely book _Eggs,_ the Michelin-starred chef Michel Roux even suggests serving them as an elegant dessert, inviting guests to spoon caramel sauce into the liquid yolk at the table and dip _batonets_ of brioche into the sweet yolky goodness.
_Mollet_ eggs can be served as a garnish on, well, just about anything, but they're especially good on salads and on warm or room-temperature vegetables.
You'll notice differences in both flavor and texture from yolks cooked to varying nonliquid consistencies, so experiment until you find just the right combination.
To achieve across-the-board consistent results for the full range of egg donenesses, we again turn to the miracle tool, water, with its built-in thermometer and a density that makes it an enormously efficient heat giver.
Some cooks recommend starting with room-temperature eggs to reduce the chance of eggs cracking during the cooking, so feel free to remove your eggs from the fridge 1 to 2 hours before cooking them. I don't bother with this. First, it's not practical—I rarely remember to do it, or I don't even know in advance that I'm going to cook eggs. Second, I don't find that they cook any differently; whether the eggs started off cold or hot, my side-by-side trials were identical.
# HOW TO MAKE PERFECT SOFT-COOKED EGGS
Put cold eggs in a pan in one layer, cover them with water by about 1 inch/2.5 centimeters, and put the pan over high heat. When the water reaches a full boil (at least 209°F/98°C), cover the pan, remove it from the heat, and follow the directions below for your preferred doneness.
**For soft-cooked eggs:** Remove the eggs from the water after 90 seconds if you want the whites to be loose, a true soft-cooked egg. For a soft-cooked egg in which the white is set but the yolk is molten, remove the egg after 3 minutes. Any time between 90 seconds and 3 minutes will yield a good soft-cooked egg. Cut the top off the eggshell and serve immediately.
**For _mollet_ eggs:** Remove the eggs from the water after 5 to 7 minutes. At 5 minutes the yolk will be beginning to set, about ¼ inch/6 millimeters around the periphery, while the center remains molten. By 7 minutes, the center will be more solid than liquid but will be darker yellow and creamy. Either serve immediately or chill in an ice bath for 10 minutes and store in a covered container in the refrigerator for up to 2 days.
**For fully cooked but creamy yolks:** Remove the eggs from the water after 9 minutes. The white will be solid and the yolk will not be fluid, but rather dark and spreadable. Either serve immediately or chill in an ice bath for 10 minutes and store in a covered container in the refrigerator for up to 2 days.
# SOFT-COOKED EGGS TO VARYING DONENESSES USING A PRESSURE COOKER
Pressure-cooking eggs is an ideal method if you want to serve soft-cooked eggs out of the shell, since it makes them so much easier to peel. As with pressure-cooking hard-cooked eggs (see here), you'll need a steamer basket or trivet to keep the eggs above the water in the pot.
1. Put the eggs in a steamer basket or on a trivet in the pot. Add 1 cup/240 milliliters water and lock the lid, turning the pressure setting to low. (Setting it to high usually results in violently cracked eggs.)
2. Put the pressure cooker over high heat. After the pressure button pops up, the steam will build and begin to whistle out of the valve. As soon as it reaches its maximum pitch, reduce the heat to medium-low and start your timer according to the next step.
3. For very soft eggs, cook for 3 minutes. For set whites and liquid yolks, 4 minutes will do (this will allow you to peel the egg without breaking it if you want to serve a whole egg out of the shell with a liquid yolk, as with the artichoke preparation here). For medium-cooked eggs, cook for 5 minutes.
4. After the appropriate time, put the pot under cold running water until the pressure button falls (a few seconds). Release any remaining steam, open the pot, and serve the eggs or chill them in an ice bath (half ice, half water) for 10 minutes, depending on how you intend to serve the eggs. Soft-cooked eggs can be stored in a covered container in the refrigerator for up to 2 days.
# Soft-Cooked Eggs on Artichoke Hearts with Creamy Lemon-Shallot Vinaigrette
SERVES 4
_This dish pairs a classic thick lemon vinaigrette with artichoke, egg, and, for crunch and color, panko bread crumbs fried in butter. It is definitely in the Impress Your Date category of dishes and 80 percent guaranteed to work. Regardless, it's delicious and fun to eat, and each of the four components can be prepared well before serving (the mayonnaise should be made within hours of serving, but the artichoke and eggs can be cooked 2 days in advance if you wish). I also like_ mollet _eggs with this, but it's always impressive to have some yolk spill out when you cut into it, so they can be anywhere between soft and_ mollet _. Serve some good toasted bread or extra-crispy fries with this. I serve this dish slightly warm, but it can be served cold or at room temperature, as you wish._
4 whole artichokes
1 large onion, thinly sliced
Salt
4 eggs
1 to 2 tablespoons fresh lemon juice
1 cup/240 milliliters Lemon-Shallot Mayonnaise (here)
Pinch of cayenne pepper
¼ cup/10 grams minced fresh chives
¼ cup/20 grams panko, browned in 2 tablespoons butter (optional)
There are many ways to cook an artichoke, and any of them will work here. This way is the easiest and can be done a day or two ahead: Using a serrated knife, saw off the top half of each artichoke (this will make the leaves easier to remove after cooking) and cut the stem from the bottom so that it has a flat base. Stand the artichokes in a pot, scatter the sliced onion among them, and add enough water so that it comes halfway up the artichokes. Bring the water to a simmer over high heat, then reduce the heat to medium-low, cover the pot, and simmer until the artichokes are tender, 45 minutes to 1 hour. Transfer the artichokes to a plate to cool. When you can handle the artichokes comfortably, remove and discard the leaves and choke. Reserve the hearts and keep warm, or wrap and refrigerate them for up to 2 days.
Soft-cook the eggs to your desired doneness; pressure-cooking for 4 minutes is recommended for easy peeling, but the standard boil-and-cover, for a total cover time of 3 minutes, is fine, too. If you're not continuing with the recipe right away, chill the eggs in an ice bath for 10 minutes, then store in a covered container in the refrigerator for up to 2 days.
While the eggs are cooking, prepare the plates. Stir 1 tablespoon of the lemon juice into the mayonnaise and taste. It should be vinaigrette-strength to pair with the egg; add another tablespoon if you wish. Spoon 1 tablespoon mayonnaise into the center of each plate. Place an artichoke heart on top of the mayo (warm them in the microwave for 30 seconds if they were refrigerated). Spoon 2 tablespoons mayonnaise into the cup of the heart.
When the eggs are done, carefully peel them and place one on each artichoke. (If you've made the eggs ahead of time, first warm them in gently simmering water for 60 to 90 seconds.) Garnish with the cayenne, chives, and, if desired, the crunchy panko.
**VARIATION:** In the 1990s Cleveland chef Parker Bosley, one of the people who, in his own way, encouraged me to educate myself about food and cooking so I didn't continue to toil in ignorance, served a soft-cooked egg on spinach. It was the first time I'd had the surprise of cutting into what looked like a hard-cooked egg only to get the rush of bright yellow yolk spilling out onto the dark green spinach. Any kind of egg, from hard-cooked to poached, is great on spinach, but I loved the surprise of this preparation. When I next went to Parker's, I went back to the kitchen to ask the _chef de cuisine_ , who did the actual cooking and peeling, how he peeled the soft-cooked eggs, as mine kept breaking when I tried it at home. He shook his head wearily and said, "No secret, man, it's a real pain," with a gravitas that I knew implied, _and I can't wait till that damn dish is off the menu._
Discovering the unique capacity of the pressure cooker to separate white from shell solved the problem (for the most part, anyhow—some eggs might break, so I always make extra; if they don't break, they can be refrigerated for a few more days and reheated or cooked further). If you want to try Parker's spinach version, replace the artichoke and vinaigrette with the spinach from the Shirred Eggs Florentine (here) and pressure-cook your egg for 4 minutes for easy peeling.
# Pork Ramen with Soft-Cooked Egg and Scallions
SERVES 4
_Hard-cooked eggs are common in ramen, but I particularly like soft-cooked eggs for this dish. They're easy to serve and come out perfectly every time._
_Use a Japanese mandoline (often referred to as a Benriner, after the company that makes them) to julienne the vegetables; the radishes will look especially cool. You can use any pork and cook it any way you wish. Grilled over coals is best, but you can also roast it. I like using the pork skirt steak (from the diaphragm of the animal), which connects to the spareribs, and either roasting or grilling it before cooking it in the ramen for the flavor boost and for setting the protein. You can cook the pork whole, or slice and stir-fry it. If you are using pork shoulder, roasting it whole will make it easier to slice thinly. This is also a great way to use leftover pork._
_It should go without saying that any meat and any good stock can be used in making this dish. Any tasty vegetables will work, too. Again, the key to excellence in this dish is a good stock and good noodles._
_If you're using dried ramen noodles, you can cook them in the stock or precook them in salted water if you don't have enough stock._
1 quart/1 liter Pork Stock (recipe here) or Miso-Kombu Broth (here)
Salt and freshly ground black pepper
1 pound/450 grams fresh ramen noodles (or 12 ounces/350 grams dried)
1 pound/450 grams cooked pork skirt steak or boneless pork shoulder, cut into ½-inch slices
4 eggs
8 ounces/225 grams baby spinach
2 carrots, peeled and julienned
4 radishes, julienned
6 scallions, sliced feather-thin on the diagonal
Put the stock in a large pot and bring it to a boil over high heat. Taste the stock and adjust the seasoning with salt and pepper if you wish. Add the ramen and cook for 1 minute, then add the pork and reduce the heat to low.
Put the eggs in a small pot and cover with water by about 1 inch/2.5 centimeters. Bring to a boil over high heat. Immediately cover the pot and remove it from the heat. Set a timer for 1 minute.
While the water is heating for the eggs, line four soup bowls with spinach. Divide the noodles, pork, and broth among the bowls. Garnish with the carrots, radishes, and scallions. Crack an egg into each bowl after they've sat in the covered pot for 1 minute, scraping out any white that may adhere to the shell. Serve immediately.
# Pork Stock
MAKES 2 QUARTS/2 LITERS
_It's important to roast or blanch any raw bones to avoid emulsifying the nasty coagulated protein that water pulls out of raw bones (for chicken stock it's enough to skim the stock once it's come to a simmer). Any pork bones with lots of connective tissue and meat, such as neck bones or joints, will do. I like to use pigs' feet because they have an abundance of all the stuff you need for great stock: meat for flavor and bones and skin for body (skin has a lot of gelatin, which is where the body comes from). They make for a very earthy, farmlike pork broth._
4 pigs' feet or 2 pounds/910 grams meaty pork bones and joints
2 Spanish onions, roughly chopped
4 carrots, roughly chopped
4 celery ribs
5 garlic cloves
¼ cup/70 grams tomato paste
2 bay leaves
2 teaspoons whole black peppercorns, roughly chopped or coarsely ground in a mortar and pestle
Several parsley sprigs (optional)
Put the pigs' feet in a pot that will hold them and allow you to cover them by a couple of inches of water (but not so big that you have to add too much water; a general rule is 2 parts bones, 3 parts water by weight). Turn the heat to high. When the water reaches a boil, strain the pork bones, and rinse them in cold water. Wipe out the pot, return the bones to it, and add water until they are covered by a couple of inches. Bring the water to a simmer. Now you can either turn the heat to low and let it cook, uncovered, for 8 to 10 hours or, better, put it in the oven, uncovered, set to its lowest setting overnight.
After the stock has cooked for at least 8 (and as many as 12) hours, add the remaining ingredients. Bring the stock back to a simmer on the stovetop, then turn the heat to low and cook for another hour or so.
Pass the stock through a fine-mesh strainer. For an even more refined stock, strain it again through cheesecloth.
This will make twice what you need for the pork ramen recipe; the rest can be stored in a covered container in the refrigerator for up to 4 days or in the freezer for up to 3 months.
# Soft-Cooked Egg with Buttered Toast for Two
SERVES 2
_My grandma once served me a soft-cooked egg by pouring and scraping it from the shell into a bowl. It looked disgusting and I'll never forget it. But there is something comforting and visually appealing about eating a very soft egg straight from the shell, and enjoying it with that other simple preparation, buttered toast. Toast made from grocery-store sandwich bread has no flavor, so I recommend using a slice of sourdough bread. I prefer to toast the bread twice. First dry, then buttered so that the butter is bubbling hot and flavorful when you serve it. You'll need eggcups and egg spoons to serve._
_I see this being eaten on a Sunday morning after a happily drunken Saturday night, freshly showered and shaved and in a smoking jacket that I'd like to own, across from my equally contented wife, each of us reading the_ Times _and pretending that all is well with the world. (Consider serving with a Bloody Mary, depending on how useless you're willing to be for the rest of the day.)_
2 slices sourdough bread or good country bread
2 eggs
A gratuitous amount of butter
Abundant strong coffee, preferably a full fresh percolator
2 Bloody Marys (optional)
Toast the bread.
Meanwhile, cover your eggs with 2 inches/5 centimeters water in a small saucepan. Bring the water to a boil over high heat. Remove from the heat and cover for 2 to 3 minutes (I think 3 is perfect, but 2 will give you a truly soft-cooked egg).
While your water is boiling, butter your toast and retoast till the butter is bubbling.
Kiss your spouse's crown and say, "Love you."
Transfer the eggs from the water to the eggcups, slice off the top ½ inch/1 centimeter with an egg cutter or knife, and serve with the toast and hot coffee, followed by a Bloody Mary, if desired.
#
# Crispy Mollet Egg with Asparagus
SERVES 4
_I often use this asparagus sauce garnished with asparagus tips for sautéed scallops, a great pairing. But it works beautifully with egg as well. I love the crispy egg preparation here, simply breaded and fried. I think a_ mollet _egg is best, but feel free to do a soft-cooked egg as for the Soft-Cooked Eggs on Artichoke Hearts (here). Both the asparagus and eggs can be cooked the day before serving._
1 pound/450 grams asparagus, woody ends trimmed
5 eggs
2 tablespoons minced shallot
2 tablespoons fresh lemon juice
About ½ cup flour
About 1 cup panko
Vegetable oil for deep-frying
¼ cup/60 grams butter, cut into 3 pieces
Salt
Grated lemon zest (optional)
Bring a large pot of heavily salted water to a boil and cook the asparagus in it till tender, a few minutes (taste one to see if it's done). Drain and plunge them into an ice bath (half ice, half water) till thoroughly chilled, then drain again.
Cut off the asparagus tips and reserve in a small dish covered with a damp paper towel and plastic wrap. Cut the stems into 1-inch/2.5-centimeter pieces and puree in a blender, adding just enough cold water or ice to get the pieces spinning. Cover and keep refrigerated till you're ready to finish the dish.
Cook all but one of the eggs to the _mollet_ stage (boil for 5 minutes, here, or use the pressure cooker method, here) and chill in an ice bath. Carefully peel them (this will be easier if you used the pressure cooker).
Meanwhile, combine the shallot and lemon juice in a small bowl and set aside.
Whisk the remaining egg in a shallow dish till it's uniformly mixed. Put the flour in a plastic bag and the panko in a bowl.
Carefully dredge the peeled eggs in the flour, then roll them in the beaten egg until the flour has absorbed it, then roll them in the panko, where they can stay till you're ready to cook them.
Heat 3 inches/7.5 centimeters oil in a high-sided saucepan over high heat. When the oil is hot, drop in the eggs. Cook until browned, 2 to 3 minutes. (Alternatively, you can pan-fry the eggs, with the oil coming halfway up the eggs, turning them till golden brown, 2 to 3 minutes.) Use a slotted spoon to transfer the crispy eggs to a plate.
Put the asparagus sauce in a small saucepan over medium-high heat. When it begins to bubble, reduce the heat to medium and add the butter pieces one at a time, whisking continuously until the butter is melted. Season the sauce with salt and remove it from the heat.
Reheat the asparagus tips in the microwave for 20 or 30 seconds.
Divide the sauce among four plates. Strain the shallot and sprinkle on top of the sauce on each plate. Scatter the asparagus tips along the periphery of the sauce. Place an egg on each plate, grate some lemon zest over the dish, if desired, and serve.
#
# **COOKING EGGS _SOUS VIDE_**
WHILE _SOUS VIDE_ LITERALLY TRANSLATES as "under vacuum," referring to vacuum-sealed food cooked in water at precise, below-boiling temperatures, we use the term _sous vide_ broadly to indicate the cooking of _any_ food at precise, below-boiling temperatures.
The devices used to maintain exact temperatures, immersion circulators and _sous vide_ water-bath systems, have been used in restaurant kitchens for some time and are now becoming affordable enough that they are found in home kitchens as well.
Eggs, already packaged by nature and ready for water, are fun to cook _sous vide_ because you can create textures in both the yolk and the white that are otherwise impossible to achieve. If you cook them at 144.5°F/62.5°C for 35 to 45 minutes, you will have a perfect soft-cooked-like egg that can finish cooking in a hot broth: the white will be quiveringly tender, with most of the proteins fully congealed and others just barely opaque, and the yolk warm but fluid. This is the perfect way to cook an egg that will be served as a garnish in another dish. It's great for entertaining since you can do a lot of eggs at once and they all come out perfect; they're also easy to serve since they more or less pour out of the shell. So these can be used in virtually any soup or stew, on hot grits, or with slow-cooked beans.
Ramen is the perfect vehicle for a soft-cooked egg. Because there are two ramen recipes here, one pork-based and one vegetarian, I'm recommending two different methods for soft-cooked eggs, one cooked as a regular soft-boiled egg and one cooked _sous vide_ for those who have _sous vide_ equipment. In both dishes the egg finishes cooking in the stock, so either egg-cooking method can be used in either recipe.
# Ramen with Soft-Cooked Eggs
RAMEN, THE JAPANESE NOODLE, HAS BEEN largely debased in America by packaged dry ramen with nasty "flavoring" packets. Great ramen has a distinctive flavor due to alkaline (as opposed to acidic) ingredients, such as sodium and/or potassium carbonate, food-grade lye, or even baking soda. If you live in a city that offers fresh ramen, it's worth seeking it out. If you like to make pasta at home, the recipes in David Chang's _Momofuku_ cookbook or Takashi Yagihashi's _Noodles_ are worth the effort. But you can also use the packaged ramen sold in every grocery store—the noodles themselves are fine, it's the flavoring packets that are just plain wrong in every way.
I'm offering two versions, one with pork because I love all things pork, and a vegetarian ramen dish made simply with onion and miso, no elaborate stock-making required.
The real star here is the soft-cooked egg added to the soup at the end. If you have an immersion circulator or _sous vide_ water bath, cook your egg at 144.5°F/62.5°C for 45 minutes. If you don't, you can get a similar effect by bringing a large pot of water to 150°F/65°C, removing it from the heat, putting your eggs in the water, covering the pot, and letting the eggs cook in the hot water, undisturbed, for 45 minutes.
Ramen is a great dish because you can do anything you want with it, give it any garnish, seasoning, or spice you want (just don't use that horrible mixture of flavored salt they put in the package!). Of course it's best when you make your own stock, because good stock is one of those things that you cannot buy at a store. Small amounts of stock are not difficult to make, and the vegetarian ramen recipe here uses a ridiculously simple concoction that can't even be called stock it's so easy.
The basic ramen recipe is simple: Heat up 1 quart/1 liter stock. Add 1 pound/450 grams fresh ramen. Add any garnish that needs cooking (such as meat or hard vegetables), then serve, garnished as you wish with scallions, mushrooms, bamboo shoots, and an egg, of course.
# Ramen with Miso-Kombu Broth, Shiitake Mushrooms, and Sous Vide Egg
SERVES 4
_This is a light, refreshing vegetarian ramen dish. I like to sear the shiitakes in a very lightly oiled cast-iron pan to brown them and intensify their flavor, but they can also simply be sliced and added to the broth raw. Save the stems to add to the Miso-Kombu Broth._
4 eggs
1 quart/1 liter Miso-Kombu Broth (recipe below)
1 pound/450 grams fresh ramen noodles (or 12 ounces/350 grams dried)
¼ pound/115 grams fresh shiitake mushrooms (approximately 20), stems removed, caps sliced and sautéed, roasted, or raw
6 scallions, sliced feather-thin on the diagonal
2 carrots, peeled and julienned
4 radishes, julienned
Preheat a _sous vide_ water bath to 144.5°F/62.5°C and add the eggs.
Shortly before the eggs are done (they should be in the water bath for at least 35 minutes and as long as 45 minutes), bring the broth to a simmer in a large pot. Add the ramen noodles and cook till they're done. Divide the broth and noodles among four bowls. Garnish with the shiitakes, scallions, carrots, and radishes.
Crack one egg into each bowl of broth. Serve immediately.
# Miso-Kombu Broth
MAKES 1 QUART/1 LITER
_This is a delicious, simple broth, not unlike miso soup. You can use white, red, or mixed miso, whatever kind you prefer. I use white, or shiro, miso. Kombu seaweed and bonito flakes are both available at Asian markets and some grocery stores. They give the broth a fresh-sea flavor, but you can omit them or substitute fish sauce for a different but not dissimilar effect. The kombu may have blooms of chalky, salty minerals—this is good stuff, so don't rinse it off. The broth should be cooked very gently, steeped rather than simmered._
1 quart/1 liter water
1 Spanish onion, roughly chopped
½ ounce/15 grams kombu (1 or 2 large sheets)
Leftover shiitake stems (optional)
1 ounce/25 grams bonito flakes (about 1 cup)
2 tablespoons miso paste
Pour the water into a large pot and add the onion. Bring the water to a simmer over high heat, then reduce the heat to low. Add the kombu and mushroom stems (if using) and cook for 1 hour. Stir in the bonito flakes and cook for another 5 minutes. Stir in the miso. When it has dispersed, pass the stock through a fine-mesh strainer into a clean vessel. Use as desired, or store in a covered container in the refrigerator for up to 1 day or in the freezer for up to 2 months.
Once we remove the egg from its lovely walls, we can have even more fun, depending on what our final goals are. We can cook it in dry heat or wet heat; high-high heat or gentle heat; free-form or in a mold; with yolk and white distinct, uniformly blended, or even in between, as with the first preparation here.
#
# Weekend Broken-Yolk Fried-Egg Sandwich
SERVES 1 HURRIED FATHER, MOTHER, OR FIFTH-GRADER
_Here eggs are gently fried in butter before all of the water has cooked out of the butter, so that they are cooking at near poaching temperatures. Cooking eggs gently will render them very tender and mildly flavored. If you heat a steel pan properly, and that pan is pristine, and you allow the butter to melt and wait until the bubbling is just beginning to subside, your eggs should not stick. But sometimes they do stick, and then you wind up with a scrambled-egg sandwich, which is OK but not as good. So, if you have a good nonstick pan, this is one of the few times to put it to use. (As a rule, cooking with nonstick pans should be avoided, as they don't allow you to develop the best browning, which is where the flavor comes from. They're always the best choice for eggs when cooking with low heat, however.)_
_I've been making fried-egg sandwiches since I was in fifth grade, and they remain a common, quick lunch, especially on a Saturday when I've got a lot of errands or projects to get done. It's quick and satisfying, and lasts me all day._
_I have a reputation for advising people to avoid processed food, like making BLTs completely from scratch, for instance: curing your own bacon, growing the lettuce and tomato, baking the bread, and so on. So a lot of people are surprised when I tell them that I keep a jar of Hellmann's mayonnaise in our fridge. I love Hellmann's, and there's nothing wrong with using it as long as you recognize that homemade mayo is a completely different product. You make homemade mayonnaise in part because you can't buy mayonnaise of the quality you can make at home. But on a busy Saturday, if I've got a long To Do list, I'm not going to take an extra 5 minutes to make mayonnaise; I'll have a quick fried-egg sandwich, with Hellmann's, on soft sandwich bread, and a glass of milk. If you want to make your own mayo, go to town; instructions arehere and you will be rewarded with a_ superlative _egg sandwich._
_I break the yolk before adding it to the pan so that it intermingles with the white as it cooks, which of course is part of the unique flavor of eggs cooked this way._
1 tablespoon butter
2 eggs, cracked into a bowl, yolks poked once to break them
Salt and freshly ground black pepper to taste
Mayonnaise
2 pieces soft sandwich bread
Put a pan (preferably nonstick) over medium-low heat and allow it to get hot, about 5 minutes. Add the butter and allow it to melt completely. As the water cooks out of it, it will froth. When the frothing seems to be at its peak, pour in the eggs and give the pan an immediate shake to prevent the eggs from sticking. Season the eggs with salt and pepper and cook for 1 minute. Flip the eggs and cook until the white is just set, about 1 minute more.
Meanwhile, spread as much or as little mayonnaise on the sandwich bread as you wish. When the eggs are done, pour them out onto the bread, folding them over so that the eggs don't flop over the edges of the bread. Cover the eggs with the other piece of bread and eat with a glass of milk. I usually eat right there next to the stove, don't even use a plate.
# Gently Fried Eggs on Grits with Bacon and Toast
SERVES 4
_This is by far one of my favorite break fasts, taking advantage of a seriously underused food, grits, which is coarsely ground corn. (Corn kernels treated with an alkaline solution and ground are called hominy grits.) Grits alone are wonderful, but something magical happens when egg yolk runs through them._
_I include this recipe not only to contrast a gently fried egg with an egg fried in superhot oil, but also as an excuse to encourage people to eat a delicacy rarely served outside of America's southern states. Please don't use instant grits, and it's worth mail-ordering very good grits such as those offered by Anson Mills, McEwen & Sons, or Adluh Flour._
_Grits should be creamy. The amount of liquid needed varies depending on how long you cook the grits, so use your common sense; you can't really overcook them. Cheese pairs beautifully with grits as an enricher; cheddar is common, but a Nashville friend is devoted to smoked Gouda in grits. Makes me hungry just thinking about it._
1 quart/1 liter water
1 cup/170 grams grits
2 cups/480 milliliters milk
Salt and freshly ground pepper
6 tablespoons/90 grams butter, plus more for the toast
½ cup/60 grams shredded cheddar cheese (optional)
8 slices bacon
4 slices bread
4 eggs
Combine the water and grits in a medium saucepan, bring to a simmer over high heat, and reduce the heat to low. Cook, stirring frequently, to the consistency you prefer, loose or stiff. Most grits will be done in 30 to 40 minutes, but they can be cooked slowly for many hours—just add more liquid as needed. To finish the grits, add as much of the milk as needed and plenty of salt and pepper. Stir in 4 tablespoons/60 grams of the butter, or more if you wish, taste again, and add more salt if necessary. Continue cooking till the consistency is to your liking and stir in the cheese just before serving.
While the grits are finishing, fry the bacon in a pan (I like to start the bacon in ½ cup water, which begins the rendering process). When the bacon is done, drain it on paper towels. Toast the bread.
Melt the remaining 2 tablespoons/30 grams butter in a large, nonstick pan over medium heat. Crack the eggs into the pan. For eggs sunny-side up, cover the pan and cook just until set, a few minutes. If you want them over easy, don't cover the pan, and flip the eggs after a few minutes, before the white is set. Meanwhile, butter the toast and place a mound of grits on each plate.
Top each mound of grits with a fried egg and serve with toast and bacon.
# Quail Eggs Croque Madame
MAKES 12 CANAPÉS
_Quail eggs are perfect for show-offy canapés and can be prepared in numerous ways. My first taste of this delicacy was in culinary school, when chef-instructor Ron DeSantis rewarded us with a quail egg and caviar pizza after a good run at the St. Andrew's Café on graduation day (an idea he took outright, he told our class, from Jeremiah Tower, who made it popular at Stars in San Francisco). I had my next quail egg in the kitchen of the French Laundry, where Thomas Keller poached them before service, then reheated them in a little butter and garnish and served them on silver spoons as a fun one-bite course. When I found some in Cleveland I wrote about them on my blog, frying them and serving them on a spoon with some wilted arugula, bacon, and English muffin croutons. You could easily lose the arugula and poach the eggs and serve them as eggs Benedict canapés. Or you can go all out and make this take on the_ croque madame: _grilled ham and cheese with Mornay sauce and a fried egg on top, one of my favorite meals._
# FOR THE MORNAY SAUCE;
1 tablespoon butter
1 shallot, minced
Salt and freshly ground black pepper to taste
1½ tablespoons flour
1 cup/240 milliliters milk
A few gratings of fresh nutmeg
¼ cup/130 grams grated Gruyère or Emmentaler cheese
# FOR THE CROQUE MADAME:
12 slices mini sandwich bread or any thinly sliced bread cut into 2-inch/5-centimeter rounds
1 to 2 tablespoons Dijon mustard
6 ounces/170 grams ham, julienned, at room temperature
4 teaspoons/40 grams grated Gruyère or Emmentaler cheese
1 tablespoon butter
12 quail eggs
_Fleur de sel_ or coarse sea salt
Preheat the broiler.
To make the Mornay sauce, combine the butter and the shallot in a small saucepan over medium heat. Season the shallot with a good pinch of salt and a few grinds of pepper. Cook until the shallot is tender, a couple of minutes, then add the flour and cook, stirring, for another minute. Whisk in the milk and bring to a simmer to thicken the milk. Turn the heat to low and grate in some nutmeg, then stir in the cheese. Set aside.
To make the croque madame, lightly toast the bread rounds. Spread thinly with the Dijon. Top each with some of the ham and the cheese. Gently broil to melt the cheese.
Reheat the sauce over low heat. Heat the butter in one or two nonstick sauté pans over medium heat. When the cheese on the croutons is melted, crack the eggs into the pan(s) and cover.
Place the croutons on a platter and spoon a bit of sauce on each. When the eggs are cooked (sunny-side up takes just a few minutes), place one on each crouton and garnish with _fleur de sel_ or coarse salt. Serve immediately.
#
# Michael Pardus's Bibimbap
SERVES 4
_When I was reporting_ The Reach of a Chef, _I spent some time with my old skills instructor and friend Michael Pardus, who was by then teaching Asian cuisines. It was Korea Day when I cooked there, and for a kind of extra-credit dish he'd assigned us something called_ bibimbap _, which translates more or less as "thrown together with rice." A_ bibimbap _can be a clean-out-the-fridge dish, but Pardus used marinated skirt steak and vegetables and topped it with a fried egg, which is not uncommon. (As you can see from many of the recipes in the whole-egg section, there are very few dishes that aren't improved by the addition of an egg.)_
_It was the eggs that gave me a bit of a problem during service. They kept sticking and breaking. While I was charged with making only ten of them, it was a complete disaster._ Bibimbap _had handed me my ass. But it made me think about the sticking issue. I eventually figured out that sticking is not an issue in a steel pan if you get it smoking hot. This also results in the egg white becoming brown and crispy and flavorful, which I don't want in my sandwich, but I very much want in my_ bibimbap _. This simple, delicious preparation has become a staple meal in our house._
**A NOTE ABOUT THE BEEF:** _I think it's best to julienne the beef, but depending on the size and cut you use, you can simply slice it as thinly as possible; if you want to cut these slices in half, that's fine. If you use flank steak, cut across the grain; if using skirt, cut with the grain along the natural striations; in either case, you'll be cutting widthwise, not lengthwise._
# FOR THE MARINATED BEEF:
1 pound/450 grams beef skirt steak or flank steak, sliced as desired
2 scallions, finely sliced
2 garlic cloves, smashed aggressively with the flat side of a knife and roughly chopped
1 tablespoon peeled and grated fresh ginger
2 tablespoons soy sauce
1 teaspoon sugar
# FOR THE KOREAN RED PEPPER SAUCE:
2 tablespoons gochujang or other Korean red pepper paste
2 tablespoons water
2 teaspoons rice vinegar
1 teaspoon sugar
1 teaspoon fish sauce
1 teaspoon toasted sesame oil
# FOR THE BIBIMBAP:
1½ cups/280 grams jasmine rice, well rinsed
¼ cup/60 milliliters vegetable oil
½ cup/100 grams peeled, julienned carrot
½ cup/100 grams julienned daikon radish
½ cup/100 grams julienned celery
½ cup/100 grams julienned head lettuce
4 eggs, each cracked into its own ramekin
Combine the sliced beef and other marinade ingredients in a covered container or zipper-top plastic bag and marinate in the refrigerator for up to 48 hours (optimally you'll want to give the meat at least 1 hour to marinate, but if you just got home from work, don't sweat the marinating time).
In a small bowl, stir together all of the ingredients for the red pepper sauce and set aside.
Put the rice in a medium saucepan and cover it with about 1 inch/2.5 centimeters water. Put the pan over high heat and boil the rice till the water has reached the level of the rice and the steam holes are releasing bubbles (aka "fish eyes"). Cover the pot and put it on a back burner over low heat.
Put a wok over high heat. Put a large sauté pan with a cover over low heat. When the wok is smoking hot, add 2 tablespoons/30 milliliters of the vegetable oil and allow it to get smoking hot. Add the beef and stir-fry until cooked, 2 minutes or so. Add the red pepper sauce and stir to combine.
Turn the burner with the sauté pan to high.
Add the vegetables to the beef in the wok, turn the heat off, and stir to combine.
When the sauté pan is hot, add the remaining 2 tablespoons/30 milliliters vegetable oil and allow it to get almost smoking. Quickly pour in the eggs, one at a time, leaving space between them, and allow them to cook for 20 seconds or so, then cover the pan and reduce the heat to low.
Divide the rice among four large bowls. Top with the beef and vegetables. When the whites of the eggs have congealed but the yolks are still fluid, place an egg atop the beef and vegetables in each bowl. Serve immediately.
#
# Frisée Salad with Bacon Vinaigrette and Deep-Fried Egg
SERVES 4
_Yes, you can crack eggs into boiling oil and in about 15 or 20 seconds have a fabulous egg, with cooked whites and a runny yolk. It doesn't look or taste oily at all, and the high heat makes some of the white turn brown and crispy—if you're lucky, you may even get a crisp comet tail on it (here). This egg could go in any of the preceding recipes, but because you're deep-frying, you should pair it with something acidic. I like to put it on a bed of frisée lettuce, which is slightly bitter and chewy-crunchy, and for a little extra flavor, bacon bits, for a different take on the French lardon salad. Use the paler parts of the frisée, which are a little less bitter. Add a few drops of red wine vinegar and maybe a dot or three of balsamic to offset the pleasant bitterness of the greens and you're good to go. This makes an impressive opening course for a bigger dinner or the perfect light lunch. The bacon can be cooked well in advance of serving if you wish, but save the fat in the pan for coating the frisée._
3 ounces/85 grams sliced bacon (2 extra-thick slices)
Vegetable oil for deep-frying (2 to 3 cups/480 to 720 milliliters)
8 ounces/225 grams frisée lettuce
1 teaspoon red wine vinegar
Balsamic vinegar
Salt and freshly ground black pepper
4 eggs, each cracked into its own ramekin
In a medium sauté pan or cast-iron skillet, sauté the bacon till it's crispy.
While the bacon is cooking, get the oil heating in a medium high-sided saucepan or, better, a wok, the shape of which allows you to use less cooking oil.
Chop the crispy bacon into bits. Pour off all but about 1 tablespoon of the fat from the pan the bacon cooked in. Return the bacon to the pan. Toss the frisée in the pan to lightly coat the lettuce with the fat (it's fine if the pan is still hot), then divide among four plates. Scatter any bacon bits left in the pan on the salads.
Sprinkle about ¼ teaspoon of the red wine vinegar over each salad, or more to taste, and a few dots of balsamic around the edges of the plates. Season to taste with salt and pepper.
Line a plate with paper towels for the eggs. When the oil reaches 375°F/190°C (or when a chopstick bubbles vigorously when inserted into the oil, but before the oil begins to smoke), pour each egg into the oil and cook, stirring the oil gently; the eggs will sink, then float. Flip them if they'll let you, or spoon oil over their tops until the whites are set, 15 or 20 seconds total. Use a slotted spoon to remove them to the paper towels to drain for a moment (try to remove the eggs in the same order they were added to the oil). Sprinkle them with a little salt, then place each atop a salad and serve immediately.
#
JUST THE SOUND OF THIS PREPARATION, coddled egg, comforts. _Coddle_ means to treat tenderly, and the word may derive from _caudle_ , a warm drink for invalids. And so a coddled egg is treated as if it were a culinary invalid. Of course, it is a culinary superhero, but here we handle it with the utmost gentleness, protected on all sides from heat, the gentlest of heat, in a covered container. (There's some disagreement on just what coddled eggs are. Some culinary authors, including James Beard and Madeleine Kamman, write that a coddled egg is cooked in the shell for a short time. But others, such as myself, call this a soft-cooked egg. I believe we should use the term _coddle_ solely to describe an egg gently cooked out of the shell, in a covered vessel.) While coddled eggs are certainly a restorative nourishment, they should in no way be limited to invalids. Indeed, it is an uncommonly elegant preparation, delicious, and virtually foolproof.
Because it's relatively uncommon, I find it to be a wonderful breakfast for weekend guests. It's also something of a no-brainer—10 to 12 minutes in hot water does the trick.
The egg alone is fine, but it should be flavored somehow. When I was considering the many possible ways, refrigerator door open, scanning its contents for ideas, my eyes stopped on a small plastic container of butter flavored with black truffles. Butter and eggs, truffle and eggs—no more felicitous pairings on earth. And it would be so simple. Crack an egg into the coddler, add a dollop of truffle butter, cover it, and put it in a water bath, just until the white is set. Season with a little _fleur de sel_ or Maldon salt, re-cover, and serve with some toast. I like to make batons of toast that I can dip into the yolk.
I remembered that Thomas Keller's Manhattan restaurant Per Se serves a coddled egg. I wrote to one of its managers, fellow Cleveland native Michael Minnillo, to ask how they serve it. He wrote back, "Just a little truffle butter." I should have known! I also asked what they serve it in, as I'd asked Donna, photographer in charge of props, to find some nice coddlers. Michael emailed a link to their coddler, I forwarded it to Donna, and she, in the next room, shouted out immediately, "That's one of the coddlers I've ordered!" I guess our long association with Thomas Keller and his restaurants has left its mark.
While truffle butter is exquisite and easy, it's by no means the only flavoring you could use. A teaspoon of cream and some tarragon would be lovely. Several drops of excellent extra-virgin olive oil and Parmigiano-Reggiano is delicious. I'm not a fan of hot sauce on eggs, but a shot of sriracha would be great if your palate demands some heat.
Coddled eggs must be cooked in a water bath, 10 minutes for loose white, 12 to 15 minutes for a firmer egg. The time will vary depending on the thickness of the vessel, so determine doneness by looking at the egg. Buy coddlers if you want to emulate Martha Stewart, but you can also coddle eggs in a ramekin loosely covered with foil or even in demitasse coffee cups. That would be quite elegant, actually. Maybe that wedding-gift china will get a little more use around our house now that I've thought of this.
# Coddled Eggs with Truffle Butter
SERVES 4
4 eggs
4 teaspoons black truffle butter (or other flavorings as noted above)
_Fleur de sel_ , gray salt, or Maldon salt
Freshly ground pepper
Preheat the oven to 300°F/150°C.
Bring a pot of water to a boil—the amount will be determined by the size of your water-bath vessel and coddlers.
Crack an egg into each coddler, add a teaspoon of truffle butter, and cover. Put the coddlers in a baking dish or high-sided pan. Pour boiling water into the dish so that it reaches the level of the egg or even goes above it, taking care not to get any water in the coddlers.
Place the baking dish in the oven and coddle the eggs until they are the way you like them, 10 to 15 minutes. Serve immediately.
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SHIRRED EGGS ARE BAKED USING DIRECT heat, as opposed to coddled eggs, which use the gentle heat of water in the form of a water bath on the outside and steam trapped within the coddler. They're equally comforting and equally delicious, but the flavors take on a bit more complexity due to the high heat and added ingredients. I think a little cream or butter makes a sweet addition. Freshly grated Parmigiano-Reggiano is an excellent finish and, slightly browned beneath a broiler, contributes color as well as flavor.
If you want to add other elements, there's no end to what you might use. You could go traditional and line the ramekin with cooked spinach, sautéed shiitake mushrooms, roasted red bell pepper, or broiled eggplant. You could add ham or bacon or duck confit. Some minced shallot cooked in a little butter is always a delicious addition.
As far as technique goes, the egg cooks a little more evenly if you heat the ramekin in a microwave, on the stove top, or in the oven first. Bake the eggs in a 350°F/180°C oven just till they're set, about 10 minutes. You can also broil them, if you wish—a good strategy if you're topping the eggs with cheese.
The size of the shirring vessel is critical—it should be wide and shallow, not a conventional round ramekin. A standard 8-ounce/240-milliliter round ramekin can be used to shirr 1 egg, but 2 eggs would be best cooked in a water bath for triple the time to ensure good texture throughout (a shirr/coddle combination).
# Shirred Eggs Florentine for My Mom
SERVES 1 MOM AND 1 SON
_My mom loves shirred eggs with spinach, but she was having trouble perfecting the preparation, so this recipe is for her. You can cook the spinach either by blanching it and then shocking it in an ice bath, or by sautéing it in butter with some shallots (my preference). Either way, wring it out and chop it before adding it to the ramekins._
2 teaspoons butter
8 ounces/225 grams spinach, cooked, thoroughly squeezed dry, coarsely chopped
4 eggs
2 tablespoons heavy cream
Salt and freshly ground black pepper
Several gratings of Parmigiano-Reggiano
Good buttered toast
Preheat the oven to 350°F/180°C.
Put 1 teaspoon butter in each of two shallow ramekins or shirring dishes. Divide the cooked spinach between the two ramekins. Heat the ramekins in a microwave so that the spinach is hot but the ramekin is not too hot to hold, 30 to 40 seconds.
Crack 2 eggs into each ramekin, add 1 tablespoon cream to each, and season to taste with salt and pepper. Sprinkle some cheese over the eggs.
Put the ramekins in the oven and cook until the whites have fully congealed, 10 to 15 minutes. Serve with toast.
#### EGGS FLORENTINE
1. / _Be sure the spinach and ramekins are gently heated before adding the eggs._
2. / _Finish the eggs with grated cheese, pop them in the oven, and cook them until they are beautiful._
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POACHING AN EGG IN WATER IS INSTRUCTIVE in the ways of the egg. In particular, it provides a dramatic display of the different parts of the egg white. While an egg white comprises several different proteins that coagulate at different temperatures, there are two visible parts of the white, the loose part and the thicker, more cohesive part. Crack an egg onto a plate and see for yourself. Usually, the fresher the egg, the more of the thicker part of the white there will be, and the thicker it will be.
The poached egg is one of the simplest, most versatile of all egg preparations. The white cooks faster than the fat-rich yolk, and the gentle heat of the water keeps the white tender. It is also infinitely variable. Just check out the list of variations on the "Eggs Benedict" page in Wikipedia—more than twenty—and those are just the most common ones. On toast, on an artichoke heart, on polenta, on a crab cake, on a _lardon_ salad, in a soup—there's pretty much no dish that can't be improved by the addition of a poached egg. Thus the need to address the following:
# HOW TO MAKE PERFECT POACHED EGGS
When you crack an egg into hot water, much of the loose white separates from the thick white, leaving something resembling a rumpled bed sheet with a yolk attached in a haze of confetti water. Countless people advise adding vinegar to the poaching water to help prevent flyaway whites, as acid can help speed coagulation. Do not do this! I repeat, do _not_ add vinegar to the water in which you're poaching eggs. It does nothing more than force you to rinse off your sour egg. The trick, as I read in Harold McGee's invaluable book _On Food and Cooking_ , is to let the runny whites drain off by putting the egg in a slotted spoon before placing it in water (it works so well that I created my Badass Perforated Egg Spoon, a slotted spoon with an extra-deep bowl, just for this purpose—see the "Shop" page at ruhlman.com). This technique gives you a gorgeously smooth, oval poached egg without all those scraggly whites adhering to it. Simply crack an egg into a ramekin, pour it from the ramekin through a deep perforated spoon into another ramekin, then return it to the first ramekin and it's ready to go.
So, to poach an egg perfectly: Bring a pot of water to a simmer, then reduce the flame to low or remove the pot from the burner completely. Give the water a stir to keep the egg from sticking to the bottom (the stirring also helps make the eggs more shapely), and drop your drained egg into the gently swirling water. When the white has completely congealed, after about 90 seconds, use your perforated spoon to lift the egg out of the water, swooshing the water underneath the egg if necessary to help you lift it (if you see that the egg isn't quite done, just return it to the water until it is). Allow any excess water to drain off; you may need to tilt the egg so that any water collecting on the white can run off, or touch a paper towel to its surface. Serve immediately.
#### POACHING AN EGG
1. / _Strain the raw egg, allowing the loose white to drain off._
2. / _Add the egg to water that has simmered, first stirring the water to prevent the white from sticking._
3. / _Notice how few flyaway whites there are in the water._
4. / _Gently lift the poached egg out of the water._
5. / _Check to make sure the egg is done to your liking and allow it to drain on a paper towel._
6. / _A poached egg can be served on virtually any dish and improve it—a salad, a sandwich, a hamburger. Here I put it on warm shaved ham and toast, before adding hollandaise (below)._
# Eggs Benedict
SERVES 4
_There is simply no reason this restaurant/hotel buffet-line brunch favorite shouldn't be served more often at home. No matter how you prepare it, this is one of the most luxurious and wonderful dishes I know of. I love that the sauce is made of yolk and butter—that lovely egg-butter pairing doubled, egg and butter on top of another egg ("Waiter, I'd like an egg, with butter and more egg on top, please")._
_I grew up expecting the ham to be in the form of Canadian bacon, which is cured and smoked pork loin, and it's likely that became standard because the circumference of the pork loin matches the English muffin. That's why I include it here, but you can use any kind of ham or cured pork, such as prosciutto or bacon, or even pulled pork or duck confit—just don't call it eggs Benedict, which is English muffin, Canadian bacon, poached egg, and hollandaise sauce. I recommend Bays brand English muffins, which can be found in the refrigerated section of most supermarkets. I prefer the larger, crustier bottom half and simply serve the top half alongside each plate; or you can toast just two English muffins and serve half to each person._
4 English muffins, halved
4 (¼-inch/1-centimeter)slices Canadian bacon
4 eggs
1 cup/240 milliliters hollandaise sauce (either traditional or blender version, here)
Salt and pepper to taste
1 tablespoon chopped fresh chives or parsley (optional)
Toast the English muffin bottoms. If using a toaster oven, put the Canadian bacon on the English muffins and keep toasting to heat the bacon. If using a pop-up toaster, heat the Canadian bacon gently in a microwave—or, better yet, sauté in butter! Then toast the English muffin tops.
Meanwhile, bring a large pot of water to a boil and prepare the eggs by draining off the whites as described here. When ready to serve, drop the eggs into the water and remove the pot from the heat. Put an English muffin bottom on each of four plates and a piece of Canadian bacon atop each muffin. When the eggs are cooked (after about 90 seconds), remove them from the water, allow them to drain, and place one atop each Canadian bacon–topped muffin. Spoon the hollandaise over the eggs, garnish with chives, if desired, and serve immediately, with the English muffin tops alongside.
# Poached Duck Egg on Duck Confit Hash
SERVES 4
_A friend of a friend connected me with some farmers who raise ducks, and they often find themselves with more duck eggs on their hands than they can eat. I am the happy recipient of them and urge you to seek out duck eggs at your local farmers' market if possible. Some Whole Foods Markets also sell duck eggs. While all bird eggs taste more or less the same, duck eggs are larger than chicken eggs and a little more oblong, with very large yolks and considerably less white relative to the yolk, thus richer._
_I love to make a duck-egg omelet (filled with duck confit and mushrooms), scrambled duck egg on toast, or, my favorite, poached duck egg nestled on a hash of rich duck confit, potatoes, and onions. I confit duck legs in olive oil to keep on hand all winter long (see my site,ruhlman.com, for how to make this). Or you can prepare the duck legs specifically for this dish by cooking them in a covered baking dish with a sliced onion at 325°F/165°C for 90 minutes._
_A duck egg (left) has a considerably larger yolk than a hen egg (right)._
¼ cup/60 milliliters duck fat, confit fat, or vegetable oil
2 russet potatoes, peeled and cut into small dice (about 1½ pounds/680 grams)
Salt and freshly ground black pepper
1 medium onion, cut into small dice
4 confit or braised duck legs, bones discarded, skin and meat coarsely chopped
2 tablespoons Dijon mustard
4 duck eggs
1 teaspoon minced garlic (optional)
1 tablespoon minced fresh parsley (optional)
Heat the fat in a large skillet over high heat. Add the potato, tossing to coat it with the fat. Reduce the heat to medium-high and cook till the potatoes begin to turn brown, about 20 minutes. Season to taste with salt and pepper. Add the onion and chopped duck meat and skin and continue cooking until the onion begins to brown, another 10 minutes or so (the cooking time will be determined by how small you've cut the onion). Stir in the mustard and reduce the heat to low.
When the hash is done, bring the poaching water to a boil and poach the duck eggs (see here; the same procedure applies for duck eggs). Stir the garlic and parsley into the hash, if desired, and divide the hash among four plates. Rest a duck egg in the center of the hash. Season the egg with salt and pepper and serve.
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WATER IS A FABULOUS MEDIUM FOR POACHING eggs, but it's not the only fluid that works. Eggs can be poached in anything that's bubbling on your stove, whether soup, stew, or sauce.
When poaching in something you're going to eat, draining off the loose egg white is even more important; you don't want it messing up a nice sauce or broth (unless of course you're making egg drop soup, here). See here for how to strain an egg before poaching.
The best-known version of eggs in broth is the Italian _uova in brodetto,_ the _brodetto_ being the diminutive form of _brodo,_ "broth." In America you often see this in the form of eggs in tomato sauce, known as eggs in purgatory, and usually served on toast. In Israeli cuisine it's called _shakshuka,_ and no doubt eggs poached in tomato sauce has a place in many cuisines because, well, it's so damned good. I like to spice it up, puttanesca style, with garlic, red pepper flakes, olives, and capers, but if you want a last-minute dish or a quick breakfast, a good jarred tomato sauce works fine, because the egg makes everything better. (If you have time, sauté a little chopped onion before adding the jarred sauce; this improves the flavor well beyond the effort required.)
The proper French version of eggs poached in broth calls for a reduction sauce of half wine and half veal or beef stock. I really think it ought to be made with homemade stock, because boxed stock isn't very good, and when you reduce something not very good, the not very good part gets intensified. This stock-wine reduction is quickly thickened with _beurre manié_ (butter with flour kneaded into it) after the eggs are poached in it, then served over the egg.
# Oeufs en Meurette (Eggs Poached in Red Wine Sauce)
SERVES 4
_This is a French classic from Burgundy, eggs poached in red wine and stock, which then becomes the sauce for the eggs. There are a number of ways to do this. At a restaurant the eggs would be poached in the wine and chilled in an ice bath while the rest of the components are prepared; the sauce would be reduced and thickened, then reheated for service. For this home version, I do the reduction first, then cook the eggs, then thicken the sauce._
_Whenever you cook with wine, use a wine you would happily drink, but not a special-occasion wine. A ten-dollar bottle will do, and what you don't use in the sauce, you can drink with the eggs._
_Veal stock is a great ingredient to have in the kitchen and really makes this dish the best it can be (there's a recipe for it on my site,ruhlman.com), but you can use a rich homemade chicken stock with good results. These eggs are best served on toast, a piece of baguette, or the bottom half of an English muffin—something with crunch that will also soak up the sauce. Choose your own side components—bacon or_ lardons _are popular, of course, and would be my choice. Be sure to have them all ready before you poach the eggs._
1 shallot, finely diced
4 tablespoons/60 grams butter, at room temperature
½ teaspoon salt
2 teaspoons tomato paste
2 cups/480 milliliters dry red wine
2 cups/480 milliliters brown veal stock or homemade roasted chicken stock
2 teaspoons sugar
1 bay leaf
Freshly ground black pepper
3 tablespoons flour
4 slices bread
4 eggs
1 tablespoon minced fresh parsley
Combine the shallot and 1 tablespoon/15 grams of the butter in a medium saucepan over medium heat. When the butter is melted, add the salt. Cook, stirring, until the shallot is translucent. Add the tomato paste and cook for 30 seconds. Add the wine, stock, sugar, bay leaf, and a few grindings of pepper and simmer until the liquid has reduced by a little more than half, 5 to 10 minutes.
While it's reducing, in a small bowl combine the remaining 3 tablespoons/45 grams butter with the flour and mash and knead it with a spoon till the flour is completely mixed into the butter. Cover and refrigerate this _beurre manié_. Toast the bread and place one slice on each of four plates.
Crack the eggs into individual ramekins and strain them to get rid of any loose egg white (see here). When the wine and stock have reduced, add the eggs and cook them gently in the sauce. Spoon the sauce over the tops of the eggs if they're poking up through the sauce. When the whites are completely set (after about 90 seconds, but always judge by sight), remove each egg and place it on a slice of toasted bread.
Strain the reduced sauce into a clean pan over high heat and bring to a simmer, then reduce the heat to medium-high. Whisk in the _beurre manié_ to thicken the sauce. Spoon the sauce over the eggs and toasted bread and garnish with the parsley. Serve immediately with bacon or other sides.
# Eggs in Puttanesca Sauce with Angel Hair Pasta
SERVES 4
_Pasta puttanesca translates as whore's pasta, allegedly because it was hastily thrown together by Sicilian hookers between clients. But what it always designates is a spicy tomato sauce with lots of salty, umami-giving ingredients. It usually includes anchovies, but I'm using fish sauce here because it's one of my invaluable pantry items, always on hand. Add red pepper flakes, kalamata olives, capers, and eggs to make a fabulous pasta dish—and you'll have some sauce left over for other uses. This dish is usually served on a thick piece of crusty toasted bread, but here I'm pairing it with pasta instead._
_You can puree canned tomatoes in a blender, or just stick an immersion blender right in the can, but you should first pour off some of the tomato juice into the pan with the onions or you'll have a mess—trust me._
1 Spanish onion, cut into small dice
4 garlic cloves, smashed with the flat side of a knife and roughly chopped
1 tablespoon extra-virgin olive oil
1 teaspoon salt, plus more to taste
1 teaspoon red pepper flakes
1 cup/240 milliliters dry red wine
1 (28-ounce/794-gram) can whole peeled tomatoes, pureed, or 10 fresh Roma tomatoes, broiled for 15 minutes and pureed
1 bay leaf or 2 teaspoons dried oregano (or both)
½ tablespoon fish sauce or 4 anchovies, roughly chopped
½ cup/90 grams pitted, chopped kalamata olives
2 tablespoons capers
4 eggs
1 pound/450 grams angel hair pasta or thin spaghetti, cooked al dente and tossed with extra-virgin olive oil or butter, kept warm in a covered pot
In a large sauté pan or skillet over medium-high heat, sauté the onion and garlic in the olive oil, adding the salt as you do. Stir until the onion and garlic are tender and translucent, then add the red pepper flakes and stir to cook them and coat them with the oil.
Add the wine and bring it to a simmer. Add the pureed tomatoes, the bay leaf and/or oregano, and bring it back to a simmer. Reduce the heat to low and cook the sauce till it's nice and thick, about 1 hour. The sauce can be prepared in advance up to this point, allowed to cool, and stored in a covered container in the refrigerator for up to 3 days.
Remove and discard the bay leaf and add the fish sauce (or anchovies), olives, and capers. If the sauce was refrigerated, bring it to a full simmer over medium heat, then turn the heat to low. Lower each egg into the sauce with a ladle, making a small well in the sauce with the ladle to contain it. Cover the pan and cook the eggs until the whites are set, 3 to 6 minutes.
Divide the warm pasta among four serving dishes. Spoon the sauce over the pasta, topping each portion with an egg and finishing each dish with more sauce as needed. Serve immediately.
#### EGGS IN PUTTANESCA SAUCE
1. / _Use a ladle to put the eggs into the sauce, pressing the bowl of the ladle down into the sauce to create a divot that will contain them._
2. / _Cook covered just until the whites are set, checking frequently to avoid overcooking._
3. / _Top the pasta with sauce, followed by an egg._
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THIS IS THE SECOND OF TWO EGG TECHNIQUES I picked up on Twitter. Why would you want to poach an egg in a plastic bag? Because you can. Because it keeps a compact shape. Because you capture the whole egg and don't lose the loose part of the egg. And because you can flavor the egg with oil (it's important to oil the bag first so that the egg doesn't stick to it). You might also do it to serve numerous poached eggs at once; pop them straight into an ice bath and reheat as needed. (If you're concerned about chemicals from the plastic getting into the food, use Glad plastic bags, which don't contain BPAs or phthalates.)
# Poached-in-a-Bag Egg Sandwich with Caramelized Onion and Roasted Red Pepper
SERVES 4
_This is simply an excuse to have an egg sandwich. The caramelized onion and roasted red pepper make it especially delicious. Or create your own variation: bacon, Parmigiano-Reggiano, and parsley (alla carbonara); butter and sautéed spinach (Florentine); truffle butter and mushrooms. Or you could just spread some mayo on the muffin and be done with it. Serving it on an English muffin is not just a tasty way to go, it's also practical: the holey crumb helps catch the yolk when you bite into it. But you definitely want to eat this over a plate; use the muffin to soak up any fallen yolk._
4 teaspoons extra-virgin olive oil
4 eggs
1 teaspoon butter, plus more for the English muffins
½ onion, thinly sliced
1 red bell pepper, charred black over a flame or under a broiler, then peeled and diced
Salt and freshly ground black pepper
¼ teaspoon red or white wine vinegar
4 English muffins
Butter
If you wish to cook your eggs ahead of time, bring a medium pot of water to a simmer over high heat, then reduce the heat so that water is gently simmering; prepare an ice bath (half ice, half water). Put 1 teaspoon olive oil into each of four small plastic bags, then crack an egg into each bag. Twist each bag closed and secure it with a twist tie. Lower the bags into the simmering water and cook for 4 minutes. Transfer the bags to the ice bath and put the whole thing in the refrigerator until you're ready to serve. At that point, return the bags to simmering water for 90 seconds to reheat before serving.
When you're ready to prepare the sandwiches, heat the butter over medium heat and sauté the onion gently till nicely browned and caramelized, 15 to 20 minutes. Add the red bell pepper to reheat, season to taste with salt and pepper, and then add the vinegar.
Toast and butter the English muffins.
If you haven't made the eggs ahead of time, cook them now as described above. Divide the onion-pepper mixture among the four muffin bottoms. Place a cooked egg on each—they will slip easily out of their oiled bags. Season the eggs with salt and pepper and top with the muffin tops. Serve immediately.
This branch of the egg cookery flowchart features recipes using the whole egg, out of its shell, and blended. The egg becomes a different entity altogether when the fatty, rich yolk is interspersed with the lean and powerful proteins of the white. A beautiful entity. Like all beautiful things, though, the blended egg can be used carelessly (when it's not fully mixed, for instance) or even badly abused, as scrambled eggs all too often are. But the blended egg can also be transformed into some of the most ethereal creations on earth.
When the yolk and white join forces, the egg becomes a dynamo, a powerhouse ingredient. The blended egg is a remarkable tool when it's whipped to a froth, leavening cakes and quick breads. When combined with liquids and cooked gently, it creates textural heaven in the form of custard. It helps to bind bread crumbs to the floured surfaces of meat and gives a glossy golden hue to breads and crusts when it's brushed on the still-baking goods. And it is an enriching ingredient in a whole continuum of flour-based food, from bread to pasta, cookies to crêpes.
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# **HOW TO MAKE PERFECT SCRAMBLED EGGS**
SCRAMBLED EGGS—A NO-BRAINER, RIGHT? Only if you think first. This is probably the most badly abused technique in the entire repertoire of the egg, and as scrambled is one of the most common forms of cooked eggs, the abuse is especially unfortunate. They are abused by being overcooked, way overcooked, cooked dry. The key to perfect scrambled eggs is gentle heat. In an earlier book, _Ruhlman's Twenty_ , I recommended scrambling eggs in what amounts to a double boiler, so that the eggs never go above 212°F/100°C. You can overcook them in a double boiler, but the cooking happens much more slowly, so you have much more control. Once you know how to properly scramble eggs, or rather what properly scrambled eggs taste like—light, silky, the delicate curds essentially sauced by warm but uncongealed egg—you can then move, if you feel comfortable doing so, to a pan over direct heat. You'll need to use very low heat and to pay careful attention the entire time, moderating the heat by lifting the pan above the burner to ensure you don't cook the eggs too quickly or too much.
The best way to learn how to properly scramble eggs yourself—eggs that make people swoon; that make people think to themselves, "These are the best scrambled eggs I've ever had!"; that make people exclaim, "I never knew scrambled eggs could be so amazing!"—is to cook them in a saucier or other round pan or in a double-boiler insert over simmering water. I use a stainless-steel saucier pan that floats in a pot of simmering water. If you have a nonstick pan, that's best, because egg will stick to most surfaces at this temperature and the pan will be a pain to scrub clean.
Thoroughly blend your eggs, two per person. By thoroughly blended I mean _absolute uniformity_. Use a whisk, a handheld immersion blender, or a countertop blender and blend well. You don't want them too frothy—you're not making a cake—but you don't want to be able to spot any white not mixed fully into the yellow. If you're using an electric appliance to mix the eggs, the bubbles and froth will go away as you cook the eggs.
You can give them a pinch of salt at this point if you wish so that it dissolves, or wait to the end if you want the salt flavor to be more distinctive.
And that's it for preparation: a pot of water simmering on the stove, a metal cooking vessel (you can use a Pyrex bowl, but heat transfer will be very slow), and eggs blended to uniformity. If you want additional flavors—seasonings such as cayenne or black pepper or curry; fats such as goat or cheddar cheese, truffle butter, olive oil; aromatic or tasty herbs or vegetables such as chives, tarragon, shallot, mushroom—have these ready, but these scrambled eggs you're about to make will be fabulous with just the eggs cooked in some butter, with salt and toast. I like to pare down to essentials, get those essentials right, and then build from there.
Float your cooking vessel in the simmering water and get your bread toasting; give the eggs a multiple-finger pinch of salt and another few whisks or a pulse of the blender. Put a couple teaspoons of butter into the floating pan as it starts to get hot from the simmering water below. When it's melted, pour in the eggs and, using a rubber spatula, begin to cook the eggs. Stir plenty and gently in the beginning as the cool eggs and the hot pan adjust to one another, 30 seconds or so. Then stir incrementally, allowing the first curds to form. Lift and fold them into the fluid egg. Now is when you might want to add cheese or sautéed mushrooms or shallot (other flavors such as fresh herbs or truffle might go later), but for now let's focus on the eggs, curdling in the pan.
Continue to fold and stir, and as you see steam rising and the whole mixture is beginning to get hot, pay close attention. Ask someone to butter the toast and to make sure the coffee is hot and ready or the Champagne is uncorked.
Continuously, more or less, stir and fold the eggs until about two-thirds of the eggs are stiff and one-third is viscous but fluid. Remove the pan from the water and continue to stir until the curds appear to be sauced by the fluid part of the eggs. Now they are ready to be served. Remember that just because you took the pan out of the water doesn't mean the eggs have stopped cooking—they're still cooking in the residual heat of the pan. So hie them hence to warm plates with buttered toast. Finish with chives, a pat of butter, _fleur de sel,_ freshly ground pepper, whatever, but be sure to do this: rejoice at the perfect scrambled eggs you've just prepared. Once you've experienced delicate egg curds lightly sauced by their more fluid self, replicate the effect using direct heat, preferably a nonstick pan on a low flame or burner, lifting the pan above the flame or burner as needed to regulate the heat. Your life will be immeasurably better.
Now begin to experiment with flavors. Use flavors that go well with eggs: mushroom, onion and its brethren (chives, scallion, shallot), or freshly shaved white truffle (I wish!). Donna loves to throw diced or shredded cheese in there, anything from mild mozzarella to acidic goat cheese, rich cheddar or smoked Gouda. I know of no cheese that doesn't work, but the better the cheese, the better the eggs. All of that is up to your taste now that you've perfected the scrambled egg.
# Scrambled Eggs with Fines Herbes
SERVES 4
_My favorite fresh herb mixture is_ fines herbes, _equal parts parsley, tarragon, chervil, and chives. I grow all of these soft, fragrant, volatile herbs, but the chervil doesn't survive long into the summer. Chervil has a lovely shape and a delicate tarragon flavor, so if I'm out, I use more tarragon—indeed, you might want to use only tarragon. But I like the combination of the tarragon and chervil with the oniony chive and floral flat-leaf parsley for its complexity and aroma when it lands on the hot eggs (you'd lose the freshness if you cooked them with the eggs; though if you want, you can stir half of the herbs into the eggs just before serving). I highly recommend serving this in the late morning with excellent buttered toast and a glass of Champagne._
1 tablespoon butter
4 eggs, whipped to a uniform pale yellow
¼ to ½ teaspoon salt
1 teaspoon _each_ minced fresh flat-leaf parsley, tarragon, chervil, and chives, combined
Bring a large pot of water to a simmer over medium-high heat. Meanwhile, ready everything you intend to serve with the eggs (toast, coffee, steamed asparagus, Champagne, up to you).
When the water is simmering, float a high-sided pan in the water and add the butter to the pan. When the pan is hot and the butter melted, add the eggs and the salt and stir more or less continuously until they are perfectly cooked (see the instructions starting here).
Divide the eggs among four plates. Sprinkle 1 teaspoon of the _fines herbes_ mixture over each portion and serve immediately. (Though they're good chilled, scrambled eggs are better eaten hot rather than lukewarm.)
# The Omelet
THE WORD _OMELET_ ORIGINALLY DERIVES from the Latin for "little plate," and omelets are usually made individually. You quickly cook one or two eggs while stirring rapidly and continuously to make the curds very fine, then stop the stirring to let the eggs set in the pan. When the omelet is just barely cooked, you grip the handle of the pan, palm up, and roll the egg from the handle side of the pan out of the pan and over the opposite edge in, one hopes, a lovely long oval of delicately pale, perfectly smooth, uniformly yellow egg. It takes practice—mistakes are delicious and successes are high-five-worthy.
The omelet becomes more visually appealing if you put some soft butter on top to give it a glossy shine and, if you wish, some minced chives, but really, an omelet is simply an elegant, shapely form of the scrambled egg.
We've come to think of omelets in America as always being stuffed with something—mushrooms and bell peppers and ham and cheese—so that there's more stuffing than egg. People debate whether to add cream or water or even olive oil, and you can, but why? Instead, it's worth pausing to appreciate the simple beauty of the omelet. All egg, enriched with a little butter and nothing more. If you have access to very fresh eggs from a friend or farmer near you and some delicious butter, an omelet with a glass of wine and maybe some charcuterie makes an excellent light meal. Go read Elizabeth David's _An Omelette and a Glass of Wine_ if you don't believe me.
Before we developed really good nonstick pans, cooks relied on creating a nonstick surface on their steel pans and reserved them only for omelets, allowing no soap anywhere near them. But I highly recommend that home cooks use a good, lovingly cared for nonstick pan for omelet making. They really do take the risk of sticking out of the equation, not to mention the omelet-pan fanaticism you'll find among traditional chefs.
Because of the omelet's simplicity, errors are glaring. Many restaurant chefs will ask an applicant to make an omelet, because omelet technique tells a chef many things about the finesse and skills of a young cook.
# HOW TO MAKE A PERFECT OMELET
Again, I'd like to reiterate that everyone should first make a plain omelet—two eggs, a pat of butter, a pinch of salt—to appreciate what an omelet is. I often add cheese or mushrooms to make the omelet more interesting and fun to eat. But it's important to first understand the foundation those garnishes are enhancing.
To make an omelet, crack the eggs into a bowl and whisk or blend till they are uniformly combined and no clear white remains floating on the surface. Give it a three-finger pinch of salt and stir in the salt.
Put a nonstick pan over medium heat and let the pan get hot, a couple minutes or so, depending on your stovetop. Have a heatproof rubber spatula ready, and warm a plate in the microwave. Add a pat of butter (about 1 tablespoon) to the pan. It should melt and bubble immediately but not brown. After the butter has melted and coats the bottom of the pan, pour in the eggs.
Shake the pan back and forth while stirring the eggs with the spatula. Stirring continuously will give you a finely textured curd. After about 30 seconds of stirring, stop and let the omelet continue cooking until there remains the thinnest liquid film on top, another 60 seconds or so. Remove it from the heat and allow it to finish cooking just sitting there in the pan (after all, you're not a line cook getting thirty-nine orders at once). With the pan handle at three o'clock, and your warm plate at nine o'clock (reverse this if you're a lefty), grip the handle from underneath and tilt the pan toward the plate. Using the spatula, encourage the front end of the omelet to slide out (you may need to give the pan a rap on a cutting board to loosen the egg, then nudge it from underneath to get it sliding) and roll the omelet over itself and onto a plate. The heat from the pan should finish cooking the omelet; it should be moist but you shouldn't have sauce on your plate. If it's not perfectly shaped, use your hands (which you've been washing continually, what with being in the kitchen, cracking eggs, etc.) to make it pretty and uniform. It's fairly pliable at this point, so you can tuck freely if needed. Run a little soft butter over the top so that it melts and gives the omelet a nice shine. Finish with some _fleur de sel_ or Maldon salt if you have it, and some chives always look nice.
Eat this immediately.
#### OMELET
1. / _When the pan is hot and the butter is melted, your pan is ready for the eggs._
2. / _Swirling both the pan and the spatula, mix the eggs in the pan to create a very fine curd on the bottom while heating the eggs throughout._
3. / _Stop stirring and allow the eggs to set. When the omelet has a very thin film of loose egg on top, it's ready to be plated._
4. / _Roll and slide the omelet onto a plate. Remember that you will be able to shape it after it's out of the pan._
5. / _Notice the pale uniform color of the omelet, with no browning._
6. / _Shape the omelet as you wish and finish with a pat of butter._
# Omelet with Creamy Morel Mushrooms
SERVES 2
_An omelet composed solely of egg and brushed with butter is great on its own, but to cut into an omelet and see gooey cheese run out is a pleasure. Omelets do make a lovely package for sautéed mushrooms. So that's what I'm offering here, a creamy mixture of morel mushrooms encased in delicately set egg. Of course any kind of mushroom will work—button mushrooms, shiitakes, or other wild mushrooms (though I like to sear buttons and portobellos to give them color and flavor). It makes the cook's work easier if the filling is very hot, so that it will contribute to the cooking of the center of the omelet when added._
_If you live where morels grow and can harvest your own or have a source for good fresh morels, lucky you! I don't, but morels dry very well, and several online sources sell them (our fresh morels came from a wonderful Michigan company called Earthy Delights). Dried morels should be reconstituted according to the instructions on the package, then halved and used as instructed here. Morels come up when ramps do, so if this is an option, use them instead of the shallot; thinly slice the whites of the ramps and sauté them with the mushrooms, then mince the greens and add them to the sauce at the last minute._
2 tablespoons butter, plus more for serving
1 tablespoon minced shallot (or 6 ramps, prepared as described above)
Salt
16 fresh morels, halved, or ½ ounce/15 grams dried morels, reconstituted in water
Freshly ground black pepper
½ cup/120 milliliters heavy cream
4 eggs, thoroughly blended
1 teaspoon minced fresh chives (optional)
Melt 1 tablespoon of the butter in a small sauté pan over medium heat and add the shallot (or whites of ramps) and a healthy pinch of salt (¼ teaspoon if you must measure). Cook the shallots in the butter till they're tender. Add the morels and stir to heat and coat with butter. Grind some pepper over the mushrooms. Add the cream and bring to a simmer, then reduce the heat and cook until the cream has thickened and coats the morels. (Cream will break if you overcook it, so don't.) Remove the pan from the heat.
In a medium sauté pan, melt the remaining 1 tablespoon butter over medium heat. Add the eggs and cook the omelet as instructed here. When the omelet is set, and there is a thin film of liquidy egg on top, scatter the hot mushrooms down the center of the omelet. Roll the omelet out of the pan and onto a warm plate. Slide a little butter over the top if you wish, garnish with chives if desired, and add more salt and pepper to taste. Cut the omelet in half crosswise and transfer one half to another warm plate. Serve immediately.
_Two equally fine ways to serve an omelet: with the garnish rolled up inside (top and bottom left) or spooned over the top (right)._
# Potato, Onion, and Cheese Frittata
SERVES 4
_This was the first dish I learned how to make on my own, when I was in the fifth grade. As the only child of working parents, I often made it as an after-school meal, to tide me over until they got home and we could eat a late dinner together. I still remember the little nonstick pan I cooked it in. In the beginning I simply used small-diced potatoes, cooked till they were soft. I poured in whisked eggs, fried the mixture over medium heat till it was almost set, inverted it onto a plate, and slipped it back into the pan to finish it. The top was mottled brown, the potatoes and eggs tender, and I'd eat it lying on the carpeting of our den watching_ Gilligan's Island _. It was simple, delicious, and nourishing._
_Now I make enough for the whole family, in a larger pan, so inverting is more difficult. You can cover it to set the top, invert it onto a plate, and slide it back into the pan; but it's easiest to finish it beneath a broiler just until the eggs are set. However you finish it, invert it onto a cutting board or plate before serving so that you present the mottled brown bottom surface._
_A frittata is simply an Italian omelet, and you can use any additional ingredients you wish. I'm making this a little more complex than my fifth-grade version, adding onion to the browned potato and cheddar cheese to the egg. When I tested this recipe, I gave a slice to my son, James, who was cutting up an avocado. We put some avocado on top and it was fabulous. The frittata is infinitely variable._
1 small potato, peeled and cut into small dice (about 1 cup/225 grams)
2 tablespoons extra-virgin olive oil
Salt
½ onion, cut into small dice (about ½ cup/100 grams)
6 eggs, thoroughly blended
½ cup/60 grams shredded cheddar cheese
Freshly ground black pepper
2 avocados, peeled, pitted, and diced (optional)
Preheat the broiler.
In a medium nonstick fry pan, combine the potatoes and olive oil over medium-high heat and stir or toss them in the pan to coat the potatoes with oil. Add a three-finger pinch of salt, just to coat the surface. When the potatoes are lightly browned, add the onions, salt to coat the onions, and continue to cook until the onions are tender, stirring or tossing the potato and onion.
Place the eggs in a medium bowl and add the cheese, along with ½ teaspoon salt and several grinds of pepper, and stir to combine and disperse the cheese. Pour the egg mixture over the potatoes and onions and reduce the heat to medium, swirling the pan so that the eggs even out. Cook until the edges are set, a couple of minutes depending on the heat level, checking to make sure that the eggs aren't sticking. Place the pan underneath the broiler until the eggs are just set, a minute or two depending on your broiler. When the top is set, invert the frittata onto a cutting board, cover with the diced avocado, if using, and cut into wedges. Serve.
#### FRITTATA
1. / _The frittata is one of the easiest egg preparations. Simply pour the eggs over the cooked ingredients—here, potato and onion—and let them cook._
2. / _Cheese makes egg and potato even better._
3. / _Run a spatula around the edge to loosen the frittata. Either invert it onto a plate and slide it back into the pan or, my preference, place the pan beneath the broiler until the egg is set and then invert onto a cutting board._
4. / _Garnish as you wish, with herbs, avocado, or more cheese, then slice and serve._
#
# Daniel Patterson's Poached Omelet
SERVES 1
_Eggs are so versatile, and so inexpensive, they're fun to experiment with. (I put some in the hot ashes of our fireplace, a technique from the Middle Ages; they were brown, overcooked, and sulfuric—but worth a try!)_
_The talented Daniel Patterson, chef-owner of the restaurant Coi in San Francisco, wrote in the_ New York Times _about an egg experiment he did that seems so obvious I'm amazed it's not simply common practice. It's been widely written about since the January 2006 article appeared; given the power of social media regarding food information, word spread rapidly._
_"Then one day," Patterson writes, tired of eating the same old scrambled eggs, "I had an idea for a new way to cook eggs—probably not new to the world, but new to me. It was a little humbling. After more than 20 years of cooking in restaurants, I had clearly failed to master basic egg cookery. I took a moment to ponder this, but then my curiosity got the better of me: what would happen, I wondered, if I beat the eggs before putting them in the water? I expected that they would act much as the intact eggs did and bind quickly, but I did not expect them to set into the lightest, most delicate scrambled eggs imaginable. I became so excited that I immediately reverted to my old ways, eating them standing up in the kitchen."_
_The technique is probably the easiest and most foolproof method devised for cooking eggs to such delicate consistency. Simply give boiling water a swirl, pour the blended egg into it, and strain after 10 or 20 seconds. (Patterson and others, including myself, recommend first straining off the loose white, which you should definitely do for whole poached eggs, as well—seehere.) Blended poached eggs are incomparably fluffy and light—the main pleasure here in fact is their ethereal texture—so I think a flavorful garnish is important. I love these with some good olive oil and truffle salt, or olive oil and Parmigiano-Reggiano. But salt and pepper and some fresh herbs are fine, too. Marlene Newell, my chief recipe tester, longed to toast an English muffin with some cheddar and top it with the eggs._
_While most of the water is expelled, they'll usually trap some water, so do your best to strain off as much water as possible. I tried different methods but couldn't improve on Patterson's, so this is taken straight from him. It's brilliant._
Salt
2 eggs, strained of the loose white and blended to uniformity
1 tablespoon extra-virgin olive oil
Several gratings of Parmigiano-Reggiano
Bring a 2-quart pot of pleasantly salted water (a teaspoon or two) to a low boil. Give the water a swirl with a spoon to create enough current to keep the egg whites from sinking to the bottom and sticking there. Pour in the eggs. After the eggs have risen to the surface and are cooked, 20 to 30 seconds, hold back the eggs with a spoon and pour off as much water as possible, then dump the eggs into a strainer to get rid of any remaining water. Serve the eggs in a bowl, with the olive oil and grated Parmigiano and more salt if needed.
# Traditional Egg Flower Soup
SERVES 4
_Here blended eggs are poached in a delicious stock and streamed into the hot liquid, rather than dumped in as with Chef Patterson's poached eggs. I include this recipe because it is so delicious, simple, and nourishing. But it's paramount that you use a chicken stock that is delicious on its own (meaning you've made it yourself, stolen it from a friend or family member who has made it, or bought it frozen from a very good source). Egg can be swirled into any delicious broth and served as a soup, but here I'd like to offer a traditional egg "drop" soup in the Chinese fashion, a dish that ought to be more common around the world._
_As Eileen Yin-Fei Lo notes in her fine book_ The Chinese Kitchen _, egg drop soup in America has become dull, unnecessarily thickened with cornstarch, and often made with badly cooked, lumpy eggs. "In China, this soup,_ don far tong _," she writes, "translates as 'egg flower soup.' " How much more lovely is that than "drop"? Our name is the literary equivalent of our actual debasing. So here it is, back to the Chinese, with fresh delicious stock, eggs, and scallion for garnish. Make as much or as little as you wish, planning on 1 egg per 1 cup broth per person. The key to getting the right texture of the eggs, the flowerlike delicacy, is in the gentle heat of the liquid and gentle whisking._
1 quart/1 liter delicious chicken broth
Salt
2 teaspoons fish sauce (optional)
4 eggs, beaten
2 scallions, sliced feather-thin on a bias, including the white and some green
Freshly ground black pepper
Bring the broth to a simmer in a large saucepan over medium heat. Taste it. Add salt, stir it, and taste again. When it is nicely seasoned, add the fish sauce, if using, and bring the soup to a boil. Turn the heat immediately to low (or remove it from the heat altogether if you're using an electric burner).
Give the broth a vigorous circular stirring with a large fork, then pour the eggs in a stream into the broth, whisking gently as you do to make a flower of the eggs.
Ladle into bowls and garnish each serving with scallions and freshly ground pepper. Serve.
**NOTE:** _For a more powerfully flavored soup, sauté scallion, minced ginger, and garlic in a little oil. Add the stock and cook gently—scarcely a bubble—for 20 minutes. Strain the infused stock into a clean pan and proceed with the above recipe._
#
# Marlene's Bacon and Sausage Breakfast Strata
SERVES 6 TO 8
_I didn't even know what a strata was—a layered casserole in the annals of American cuisine—until my chief recipe tester, Marlene Newell, a Canadian, pleaded to include one. I said give me a killer strata and it's in. This one rocks. It's basically a bacon-sausage-onion bread pudding (or dressing—it would go great with a big roasted bird!). I like it as a hearty break fast when you have lots of mouths to feed. It's a go-to for weekend brunch when guests are in town, because it can be made the night before with the leftover bread from dinner, and can sit in the fridge all night soaking up the custardy goodness, then baked in the morning straight from the fridge while someone mixes the Bloodies._
_Like a quiche, it's excellent cold as well, and it reheats beautifully, so it could be made a day or two in advance and reheated as needed._
_A strata is in effect a savory bread pudding. A custard is poured over bread and other tasty ingredients and then baked until the custard is set._
8 ounces/240 grams sliced bacon
8 ounces/240 grams bulk breakfast sausage
½ Spanish onion, cut into medium dice
6 eggs
1 cup/240 milliliters milk
1 cup/240 milliliters heavy cream
1 teaspoon mustard powder
1 teaspoon salt
1 teaspoon freshly ground black pepper
¼ teaspoon cayenne pepper
7 to 9 slices day-old French or Italian bread, cut into ½-inch slices or cubes, toasted or allowed to sit out for a day or two (enough for two layers in your dish)
3 cups/300 grams grated hard cheese (such as cheddar, Gruyère, or Emmentaler)
Preheat the oven to 325°F/165°C. Butter a 9-by-13-inch/23-by-33-centimeter baking dish and set aside.
Place the bacon slices in a large skillet, cover them with water, and put the skillet over high heat. When the water has cooked off, reduce the heat to medium and cook the bacon until nicely browned and crisp. Remove to a plate lined with paper towels. Add the sausage to the bacon fat in the pan. Cook the sausage, breaking it up as it cooks. Meanwhile, chop the bacon coarsely. When the sausage is nearly cooked through, add the onion and continue to cook until the onion is tender, another 5 minutes or so. Use a slotted spoon to transfer the sausage-onion mixture to the plate with the bacon.
Combine the eggs, milk, cream, mustard powder, salt, pepper, and cayenne in a large bowl or glass measuring cup. Whisk or blend with a hand blender until the custard is uniform.
Put half of the bread cubes in an even layer in the prepared baking dish. Sprinkle half of the bacon-sausage-onion mixture evenly across the bread. Sprinkle half of the cheese evenly over this. Repeat the layers. Pour the custard over them. The dish can be baked right away or covered and refrigerated for up to a day. Bake until the center is hot and the top is invitingly browned, 45 minutes to 1 hour.
Cut and serve hot. Alternatively, let it cool, then chill it in the refrigerator and eat it cold, or cut the chilled strata into serving-size pieces and reheat in a 350°F/180°C oven to serve.
#
BLENDED EGG IS USEFUL IN MANY WAYS when applied to the exterior of food. There are numerous variations, including batter for onion rings and tempura. (I will get to the latter, as I am a great fan of deep-fried foods.)
But here I am discussing something widely referred to as egg wash. Egg wash is nothing more than blended egg (often with water or milk added to thin it out). If you don't want a baked product to look dry, adding some egg wash to the surface is usually a good solution. Brushing beaten egg on dough (see the eggy challah recipe, here) results in an appealing shiny and browned surface. The protein in the white becomes shiny when cooked; the water in the eggs also promotes the gelatinization of starch, which enhances the shininess. The glucose in the egg browns. Water, milk, or cream can be added to the wash to thin it out and to contribute slightly to browning. Yolk-only washes result in a more deeply browned finish, but they can burn in high-heat baking. White-only washes result in a lighter, glossy finish and are often used on delicate pastries, such as tuiles.
Egg wash also creates a wet, tacky surface to which dry things will stick. My parents used to love chicken coated in crushed cornflakes (I couldn't stand it). I prefer panko, which are coarse Japanese-style bread crumbs that yield a very crispy crust. But there's no reason any wheat, corn, or rice products won't work.
Dry sticks to wet and wet sticks to dry. This is the nature of things. So when you're breading food, no matter what you're using, it's best to have a completely dry surface that the egg wash can adhere to. Flour is typical, but you could try fine cornmeal or even, say, pulverized pretzels. One thing I don't like to do with egg wash is to season it. Sure, you could put rosemary or pepper in it, but I prefer to apply dry seasonings to whatever it is I'm coating. I could countenance adding a liquid seasoning, such as sriracha or some other hot sauce. That's up to you. Here we're doing what's referred to in culinary school as "standard breading procedure." Flour, followed by egg wash, followed by bread crumbs.
# Breaded Chicken Cutlets with Dijon and Thyme
SERVES 4
_This recipe uses my least favorite cut of meat, which also happens to be the most popular cut of meat in the American home kitchen. It's not a bad cut, but it is so lean that it is easily overcooked and has almost no flavor—it's the skim milk of the meat world. America's laziness and pervasive fear of fat has made this its go-to cut of meat for all the wrong reasons._
_This is the only way I use the boneless, skinless chicken breast, and it's a sure-fire way to make it delicious and fun to eat, with its crispy exterior and tender juicy interior. I pound the breasts between sheets of plastic wrap to make the thickest part tender and, importantly, to give them a more uniform thickness so that the thin, tapered end doesn't overcook._
_With a barrier of crumbs attached to the exterior by the miraculous egg, the meat stays tender and juicy, and it is easy to cook perfectly every time. There are any number of ways to flour the meat and dredge it in bread crumbs. I put flour and any seasonings (here, plenty of pepper) in a plastic bag to coat the chicken Shake 'n Bake style. I slip them through the beaten egg in a broad bowl and use a 9-by-13-inch/23-by-33-centimeter baking dish for dredging in crumbs._
_Chief recipe tester Marlene Newell wanted to liven up the breading by putting Dijon and fresh thyme leaves in the egg wash. I liked the idea of additional flavoring but feared that plenty of good Dijon would be left in the egg wash. I suggested she instead try applying the mustard and thyme directly to the meat before breading it. She did and it worked fabulously well._
_This same method can be used for any tender cut of meat, such as a pork cutlet or chop, veal, or frying fish (flounder, hake, cod, orange roughy). And feel free to vary the seasoning used for the flour—you might try 1 tablespoon paprika, garlic salt, or onion powder, or 1 teaspoon cayenne pepper._
1 cup/140 grams flour
1 tablespoon freshly ground black pepper or seasoning of your choice (optional), plus more for seasoning the chicken
2 eggs, blended to uniformity in a broad bowl that will accommodate the chicken
1½ to 2 cups/162 to 220 grams panko
4 boneless, skinless chicken breasts (halves)
Salt
2 tablespoons Dijon mustard
1 tablespoon fresh thyme leaves
About ⅓ cup/75 milliliters vegetable oil, depending on the size of your pan
4 lemon wedges, for serving (optional)
First set up your breading _mise en place_. Put the flour in a plastic bag and add the pepper or other seasoning; shake to distribute throughout the flour. Next to this, set the bowl of egg wash. Next to this, set a plate or baking dish with 1 cup of the panko, with more at the ready as needed. Have also on your counter a wire rack on which to set the breaded chicken (a rack will allow the underside of the chicken to stay as dry as possible, but you could use a baking dish, plate, or sheet of parchment paper).
Put each chicken breast between two sheets of plastic wrap and pound with a meat mallet or sauté pan until the meat is more or less flat and uniform in thickness, concentrating mainly on thinning out the fat end.
Give each breast a liberal sprinkle of salt and some grinds of black pepper on each side. Spread the top of each breast with a couple of teaspoons Dijon, then sprinkle the thyme over each.
Gently put two breasts in the bag of flour and flip the bag or the breasts to dust the other side of the breasts. Remove them from the bag, shaking off any excess flour, and place them on the rack. Repeat with the remaining two breasts.
Slip one floured breast into the egg wash and remove it when it is evenly coated with egg, allowing any excess egg to fall back into the bowl. Press the breast into the panko. Shake the pan or plate back and forth with your dry hand to toss the crumbs onto the top of the breast, then flip the breast and press it firmly into the bread crumbs. Continue to flip and press it until it is uniformly coated with bread crumbs. Place it on the rack. Repeat with the remaining chicken breasts.
Pour about ¼ inch/6 millimeters oil into a sauté pan large enough to hold the four breasts (see the cooking note below) over high heat. Just before the oil begins to smoke, lay the breasts into the pan (away from you so you don't splash oil on yourself). Cook on each side till they're golden brown and delicious-looking, a few minutes per side. While they're cooking, set out a clean wire rack or a plate lined with paper towels.
Remove the finished breasts to the rack or paper towel–lined plate. Allow them to rest for a few minutes before serving. Serve with lemon wedges to squeeze over them if you wish.
**NOTE:** _If you don't have a pan big enough to cook all four breasts at once, preheat your oven to 250°F/120°C and place a baking pan with a rack insert in the oven (the rack will keep the breading on the bottom from getting soggy). You can cook as many as will fit in your pan and keep them warm in the oven for up to 20 minutes (if this is the plan, remove them from the oil as soon as they're golden brown, as they'll keep cooking a bit in the oven). You can cook all four breasts and keep them warm this way while you finish other components of the meal._
#
# Picadillo Meatballs
MAKES 30 TO 40 (1-OUNCE/30-GRAM) MEATBALLS
_Whole eggs can serve as a binding device, and meatballs are the perfect example. Inspired by my friend Annie's love of the meatball shops in Manhattan, and by my friend Nathan's passion for a Cuban stew called_ picadillo _, I decided to create a spicy meatball that features the egg as both binder and enricher. These meatballs are spiked with chipotle peppers and flavored by annatto seeds, which color the oil a deep red and impart a slightly bitter flavor to balance the sweetness of the onions and peppers. (Annatto seeds can be found in Mexican or Latin grocery stores or specialty spice stores, and in many well-stocked supermarkets.) This recipe will yield four to six meal-size portions—make yellow rice by sautéing some onion in more annatto oil, then adding the rice and liquid—but they also make great hors d'oeuvres, with some hot sauce on the side for dipping._
_Note that annatto oil stains countertops and finger-nails, so be sure to lay a large sheet of parchment on your work surface, and manicured ladies may want to don gloves or use a scoop to shape the meatballs (nonpowdered latex gloves are inexpensive and handy to have generally—to wear when cutting hot chile peppers, for instance)._
¼ cup/60 milliliters vegetable oil, plus more for pan-frying
3 tablespoons annatto seeds
1 medium red bell pepper, seeded and cut into small dice
1 medium Spanish onion, cut into small dice
4 garlic cloves, minced
2 teaspoons salt
⅓ cup/60 grams pitted, chopped manzanilla olives
¼ cup/30 grams chopped capers
2 chipotle peppers in adobo sauce, seeded and minced
1 pound/450 grams ground beef
1 pound/450 grams ground pork
1 egg, beaten
Flour, for dredging
Hot sauce, for serving
Combine the oil and annatto seeds in a small saucepan over medium heat, bring the oil to a simmer, reduce the heat to medium-low, and continue to cook for about 5 minutes to infuse the oil.
Strain the oil into a sauté pan over medium-high heat, add the pepper, onion, and garlic, and then sprinkle them with 1 teaspoon of the salt. Sauté the vegetables, reducing the heat to medium-low after a minute or two, until they're very tender, 15 minutes or so. Remove the vegetables, along with any residual oil, to a plate and set aside to cool, then cover and refrigerate until completely chilled, 20 to 30 minutes.
In a large mixing bowl, combine the chilled onion-pepper mixture with the olives, capers, chipotle peppers, beef, pork, and egg. Sprinkle it all with the remaining 1 teaspoon salt and mix thoroughly by hand (or use a standing mixer with a paddle attachment, but be aware that the more you paddle, the more tightly bound the meatballs will be when cooked).
Put some flour in a shallow bowl. Form the meat mixture into balls the size of golf balls, dredge them in the flour, and shake off the excess. Pour about ¾ inch/2 centimeters oil into a large skillet over high heat—the oil should come halfway up the sides of the meatballs. When the oil is hot, add as many meatballs as will fit comfortably without crowding, and pan-fry them, rolling them around in the oil until they are nicely browned on all sides and just cooked through, 7 minutes or so. Transfer the cooked meatballs to a wire rack or paper towel–lined plate and continue cooking the remaining meatballs.
Serve with hot sauce.
#
CUSTARD IS WITHOUT DOUBT ONE OF THE supreme transformations of the egg. A plain vanilla custard, such as crème caramel, is a textural marvel, quivering on the spoon, delicate on the palate. And a big, thick quiche—which Thomas Keller rightly calls the sexiest pie—is a silky, satiny wonder when baked in a proper 2-inch ring. A custard cooked in a piecrust or tart ring is a different kind of textural experience: the pleasure is in the contrast of the creamy custard and the crunch of the crust. A true fat quiche or sweet vanilla custard is _voluptuous_.
Custards can be sweet or savory. Thick or thin. Served standing alone, or supported by a crust. They can be made with yolks alone, as in crème brûlée, or with whites only, as in panna cotta. The whites are composed mainly of protein, which sets up very solidly, so that a custard made with whole eggs can be sliceable, while one made with only yolks is so creamy and adhering you'd be unable to remove it intact from its cooking vessel.
Custards are also a tool, a binder that can add satisfying fat and flavor. The best bread puddings are simply bread soaked in a flavorful custard and then baked. You can make a sweet custard with cream and cinnamon and sugar and vanilla for a dessert bread pudding, or you can make it savory. Last Thanksgiving, I was in charge of the dressing (we've long stopped stuffing the turkey), and I made a custard using eggs and turkey stock, then poured it over the cubed day-old bread. It made a fabulous, flavorful dressing.
And custard is an easy ratio, as well: two parts liquid to one part egg. Large eggs weigh about 2 ounces/60 grams each. Double that quantity for the liquid, blend, then mix with your ingredients.
# Classic French Quiche with Chorizo and Roasted Peppers
SERVES 10 TO 12
_I was a youngster in the 1970s when quiche made its appearance on the Midwestern scene, only to be quickly lampooned as girly-man food. It was a time when French food was beginning to make its way into American home kitchens, thanks to Julia Child and the popularity of French restaurants in New York City in the previous decade. But a lot of things got lost in translation. Spinach salad with bacon vinaigrette was the only way we Midwesterners could re-create a classic_ salade frisée aux lardons _. And we cooked quiche in a piecrust—too thin to be of any deep pleasure._
_It wasn't until decades later, when I was working with Thomas Keller and Jeffrey Cerciello on the_ Bouchon _cookbook, that I learned all that a quiche might be. It is little more than an egg pie, a custard pie, but when it's cooked at a proper thickness, it can become utterly seductive. Again, it's all about the power of the egg to transform the ingredients it joins._
_In France, a common quiche is Lorraine, with bacon and onions—and if you're sticking with its peasant roots, no cheese. Or Florentine, with spinach. Here, to show off the versatility of quiche, I'm usingsausage, onion, and roasted red peppers. But you can use any flavorings you wish as long as you bake the quiche in a 2-inch/5-centimeter ring mold or cake pan (alas, a springform pan will leak). I treat a quiche like a cake, pressing the dough into a parchment-lined cake pan, so even if it does leak through the crust, you'll still have a thick, creamy quiche. The amount of dough in this recipe will fit a 9-inch/23-centimeter mold with plenty left over for patching should you need it. If you have only an 8-inch/20-centimeter mold, you'll have even more dough left over, which can be frozen for later use, or you can reduce the amounts of the crust ingredients by one-third if you're measuring by weight; you'll also have some custard left over, which can be baked separately in a large ramekin (it will take 20 to 30 minutes to cook)._
# FOR THE CRUST:
3 cups/420 grams flour
1 cup/225 grams cold or frozen butter, lard, shortening, or any combination thereof, cut into small pieces
½ teaspoon salt
¼ to ½ cup/60 to 120 grams ice water
(The quantity depends on the fat—whole butter has water in it so you need only a couple of ounces; shortening and lard do not contain water and thus need more.)
# FOR THE QUICHE:
Vegetable oil for sautéing
8 ounces/225 grams cured Spanish chorizo, cut into medium dice
½ Spanish onion, diced
3 teaspoons salt
1 red, orange, or yellow bell pepper, charred, peeled, seeded, and cut into medium dice (see note below on roasting peppers)
2 cups/480 milliliters milk
1 cup/240 milliliters heavy cream
6 eggs
1 teaspoon salt
½ teaspoon freshly ground black pepper
5 to 6 gratings of nutmeg
2 cups/170 grams grated cheddar cheese
To make the crust, combine the flour, fat, and salt in a mixing bowl and rub the fat between your fingers until you have small beads of fat and plenty of pea-sized chunks. Gradually add the ice water and then the salt and mix gently, just until combined—if you work the dough too hard it will become tough. (If you're making a bigger batch, you can use a standing mixer with a paddle attachment, but remember not to paddle too much after you add the water, just enough so that it comes together.) Shape the dough into a disc, wrap in plastic, and refrigerate for at least 15 minutes or up to 24 hours.
The dough can be used raw for some recipes, as with an apple pie (it's enough for a double-crusted pie). But for a quiche or other pie with a liquid batter, you'll need to bake the shell first (known as blind baking).
Preheat the oven to 325°F/165°C. Line a 9-inch/23-centimeter cake pan (or a ring mold placed on a baking sheet) with parchment. Roll the dough into a circle about ¼ inch/6 millimeters thick. Roll it over the rolling pin to lift it and unroll it over the parchment-lined cake pan, pressing the dough into the corners (use a scrap of dough to do this to avoid tearing the dough with your fingers).
To blind bake a crust, you need to fill the shell with something heavy to prevent the bottom from buckling up. Pie weights are made specifically for this, but a layer of aluminum foil and a pound of dried beans or rice reserved for this purpose does the job just as well. Line the bottom of your shell with another layer of parchment, then add the pie weights or beans and bake for 20 minutes. Remove the parchment and the weights or beans and continue baking until the crust is golden brown and cooked through, another 10 to 15 minutes. Cool completely.
To make the quiche, heat 2 teaspoons of the vegetable oil in a sauté pan over medium heat. When hot, add the chorizo and sauté for a few minutes till warm; remove to a plate lined with paper towels. Add the onions to the same pan with the chorizo fat, along with a four-finger pinch of salt, and sauté until softened, about 10 minutes. Add the roasted peppers and stir just to combine (they don't need further cooking). Remove from the heat and add to the plate with the chorizo. (The filling can be made up to 1 day ahead, as can the quiche shell—either raw or blind-baked).
Preheat the oven to 325°F/165°C. Use raw leftover dough to patch any cracks that opened in the quiche shell as it baked. Place it on a baking sheet.
In a large liquid measure or mixing bowl, combine the milk, cream, eggs, salt, pepper, and nutmeg and, using a hand blender, blend until frothy. This can be done in a standing blender as well, though you may need to do it in two batches, depending on the size of your blender. Or you could even mix the batter in a large bowl using a whisk—in this case, beat the eggs first, then add the rest of the ingredients. The idea will be to add the ingredients in two layers, using the froth to help keep the ingredients suspended.
Layer half of the chorizo mixture into the shell. Pour half of the frothy custard over the mixture. Sprinkle with half of the cheese. Layer with the remaining chorizo mixture. Refroth the batter and pour the rest into the shell. (You may want to put the baking sheet with the quiche shell into the oven and pour the remaining batter into it there so you can get every bit into the shell. You can even let it overflow to make sure it's up to the very top.) Sprinkle the remaining cheese over the top. Bake until the center of the quiche is just set, about 1½ hours (it may take as long as 2 hours, but don't overcook it; there should still be some jiggle in the center when you take it out of the oven).
Allow the quiche to cool, then cover it with plastic wrap and refrigerate until it's completely chilled. The quiche will keep for up to 5 days.
To remove and serve the quiche, use a knife to cut off the top edges of the crust along the rim or simply break them off by hand. Tug the parchment gently and lift the quiche from the cake pan; if using a ring simply press gently on the bottom once the sides are loosened.
Slice and serve cold or, to serve hot, slice and reheat for 15 minutes in a 350°F/180°C oven on lightly oiled parchment or foil, or cover with plastic and microwave for 1 minute.
**NOTE:** _To roast bell peppers, set them directly over a gas flame and cook all surfaces until they're black. You can also halve them and broil them, cut sides down, till black. Remove them to a paper bag or put in a bowl and cover with plastic wrap till cooled. Remove the charred skin under cold running water. To use, remove the stem and seeds and cut as directed._
#### FRENCH QUICHE
1. / _Roll the dough around a rolling pin to transport it to the pan or ring you've lined with parchment._
2. / _Use a scrap of dough to press the quiche dough into the corners of the pan or mold._
3. / _Place beans on a layer of parchment and bake. About midway through baking, remove the beans and parchment to finish the crust._
4. / _Put half of the sausage-onion mixture in the bottom of the shell. Pour your very frothy custard over it to about halfway up the pan._
5. / _Spoon the remaining sausage mixture gently over the frothy surface._
6. / _Refroth the custard and finish filling the shell._
7. / _The quiche, fresh out of the oven. Allow it to cool, then refrigerate._
8. / _After the quiche has thoroughly chilled, remove it from the mold. If you do so by turning it upside down, that's fine._
9. / _The whole quiche can be reheated to serve a group, or individual slices can be cut and served hot or cold._
10. / _The quiche is delicious cold, but my favorite way to eat it is hot._
11. / _Quiche and salad make an enormously satisfying and healthy meal._
# Chawanmushi
SERVES 4
_This is a lovely savory custard by one of our key recipe testers, Matthew Kayahara, a French-English translator in Ontario. Matthew spends many off-hours apprenticing at restaurants and cooking at home—particularly Japanese cuisine, since his grandfather was Japanese. This dish—chicken, shrimp, and mushroom bound in a light custard—makes a lovely first course or light meal. The custard uses just enough egg to set the top, but it is so fragile that it breaks as you eat it, and the clarified liquid is almost like a consommé._
_Like the Shrimp Tempura (here), this recipe calls for dashi, the excellent all-purpose Japanese stock made from kombu seaweed and bonito flakes. It also uses a little sake, so little it's not worth using an expensive one, unless you're also drinking it with the meal. Matthew's preference for everyday use is sake made by Hakutsuru, which is drinkable and reasonably priced. As for the soy sauce, if you can find the light-colored Japanese_ usukuchi shoyu _, use that._
2 ounces/60 grams chicken thigh or breast meat, cut into ½-inch dice
1 teaspoon sake
1 tablespoon plus 1 teaspoon soy sauce
4 small or 2 large shrimp, peeled and deveined, halved lengthwise if large
A few mushroom pieces for each serving (enoki, maitake, shiitake, cut bite-size as necessary)
2 eggs
1½ cups/360 milliliters dashi (here)
2 teaspoons mirin
½ teaspoon salt
½ scallion, sliced feather-thin on the diagonal, or 4 small watercress sprigs, for garnish
Grated lemon zest, for garnish (optional)
Combine the diced chicken with the sake and 1 teaspoon of the soy sauce in a small bowl and marinate for 15 minutes. Drain.
Divide the chicken, shrimp, and mushrooms among four 4- to 5-ounce/120- to 150-milliliter ramekins.
In a medium bowl, beat the eggs until smooth. Add the dashi, remaining 1 tablespoon soy sauce, mirin, and salt, and stir to combine well. Divide the mixture among the ramekins. Cover each ramekin with foil.
Place a rack or steamer basket in a pot large enough to hold all four ramekins. Pour in enough water to just reach the rack and bring it to a boil over high heat. Once the water is boiling vigorously, reduce the heat to medium-high, add the ramekins, cover the pot, and steam until the custard is set but still jiggly and the chicken and shrimp are cooked through, 15 to 17 minutes. Carefully remove the foil from the ramekins, garnish each with the scallions or watercress, cover the pot again, and steam until the garnish is just wilted, about 1 more minute. Remove the ramekins, grate a little lemon zest over each if desired, and serve immediately.
# Crème Caramel
SERVES 4
_This is one of my favorite desserts, notable for its simplicity and elegance. A quick caramel of cooked sugar is poured into a dish and cooled till it's as hard as candy. Then custard is poured into the dish and cooked in a water bath. The liquid custard sets but also transforms the hardened caramel into a sweet syrup that releases from the dish and runs down the sides of the custard. It's a great dessert for a dinner party because it can be made ahead of time and is easy to serve. It is usually prepared in individual ramekins, but it can also be made in a large baking dish, to be served family style, which is how the young Thomas Keller prepared and served it when he was chef at La Rive in New York's Hudson Valley in the early 1980s, a decade before opening the French Laundry in California's Napa Valley._
_I prefer my crème caramel straightforward, adhering to its simple roots—a milk custard flavored with vanilla only—though you could infuse it with aromatics, as I do for the Orange-Ginger Panna Cotta (here). If you want a richer custard, substitute half-and-half for half of the milk._
# FOR THE CARAMEL:
½ cup/100 grams sugar
2 tablespoons water
# FOR THE CUSTARD:
2 cups/480 milliliters milk
4 eggs
½ cup/100 grams sugar
Seeds scraped from 1 vanilla bean or 2 teaspoons pure vanilla extract
¼ teaspoon salt
To make the caramel, combine the sugar and water in a small pan over medium heat and cook until the sugar has melted and the mixture turns an appealing shade of brown, gently swirling the sugar (you can stir but do so gently or else the sugar may crystallize). If it foams up, take it off the heat and let it calm down so that you can evaluate the color and doneness. Pour the caramel into four 4- to 5-ounce/120- to 150-milliliter ramekins; it should coat the bottom to about **⅛** inch/3 millimeters. Allow it to cool completely into a hard candy.
Preheat the oven to 325°F/165°C.
Place the ramekins in a large roasting pan and pour water into the pan so that it comes three-quarters of the way up the sides of the ramekins. Remove the ramekins and place the pan of water in the oven.
Combine the custard ingredients and blend them till the mixture is uniform (this can be done with a whisk, a hand blender, or a standing blender, but the whisk is best so that you don't get too much froth). Divide the custard evenly among the ramekins. Place the ramekins in the water bath in the oven. Bake until the custards are almost set, with just a little jiggle at the center, 30 to 40 minutes. Remove them to a rack to cool, then cover the ramekins and refrigerate until thoroughly chilled, several hours at least.
To serve, use the tip of a knife to loosen the edges where the custard adheres to the ramekin and invert them onto plates. Serve immediately.
# Basic Bread Pudding (and French Toast)
BREAD PUDDING IS NOTHING MORE THAN A custard poured over bread and baked. When you remember that a custard is two parts liquid and one part egg, then you have a simple and delicious formula to use up leftover bread for either a savory dish, as with Thanksgiving dressing, or a sweet dessert. Bread pudding is especially good when made with leftover Challah (here) or Brioche (here). And French toast is essentially the same thing—bread soaked in a custard mixture—then sautéed in butter or oil.
This preparation depends on how much bread you have and how many people you need to serve. For each serving, plan on about 1 cup/90 grams stale bread cubes and 6 ounces/170 grams custard (1 egg, ½ cup/120 milliliters milk or half-and-half, 1 to 2 tablespoons sugar, plus flavoring such as vanilla or citrus zest). For savory versions, omit the sugar and replace the dairy with stock or flavor the dairy with sautéed onions, then top the custard-soaked bread with grated cheese.
# Bourbon Brioche Bread Pudding
SERVES 6 TO 8
_This is a delicious boozy variation on classic bread pudding, using a full loaf of rich brioche._
1½ cups/360 milliliters heavy cream
1 cup/240 milliliters milk
½ cup/120 milliliters bourbon
6 eggs
1¼ packed cups/250 grams dark brown sugar
1 teaspoon pure vanilla extract
1 teaspoon ground cinnamon
½ teaspoon ground nutmeg
¼ teaspoon salt
Pinch of cayenne pepper
10 cups diced brioche (1-inch/2.5-centimeter dice), left out overnight or baked in a 200°F/95°C oven till dry
Preheat the oven to 350°F/180°C.
In a large mixing bowl, combine all of the ingredients except the brioche cubes. Whisk until well combined.
Add the brioche cubes and toss them in the liquid, pressing them down so they absorb it. Allow the mixture to sit for 1 hour.
Butter a 9-inch (23-centimeter) square baking pan or standard loaf pan. Press the mixture into the pan.
Bake until a toothpick inserted into the center comes out clean, about 1 hour. Slice and serve with Crème Anglaise (here) or your favorite ice cream.
# Almond Challah French Toast
SERVES 4 TO 8
_If you don't have leftover challah, then you haven't made enough! But neither do you want to waste it. Wrapped in foil, challah freezes well, but a better use is simply to slice it thick and let the pieces dry overnight, then use it to make a delicious break fast by soaking it in a tasty custard mixture—here equal parts egg and liquid—and then sautéing it. These finish in the oven, so it's an easy dish to serve to a hungry family. You can certainly slather this French toast in butter and maple syrup, but if you want to make it an especially extravagant plate, I suggest serving it with toasted almonds, whipped cream, and a simple cherry sauce._
¾ cup/180 milliliters half-and-half
3 eggs
2 tablespoons kirsch
2 tablespoons sugar
1 teaspoon pure almond extract
1 teaspoon ground cinnamon
Pinch of ground nutmeg
Pinch of salt
6 to 8 thick (1 inch/2.5 centimeters) slices challah, left out overnight or baked in a 200°F/95°C oven till dry
Vegetable oil for pan-frying
Cherry Sauce (recipe below), for serving
¾ cup/180 milliliters whipping cream, whipped to firm peaks, for serving
½ cup/54 grams toasted slivered almonds, for serving
Preheat the oven to 325°F/165°C.
In a shallow mixing bowl, combine the half-and-half, eggs, kirsch, sugar, almond extract, cinnamon, nutmeg, and salt. Whisk until well combined.
Place the challah slices in the batter for 5 minutes, then flip to the other side.
Heat a large sauté pan over medium-high heat with a coating of vegetable oil. Working in batches, remove the challah slices from the batter and fry them until golden brown, roughly 3 minutes per side, then transfer them to a baking sheet.
Once all of the slices are cooked, bake them for 10 minutes.
Serve with the warm cherry sauce, whipped cream, and toasted almonds.
# Cherry Sauce
MAKES 2 CUPS/480 MILLILITERS
2 cups/250 grams sweet cherries, pitted and halved
¼ cup/50 grams sugar
¼ cup/60 milliliters plus 2 teaspoons water
Pinch of salt
A few drops of fresh lemon juice
1 teaspoon cornstarch
2 tablespoons kirsch
In a small pot combine the cherries, sugar, ¼ cup of the water, and the salt and bring to a simmer over medium-high heat. Add the lemon juice and allow to cook for a few minutes more.
Put the cornstarch in a small bowl and stir in the remaining 2 teaspoons water to make a slurry. Pour the slurry into the cherry mixture and simmer until the sauce has thickened, 20 to 30 seconds, then remove it from the heat. Stir in the kirsch.
Serve warm, poured over the French toast.
#
# Nineteenth-Century Ale and Rum Flip
SERVES 1
_Flips, combining ale, rum, and sugar, may have originated with British sailors; the term_ flip _refers to the froth created. Some descriptions are of hot flips, heated with the insertion of a red-hot poker, which would have generated froth. Eggs appear to have been introduced in Jerry Thomas's nineteenth-century book_ How to Mix Drinks _. Today's flips are typically sherry based. This cocktail is practically a meal in itself and a great libation to serve hot or cold, but I prefer it hot. Instead of adding my own spices, I simply use a spiced ale, which is ubiquitous around the winter holidays. This recipe makes one flip; if you're serving a group (which you should), use a standing blender._
3 ounces/85 milliliters spiced Christmas or pumpkin ale
2 ounces/55 milliliters dark rum
1 egg
Pinch of ground coriander (or whole coriander seeds shaved on a Microplane) or ground ginger (optional)
Grated orange zest (optional)
Combine the ale, rum, and egg in a large mug. Whisk or blend with a hand mixer. Or toss the drink back and forth between two large mugs, as Jerry Thomas might have done. Heat it for 40 to 50 seconds in the microwave (or use a red-hot poker). If desired, top with a pinch of coriander or ginger and some grated orange zest.
It was as close to a spiritual revelation as I've had when writing about food. I was at work on _Ratio: The Simple Codes Behind the Craft of Everyday Cooking,_ which looks at basic food preparations—bread, cake, stock, custard, sauces—in terms of the proportions of their key ingredients, the ingredients without which they would cease to be what they are. That is, I didn't concern myself with how much curry was in the mayonnaise but rather with the ratio of oil to water and yolk that made the mayonnaise _mayonnaise_. You need some yeast for leavening and salt for flavor in a bread, but what is critical is the ratio of the flour to the water. It doesn't matter what flavorings go into a muffin—cranberries or chocolate chips or walnuts—but rather the architecture of the muffin itself, equal parts flour and liquid by weight, half as much egg and butter. Which is basically what pancake batter is, whether plain or flavored with bananas or blueberries, just poured onto a flat hot surface rather than into a muffin tin. And then I realized that if you poured pancake batter over corn or peas, just to coat, and then fried them, you'd have a fritter. And that if you simply added more liquid to your fritter batter you had crêpe batter, not muffin-pancake-fritter batter.
So in my head, over a period of about twenty minutes, I'd gone from a dense bread dough to a loose crêpe batter, and it occurred to me that these preparations were essentially the same thing except for how much liquid they contained. It's an important fact for all cooks to know. There are, of course, small differences, most notably the quantity of fat used. For instance, I use more fat in muffins or quick breads than in pancakes; and since fritters are cooked in tons of fat you don't really need to add fat to the batter—it's the liquid-egg-flour combo that makes it what it is. What's critical here is the mighty role that eggs play in this continuum—so critical that I'm devoting a whole section to the range of flour-liquid preparations that rely on egg for their existence.
It is obvious that the egg is essential to egg salad, or eggs Benedict, or a fried-egg sandwich, but it is equally essential when hidden within a wide range of preparations, its effects at once beguiling and wondrous. Challah isn't challah without the egg, for example. A crêpe batter without egg is paste.
Because I'd like to move from dough to batter, stiff to loose, we begin with the great egg breads.
Brioche and challah are distinguished from one another—that is, made what they are—by the egg and the fat. Whole eggs are used in brioche; challah can be made with whole eggs or yolks only. The addition of eggs makes both doughs rich and flavorful; take out the eggs and they become other types of bread entirely. The brioche gets its tender crumb from the fat added to the dough; the fat prevents the gluten from forming, resulting in a cakelike bread. The challah uses less fat, so it is chewier than brioche; it tears rather than breaks or crumbles. Each bread is simple to make, the most basic of foods, comprising fundamental ingredients: flour, water, salt, yeast, egg, and fat. The classic brioche is made with luxurious butter, though there's no reason why you can't use any fat, such as olive oil or, as in one of my recent projects, _The Book of Schmaltz_ , rendered chicken fat, which results in a splendid savory brioche.
#
# Brioche
MAKES 1 (2-POUND/900-GRAM) LOAF
_I created this bread for my daughter, who, at an early age, adored the tender richness of brioche. I used my trusty_ New Professional Chef _as my guide, the CIA textbook that I remain devoted to (despite the fact that it calls for 5 pounds of flour, 26 eggs, and 3½ pounds of sugar—not exactly home-kitchen friendly; also it uses fresh yeast, which I no longer use_ * _). This brioche is a staple in our house for Christmas morning, and the leftover loaf can be frozen for a week and eaten sliced and toasted with foie gras to celebrate the New Year. Extra, left out to dry, will make outstanding French toast (here) or bread pudding (here)._
¼ cup/60 milliliters milk
¼ cup/75 grams honey
1 teaspoon/7 grams instant dry yeast
3 cups/420 grams bread flour
6 eggs
¼ cup/50 grams sugar
½ teaspoon salt
1½ cups/340 grams butter, cut into 12 pieces and brought to room temperature
Combine the milk, honey, and yeast in the bowl of a standing mixer and stir until the yeast is dissolved. Add about one-quarter of the flour and mix it well with the dough hook. Allow this mixture to sit for 1 hour to ferment.
Add the eggs, sugar, salt, and the remaining flour. Mix until the dough forms and pulls cleanly from the sides of the bowl.
Add the butter, one chunk at a time, until it's all been incorporated and a smooth, soft dough forms. Cover the mixing bowl with plastic wrap and let it rise to double its volume, 2 to 4 hours depending on the temperature of the kitchen.
Turn the dough out onto a floured surface and knead it to deflate the dough and redistribute the yeast. Shape it to fit in whatever mold you are using (anything from a terrine mold to a traditional bread-baking dish), cover loosely with plastic, and refrigerate overnight.
Remove the dough from the refrigerator 1 to 2 hours before baking. Preheat the oven to 350°F/180°C. Bake until the crust is golden brown and the dough is cooked through (the interior temperature should read around 200°F/95°C on an instant-read thermometer), about 45 minutes.
# Challah
MAKES 1 (1½-POUND/680-GRAM) LOAF
_Challah is among the most delectable breads you can make and is very easy to do at home. It derives its great color, texture, and flavor from the egg, in this case whole eggs. Challah doesn't have nearly as much fat as brioche, but it has some. My preferred fat is schmaltz—chicken fat and skin rendered with onion—which is what would have been added in the Ashkenazi Jewish culinary tradition from which challah derives (it's traditionally eaten on the Sabbath and at the two primary holiday times of the year). This fat gives the challah a great depth of flavor. Butter is my second choice. The dough should be mixed a day before baking, which helps it develop flavor. If you want a gigantic, celebratory six-braid bread, you'll need to weigh out your ingredients (which you should do anyhow) and double them. I'll explain how to create a three-braid loaf (see photoshere), but there are many good videos and pictorials online that demonstrate how to braid larger loaves._
3½ cups/490 grams flour
¼ cup/50 grams sugar
2 tablespoons honey
1 tablespoon/21 grams instant dry yeast (or 3 tablespoons/63 grams if you'd like to bake the bread on the same day you mix it)
7 eggs
¼ cup/60 grams butter or schmaltz, at room temperature
1 tablespoon salt
2 tablespoons sesame seeds or poppy seeds, for garnish (optional)
In the bowl of a standing mixer fitted with a dough hook, combine the flour, sugar, honey, yeast, and 6 of the eggs. Mix on medium until the dough comes together. Add the butter and continue to mix. Add the salt. Mix until the dough is well kneaded, 8 to 10 minutes in all. Turn the dough out onto a floured board and knead for a minute or so by hand to develop more structure. Put the dough into a bowl wiped with vegetable or olive oil.
Cover the bowl and refrigerate overnight or, if you want to bake it that day, cover and let it rise at room temperature for a few hours (it won't quite double in size).
Turn the dough out onto a floured board. Knead it to press out air and redistribute the yeast. Divide the dough into three equal pieces (about 300 grams apiece; weighing the dough will ensure accuracy) and shape them into rectangles. Cover the dough pieces with a towel and let them rest for 30 minutes.
To shape the bread, fold the top of each rectangle down, sealing it with the heel of your hand as you do. Continue sealing until you have a squat cylinder, then roll the dough to elongate it so that it's about 12 to 14 inches/31 to 36 centimeters long.
For a three-braid challah, work from the middle out. Set the pieces parallel on a cutting board or a piece of parchment paper. Put the center of the top piece over the middle piece. Put the center of the bottom piece over the center of the top piece (which is crossing the middle piece). Now, finish this half of the bread by folding the top piece into the center, followed by the bottom piece, until you reach the end, pinching the three tips together.
Turn the dough around and do the same to the other side. Whip the last egg till it's uniformly mixed and brush the dough with it; reserve the remaining egg wash. Allow the dough to rise for 1 hour.
Put a baking sheet or baking stone in the oven and preheat the oven to 350°F/180°C.
Brush the dough again with egg wash and transfer the dough, on the parchment paper, to the baking sheet or stone in the oven. Bake until cooked through (the interior temperature should read around 200°F/95°C on an instant-read thermometer), about 40 minutes. It should be gorgeously brown and shiny on top. Allow it to cool for at least 30 minutes before cutting.
#### CHALLAH
1. / _When the dough has risen, turn out onto a floured board and knead to deflate it and redistribute the yeast._
2. / _Divide the dough in thirds and shape them into rectangles._
3. / _Fold the top of each cylinder down by an inch or two and seal with the heel of your hand. Continue sealing until you have a squat cylinder._
4. / _After rolling out the three cylinders till they're about 12 inches long, lay them parallel on a cutting board or piece of parchment._
5. / _Working from the center out, cross the top piece over the middle one, the bottom piece over the top, and repeat._
6. / _After pinching the ends together, rotate your board 180 degrees and braid the remaining half of the loaf._
7. / _Brush the dough with egg wash. Feel free to sprinkle the loaf with sesame seeds or poppy seeds after brushing._
8. / _The finished dough should be light and rich, and the crust appealing and golden-brown._
# All-Purpose Pasta
SERVES 4
_Needless to say, making your own pasta is more time-consuming than cracking open a cardboard box and dumping the contents into boiling water. But dry pasta—and, indeed, even "fresh" pasta sold in stores—is a different product than what you can make on your own. (It's also a lot more fun, if you like to cook.) As with homemade mayonnaise, you simply cannot buy anything like what you can make at home, which is reason enough to make it. Also, it's not that difficult if you plan ahead, clear your work surface, and stay organized. When you make your own pasta, not only is it better tasting, but it allows you to cut it to whatever shape you wish—fettuccine, tagliatelle, or flat sheets for ravioli or lasagna. I worked for a chef who rolled big sheets of cooked pasta around pastry cream for a pasta dessert, slicing off portions as if it were a tortilla sandwich wrap._
_Pasta can be made with just flour and water, but add whole egg to the equation and it becomes a flavorful wonder. The flour needs some water to create the gluten that allows it to become elastic, but there's enough water in the egg itself to cover this duty. Again, weighing your ingredients is highly recommended, but you can always adjust the consistency of your pasta dough by adding water if it's too stiff or more flour if it's too sticky (try to avoid the former, because it's harder to incorporate liquid into a stiff dough than flour into a wet one)._
_Once again, egg turns ordinary flour into a heavenly food. And frankly, it doesn't matter what flour you use. The hunger for homemade noodles usually strikes me from out of the blue, and because all-purpose is always on hand in my house, that's what I use. But if you have access to finely ground "00" Italian flour, terrific. Want to try the high-protein semolina flour from durum wheat? Go for it. You'll find various recipes out there, but it boils down to a basic ratio of three parts flour to two parts egg by weight. Using a kitchen scale makes this a breeze. If you don't have one, figure on a generous ½ cup flour per egg per serving._
_I have a pasta roller, but I find that my 4-foot-long pasta rolling pin is the easiest, fastest, and cleanest method (a standard rolling pin can be used as well). The thinner you roll the dough, the better, as it does expand and thicken when cooked._
2½ cups/350 grams flour
4 eggs
In a mixing bowl, combine the flour and eggs and stir till the eggs are incorporated. When the dough is kneadable, turn it out onto a work surface and knead until all of the flour is incorporated into the dough and the dough is luxuriously smooth (this should take about 10 minutes, the cook's version of transcendental meditation). Shape the dough into a rectangle about 1 inch/2.5 centimeters thick, cover with a kitchen towel, and let rest at room temperature for at least 20 minutes, or wrap it in plastic and refrigerate for up to a day.
Cut the dough into thirds and roll it using a machine. Or, if you have a pasta rolling pin, search online for any of numerous videos demonstrating how to roll; still photos can't capture the technique required as you roll, pressing both down and out to stretch the dough. You'll need to roll patiently, as the dough will resist you (part of its goodness in the end), and allow it to rest as needed.
Cut as desired and cook in boiling salted water until done, 3 to 5 minutes, depending on the size and shape.
#### PASTA DOUGH
#### ALL-PURPOSE PASTA
1. / _The easiest, cleanest way to make pasta is to weigh your eggs before adding the appropriate amount of flour._
2. / _The ideal pasta dough is a 2:3 egg-flour ratio (e.g., 6 ounces egg and 9 ounces flour, or 200 grams egg and 300 grams flour)._
3. / _I begin mixing the dough in a bowl because the traditional method of adding eggs to a well of flour can be messy if the eggs overflow the well._
4. / _Once the egg is incorporated, dump the dough onto a work surface._
5. / _Knead the dough into itself (the yolks make it very sticky)._
6. / _Continue kneading until the dough is smooth and no longer sticky, about 10 minutes. Take a moment to reflect on the lovely texture._
7. / _Divide the dough in thirds._
8. / _Run each piece of dough through the largest setting of a pasta machine._
9. / _Fold each piece in thirds so that it matches the maximum width of the machine and roll it through the largest setting again. Repeat with all the pieces._
10. / _Roll each piece through consecutively thinner settings._
11. / _As the pasta gets thinner and thinner, add more flour as necessary to prevent sticking._
12. / _The rolled pasta can now be cut to whatever shape you wish._
# Yolk-Only Pasta
SERVES 2 TO 4
_In years of writing about, and for, chefs, as well as cooking and hanging out in kitchens, I've found that certain odd bits of information catch in the brain and remain stuck there. During Thomas Keller's travels in Italy, he worked in a kitchen in the Piedmont region where he learned to make the ingenious self-sealing ravioli called_ agnolotti _. But what stuck with me was his comment about how every chef wanted to see how many yolks he could get into a kilo of flour. The general consensus was thirty. Here I'll make the ratio even easier: one part flour to three-quarters part yolk, with olive oil to ease the stickiness (so if you're using a scale to weigh out your flour, multiply that weight by 0.75 for the weight of yolks required; or weigh out the yolks and multiply it by 1.33 to determine the weight of the flour)._
_Yolk-only dough results in a delicious, rich pasta, excellent for ravioli or agnolotti or simply cut as spaghetti and finished with some excellent olive oil. Unlike whole-egg pasta dough, this yolk-only dough—via chefs Michael Symon and Marc Vetri—requires very little kneading, just enough to bring it all together. But it does require a pasta roller, which serves as the kneading device. I find this dough to be so dry and flaky that I don't need extra flour or cornmeal to keep it from sticking, but it depends on your environment; use your common sense. If you have access to very finely ground "00" Italian flour, use that, but all-purpose will work fine._
1½ cups/210 grams flour
8 to 10 egg yolks
1 tablespoon flavorful extra-virgin olive oil
Combine all of the ingredients in a bowl and stir to mix. Pour out onto a work surface and knead the dough just till it comes together. Shape it into a rectangle about ½ inch/1 centimeter thick, cover it with a kitchen towel, and let it rest for 20 to 30 minutes.
Cut the dough in half and roll it through the widest setting of your machine several times. Then decrease the width by one click and continue to roll until you've reached the penultimate setting. Cut as desired.
Cook in boiling water (which you've salted until it tastes seasoned) until the pasta is tender, 3 to 5 minutes, depending on the size and shape.
# Italian Drop Cookies with Lemon Glaze
MAKES 20 LARGE OR 40 SMALL COOKIES
_Here I put the egg into action to make cookies. The richness of the dough comes from the yolks, and the water in the egg allows some gluten to form, giving the cookies a cakey bite rather than the crumbly texture of a shortbread-style cookie. These are not sweet cookies, but they get sweetness from a finishing lemony glaze._
# FOR THE COOKIES:
½ cup/100 grams granulated sugar
¼ cup/60 grams unsalted butter, at room temperature
Zest and juice of ½ lemon
3 eggs
2 cups/280 grams flour
2 teaspoons baking powder
¼ teaspoon salt
# FOR THE GLAZE:
2½ cups/250 grams confectioners' sugar
¼ cup/60 milliliters fresh lemon juice
Zest of 1 lemon
Preheat the oven to 350°F/180°C.
Combine the sugar, butter, and lemon zest in a standing mixer and paddle on medium-high till thoroughly creamed, about 5 minutes. Reduce the speed to medium and add the eggs one by one while paddling, followed by the lemon juice.
In a small mixing bowl, whisk together the flour, baking powder (press it through a strainer if it's chunky), and salt. Add the dry ingredients to the egg-butter mixture while paddling, just till the dough comes together. The dough should be pale yellow and sticky.
Line a baking sheet with a Silpat or parchment paper. Using two spoons to scrape the dough off each other, drop the sticky cookie dough onto the lined baking sheet. You can make the cookies teaspoon- or tablespoon-size. Leave about 2 inches/5 centimeters between the cookies; they will spread a bit.
Bake until the cookies are lightly browned, 10 to 12 minutes. Remove the cookies from the oven and transfer them to a wire rack to cool.
To make the glaze, in a small bowl, whisk together the confectioners' sugar, lemon juice, and half of the zest until the sugar is completely dissolved. The icing should be nearly a paste that falls in ribbons when you lift the whisk from the bowl.
When the cookies are cool, hold each one by the edges and dunk just the very top in the glaze. Once they're glazed, place the cookies on a wire rack to allow any excess glaze to drip off. Garnish with the remaining zest while the glaze is still soft and not yet set.
# Norwegian Berlinerkranser
MAKES ABOUT 40 COOKIES
_My assistant Emilia, a true cookie hound, sent me this recipe. I found it irresistible for its use of whole eggs and also two hard-cooked yolks, which I've never seen elsewhere. These are crisp shortbread-style cookies, enriched with the yolk and shaped into folded ribbons before baking._
2 hard-cooked egg yolks
2 eggs, separated
½ cup/60 grams sugar
1 cup/225 grams butter, cut into chunks and brought to room temperature
Pinch of salt
2 cups/280 grams flour
Pearl or decorating sugar, for garnish
Preheat the oven to 350°F/180°C.
Press the hard-cooked yolks through a sieve into the bowl of a standing mixer. Add the raw egg yolks and sugar to the sieved yolks. Using the paddle attachment, beat on medium-high speed until smooth, light, and creamy, about 5 minutes.
Paddle about one-quarter of the butter into the yolk batter. Once combined, add the salt, then add the remaining butter and the flour alternately, until the ingredients come together into a tight ball. This will become a very thick dough.
Scrape the dough out onto a sheet of plastic wrap. Form the dough into a rectangle 10 inches/25 centimeters long, 4 to 5 inches/10 to 12 centimeters wide, and **⅜** inch/1 centimeter thick. Cover with another sheet of plastic, wrap tightly, and refrigerate for at least an hour or overnight.
Slice the dough crosswise into strips about inch/5 millimeters thick. Roll the strips into cylinders about inch/5 millimeters thick and 6 inches/15 centimeters long. Form each cylinder into a loop and cross the ends over each other. Place the cookies on two parchment- or Silpat-lined baking sheets, spacing them about 2 inches/5 centimeters apart. Brush the tops with the egg whites and sprinkle with pearl sugar.
Bake until slightly golden, 12 to 15 minutes. Transfer to a wire rack to cool.
# Gougères
MAKES ABOUT 24 GOUGÈRES
_Cheese puffs are the savory version of cream puffs (or profiteroles, which follow). They're passed around at wine tastings, popped into the mouth without a thought—but mmm, so good. Truly these little creations are one of the great manifestations of how the egg unites the fundamental ingredients of our kitchen—butter, flour, and water—and transforms them into airy little wonders. Take away the egg and you've got a projectile—"lethal when thrown," according to a professional baker on Twitter._
_Despite the elegant results, the preparation is simple and also unique in that the flour is first cooked in the water and butter, then the eggs are whipped in. Then this dough-batter hybrid can be baked (it puffs!), deep-fried, or boiled, all with different and wonderful results._
_But here we simply want puffballs. One part water and half as much butter is brought to a simmer; an amount of flour equal to the weight of the butter is then stirred in. The flour absorbs the water and begins to gel, but it is also shortened by the butter (that is, gluten is prevented from forming, which results in tenderness). Next, eggs are whipped into this paste, the yolks giving their richness to it, the whites eventually spreading out to trap steam as the paste bakes. This is basic_ pâte à choux _dough. The dough is piped into balls and popped into a very hot oven. They begin to puff as the water expands into steam, but the exterior crusts over while the creamy interior—the starch gelling, the egg-white proteins setting—continues to expand. The oven temperature is lowered so that the exterior crust doesn't overcook or burn before the interior is fully set. The result is a delectable puff._
_How to make them even easier? Pipe out the balls, freeze them, then pop them in the oven whenever you want them._
_Now the question becomes how to flavor them._ Gougères _are flavored with cheese, typically Gruyère, Emmentaler, or Comté. Chef Shuna Fish Lydon, who has made thousands, probably tens of thousands, of_ gougères _, feels that because they are a French preparation, one should use only a French cheese (Comté). I say we're in America, a melting pot, so it's fine to add a little Parmigiano-Reggiano to the mix, and cheddar would surely work as well. If you want to take this one step further, pipe a cheese sauce into the hollow puffs. (See Quail Eggs Croque Madame,here, for the Mornay sauce recipe, increasing the amount of flour by a few tablespoons.)_
1 cup/240 milliliters water
½ cup/110 grams butter
½ teaspoon salt
1 scant cup/130 grams flour
4 eggs
1 cup/120 grams grated Gruyère (or other cheese noted above)
½ cup/50 grams grated Parmigiano-Reggiano
Preheat the oven to 425F°/220°C. Line a baking sheet with parchment paper or a Silpat and set aside.
Combine the water, butter, and salt in a small saucepan and bring the water to a simmer over high heat. When the butter has melted, add the flour, lower the heat to medium-low, and stir continuously until a paste forms, about 30 seconds. Continue to cook and stir for another 30 seconds.
To get maximum rise, transfer the paste to a standing mixer fitted with the paddle attachment, and paddle on medium-high, adding one egg at a time. Or, leaving the paste in the saucepan, stir in the eggs one at a time with a wooden spoon, vigorously, till incorporated. Add the Gruyère and half the Parmigiano and paddle or stir till incorporated.
Transfer the mixture to a pastry bag fitted with a ½-inch/1.25-centimeter straight tip (or use a sturdy plastic bag, cutting a ½-inch/1.25-centimeter hole at the corner). Pipe out 1½-inch/4-centimeter orbs of paste onto the prepared baking sheet, leaving an inch or two between them. With a wet finger tap down any tips that you may have left from the piping. Sprinkle each orb with a little of the remaining Parmigiano. (The raw puffs can be frozen like this, then dumped into a plastic bag and stored in the freezer for up to a month. You may want to add more cheese before baking.) Slide the baking sheet into the oven.
Bake for 10 minutes, then lower the heat to 350°F/180°C and continue to bake till cooked inside, 25 to 30 minutes more (you'll have to sacrifice one to yourself to know for sure whether they're done). Serve within the hour.
# Profiteroles with Vanilla Ice Cream and Chocolate
MAKES ABOUT 24 PROFITEROLES, SERVES 8
_Profiteroles are simply the sweet version of_ gougères _. I use milk instead of water, and a little sugar instead of salt. There's no reason you couldn't infuse the milk with vanilla beans if you want to up the flavor._
_A traditional French bistro dessert is profiteroles with vanilla ice cream and chocolate sauce. Because these puffs are largely hollow, they are often filled with pastry cream (cream puffs). And elaborate statues composed of profiteroles are often created by sticking them together with caramel; these structures are called croquembouches._
_As with_ gougères, _you can pipe these out several weeks in advance and freeze them, making them great for last-minute desserts and great for entertaining. For an all-homemade dessert, go all out and make Vanilla Ice Cream (here) and Chocolate Glaze (here), or throw together a quick chocolate sauce by pouring piping-hot cream over an equal amount of chopped chocolate, waiting a few minutes and whisking to satiny uniformity._
1 cup/240 milliliters milk
½ cup /110 grams butter
1 tablespoon sugar
1 scant cup/130 grams flour
4 eggs
Preheat the oven to 425°F/220°C. Line a baking sheet with parchment paper or a Silpat and set aside.
Combine the milk, butter, and sugar in a small saucepan and bring to a simmer over high heat. When the butter has melted, add the flour, lower the heat to medium-low, and stir continuously until a paste forms, about 30 seconds. Continue to cook and stir for another 30 seconds.
To get maximum rise, transfer the paste to a standing mixer fitted with the paddle attachment, and paddle on medium-high, adding one egg at a time. Or, leaving the paste in the saucepan, stir in the eggs one at a time with a wooden spoon, vigorously, till incorporated.
Transfer the mixture to a pastry bag fitted with a ½-inch/1.25-centimeter straight tip (or use a sturdy plastic bag, cutting a ½-inch/1.25-centimeter hole at the corner). Pipe out 1½-inch/4-centimeter orbs of paste onto the prepared baking sheet, leaving an inch or two between them. With a wet finger, tap down any tips that you may have left from the piping. (The raw puffs can be frozen like this, then dumped into a plastic bag and stored in the freezer for a month at this point.) Slide the baking sheet into the oven.
Bake for 10 minutes, then lower the heat to 350°F/180°C and continue to bake till cooked inside, 25 to 30 minutes more (you'll have to sacrifice one to yourself to know for sure whether they're done). Serve within the hour.
# To finish the dessert:
12 profiteroles
1 cup/240 milliliters Vanilla Ice Cream (here)
1 cup/240 milliliters Chocolate Glaze (here)
Halve each profiterole through the equator (to make ice cream sandwiches) and place three on each plate. Scoop ice cream onto the bottoms of the profiteroles, put the tops on, and finish each by drizzling warm chocolate sauce over the profiteroles.
# Crisp and Fluffy Potato Pancakes
SERVES 4
_One of the great uses for_ pâte à choux _, the eggy cooked flour dough-batter, is that when added to leftover mashed potatoes, it creates sensational potato pancakes unlike any other. The_ choux _paste serves as a binder and a leavener, once again thanks to the egg and the trapping of expanding steam. As you should make mashed potatoes often, and always enough so that you have leftovers, you should be able to make these potato pancakes regularly. Use an equal volume of potatoes and_ pâte à choux _for the pancakes and sauté. I like them very crispy, so I press both sides in panko bread crumbs before sautéing. It's not necessary, but I love the contrasting crunch. Marlene Newell, chief recipe tester, begged me to include herbs and cheese to further amp up the flavor. OK, bend my arm._
1 cup/240 milliliters mashed potatoes
1 cup/240 milliliters _pâte à choux_ (½ recipe, here)
Salt
2 scallions, finely chopped (all the white and some green), or 4 tablespoons minced fresh chives
¼ cup/30 grams grated cheddar cheese (optional)
About 1 cup panko
Vegetable oil for sautéing
Salt and freshly ground black pepper
Stir together the mashed potatoes and _pâte à choux_ , salting to taste, along with the scallions and cheese, if using, until they're uniformly distributed. Shape into patties as desired. You should have about eight 3-inch/7.5-centimeter pancakes. Press both sides of each pancake into the panko.
Pour about ¼ inch/6 millimeters oil into a large sauté pan over medium-high heat. When the oil is hot, add the pancakes in batches. Cook until they're nicely browned on the bottom. Flip them and cook until the other side is likewise browned. Remove to a plate lined with paper towels, season to taste with salt and pepper, and serve immediately.
# Cake and the Egg
THERE ARE MANY RECIPES ONLINE FOR eggless cakes, and surely you can rely entirely on chemical leaveners to create an airy crumb, and probably use a variety of modernist techniques to set a foam into a cakelike sponge. But cake could only have come to be because of eggs. Flour and liquid make dough; flour and egg make cake. Indeed, it's the most elementary of culinary creations and, given electric mixers, the easiest. Of course we almost always add sugar for sweetness. You can then flavor it any way you want, with vanilla, almond, or other extracts, or cocoa powder. You can enrich it with butter or, commonly, oil. And cheesecakes and certain chocolate cakes omit the flour entirely, relying only on egg, fat, and sugar. The egg is the single essential.
Here I'm going to focus on the basic cake, where the egg's primacy is unquestionable. But I also want to focus on cake for another reason: it's a preparation that has been replaced all but entirely in the home kitchen by a packaged mix. The processed food companies created the dump-and-stir cake early in the 1900s, and all you had to do was add water, since the mix included powdered eggs. But it didn't sell. Not long after World War II, when the food processing industry got into gear, the marketers figured out that they'd made it _too_ easy. People wouldn't buy it if they didn't at least have the _illusion_ of actually doing something on their own to create the cake. "Let's have them add eggs!" the marketers said. It worked. The life-giving eggs gave life to the boxed cake mix.
I say it's time to take back the cake. And I know all of you cooks who already love to make your own cakes will join me in encouraging others to bake a cake—yolks and sugar whipped to a foam, flour folded in, whites beaten to peaks and folded in, then baked.
That's all there is to it.
The "cake" part, anyway. Remember that a great cake is a composition; what makes it delicious is all the stuff you eat with it. A plate of boiled noodles is one thing; but boiled noodles tossed with egg yolk and bacon and Parmesan becomes a heavenly pasta carbonara. You could put two pieces of fried bread together and eat them and, well, consider how enticing that idea is. But: those two pieces of bread become something else if there's some great farmhouse cheddar melting between them. Add some ham? Yum. Use a Gruyère cheese instead, then put a fried egg on top with some Mornay sauce and you have a _croque madame_ , a French classic that's out of this world.
Cake is no different from pasta or a slice of bread and should be thought of in exactly the same way. While your homemade sponge cake is going to taste delicious as is, when you add buttercream, lemon curd, and jam inside and cover it with a simple whipped cream concoction, you create an event.
I co-wrote the _Bouchon Bakery_ cookbook, in which chef Sebastien Rouxel offers some astonishing cakes (and stresses the importance of the freezer, about which more below), but for this book I wanted to do some very basic cakes that can be used as launch points and make people feel comfortable with cakes, something I myself never really have. I'm a savory cook, more at home working the grill than the pastry station. So I brought in a ringer for one of the most important egg preparations in the entire egg pantheon, the cake, our great culinary symbol of celebration. I happen to have a talented wife, who happens to have a talented sister, Regina Simmons, who happens to be a professional baker in New York's Hudson Valley. Her forte is cake, specifically wedding cakes and other special-event cakes.
Regina agreed to fly to Cleveland and bake in my kitchen. What I like about Regina's techniques is that they are based on speed, resourcefulness, and economy. She's not lucky enough to have a fancy French oven, her mixer is twenty years old, and she'll make do with any circular mold, dented cake pan, or wobbling cake ring she can find—hell, she could probably bake amazing cakes with a couple of hubcaps. It hardly matters because, as she says, "Cakes start out ugly, they all do. The fun is in making them beautiful."
The following are a few cake basics and three of Regina's signature cakes, as well as some creams and frostings. I'm not going to claim that these are not undertakings, but while they involve numerous steps, if you plan ahead and spread out the process over several days, they're all dead simple and absolutely delicious—and with just a little patience and practice, gorgeous as well. The coconut cake is probably the easiest, so if you're new to cakes, that's a good place to start.
# THE KEYS TO FABULOUS CAKE
**The importance of the freezer cannot be overstated.** The first and perhaps most useful knowledge to have under your cap is how important it is to freeze cakes. They taste better after they've been frozen, their texture is improved, they're easier to work with and to frost or ice, and they cut more cleanly for a great presentation. Best of all, since these are often special-occasion creations, they can be made many days, even weeks, before you want to serve them. And they are very easy to transport when frozen, the perfect gift to bring to a friend. I've kept a well-wrapped cake in the freezer for more than two months without compromising it. You can make the sponge one weekend and freeze it, make the creams the next (they'll keep for a week in the fridge), and put the cake together at your leisure.
**Parchment is your friend.** By all means, bake in a springform pan if you wish. If you must have perfect edges, line only the bottom of your cake pan or use a proper cake ring. Regina lines whatever is at hand with parchment, sides and all, just smashes it down into the pan (making it stick to the sides with shortening or oil), not worrying about imperfections in the exterior of the cake—in fact, all those wrinkles and creases will just take in more tasty icing.
**The icings/frostings/fillings are what make a cake truly delicious.** Don't skimp on these, and consider using more than one in the same cake for more complex flavors, just as you would with any other dessert or any dish, period. These are also the most time-consuming elements in creating a great cake.
From a baking standpoint, the main thing to be is organized so that you get as much rise out of the sponge as possible. This means primarily two things: **prep your pans and preheat your oven before you start whipping your eggs.**
And one final note about pastry cream and buttercream: Here is where **it's important to know the salt level of whatever butter you use.** If you use unsalted butter, everything benefits from at least a pinch of salt, and anything chocolate benefits big-time from the right salt level. If you use salted butter, you often don't need to add extra salt to the creams.
# Regina's Lemon Cream Cake
MAKES 1 LARGE CAKE, SERVES 20
_This is one of Regina's most popular cakes, and I love it for the mixture of flavors. It's also relatively economical; while you have to make three different fillings, two others can be store-bought: jam and frozen fruit (which you pulverize). She makes a simple lemon curd, which becomes part of the layering and is also mixed into chantilly cream (whipped cream sweetened with confectioners' sugar); this lemon-chantilly is used to ice the finished cake and is also the main icing between the layers. The fat-rich buttercream prevents the fruit liquids from seeping into the cake itself._
_The basic sponge can be used to make just about any kind of cake you wish, and is a standard_ biscuit _, a French term designating that a meringue is folded into the base ingredients (yolks, sugar, flavoring), then baked._
_Again, I'd like to stress that if you've got a lot going on in your life and don't have a half day to mosey in the kitchen, you should spread this process out over days or weeks, baking the sponge cakes first and freezing them, then making the three creams some other day. This way you can put it together when it's convenient and not feel harried._
1 recipe Basic Sponge Cake (here)
1 recipe Lemon Curd (here)
1 recipe Chantilly Cream (here)
1 recipe French Buttercream (here)
½ cup/150 grams raspberry jam
½ cup/120 grams frozen mixed berries that have been pulverized, while frozen, in a food processor
1 cup/100 grams cookie crumbs (any kind of plain butter or almond cookies or graham crackers, coarsely pulverized in a food processor), for decoration (optional)
A few sliced fresh strawberries, blackberries, or blueberries, for garnish
When you're ready to assemble and finish the cake, flip each sponge cake layer onto a cutting board, bottom side up. Using a serrated knife, cut the cake into two equal layers by beginning to slice horizontally into the side of the cake about an inch deep, then rotating the cake (or the board) counterclockwise (or clockwise if you're left-handed), continuing to slice in a circular motion toward the center until you are all the way through. This should give you two even layers without the aid of a jig. (I once cut through the top of one layer and Regina said, "Don't worry about it, you won't see it in the finished cake—the cream will hold it together." I love her easy, cavalier attitude.)
Ready a cake round (cut from heavy cardboard) slightly larger than the cake, or a large plate with no lip, or a cake platter, or a cutting board—really any flat surface that will allow you to move the cake as you ice it.
Combine two-thirds of the lemon curd with the chantilly cream and fold until combined. Place one of the bottom cake layers on your cake round, cut side up (if this is your first time, put down a sheet of parchment paper to make the cleanup easier). For each side of each layer (except the bottom), you're going to spread a thin coat of buttercream first; this layer of fat prevents any berry juice from leaching into the cake. (For extra flavor and moisture, Regina sometimes first brushes the cake with a simple flavored syrup—sugar and water flavored with almond extract, for instance. Feel free to do this!) Next, spread a thin layer of lemon curd, 2 tablespoons or so, followed by a thin layer of jam, about 1 tablespoon or to taste.
Spoon out enough lemon-chantilly cream to create a ½-inch/1-centimeter layer and spread it evenly across the surface. Sprinkle half of the frozen berries on top.
The next layer will be one of the cake tops; before you place it on top of the frozen berries and cream, you need to prepare the underside as you did with the first coating of the sponge: spread a layer of buttercream, then lemon curd, then jam. Then flip it, coated-side down, onto the first frosted layer, sandwiching the lemon-chantilly between the two cake layers.
You should now be looking at the un-iced top of the second layer. Repeat the process with the following layers: buttercream, lemon curd, jam, chantilly cream, frozen berries. For the illustrations (here and here) we made a three-layer cake, but you could make one with four layers or as many as you wish.
When the layered cake has been constructed, an optional step is to coat the entire exterior with a thin layer of buttercream, called a crumb coat (not pictured). If you have time, chill the cake in the refrigerator so that the buttercream firms up; this will make icing the cake easier.
To ice the cake, use a cake spatula, or any long, flat-sided implement. Apply the lemon-chantilly cream to the sides, creating a ¾-inch/2-centimeter layer. Turn the board or plate toward you as you add the icing and smooth out the sides, holding the spatula still (a rotating cake stand or lazy Susan makes this very easy). When icing has been applied all around, continue to turn the plate to create a smooth surface. Don't worry about the top or bottom edges—you can give the bottom a coating of cookie crumbs and the top a decorative piping.
When the sides have been iced, apply a similar layer across the top. Again, spin the plate while holding the spatula still to achieve a perfectly smooth surface. If you don't have a long spatula, or have trouble creating a perfectly uniform top, feel free to make it wavy, or use the teeth of a serrated knife to make a decorative pattern. This can be done on the sides as well.
Now, as an optional but recommended step, make a cookie-crumb band around the cake by tilting the plate or board and pouring the crumbs around the base and halfway up the sides of the cake. Using a bench scraper or the flat side of a knife, press the crumbs gently into the icing so that they stick. Scrape away any excess crumbs for a clean appearance, or leave them for a more rustic appearance.
Put the remaining buttercream into a piping bag fitted with a large star tip (you can also use a sturdy plastic bag with a hole cut in one corner). Make a decorative border along the edges. In the illustrations, Regina is using a wave motion, down and up, down and up, as she circles the cake.
Pipe three small swirls in the center of the cake and garnish them with the fresh berries.
Freeze the cake, then wrap it and keep frozen until ready to serve or transport. Unwrap the cake when you remove it from the freezer and leave it out for an hour or two before serving, less if the weather or kitchen is particularly warm. Any leftovers can be wrapped carefully and frozen (or refrozen).
#### REGINA'S LEMON CREAM CAKE
1. / Mise en place _(clockwise from mixer): lemon curd, heavy cream, pulverized frozen berries, raspberry jam, sponge cake, powdered sugar, and buttercream._
2. / _Chantilly cream is simply whipping cream sweetened with confectioners' sugar and often some kind of extract, here vanilla._
3. / _Chantilly cream will be the main icing in this cake._
4. / _It's further flavored with some lemon curd._
5. / _Making the first layer, which is put together like all the rest: a film of fatty buttercream first, which prevents liquids from leaching into the sponge, followed by lemon curd._
6. / _The layering of flavor continues with jam._
7. / _The lemon-chantilly cream tops the jam._
8. / _Pulverized frozen berries are the final element of the interior garnish. Never waste an opportunity to flavor cake._
9. / _A sponge with jam, lemon curd, and buttercream tops the crushed berries._
10. / _When all the layers have been completed, the cake is iced with the lemon-chantilly cream, sides first._
11. / _The top is coated._
12. / _Holding the spatula still, rotate the cake to smooth the chantilly cream._
13. / _You'll be amazed how well rotating the cake works. It takes some practice and care, but a professional finish is within your reach._
14. / _Cookie crumbs add crunch, flavor, and color, and also conceal the bottom edge of the cake._
15. / _Use a flat implement to press the cookie crumbs into the chantilly cream._
16. / _Piping the last of the buttercream through a large star tip around the outer edge of the cake._
17. / _The finished cake, with multiple colors, textures, and flavors, is worthy of any kind of celebration._
# Basic Sponge Cake
MAKES 2 (8- OR 9-INCH) CAKE LAYERS
8 eggs, separated
2 cups/400 grams sugar
Pinch of salt
1½ cups/210 grams flour
Preheat the oven to 350°F/180°C. Grease the bottoms and sides of two 8- or 9-inch/20- or 23-centimeter cake pans or cake rings and line them with parchment. Set aside.
Put the yolks in a standing mixer fitted with a whisk attachment. (If you don't have a standing mixer, a hand blender with a whisk attachment or electric beaters will work, but they'll take a little more time. You can also use a big whisk, which, while requiring some effort, includes the advantage of burning off some of the calories you'll consume in the eating of this cake.) Turn the mixer on high and slowly add about two-thirds of the sugar, beating the yolks till they've doubled in volume and are light and fluffy, 3 to 5 minutes. (If you want to add flavorful extracts to your sponge, add them while mixing the yolks.) Pour them into a large mixing bowl.
Clean the whisk and bowl with soap and hot water, rinse thoroughly, dry, and return both to the mixer (errant yolk can prevent the whites from becoming a meringue). Add the whites and beat on high till they're frothy, then add the rest of the sugar slowly, about a tablespoon at a time, so that it has time to dissolve into the whites. Continue to beat the whites until all of the sugar is added and they form stiff peaks, 5 minutes or so depending on your mixer.
Add the salt to the flour and whisk thoroughly or, if it has been sitting for a long time, pulse it in a food processor or pass it through a sifter. Rain the flour over the egg yolks, using large sweeping circular motions with a wooden spoon to fold it in. When the flour is mostly incorporated, add one-quarter of the egg whites and fold these in the same way (fold slowly and gently to pop as few air bubbles in the egg whites as possible). Add the remaining egg whites in two additions and continue to fold until all of the ingredients are uniformly incorporated and you have a thick, just-pourable batter.
Divide the batter between the two lined pans, filling them two-thirds to three-quarters full. (If using 8-inch/20-centimeter pans, you'll have some batter left over. This can be piped or spooned onto a parchment-lined baking sheet and baked as you would cookies; to serve, dust with powdered sugar or glaze with chocolate—see here; these are sometimes referred to as Champagne cookies.)
Put the cakes in the oven immediately and bake until the centers are set, 20 to 30 minutes. You can use a cake tester or paring knife inserted into the center to judge doneness, but you can also simply tap the top of the cake; if it's not done, you will feel that the center is liquidy, like tapping a water bed. If you tap and the cake is clearly firm, it's done.
Remove the cakes from the oven and set them on a wire rack to cool completely. Lift them out of their molds and peel off the parchment paper. They can now be double-wrapped with plastic wrap and frozen. They can also be sliced and frozen, similarly wrapped, or they can be used right away.
#### BASIC SPONGE CAKE
1. / _For a basic sponge cake, a cake pan or ring is lined with parchment, its edges trimmed for neatness._
2. / _Egg yolks and sugar are foamed. Flour will be folded into this mixture._
3. / _Last, the egg whites, whipped to stiff glossy peaks, are folded into the yolk-sugar-flour mixture._
4. / _The batter is spread in the lined cake pan._
5. / _The finished cake, removed from the pan._
6. / _The cake can be sliced by rotating it at the same time that you gradually move a serrated knife through its center._
# Lemon Curd
MAKES ABOUT 2½ CUPS/600 MILLILITERS
_There are all kinds of ways of making lemon curd. Yolks only. Varying butter amounts. Whipping over a water bath. Even cooking it in a microwave. This is how Auntie Regina does it because, well, for a working mom of four with too much to do, this way works fast and is idiot-proof. (It is, of course, also delicious.) A little cornstarch helps to ensure that it sets up nicely so you don't need to worry about overcooking the eggs. You can also use this preparation for the Key Lime Tart (here) or any lemon tart—it's a dream._
1 cup/240 milliliters fresh lemon juice
1 tablespoon cornstarch
5 eggs
1 cup/200 grams sugar
½ cup/110 grams butter, cut into four or five pieces
Stir 2 tablespoons of the lemon juice into the cornstarch to make a slurry. Set aside.
Combine the eggs, the remaining lemon juice, and the sugar in a medium saucepan over medium-high heat (or high heat if you want it to go faster). Cook the mixture, whisking continuously, until it just starts to simmer and thicken. Add the lemon-cornstarch slurry and continue whisking until it thickens further and begins to simmer. Remove the pan from the heat and whisk in the butter, one chunk at a time. When the butter is melted, transfer the curd to a bowl and let it cool completely. Use right away or refrigerate for up to 5 days with plastic wrap pressed down onto its surface and a second layer of plastic covering the bowl.
# Chantilly Cream
MAKES ABOUT 4 CUPS/1 LITER
_This is a great all-purpose cream, as is or mixed with other ingredients._
2 cups/480 milliliters heavy whipping cream
½ to ⅔ cup/50 to 75 grams confectioners' sugar (depending on how sweet you want it)
Pinch of salt
1 tablespoon pure vanilla extract
Put the cream in the bowl of a standing mixer fitted with a whisk attachment and mix on high, sprinkling in the sugar as you do. Add the salt and vanilla and mix until incorporated and stiff peaks form. Refrigerate till chilled; this can be made up to 4 hours before using.
# French Buttercream
MAKES ABOUT 4 CUPS/1 LITER
_This is a simple buttercream, a cream whose consistency is dependent on the temperature you bring the sugar to, 235° to 240°F/113°C to 116°C, called the soft ball stage because when cool it will form a soft ball—that is, it will not have cooked to the point that it will harden when cooled. You'll need a candy thermometer for this. Regina stresses the importance of whipping the buttercream until it is room temperature so that the butter doesn't melt._
2 cups/400 grams sugar
½ cup/120 milliliters water
1 whole egg
3 egg yolks
2 cups/450 grams room-temperature butter, cut into 16 chunks
½ cup/225 grams vegetable shortening
2 teaspoons pure vanilla extract
Combine the sugar and water in a medium saucepan over high heat until the sugar has dissolved and begun to simmer. Lower the heat to medium and insert a candy thermometer; you'll need to keep an eye on the temperature while you whip the eggs.
Put the egg and yolks in the bowl of a standing mixer fitted with a whisk attachment and whip on high till the eggs have doubled in volume.
When the temperature of the sugar has exceeded 235°F/113°C (it can go as high as 240°F/116°C), pour it slowly into the eggs, with the mixer still running. Continue to whip the buttercream on high until the mixture is below 80°F/27°C, about 10 minutes (you can speed the cooling by holding a bag of ice against the mixing bowl). When the eggs and sugar have cooled, add the butter, a few chunks at a time, followed by the shortening and vanilla. Beat until all of the ingredients are incorporated. Use right away or cover with plastic wrap and refrigerate for up to a week, then allow it to come to room temperature before using.
#### FRENCH BUTTERCREAM
1. / _Buttercream is nothing more than eggs, sugar (next photo), and butter._
2. / _Eggs are whipped, and cooked sugar is added._
3. / _Fat is added after the sugar has cooled. Here we add a small amount of shortening for improved texture._
4. / _Chocolate buttercream is simply buttercream with chocolate whipped in._
# Coconut Cream Cake
MAKES 1 LARGE CAKE, SERVES 20
_This is a crowd pleaser and one of my son's new favorites—and one of mine, too, which is saying something because I've never really been a fan of coconut. What's great about this cake beyond the pleasure of eating it is that it's so easy to make, and to make pretty. It's supposed to be shaggy. And unlike the carpets of my 1970s Midwestern childhood, it won't go out of fashion._
_This cake uses the same sponge recipe for the fancy-pants lemon cream cake (here), but we used all four layers, mainly because it gives you an excuse to eat more of the delicious coconut cream. Regina also uses the technique of painting each layer with coconut syrup (I'd gone with her to our local grocery store, where she spotted a can in the cocktail mixers section—any decent coconut syrup will do). This adds flavor and moisture to the sponge._
_The sponge can be made a week ahead and frozen. The creams can be made up to 2 days before completing the cake._
1 recipe Basic Sponge Cake (here)
1 recipe Chantilly Cream (here)
1 recipe Vanilla Pastry Cream (here)
3 cups sweetened shredded coconut
½ cup/120 milliliters coconut syrup
When you're ready to assemble and finish the cake, flip each sponge cake layer onto a cutting board, bottom side up. Using a serrated knife, cut the cake into two equal layers by beginning to slice horizontally into the side of the cake about an inch deep, then rotating the cake (or the board) counterclockwise (or clockwise if you're left-handed), continuing to slice in a circular motion toward the center until you are all the way through. This should give you two even layers.
Ready a cake round (cut from heavy cardboard) slightly larger than the cake, or a large plate with no lip, or a cake platter, or a cutting board—really any flat surface that will allow you to move the cake as you ice it.
Combine the chantilly cream and pastry cream and fold to mix. Add 1 cup of the shredded coconut and fold till everything is uniformly combined.
Place one of the bottom cake layers on the cake round, cut side up, and brush it with coconut syrup, then spread it with about ½ inch/1.25 centimeters of the coconut-laced cream. Brush the underside of one of the top cake layers with coconut syrup and place it on top of the layer of coconut cream. Repeat this process with the remaining two layers. Brush the top and sides of the cake with more coconut syrup, if you have any left.
Spread the remaining coconut cream evenly along the sides and then across the top of the cake, smoothing all surfaces. Coat the entire cake with the remaining shredded coconut. Press the coconut gently into the cream with a bench scraper or flat side of a knife so that it adheres. Serve right away, or refrigerate or freeze.
#### COCONUT CREAM CAKE
1. / _The coconut cake is the most straightforward of the cakes here._
2. / _Unlike our other layered cakes, this one uses a single icing for both the filling and covering the exterior. It's finished with a coating of coconut._
# Vanilla Pastry Cream
MAKES ABOUT 2½ CUPS/600 MILLILITERS
_As notedhere, pastry cream is simply crème anglaise thickened with either flour or cornstarch. I prefer cornstarch because it's quicker and easier and there's no difference in texture or flavor. Regina's pastry cream is a little lighter than mine; for the Coconut Cream Cake (here), we used her superb version._
2 tablespoons cornstarch
1½ cups/360 milliliters milk
½ cup/120 milliliters heavy cream
2 egg yolks
½ packed cup/100 grams light brown sugar
Pinch of salt
¼ cup/60 grams butter, at room temperature, cut into 2 pieces
1 tablespoon pure vanilla extract
Mix the cornstarch with about 3 tablespoons of the milk to make a slurry and set aside.
Combine the remaining milk and the cream in a small saucepan and bring to a simmer over high heat—be careful not to let it boil.
Whisk the yolks with the sugar and salt in a bowl until the mixture is creamy.
When the milk-cream has come to a simmer, reduce the heat to medium and pour half of it into the egg-sugar mixture, whisking continuously. Put the pan back on the heat and pour the egg mixture into it. Whisk just until it comes up to a simmer, then add the cornstarch slurry (stir it first if the starch has sunk to the bottom). When the sauce comes back up to heat and has thickened, remove it from the heat. Whisk in the butter and vanilla and strain into a clean bowl. Cover and refrigerate until completely chilled; this will keep for several days.
#### VANILLA PASTRY CREAM
1. / _Vanilla pastry cream_ mise en place: _dairy, yolks, sugar, butter, milk thickened with cornstarch._
2. / _Hot milk and cream are added to the yolks and sugar to temper the eggs; then, top right, the egg mixture is returned to the pan._
3. / _The pastry cream, once it has cooked and thickened, is passed through a strainer to filter out any egg that may have coagulated._
4. / _The pastry cream sets up as it cools._
# Chocolate Mocha Cake
MAKES 1 LARGE CAKE, SERVES 20
_There's no mystery to chocolate cake—it's simply sponge cake with chocolate, usually cocoa powder. And it's a great cake to have in your repertoire. This recipe will work with regular old inexpensive cocoa, but if you're going to go to the effort, I urge you to use the best you can find. I was introduced to Guittard Cocoa Rouge, which is especially well suited to baking. Be aware that if you're using alkalized (usually labeled "Dutch process") cocoa powder, it won't interact with baking soda, which reacts to acid. We use three flavoring devices here: a basic all-purpose chocolate glaze, a mocha pastry cream (flavored with espresso powder), and chocolate buttercream for decoration._
1 recipe Chocolate Sponge Cake (here)
½ recipe Chantilly Cream (here)
1 recipe Mocha Pastry Cream (here)
1 recipe Chocolate Buttercream (here)
1 recipe Chocolate Glaze (here)
Cocoa powder, for garnish
When you're ready to assemble and finish the cake, flip each sponge cake layer onto a cutting board, bottom side up. Using a serrated knife, cut the cake into two equal layers by beginning to slice horizontally into the side of the cake about an inch deep, then rotating the cake (or the board) counterclockwise (or clockwise if you're left-handed), continuing to slice in a circular motion toward the center until you are all the way through. This should give you two even layers.
Ready a cake round (cut from heavy cardboard) slightly larger than the cake, or a large plate with no lip, a cake platter, or a cutting board—really any flat surface that will allow you to move the cake as you ice it.
Fold together the chantilly cream and half of the mocha pastry cream till they're uniformly mixed.
Place one of the bottom cake layers on the cake round, cut side up. Spread a thin layer of chocolate buttercream on top, then a layer of plain mocha pastry cream, followed by a layer of mocha-chantilly cream. Spread a layer of chocolate buttercream on the underside of one of the top cake layers and place it on top of the mocha-chantilly cream. Repeat with the remaining layers.
Warm the chocolate glaze in the microwave in increments, stirring to make it uniformly warm and pourable. Pour it over the top of the cake, all the way to the edges.
Ice the sides of the cake with the remaining mocha-chantilly cream.
Put the remaining buttercream into a piping bag fitted with a large star tip (or a sturdy plastic bag with a hole cut in one corner). Pipe decorative florets around the edges of the cake and dust them by sifting cocoa powder over them.
This cake can be frozen and wrapped until you're ready to cut and serve it.
_When serving a large cake to a big group, don't limit yourself to cutting wedges—squares are perfectly fine._
# Chocolate Sponge Cake
MAKES 2 (8- OR 9-INCH) CAKE LAYERS
1¾ cups/245 grams flour
¾ cup/75 grams unsweetened cocoa powder
1½ teaspoons baking powder
1½ teaspoons baking soda
2 eggs
2 cups/400 grams sugar
½ teaspoon salt
1 cup/240 milliliters hot coffee
1 cup/240 milliliters milk
½ cup/120 milliliters vegetable oil
2 teaspoons pure vanilla extract
Preheat the oven to 350°F/180°C. Grease the bottoms and sides of two 8- or 9-inch/20- or 23-centimeter cake pans or rings and line them with parchment paper.
Sift the flour, cocoa, baking powder, and baking soda into a medium bowl.
Combine the eggs, sugar, and salt in the bowl of a standing mixer fitted with a whisk attachment and whip on high till the eggs are creamy, a minute or two.
Add the flour-cocoa mixture and mix on high till combined. Add the remaining ingredients and continue to mix on high for another 2 minutes or so.
Pour the batter into the prepared pans or rings and bake till set in the middle, 20 to 25 minutes.
Remove the cakes from the oven and set them on a wire rack to cool completely. Lift them out of their molds and peel off the parchment paper. They can now be double-wrapped with plastic wrap and frozen. They can also be sliced and frozen, similarly wrapped. Or they can be used right away.
# Mocha Pastry Cream
MAKES ABOUT 3 CUPS/720 MILLILITERS
_This is best made a day or two before you need it so that it's thoroughly chilled._
3 tablespoons cornstarch
1½ cups/360 milliliters milk
½ cup/120 milliliters heavy cream
4 egg yolks
½ packed cup/100 grams light brown sugar
½ teaspoon salt
¼ cup/25 grams unsweetened cocoa powder
2 tablespoons instant espresso powder
¼ cup/60 grams butter, at room temperature, cut into 2 pieces
1 tablespoon pure vanilla extract
Mix the cornstarch with about 3 tablespoons of the milk to make a slurry and set aside.
Combine the remaining milk and the cream in a small saucepan and bring to a simmer over high heat—be careful not to let it boil.
Whisk the yolks with the sugar and salt in a bowl until the mixture is creamy.
When the milk-cream has come to a simmer, reduce the heat to medium and pour half of it into the egg-sugar mixture, whisking continuously. Put the pan back on the heat and pour the egg mixture into it. Whisk in the cocoa and espresso powders and continue to whisk just until it comes up to a simmer, then add the cornstarch slurry (stir it first if the starch has sunk to the bottom). When the mixture comes back up to a simmer and has thickened, remove it from the heat. Whisk in the butter and vanilla and strain into a clean bowl. Cover and refrigerate until it is completely chilled; this can be made several days before using.
# Chocolate Buttercream
MAKES ABOUT 2½ CUPS/600 MILLILITERS
_This rich, all-purpose buttercream can be used as an icing itself; here it's used for filling and decorating the chocolate mocha cake._
1 cup/200 grams sugar
¼ cup/60 milliliters water
3 egg yolks
1 whole egg
1 cup/225 grams butter, cut into about 15 pieces and brought to room temperature
2 teaspoons pure vanilla extract
1 cup/200 grams semisweet or bittersweet chocolate chips or chopped chocolate, melted and slightly cooled
In a small saucepan, combine the sugar and water over high heat. Bring to a boil and cook for 3 to 5 minutes. (The sugar syrup should register between 235° and 240°F/113° and 116°C on a candy thermometer.)
While the sugar syrup cooks, combine the egg yolks and whole egg in the bowl of a standing mixer fitted with the whisk attachment. Whip the eggs on high speed until tripled in volume. This will take about as long as needed to cook the sugar syrup.
With the mixer still running, pour the sugar syrup slowly into the beaten eggs. Continue to whip until the outside of the bowl has cooled, 8 to 10 minutes. Reduce the speed to medium and add a piece of the butter. After it begins to become incorporated, add the remaining butter, one piece at a time. The butter may look as if it's breaking, but keep whipping it, and the mixture will come together.
When all of the butter is incorporated, add the vanilla and chocolate, return the speed to high, and beat until the cream becomes smooth and luscious. Use right away or refrigerate till needed (and allow to warm till it's pliable before using).
# Chocolate Glaze
MAKES ABOUT 2½ CUPS/600 MILLILITERS
_Regina adapted this easy all-purpose chocolate glaze from a recipe by Dolores Casella. It is a liquid when heated, but sets up soft and firm at room temperature. We drizzle it on coconut cookies, pour it over ice cream, and use it for the glossy top of Regina's Chocolate Mocha Cake (here)._
½ cup/120 milliliters evaporated milk
½ cup/100 grams sugar
Pinch of salt
1 cup/200 grams semisweet chocolate chips
¼ cup/60 grams butter, cut into pieces
2 tablespoons pure maple syrup or corn syrup
2 teaspoons pure vanilla extract
Combine the evaporated milk, sugar, and salt in a medium saucepan over medium-high heat. Rub a little butter around the inner rim of the pan to help prevent the milk from boiling over. Bring the mixture to a simmer, reduce the heat to medium-low, and cook for 5 minutes. Remove the pot from the heat and whisk in half of the chocolate. When it begins to melt, add the rest of the chocolate, followed by the butter, syrup, and vanilla. Whisk until all of the ingredients are incorporated.
Pour the mixture into a clean bowl to cool. When ready to use, microwave until it's pourable, about 30 seconds (take it out midway and stir it).
#### CHOCOLATE GLAZE
1. / Mise en place _for a simple all-purpose glaze: evaporated milk, butter, sugar, and chocolate, with additional flavorings of maple syrup and vanilla._
2. / _The evaporated milk and sugar are simmered for 5 minutes. Then the pan is removed from the heat, and the chocolate and other ingredients are stirred in._
3. / _The finished chocolate glaze._
# Rum-Soaked Cherry Bread (or Muffins) with Almond-Sugar Topping
MAKES 12 MUFFINS OR 1 STANDARD LOAF
_I'm including a quick bread to show how easy they are when you know the ratio. The basic quick bread (or muffin) ratio is two parts each flour and liquid, one part each eggs and butter. If they're sweet, and most are, you can add sugar, up to the amount of the eggs by weight. Here I'm simply making a sweet quick bread flavored with almond and garnished with cherries, with the addition of browned butter and an almond topping, but really you can take this basic ratio—equal parts flour and milk, half as much egg and butter—in any direction you wish. If you have a scale, weigh your eggs first and adjust accordingly, then place your mixing bowl on the scale and pour the ingredients directly in (no need for measuring cups). This is the neatest and most reliable way to mix and bake._
_Toppings are not required, but they do add flavor, texture, and sweetness. This streusel topping, which I learned from Sebastien Rouxel while working on the_ Bouchon Bakery _cookbook, is basically equal parts all-purpose flour, almond flour (also known as almond meal), sugar, and butter. But you can also change the topping to something more like a crumble, with equal parts brown sugar, flour, oats, and butter, or simply use chopped nuts. Chef Rouxel also likes his batters to rest overnight to give the batter time to hydrate for a better interior crumb, but you can omit this step if you're pressed for time. Pastry chef and instructor Cory Barrett was the first to teach me to flour berries or other fruit before adding them to a batter; this will prevent them from sinking to the bottom._
# FOR THE STREUSEL TOPPING:
½ cup/70 grams all-purpose flour
½ cup/70 grams almond flour (almond meal)
⅓ cup/70 grams sugar
3½ tablespoons/50 grams butter, chilled and diced
# FOR THE BATTER:
3 eggs
1 cup/240 milliliters milk
2 cups/280 grams all-purpose flour, plus more for dusting the cherries
½ cup/100 grams sugar
½ cup/110 grams butter
1 teaspoon pure almond extract
1 teaspoon pure vanilla extract
2 teaspoons baking powder
½ cup/55 grams dried tart cherries, soaked for at least 8 hours in light rum
Combine all of the topping ingredients as if making a pie dough; pinch the butter until it is in tiny chunks. Cover and refrigerate till ready to use.
For the cake, mix the eggs, milk, flour, and sugar in a large mixing bowl. Whisk just till combined.
Melt the butter in a small saucepan over medium-high heat. When the crackling sound subsides and all that's visible is froth, the water has cooked out and the butter will begin to brown. When it has a nutty aroma, pour it immediately into the batter, whisking continuously until the batter is smooth and uniform. Add the extracts and whisk to incorporate.
Cover and refrigerate for 8 to 24 hours if possible.
Preheat the oven to 350°F/180°C. Butter a standard loaf pan or 12-cup standard muffin tin or coat it with nonstick baking spray.
Whisk the baking powder into the chilled batter. Put about ½ cup flour in a plastic bag. Drain the cherries (save the cherry-flavored rum for a cocktail!). Add the cherries to the flour and shake to coat, then strain out the excess flour through a basket strainer.
Fold the cherries into the batter. Pour the batter into the prepared loaf pan or muffin cups. Top with the streusel mixture. Bake until a paring knife inserted in the center comes out clean, about 40 minutes for muffins, 1 hour for a loaf.
# Emilia's Carrot Cake
MAKES 1 LOAF OR 12 CUPCAKES
_Don't let the long list of ingredients (mainly spices) frighten you. This is still a quick bread, a simple cake anchored by flour and egg. It's also virtually an American tradition, especially with the cream cheese icing._
# FOR THE CAKE:
¼ cup/60 milliliters dark rum
⅓ cup/55 grams raisins
1⅓ cups/185 grams flour
1¼ teaspoons baking powder
1 teaspoon baking soda
1 teaspoon salt
2 teaspoons ground cinnamon
½ teaspoon ground allspice
½ teaspoon ground nutmeg
½ teaspoon ground cardamom
½ teaspoon ground ginger
3 eggs
1 teaspoon pure vanilla extract
1 cup/200 grams granulated sugar
¾ cup/180 milliliters vegetable oil
⅓ cup/60 grams applesauce
8 ounces/225 grams peeled, grated carrots (about 5 large carrots)
½ cup/50 grams chopped walnuts (optional)
# FOR THE CREAM CHEESE FROSTING:
8 ounces/225 grams cream cheese, at room temperature
1½ cups/150 grams confectioners' sugar
¼ cup/60 grams butter, at room temperature
1 teaspoon fresh lemon juice
First, make the cake. In a small saucepan, combine the rum and the raisins and bring to a boil over medium-high heat. Remove the pan from the heat, and set aside to steep.
Preheat the oven to 350°F/180°C. Butter a standard loaf pan or coat it with nonstick baking spray.
In a medium bowl, combine the flour, baking powder, baking soda, salt, and spices.
In another medium bowl, whisk together the eggs and vanilla until well combined. Then whisk in the sugar and beat until incorporated. Next, whisk in the oil, followed by the applesauce. You should have a homogeneous yellow batter.
Now slowly begin to rain the dry ingredients into the batter. Whisk just until the flour is absorbed. Do not overwork the batter.
Squeeze out any excess liquid from the carrots and fold them into the batter. Next, fold in the walnuts (if using) and the raisins, along with any rum that may be left in the saucepan.
Pour the batter into the prepared loaf pan and bake until a toothpick inserted into the center of the loaf comes out clean, about 1 hour (check after 50 minutes).
Remove from the oven and allow to cool.
While the loaf is cooling, combine all of the frosting ingredients in a medium bowl and mash and whip until everything is incorporated. You can also use a standing mixer with a paddle attachment, but if the ingredients are not chilled, a whisk results in the best texture.
Turn the cooled loaf out of the pan and top with the cream cheese frosting.
# Make-Ahead Brownies
MAKES 24 (3-INCH/7.5-CENTIMETER) SQUARE BROWNIES
_I used to make brownies in a 9-by-13-inch/23-by-33-centimeter baking dish, but they disappeared too quickly. Not anymore. Now I make a big batch of these fudgy, delicious brownies, to freeze and have on hand whenever the urge strikes. My sister-in-law and baking maven, Regina, makes these brownies routinely because they sell out where she bakes, Verdigris, in the lovely town of Hudson, New York. I love taking practices common to the professional kitchen and introducing them to the home, where they are especially useful._
_As a kid, when I got my hands on a fabulous brownie and found it packed with walnuts, I felt cheated. Why, I wondered, did someone ruin a perfectly good brownie by putting nuts in it? But if you're a nut lover, feel free to add 1½ cups/130 grams chopped walnuts at the end. If not, and you want even more chocolate, add more chocolate chips! (One caveat: If you're like me, you're going to be eating these while they're still frozen; they're excellent this way, but the chips are particularly crunchy, so omit the chips if you intend to serve them frozen. That said, what a brilliant idea, putting chocolate chips into fudgy brownies! Regina uses the Ghirardelli brand, which is widely available and good quality.)_
_These are as easy to make as pancakes: combine the dry ingredients, combine the wet ingredients, stir, and pour. If you don't have what is known in the industry as a half sheet pan (13 by 18 inches/33 by 46 centimeters), halve the recipe or use an 11-by-15-inch/28-by-38-centimeter jelly roll pan. The 1-inch depth results in a fudgy, dense texture._
2 cups/280 grams flour
2 cups/200 grams unsweetened cocoa powder
2 teaspoons baking powder
1 teaspoon salt
8 eggs
4 cups/800 grams sugar
4 teaspoons pure vanilla extract
1 pound/450 grams butter, melted
2 cups/340 grams semisweet chocolate chips
Preheat the oven to 350°F/180°C. Line a half sheet pan or jelly roll pan with parchment paper and set aside.
Sift together the flour, cocoa, and baking powder in a large mixing bowl, and add the salt.
In another large mixing bowl, combine the eggs, sugar, and vanilla and whisk till well creamed, then pour in the melted butter in a thin stream, whisking continuously.
Combine the wet and the dry ingredients and stir or whisk to incorporate. Stir in the chocolate chips.
Pour the batter into the parchment-lined pan and bake until just set, 20 to 25 minutes.
Cool, turn out onto a cutting board, and cut the brownies to the desired size. Wrap each brownie individually in plastic and store in the freezer (but you'd better eat one or two right away just to make sure they're good).
#### MAKE-AHEAD BROWNIES
1. / _Making brownies like the pros, in a baking sheet tray._
2. / _Line the sheet tray with parchment to make the brownies easy to remove._
3. / _Mark your lines before cutting._
4. / _It helps to have a long knife, but a pizza cutter will also work well here._
5. / _Cut into squares._
6. / _Wrapped individually in plastic and frozen, the brownies keep for months, a great snack and a worthy dessert for impromptu dinner parties._
# Pancakes
MAKES ABOUT 8 (6-INCH/15-CENTIMETER) PANCAKES
_Pancakes are so easy and delicious when made from scratch, especially if you have a scale. Whenever I see a pancake "mix" in a cute little cloth sack for $7.99 I want to pull my hair out. If your kitchen is well organized, with all of the ingredients you use on a regular basis in one place, it should be one quick step to gather the dry ingredients (flour, sugar, baking powder, salt), and another 15 seconds to set out the wet ingredients (milk, eggs, butter, vanilla). Combine the wet, combine the dry, add the dry to the wet, and stir (the less the better). You can do this faster than it takes your pan or griddle to get hot._
_I love pancakes for their simplicity, of course, but they are an extraordinary window into the power of the egg. Without eggs, you wouldn't want to eat them. With eggs, they are one of the great staples of the American kitchen._
1 cup/240 milliliters milk
2 eggs
¼ cup/60 grams butter, melted
1 teaspoon pure vanilla extract
1⅓ to 1½ cups/200 grams flour
2 tablespoons sugar
2 teaspoons baking powder
½ teaspoon salt
Combine the milk, eggs, butter, and vanilla in a large bowl and whisk well till they're thoroughly combined.
In a medium bowl, combine the flour, sugar, baking powder (pressed through a strainer if it's pebbly), and salt.
Pour the dry ingredients into the wet mixture and whisk or stir till the batter is smooth. This ratio results in a fairly thick batter, and thick, cakey pancakes. If you like them thinner, add a little more milk.
Cook on a lightly oiled griddle or pan over medium heat until done, a few minutes per side.
# Corn and Sweet Pepper Fritters with Chipotle-Lime Mayonnaise
MAKES ABOUT 25 FRITTERS
_I love fritters in part because the ease of making them is so disproportionate to their deliciousness. Surely something so delicious and special shouldn't be so easy to make! But when you realize that fritter batter is simply pancake batter without the butter and sugar, then you can see not only why they're so easy but how infinitely variable they are. Pour the batter over any tender sweet vegetable and you have a great fritter. I've done all kinds of fritters—curried pea, salt cod, zucchini, apple, you name it. Choose your favorites and you can't go wrong. Here, it's corn, sweet pepper, and onion, dipped in a lime-chipotle mayonnaise, a flavor combination I return to again and again._
# FOR THE BATTER:
1 cup/140 grams flour
1 teaspoon baking powder
1 teaspoon ground cumin
½ teaspoon ground coriander
½ teaspoon salt
½ teaspoon freshly ground black pepper
½ cup/120 milliliters milk (or water or stock)
1 egg
# FOR THE DIPPING SAUCE:
¾ cup/180 milliliters mayonnaise
1 chipotle pepper in adobo sauce, seeded and minced
1 tablespoon fresh lime juice
Cayenne pepper (optional)
# FOR THE FRITTERS:
1 cup/170 grams corn kernels (preferably fresh, but thawed frozen will work)
1 cup/130 grams diced red bell pepper
½ cup/50 grams small-diced sweet onion
¼ cup/7 grams minced fresh cilantro
Vegetable oil for pan-frying
Stir together all of the batter ingredients in a small bowl. Set aside.
Stir together all of the dipping sauce ingredients, taste, and adjust the seasonings as you wish—you may want to add more lime juice or cayenne. Set aside.
For the fritters, combine the corn, bell pepper, onion, and cilantro in a medium bowl and stir till the ingredients are uniformly dispersed. Pour just enough batter over the mixture to coat and hold them together (you may not need all the batter) and stir.
Preheat the oven to 200°F/95°C. Line a plate with paper towels.
Pour ½ inch/1 centimeter oil into a skillet over high heat. When the oil is hot (350°F/180°C) drop tablespoonfuls of the vegetable-batter mixture into the oil (as the temperature of the oil comes back up, you may need to reduce the heat to medium-high). Cook the fritters for a few minutes on each side, turning them as necessary for even browning.
Remove a fritter and cut it in half. If it's cooked through (the batter is not runny inside), remove all of the fritters to the paper towel–lined plate. Keep these in the oven while you make successive batches.
Serve hot, with the dipping sauce.
#### CORN AND SWEET PEPPER FRITTERS
1. / _Fritters are nothing more than tasty ingredients bound with what amounts to pancake batter and then fried._
2. / _Use only as much batter as you need to bind the ingredients; the batter will puff and expand._
3. / _These employ the pan-fry technique, with just enough oil in the pan to come halfway up the sides of the fritters (though they could be deep-fried as well)._
4. / _This cross section shows a perfect dispersion of vegetable within the right amount of batter, a sight that never fails to make me lust for fritters._
# Shrimp Tempura with Tentsuyu Dipping Sauce
SERVES 4 AS A FIRST COURSE
_In the United States most tempura batter seems to be a mixture of low-protein flour and non-gluten starch for simple crispness. But in fact, when tempura traveled from Portugal to Japan (via missionaries in the sixteenth century), Japan elevated it to great heights, with the addition of a whole egg. The egg adds flavor, but if you're not careful, it can result in a soft rather than crisp batter._
_One of my recipe testers, Matt Kayahara, is interested in Japanese cuisine (he's one-quarter Japanese himself), so I asked him to create this entire recipe—not just the batter but the dashi (an elegant but very easy stock made of seaweed and bonito flakes) for the traditional dipping sauce called_ tentsuyu _. Matt says that soft water is a must for the dashi, so if you live in an area with hard water (water with more than 60 ppm calcium carbonate), use bottled water. He also notes that the dashi is very volatile and should be made the day of (it's quick and easy, so this shouldn't be a problem). You'll have a little more than you need; stir in a couple tablespoons of miso paste for a refreshing soup to accompany the shrimp._
_We found that the steam from cooked vegetables softened the fried batter too much, so we went with shrimp instead. The key is to mix the batter just before frying, but barely stirring it so that you develop as little gluten as possible and so that you have little clumps of dry flour floating in the batter (this helps to ensure maximum crispness, as does using low-gluten cake flour). For even more crispness, some chefs sprinkle a little extra batter over the shrimp after it's in the oil._
# FOR THE DASHI:
1 quart/1 liter water (see headnote)
½ ounce/15 grams kombu (four small sheets or one large sheet)
1 ounce/25 grams bonito flakes (about 1 cup)
# FOR THE DIPPING SAUCE:
½ cup/120 milliliters soy sauce, preferably Japanese
½ cup/120 milliliters mirin
2 to 3 tablespoons/60 to 90 milliliters rice-wine vinegar
¼ cup/60 grams grated daikon, drained of excess liquid
1 tablespoon grated fresh ginger, drained of excess liquid
# FOR THE TEMPURA:
16 large shrimp (about 1 pound/450 grams), tails on, peeled and deveined
Vegetable oil for deep-frying
½ cup/70 grams cake flour
2 tablespoons/15 grams cornstarch
1 egg, beaten
Ice-cold water, enough to bring the volume of the egg to 150 milliliters
To make the dashi, put the water and kombu in a medium pot over high heat, bring to 140°F/60°C, and reduce the heat to maintain this temperature for 1 hour. Remove the kombu and raise the temperature to 175°F/80°C. Add the bonito flakes, steep for about 20 seconds, then strain through a coffee filter set in a mesh strainer.
To make the dipping sauce, combine 2 cups/480 milliliters of the dashi with the soy sauce, mirin, and vinegar in a small saucepan over medium heat. Bring to a simmer, then remove the pan from the heat and add the daikon and ginger. Cover and set aside.
To prevent the shrimp from curling tightly as they fry, squeeze them and press them down on a cutting board till you feel their shape give way and they flop limply when held by the tail.
In a large pot with high sides, heat 3 inches/7.5 centimeters oil to 350°F/180°C. Line a plate with paper towels and set aside.
When the oil is hot, whisk the flour and cornstarch together. In a separate bowl, beat the egg and water together, then add it to the flour mixture and whisk or stir gently with chopsticks until a lumpy batter forms. Dip the shrimp in the batter and deep-fry until brown and crisp and cooked through, 2 to 3 minutes. Drain them briefly on the paper towel–lined plate and serve immediately with the dipping sauce.
# Popovers with Raspberry Jam and Powdered Sugar
MAKES 2 LARGE OR 4 MINI POPOVERS
_This is another miraculous egg-dependent dish. It's very similar to the_ gougère _and profiterole, though it doesn't contain fat; rather it is cooked in fat. Because there is no fat to prevent gluten from forming, popovers are chewier and breadier than profiteroles. And yet the mechanics are similar—the steam releasing within the batter, the egg protein containing the steam and allowing them to puff. Indeed, they can puff even more than_ gougères _because the gluten makes the flour elastic, so the inside remains delicate and custardy while the exterior browns._
_Because the steam remains trapped inside, the popover will contract as it cools, causing the lovely popover towers to fall, so they should be served straight from the oven. If they're going to sit for several minutes before being served, it's a good idea to remove them from the oven 5 minutes before they're done, pierce each one straight down through the center with a paring knife, and return them to the oven; this allows steam to escape while the heat solidifies the interior structure._
_Popovers are a weekend treat, served with jam and powdered sugar. Make the batter before you go to bed, cover it, and let it sit out on the counter so that the flour is fully hydrated in the morning when you go to cook them. They get the most lift when cooked in a popover or mini popover pan (I prefer the latter), but you can use any kind of mold._
1 cup/240 milliliters milk
2 eggs
1 scant cup/130 grams flour
½ teaspoon salt (if using unsalted butter)
¼ cup/60 grams butter, melted
Raspberry jam, for serving
Confectioners' sugar, for serving
Combine the milk, eggs, flour, and salt (if using) in a medium bowl and whisk till thoroughly combined. Cover and allow to rest for at least 1 hour or up to 12 hours.
Preheat the oven to 450°F/230°C (or 425°F/220°C if your oven will smoke at 450°F/230°C) and put a popover pan in the oven. When you're ready to cook the popovers, remove the pan and divide the melted butter among the cups you're using. Pour the batter into each cup, about three-quarters full. Bake for 10 minutes, reduce the oven temperature to 375°F/190°C, and continue baking for 25 to 30 minutes. Try one—they should be solid but creamy in the center.
Dust the popovers with confectioners' sugar and serve immediately, with jam on the side.
# Crêpes
MAKES 8 TO 10 CRÊPES
_The crêpe, featuring the thinnest of our egg batters, is one of the many underused preparations in the home kitchen. At the most liquidy end of the dough-batter continuum, the crêpe is a pancake made possible by the miraculous egg. Crêpes are simple to make, and I like to use them as a tool to turn leftovers into a new and fresh meal. This is a recipe to use when you have, say, some leftover Blanquette de Veau (here) or Chicken Fricassee (here). Or, if you have a hankering for creamy morels but don't want to make an omelet (here), crêpes are perfect packages for the mushrooms—simply add chives or other herbs to the crêpe batter and use stock as the liquid. If you want a quick sweet dessert, instructions for crêpes suzette follow the basic recipe here. For a cold preparation, fill them with pastry cream. They can also be made ahead and reheated._
_It's a simple ratio: two parts each liquid and egg to one part flour by weight. So if you just wanted a couple of crêpes for yourself, you could weigh 1 egg and stir in the same weight of liquid and half as much flour. One egg yields two 8-inch/20-centimeter crêpes._
4 eggs
1 cup/240 milliliters milk, water, or stock
Pinch of salt
1 scant cup/130 grams flour
Combine the ingredients and stir or whisk until the mixture is uniform. Let the batter rest for at least 15 minutes or up to several hours.
Heat a pan over medium-low heat (use a pan with an 8-inch/20-centimeter surface, preferably nonstick; of course, if you own a crêpe pan, use that). Wipe the pan with a little oil or butter and pour in a little less than ½ cup batter, just enough to coat the bottom.
You can either let the crêpe cook through without flipping it or flip the crêpe to cook both sides. If you'd like some browning on the crêpe, cook it over medium heat. The whole thing will take just a minute or so.
To serve the crêpe, spoon your desired filling down the center, fold the crêpe over it, and garnish with fresh herbs.
# To make crêpes suzette:
¼ cup/60 grams butter
¼ cup/50 grams sugar
¼ cup/60 milliliters fresh orange juice
Zest of 1 orange
8 to 10 crêpes
⅓ cup/75 milliliters Grand Marnier or other orange liqueur
Combine the butter, sugar, and orange juice and zest in a medium sauté pan. Bring the mixture to a simmer over medium-high heat and then reduce the heat to low. Reheat the crêpes in the liquid, one by one, folding them in half and then in half again into a quarter circle. Keep each one on the side of the sauté pan. When they are all hot and folded, arrange them overlapping in the pan, pull the pan off the heat, and add the Grand Marnier (off the burner because, while not likely, the fumes can ignite as you pour, which could then detonate the bottle). Return the heat to medium-high and ignite the liqueur by tilting the pan toward the gas flame, or ignite the alcohol fumes with a match or lighter.
Serve immediately.
So we come now to consider the white and the yolk, not divorced, because they will always be one and are always best that way, but, for now, separated.
The white is the man of the house, Mr. Protein, transparent, guileless, extremely useful, but bland and straightforward. The yolk is the woman, voluptuous, manipulative, brilliant, rich, nutritious, seductive, the center of life. That's one guy's view of the egg, anyhow.
The yolk is my favorite part of the egg. It is a wondrous substance for all the reasons noted above. And in a custard, the difference is revealing. A custard made from whole egg is smooth and silky if prepared correctly, but firm, with something close to a bite. A custard made with egg yolk alone, on the other hand, is deeply creamy and satisfying. You couldn't slice it cleanly, nor should you want to. It's meant to be scooped, soft and ethereal on the tongue.
And because it's soft, it's always best paired with something crisp, crunchy, shattering. This is why the crème brûlée is perhaps the perfect dessert, not just for its simplicity but for its capacity, given the bittersweet brittle surface and creamy depths, to satisfy our childish love of comfort and smoothness and sweets, as well as our adult appreciation for contrast and refinement.
For those who love the craft of cooking, the yolk is an extraordinary tool, capable of turning vegetable oil into a semisolid and extraordinary condiment, butter into a rapturous sauce, and cream into an unparalleled emollient at the table.
#
HOME COOKS TEND TO THINK OF GARNISH as something put on top of food once the food has been plated to prettify it, impart an aesthetic, and add flavor. Less frequently addressed is what is referred to as "internal garnish." For instance if you see pistachios and dried cherries in a pork _pâté en terrine_ , those red and green hits of flavor are indeed a garnish—they're just on the inside of the food, not on top of it. You might fill a meatloaf with sun-dried tomatoes; these would be considered interior garnish. It's more than professional jargon; it helps us to think about what the food is doing. I like the term and use it here with the egg-yolk ravioli here. The yolk is of course on the inside, and it adds flavor and visual drama to the finished dish when you cut into it.
# Egg Ravioli with Chèvre, Brown Butter, and Thyme
SERVES 8 AS A FIRST COURSE OR 4 AS A LIGHT ENTRÉE
_This is truly an egg-centric dish, with egg pasta, a yolk inside, and the ravioli sealed with the egg. I love putting a yolk inside pasta, perhaps my favorite way of serving the yolk on its own. It's a surprise to see the yolk spill out, and it flavors and enriches the dish. This is not a weekday meal preparation, but both the ravioli and sauce can be prepared ahead, refrigerated, and finished at the last minute. You can put anything you wish inside, or nothing. Cleveland chef Michael Symon used to serve an egg-yolk ravioli with ricotta, and he was the one who taught me the trick of putting a dot of cheese on top, which helps keep the yolk from breaking. Here I'm using a flavorful tangy cheese, but you could just as easily use caramelized onion, wilted spinach, or diced sautéed mushrooms—anything soft. One ravioli makes a great starting course; serve two for a more complete dish._
_Fresh sheets of pasta for making ravioli are available in some markets, but you'll want to make your own because homemade is best. Some people find that the thinnest machine setting results in pasta too difficult to handle, so feel free to take the pasta only to the second-to-last setting. You can use a glass or ring cutter to turn these into attractive circular ravioli, but I make squares because I don't like to waste any pasta._
_An important reminder: All yolks are not the same. Supermarket egg yolks are often incredibly delicate and quick to break. If you have these eggs, be gentle and make nice soft nests of cheese and onion._
2 teaspoons vegetable oil
1 Spanish onion, thinly sliced
2 ounces/60 grams chèvre, at room temperature
1 tablespoon milk or cream, at room temperature
Salt and freshly ground black pepper
½ recipe All-Purpose Pasta (here), rolled to the thinnest pasta machine setting and cut into four sheets (18 by 4½ inches/46 by 11 centimeters)
8 egg yolks (reserve some egg white for brushing the pasta; freeze the rest for another use)
¼ cup/60 grams butter
½ cup/60 grams slivered almonds
2 garlic cloves, minced
1 tablespoon fresh thyme, plus a few branches
Heat the oil in a small saucepan over medium-low heat and add the onion. Cook, stirring occasionally, until the onion is deeply caramelized, even mushy. This can take a couple of hours depending on how you wish to monitor heat. (You can make a big batch for soup and other uses and freeze them in individual containers ahead of time, if you wish.)
Combine the chèvre with the milk and salt and pepper to taste, and mash the cheese with a fork to combine the ingredients and make the cheese more spreadable.
Lay two sheets of pasta on a well-floured surface. Make a mark every 4½ inches/11 centimeters along the length of the pasta to designate each ravioli (for a total of 4 bottoms on one sheet and 4 tops on the other). Spread a teaspoon or two of the chèvre mixture in the center of each ravioli on one sheet only. Spread a teaspoon or two of caramelized onion on top of the chèvre, making a deep divot in the center where the yolk will rest (the yolk will slip off if you don't make the nest deep enough). Place a yolk on each bed of cheese and onion. Put another dab of the chèvre on top of the yolk.
Brush a little egg white on the pasta around the garnish so that the top will adhere. Lay the remaining sheet of pasta over the yolks, pressing down in the center and working outward to seal in the yolk while pressing out air bubbles. Using a pastry wheel (fluted if you have one) or a knife, cut the ravioli as you wish. If necessary, use a needle to poke holes in the ravioli and press the air out (otherwise the air, expanding from the heat of the boiling water, will make the ravioli float; if this happens, flip them carefully or press them below the surface so that the pasta cooks evenly). Repeat with the remaining two sheets of pasta and filling ingredients.
Bring a pot of water to a boil over high heat, adding enough salt so that it tastes like seasoned soup (2 to 3 tablespoons per gallon). Boil the ravioli till tender, 2½ to 3 minutes. Remove with a slotted spoon and transfer to a colander to drain. (The pasta can be made to this point a day in advance. Chill the ravioli completely in an ice bath, then transfer to a paper towel–lined plate. Cover with plastic wrap and store in the refrigerator. Bring the ravioli to room temperature before proceeding.)
Combine the butter and almonds in a large sauté pan, preferably nonstick, over medium-high heat. When the butter is melted, add the garlic and thyme branches. Cook until the butter begins to froth. Add the cooked ravioli, in batches if necessary, and carefully flip the pasta or spoon the melted butter over the top. When the ravioli are heated through, remove them to serving plates, spoon some browned butter and almonds over the top, and garnish with thyme leaves. Serve immediately.
#### EGG RAVIOLI
1. / _Make a bed of cheese and caramelized onion on the pasta, leaving a divot in it for the egg to nestle in._
2. / _Place a yolk on the cheese and onion, and add another dab of cheese on top to prevent the yolk from breaking when you lay the top sheet of pasta on it._
3. / _Lay the top sheet of pasta over the yolks._
4. / _I like fluted edges and so trim the ravioli with a fluted pastry wheel, but a knife will also do the trick._
5. / _The filled and shaped ravioli, ready to cook._
6. / _Boil the ravioli._
7. / _Lift the ravioli out of the water with a large strainer or spider._
8. / _Chill them in a water bath if you aren't going to serve them right away. Otherwise they can go straight into the finishing pan._
9. / _The ravioli are finished in brown butter, with garlic, almonds, and thyme._
#
IN THE FOLLOWING RECIPES THE YOLK IS considered an ingredient because it enhances the other ingredients that accompany it. Rather than transforming the other ingredients, here it blends with them to add depth and richness. First, used raw, it enriches a Caesar dressing—though as I note, the egg yolk can be seen as both ingredient and tool in this form of vinaigrette. It can also be cooked and used to enrich a sauce, as in carbonara or in _blanquette de veau_ , where it's combined with cream to create what's called a liaison. It's important not to cook the dish too hard after the liaison is added or the yolk will solidify and make your dish look curdled.
# Caesar Dressing for Crisp Romaine Lettuce and Croutons
SERVES 4
_I had my first real Caesar salad at the old Escoffier Room at the Culinary Institute of America, a restaurant dedicated to the classics. A Caesar salad for two was prepared tableside by a culinary student, who mixed the dressing in the wooden bowl before adding the lettuce. She mashed the garlic and anchovy with lemon juice to a paste, added the egg yolk and oil. She did an expert job of it, demonstrating how simple a vinaigrette can and should be. She also showed me the best way to dress a salad, by adding the lettuce to the dressing rather than pouring the dressing onto the lettuce (alternatively, you can pour the vinaigrette around the sides of the bowl while tossing the lettuce to avoid overdressing it; in either case, you want to allow the lettuce to pick up the dressing rather than get drowned in it). My cousin Rob makes a similar show of the Caesar, grandly dropping the yolk into the bowl. So it is both a show-offy salad and an easy salad for daily consumption. And it is the yolk that is the main feature of this dressing, enriching the dressing and, depending on how you mix it, making it very creamy by emulsifying the oil into the lemon juice. The flavors of garlic and anchovy should be subtle, present but not overpowering._
_This is a staple dressing in our house. I use a hand blender with a chopper attachment for the mixing, allowing the salt to dissolve and the garlic to macerate in the lemon juice to soften its flavor. I don't measure; I just mix, taste, and add more oil till it's right. If I want it very creamy, I add a drop of oil first and thoroughly blend it, which establishes the emulsification, then add the remaining oil as necessary. The anchovies can be replaced in a pinch with ½ teaspoon good fish sauce._
Juice of ½ lemon (about 2 tablespoons)
1 large garlic clove, minced to a paste
Aggressive pinch of salt
Freshly ground black pepper
2 anchovies, minced to a paste
1 egg yolk
½ cup/120 milliliters vegetable oil, plus more if needed
1 pound/450 grams romaine lettuce, cut or torn into bite-size pieces
1 to 1½ cups/30 to 45 grams croutons
For a traditional hand-blended Caesar dressing, combine the lemon juice, garlic, and salt in a large salad bowl and let sit for 5 to 45 minutes. Whisk in the pepper, anchovies, egg yolk, and oil until they are thoroughly incorporated. Taste and add more oil if necessary.
For a very creamy Caesar, combine the lemon juice, garlic, and salt in the chopper attachment of a hand blender and let sit for 5 to 45 minutes. Add plenty of freshly ground pepper, the anchovies, and the yolk and pulse once or twice to combine. Add a single drop of oil and pulse and blend thoroughly to disperse it. Add half of the remaining oil and blend till it's well incorporated. Add the remaining oil and blend till it's thick. Taste and add more oil if necessary.
Add the lettuce to the dressing in the bowl (or pour the dressing made with a hand blender along the sides of the bowl as described in the headnote). Toss to distribute the dressing, garnish with the croutons, and serve.
# Blanquette de Veau
SERVES 4
_Veal stew in a white sauce, that's all_ blanquette de veau _is. And it's fantastic. My love of it again harks back to my days in the kitchens of the Culinary Institute of America. The class I was covering was about to prepare food for a grand buffet, the pre-graduation dinner where the revered president, Ferdinand Metz, made it a point to dine. Chef-instructor Rudy Smith, an excellent teacher and gifted cook, said that when Mr. Metz ate stew, it was always this highly refined stew. "Nobody eats before Mr. Metz," Chef Smith told us. "Make sure you know the whole menu. Be proud of what you know; share it with him. One thing you should not do is try to bullshit the man."_
_So from that point on,_ blanquette de veau _became a dish of great distinction for me, a humble but refined stew preferred by this eminent chef, who had once plied his trade at Le Pavillon, the restaurant that made French cuisine the benchmark of fine dining in America. The delicate veal is first blanched to prevent coagulated protein from compromising the white beef stock–based sauce, which is enriched with a mixture of cream and our featured ingredient, the yolk—together known as a liaison. The liaison has the same effect as finishing a sauce with butter_ (monté au beurre), _but it is looser and lighter._
_I once thought that the yolk thickened the sauce as it heated, but it doesn't—something I learned when I demonstrated this dish in public, watching the egg cook and curdle in my lovely sauce. It only enriches, so the sauce should be as thick as you wish it to be before you finish it with the liaison; the liaison enacts a textural change but not a thickening one. It's a satin gown._
_I don't recommend making_ blanquette de veau _with store-bought broth—the quality of the stock is just too important to this dish. I prefer white beef or veal stock—that is, stock from blanched rather than roasted bones—though of course it works with stock made from roasted bones._
1½ pounds/675 grams veal stew meat
1 Spanish onion, cut into medium dice
4 tablespoons/60 grams butter
Salt
½ cup/120 milliliters dry white wine
1 quart/1 liter rich white beef stock
1 pound/450 grams button mushrooms, quartered
16 to 20 pearl onions, blanched and peeled (about 1 cup)
¼ cup/35 grams flour
About 1 teaspoon fresh lemon juice or white wine vinegar
½ cup/120 milliliters heavy cream
3 egg yolks
Hot buttered egg noodles, for serving
1 tablespoon minced fresh parsley, for garnish
Put the veal in a pot and cover with water. Bring the water to a boil over high heat, then strain immediately and rinse the veal under cold water till clean and chilled.
In a medium saucepan over medium heat, sauté the Spanish onion in ½ tablespoon of the butter, giving it a four-finger pinch of salt, till it's completely tender but not browned, a few minutes. Add the wine and bring it to a simmer.
Add the blanched veal and stock, add a couple four-finger pinches of salt, and simmer until the meat is tender, 60 to 90 minutes. (This can be done 3 days before serving and refrigerated till you want to finish it.) The stock should reduce by about a quarter during this time. Cover and keep warm till you're ready to finish the dish.
While the veal is cooking, sauté the mushrooms over medium heat in ½ tablespoon of the butter till tender. Add the mushrooms and pearl onions to the stew 20 minutes before it finishes.
Meanwhile, melt the remaining 3 tablespoons butter in a small sauté pan over medium heat. When some of the water has cooked off, bubbling out of the butter, add the flour and cook, stirring gently, till it forms a loose paste (a roux) and some of the raw flour smell is gone. You want to cook it, without coloring it, till it smells like piecrust. Allow the roux to cool.
Return the meat to a simmer if it's not already there. Whisk in the roux and simmer, skimming any foam that collects on the surface, for 5 minutes. Taste for seasoning. Add more salt if it needs it and a few drops of lemon juice or white wine vinegar.
Whisk together the cream and the yolks and stir the mixture gently into the thickened liquid. Allow the stew to come back up to heat, just below a simmer, and serve immediately over egg noodles, garnishing with the parsley.
# Chicken Fricassee
SERVES 4
_A fricassee is similar to a blanquette in that the meat is sautéed first, then finished in the sauce. The difference is that no pains are taken for absolute refinement in a fricassee. Chicken and onion are cooked together, stock or water is added and thickened, and the dish is finished with the liaison. I like this dish precisely because it's a simple, rustic stew made voluptuous by the refined cream-yolk liaison. The all-purpose sauce can be varied any number of ways by adding spices—curry, paprika—to the chicken before adding the stock._
1 Spanish onion, cut into medium dice
2 carrots, peeled and cut into medium dice (optional if using stock)
4 tablespoons/60 grams butter
Salt
8 boneless skinless chicken thighs, cut into bite-size pieces
1 cup/240 milliliters dry white wine
1 quart/1 liter homemade chicken stock or water
2 bay leaves (optional if using stock)
1 pound/450 grams button mushrooms, quartered
16 to 20 pearl onions, blanched and peeled (about 1 cup)
¼ cup/35 grams flour
½ cup/120 milliliters heavy cream
3 egg yolks
Hot buttered egg noodles, for serving
1 tablespoon minced fresh parsley, for garnish
In a medium saucepan over medium heat, sauté the onion and carrot (if using) in ½ tablespoon of the butter, giving them a four-finger pinch of salt, until tender but not browned. Add the chicken and cook until the meat has lost all its pinkness, 3 to 5 minutes. Add the wine and bring to a simmer, then add the stock, bay leaves (if using), and another four-finger pinch of salt. Simmer until the chicken is tender, 30 minutes or so.
While the chicken is cooking, sauté the mushrooms over medium heat in ½ tablespoon of the butter till tender. Add the mushrooms and pearl onions to the stew.
Meanwhile, melt the remaining 3 tablespoons butter in a small sauté pan over medium heat. When some of the water has cooked off, bubbling out of the butter, add the flour and cook, stirring gently, till it forms a loose paste (a roux), and some of the raw flour smell is gone. You want to cook it, without coloring it, till it smells like piecrust. Allow the roux to cool.
Return the meat to a simmer if it's not already there. Whisk in the roux and simmer, skimming any foam that collects on the surface, for 5 minutes. Taste for seasoning and add more salt if it needs it.
Whisk together the cream and the yolks and stir the mixture gently into the thickened liquid. Allow the stew to come back up to heat, just below a simmer, and serve immediately over egg noodles, garnishing with the parsley.
#
# Mayonnaise
ONE OF THE EGG YOLK'S STAR TURNS IS ITS capacity to emulsify fat into water—that is, separate the infinitesimal orbs of oil (created by a mechanical beating or blending of that oil) between thin sheets of water. This is how a fluid, translucent vegetable oil becomes an opaque, semisolid sauce. When flavor is added to it, it becomes a sauce of wonder—simply lemon juice and salt is enough to make it worthy of the gods. Spike it with cayenne, or use lime juice in place of the lemon, and you're riffing like a jazz artist. Get the yolk hot and fluffy before adding the fat and you're moving into the territory of the great French emulsified butter sauces.
But we're getting ahead of ourselves.
First, the mayonnaise basics. Foremost is to recognize that the jar of Hellmann's you get off the grocery store shelf is indeed mayonnaise, but it is different enough from homemade mayonnaise that I feel it's incumbent upon cooks to refer to the mayonnaise we buy by its brand name, and to homemade mayonnaise as "homemade mayonnaise" or "my own mayonnaise." Or maybe we could just shorten it to "my-own-aise." They are different products. I personally like Hellmann's for some uses, such as a fried-egg sandwich (here). But nothing like the mayonnaise you make at home is available at any store.
Second, there are several ways to make mayonnaise: whisk and bowl, mortar and pestle, hand blender with blade attachment, hand blender with whisk attachment.
Finally, the key to mayonnaise is twofold: you must have yolk, but the amount isn't critical, and you must have water or something water-based (e.g., lemon juice), and that amount _is_ critical; it's what holds the sauce together. Too much oil relative to the water and the sauce can break, turning back into the consistency of oil.
What the yolk adds is lecithin. As I wrote earlier, this is one badass little molecule. It's got a fat-friendly side and a water-friendly side. The fat-friendly side embeds in the oil droplets and the water-friendly side connects with the water, strengthening the barrier that maintains the emulsion. Harold McGee notes in _On Food and Cooking_ that a single yolk has enough lecithin to emulsify many, many cups of oil (provided there is enough water). The amount of water needed is also somewhat variable, depending on the power of your emulsifying tool. I've found that you need about one part water for every twenty parts oil, or 1 ounce water for every 2½ cups oil, or 10 grams water for 200 grams oil (see how easy recipes become when you use metric measurements?). If you see your mayonnaise looking like it's going to break on you, drizzle in a little water (knowing whether you need a little more water is a matter of intuition, experience, and fear). You can't break your mayonnaise by adding too much water, but you can make it too thin—in which case you'll have to emulsify more oil into it and adjust the seasonings till it returns to the consistency and flavor you want.
The only other point that should be noted is that salt doesn't dissolve readily in fat. I always add salt to the water-based element so that it dissolves there and spreads evenly through the sauce.
Other than that, one of the world's greatest sauces is, start to finish, minutes away.
The most controlled way to make mayonnaise successfully is with a whisk, but I almost never do that because it takes more time. The fastest way to make mayonnaise is with a hand blender in a 2-cup/480-milliliter glass measuring cup. Provided the blade reaches near enough to the bottom of the measuring cup, a mayonnaise can be created 30 seconds after assembling your ingredients. The only drawback is that you're limited to making about ¾ cup/180 milliliters of mayonnaise before it breaks. To make it this way, put the water, salt, and yolk in the measuring glass and, with the blade running, allow a drop or two of oil into the cup, then slowly pour in the rest of the oil, pumping the running blade up and down until you've added all your oil. You should have a thick mayonnaise that stands in peaks.
Using the whisk attachment on a hand blender in a 1-quart/1-liter glass measuring cup is just as easy, and you can make plenty at a time.
# Basic Mayonnaise
MAKES 1 CUP/240 MILLILITERS
_This is a simple, delicious, all-purpose mayonnaise that makes everything better, from egg salad to spreads on sandwiches (a basic BLT is out of this world when made with your own mayonnaise). Spoon it over asparagus or boiled cauliflower, eat it with artichokes, mix it into chopped hard-cooked egg, make a lobster roll with it. Always taste and adjust the seasoning to your liking, paying special attention to the acidity level and how you may be using it._
1 tablespoon fresh lemon juice
1 teaspoon water
Salt
1 egg yolk
1 cup/240 milliliters vegetable oil
Put the lemon juice, water, and a four-finger pinch of salt in a 2-cup/480-milliliter glass measuring cup. Whisk them together to dissolve the salt. Add the yolk and whisk to combine. While whisking, vigorously add one, two, then three drops of oil, then add the remaining oil in a steady stream, whisking continuously. You can stop midway through if your arm gets tired; the emulsion won't fall apart.
Alternatively, you can use a hand blender. Let a drop or two of the oil fall off the tip of a spoon into the moving yolk-lemon mixture to establish the emulsion. Once it's fairly stable, add the rest of the oil in a thin, steady stream. If it starts to get too thick, add a sprinkle of water.
Use right away, or store in a covered container in the refrigerator for up to 3 days.
**VARIATION:** To make aioli, add one or two smashed garlic cloves and use olive oil or a mixture of vegetable and olive oils. Use it the same day you make it.
# Lemon-Shallot Mayonnaise
MAKES ABOUT ¾ CUP/180 MILLILITERS
_This is my go-to mayonnaise for just about anything that mayonnaise goes well with. It's so good you almost want to eat it with a spoon—and my mom sometimes does! I love it with artichokes (here) and egg salad (here), but it goes with pretty much anything—fish, chicken, or any number of vegetables. It should be made the day you want to eat it; it can sit for a day in the fridge, but the shallots give it an off flavor after too many unappreciated hours._
1 tablespoon plus 2 teaspoons fresh lemon juice
1 tablespoon minced shallot
½ teaspoon salt
1 teaspoon water
1 egg yolk
Pinch of cayenne pepper
¾ cup/180 milliliters vegetable oil
Combine the shallot with 1 tablespoon of the lemon juice and set aside while you make the mayonnaise.
Combine the remaining 2 teaspoons lemon juice, the salt, water, yolk, and cayenne in a bowl (or in a large glass measuring cup if using a hand blender, here, which is my preferred method). Whisk them together to dissolve the salt. Whisking continuously, add one, two, then three drops of oil to establish the emulsion, then add the remaining oil in a thin, steady stream, until all of the oil has been incorporated and the mayonnaise is thick and sumptuous. Stir in the shallot and lemon juice. If using the hand-blender method, you can add the shallot after mixing for a chunkier mayo, or blend the shallot with the yolk for a smooth mayo.
# FIXING A BROKEN MAYONNAISE
If you make mayonnaise, every now and then it's going to break on you. It may happen quickly, go from thick to soupy in a blink, or it may just look a little thin and grainy. But that's no reason to throw it away. Put 2 teaspoons water into a clean bowl or vessel, and pour the broken mayonnaise—a drop or two at first, then in a thin stream—into the water, whisking continuously. If you want to add more yolk to the 2 teaspoons water before re-emulsifying, for safety, that's fine, but it's not usually necessary.
# Grilled Chicken Salad with Chipotle-Lime Mayonnaise
SERVES 4
_In the same way that deviled eggs are better when you use your own mayonnaise, so is an ordinary chicken salad elevated to exquisite heights when you bend your mayo to your will, as with a dynamic chipotle-lime version, scented with cumin. Of course the mayo features minced shallot, which is a small kitchen miracle in its versatility; when macerated in citrus it is a beguiling aromatic in all emulsified sauces._
_We grill a lot of chicken in the summer, and we often have leftovers, to which this is put to use. The flavors of the grill mix well with the smoky heat of the chipotle and lime. Of course any cooked chicken will work—even, sigh, baked boneless chicken breast. (Indeed, this may be one of a handful of ways of redeeming this sorry staple of American culinary culture.) But the meat of preference here would be skin-on thighs, grilled. Remember that while a layer of fat undergirds the skin, the skin itself is mainly protein, and very flavorful, and can be finely minced and added to the salad._
_This makes a terrific lunch and is an excellent dinner when the weather is so hot you don't want to use the stove. It's also a great example of how leftovers become standout main courses with the help of mayonnaise you make and flavor yourself. The chipotle-lime mayo would also work well with roasted or grilled beef, pork, or lamb. A vegetarian version could be made by swapping in baked potatoes for the meat._
_The one element that would be missing from this rich, salty, acidic, spicy flavor combination is sweetness, so I'm adding sautéed onion for this final element of balance. If you're making this on the fly, you can omit the onion, but it's really amazing how much depth of flavor a cooked onion brings to just about any dish. I season the mayo with a little fish sauce, another umami ingredient that provides additional depth (but which you shouldn't be able to actually taste)._
_Serve the chicken salad on kaiser rolls or on butter lettuce with a toasted baguette on the side._
1 tablespoon fresh lime juice
1 teaspoon water
½ teaspoon salt
¼ teaspoon fish sauce (optional)
2 chipotle peppers in adobo sauce, seeded and finely minced
1 egg yolk
¾ cup/180 milliliters vegetable oil
1 pound/450 grams grilled chicken thighs, boneless, roughly chopped, and at room temperature
½ Spanish onion, cut into medium dice and cooked in a little oil over medium-high heat till tender and a little browned at the edges, then cooled (optional but recommended)
2 to 3 celery ribs, cut into small dice
Butter lettuce
4 kaiser rolls or 1 baguette, toasted if you wish
To make the mayonnaise, combine the lime juice, water, salt, fish sauce (if using), chipotles, and yolk in a bowl (or in a large glass measuring cup if using a hand blender, here, which is my preferred method). Whisk them together to dissolve the salt. Whisking continuously, add one, two, then three drops of oil to establish the emulsion, then add the remaining oil in a steady stream, until all of the oil has been incorporated and the mayonnaise is thick and sumptuous. Taste. It should be almost like a vinaigrette and plenty spicy, so add more lime juice or extra chipotles if you wish.
Combine the chicken, onion (if using), celery, and mayonnaise and stir to mix. Taste and season with more salt, chipotles, or lime juice until it's perfect. Serve with the lettuce and rolls or bread.
#
# Pasta Carbonara
SERVES 4
_When people tell me they don't have time to cook dinner for the family, I suggest this staple of the Italian kitchen and all-around favorite for its simplicity and deeply satisfying impact. This is one of my go-to meals when I want something fast, delicious, and satisfying. All of the elements—salty smoky bacon, bacon fat, pasta, cheese—are bound together by the yolk. Reputed to be served to coal-dusty miners returning from work for their meal (hence the name), this dish is a paragon of Italian simplicity and economy. The cured pork in Italy would be pancetta or_ guanciale _, salted, dry-cured hog jowl; both work well and each alters the finished dish slightly. Pancetta lacks smokiness;_ guanciale _has a rich flavor, but less meat and more fat. Bacon is what is common here in the States. All are delicious. I prefer pork belly for its perfect meat-to-fat ratio._
_Pasta carbonara luxuriates in the amazing egg yolk, which is the primary sauce ingredient. Traditionally, there's no other liquid, just yolk, but I like to add a little half-and-half; this goes outside tradition and makes purists bristle, but the half-and-half facilitates the spreading of the yolk throughout the pasta and keeps the pasta juicy rather than sticky. It should be a little liquidy when it receives the grated cheese, with the heat from the fat and pasta lightly cooking the yolk._
_As a rule of thumb, I use 2 yolks per serving and an equal quantity of half-and-half by weight (I often make this just for Donna and myself—it's a great quick lunch). And remember that the bacon can be rendered as many as 3 days before finishing the dish, cooled, and stored, covered, in the refrigerator, so that you could literally finish the entire dish in the time it takes to boil water and cook the pasta. I like to use thin spaghetti, but any shape will work. I usually just cook a whole box and save one-quarter of it for another use. An interesting variation on this dish, which we did for Eric Ripert's book_ A Return to Cooking _, was to make a risotto in the bacon pan and simply finish with a garnish of one whole yolk in the center._
8 ounces/225 grams bacon (or pancetta or _guanciale_ ), cut into strips
12 ounces/340 grams dried thin spaghetti
8 egg yolks
½ to 1 cup/170 to 225 milliliters half-and-half
Salt to taste (about ½ teaspoon, but less to none if the bacon is very salty)
2 to 2¼ cups/170 to 225 grams grated Parmigiano-Reggiano
Freshly ground black pepper
1 to 2 tablespoons minced fresh parsley (optional)
In a large sauté pan, render the bacon till it's crisp on the outside and tender inside (the process is easiest if you start the bacon in water over high heat, then turn the heat to medium-low when the water cooks off).
Meanwhile, bring a large pot of salted water to a boil and cook the pasta till it's done to your liking; drain.
Remove the sizzling bacon from the heat. Add the pasta and toss it in the bacon fat till it's nicely coated. Whisk the yolks and half-and-half to combine; pour the mixture over the pasta, continuing to toss it. Taste it and add salt if needed. When the pasta is uniformly coated with the sauce, add two-thirds of the cheese and toss again.
Serve, garnishing with the remaining cheese, plenty of freshly ground pepper, and parsley, if desired.
#
# Hollandaise and Béarnaise
IN THE SAME WAY THAT AN EGG YOLK (combined with some water) can transform ordinary vegetable oil into a creamy, ethereal sauce, so too can it work its magic on butter. I'm talking about the emulsified butter sauces so popular in the French repertoire that most carry French names (such as their word for Holland, or the name of the mountainous region of Gascony in southwestern France, Béarn, home of the legendary d'Artagnan).
I grew up watching my mom make béarnaise sauce straight out of Julia Child's seminal _Mastering the Art of French Cooking_. She raised it to the level of a one-on-one sport, my mom against the sauce. When she got into the ring with tarragon and yolks and butter, we all drew near, watching her whip that sauce. At the time we knew only that the more butter you got in there the better the sauce, but too much and it could break on you for whatever reasons decreed by the cooking gods. So when the sauce was perfect, and she risked yet another chunk of butter, you could hear gasps from the crowd.
Which is why I love béarnaise sauce.
When my parents' parties went on so long that I woke and descended the stairs to Doris Grayson's laughter, signaling to the guests that it was time to leave (as if the daylight weren't enough), Mom would sensibly repair to bed, but Dad would make the two of us eggs Benedict, poached eggs with hollandaise sauce, the famed dish created by the chef of the Waldorf Hotel as a restorative for a hungover Mr. Benedict. After breakfast, instead of sleeping after the long night of bibulous partying, Dad would be out there mowing the lawn, thereby teaching me at an early age the restorative powers conveyed by a proper eggs Benedict.
Which is why I love hollandaise sauce.
The emulsified butter sauce strikes fear into the hearts of home cooks. It should not—it's nothing more than a hot mayonnaise with more yolk for richness. It can be made exactly like a mayonnaise, with a hand blender, a whisk, or a countertop blender. And, really, any kind of fat can be used. To do it the classical French way, use clarified butter. In my last book, I used schmaltz (rendered chicken fat with onion) to make a variation on béarnaise. If someone is allergic to dairy products, pity the poor soul, but you can make a hollandaise with warm vegetable oil instead of butter.
The single element that distinguishes a hot emulsified sauce from a cold emulsified sauce is that the heat is actively expelling water from the sauce; as water is the critical element holding everything together, losing too much of it can allow all the fat droplets to group together, breaking your sauce. Of course the heat can also cook the egg, turning it to curds in your sauce, which is why many recipes recommend cooking the sauce over simmering water as opposed to the direct dry heat of flame against metal.
So, now that you know all of the forces working for and against you before you step into the ring with an emulsified butter sauce, you can relax. Have a glass of wine. About 20 minutes or up to an hour before you are ready to serve, make your sauce, as guests gather around, not like sports fans at a boxing match, but rather as an audience amazed by your cool ease with the notoriously "difficult" French butter sauce.
There are three components to these sauces: the flavoring, the egg-water mixture, and the fat. How you use each determines the nature of the sauce. The classical method is to make a vinegar and aromatic reduction first, simmering herbs, shallot, and peppercorns and straining this into your pot with the yolks. But you can simply cook shallots and freshly ground pepper and vinegar and add this without straining. Next, cook the yolks and flavoring liquid, whipping them over gentle heat until they're warm and fluffy (when in doubt, it's better to undercook; if they go past 180°F/85°C, they're going to scramble). When they're done, slowly add your butter, whipping continuously. Mom used to throw in cold chunks of whole butter; these both cooled the sauce and added more water to it. I like to melt the butter and add it warm to the warm eggs.
I'll give an example of two methods here (though the hand-blender method used for mayonnaise can also be used): a blender hollandaise, first written about, to my knowledge, by Julia in her aforementioned book, and then a classical béarnaise sauce with a true reduction.
# REUSING LEFTOVER BÉARNAISE, HOLLANDAISE, OR ANY EMULSIFIED BUTTER SAUCE
If you have leftover sauce, don't blame the sauce; I'm sure it was excellent. But don't throw it out! Put it in a glass measuring cup, cover with plastic, and refrigerate for up to 7 days. When you want to reuse, melt it in a microwave, and whisk into it 1 teaspoon water for every **⅓** cup/75 milliliters leftover sauce you are re-emulsifying. It should return to its luxurious delicacy in moments. If you need to heat it further, do so gently over low heat, and be careful not to cook it. Serve it over scrambled eggs, toast, or a steak sandwich.
# Blender Hollandaise Sauce
MAKES ABOUT ¾ CUP/180 MILLILITERS
_This quick and nearly foolproof method is just what the doctor orders on a Sunday morning when efficiency is to be desired in getting your Eggs Benedict (here) on the table. However, if you want to prepare hollandaise in the traditional fashion, follow the recipe for béarnaise sauce on here, omitting the dried tarragon from the reduction and adding a teaspoon or two of lemon juice to the finished sauce instead of fresh tarragon._
2 tablespoons fresh lemon juice
¼ teaspoon salt (½ teaspoon if using unsalted butter)
3 egg yolks
½ cup/110 grams butter, melted
Put the lemon juice, salt, and yolks in a blender. Turn the blender to medium-high, then pour the hot butter in a steady stream into the running blender until all the butter is added and the sauce is thick.
#### SEPARATING EGGS BY HAND
1. / _By far the easiest and fastest way to separate an egg is by hand. (Please don't get me started on those egg separators for sale; the best one ever invented is at the end of your arm.) This method works especially well when you're doing a lot of eggs, as here, with a recipe that calls for eight. Gently scoop up one egg._
2. / _Separate your fingers enough for the white to slip through but not the yolk. It's fine to leave on the white coils, or chalazae. If you want to pinch them off, be aware that the yolk usually breaks when you do, so hold your hand over a bowl._
3. / _Pass the yolk back and forth between your hands until all the white has fallen away, then drop the yolk into another bowl. Separating the eight eggs should take less than a minute._
# Traditional Béarnaise Sauce
MAKES ABOUT 1 CUP/240 MILLILITERS
_This is my favorite sauce in the world. It wouldn't be out of place on baked white fish, but I like to serve this elegant French sauce on a great American hamburger. I recommend going to the trouble of grinding your own beef chuck, which has a goodly amount of fat in it. Rinse it and pat it dry, then give it a good salting an hour before grinding (as for Steak Tartare,here). Garnish the burger with butter lettuce and caramelized onion, or serve as is. It's OK to use store-ground chuck, too—the focus here is on the sauce and how well it complements the grilled beef. (Needless to say, you can use this sauce on any cut of grilled beef.)_
_I'm going to give the premier method and ingredients here for the finest béarnaise (minus the clarified butter, which is traditional but unnecessary), but you could also just adapt the blender hollandaise recipe onhere by adding 1 tablespoon dried tarragon to the butter before melting it (which is what I do when there is no fresh tarragon to be had)._
# FOR THE REDUCTION:
10 black peppercorns, cracked with the flat side of a knife
¼ cup/60 milliliters white wine vinegar or tarragon vinegar
1½ tablespoons dried tarragon
1 tablespoon minced shallot
¼ cup/60 milliliters water
# FOR THE BÉARNAISE:
2 egg yolks
2 teaspoons water
1 teaspoon fresh lemon juice
1 teaspoon salt
1 cup/225 grams butter, melted
⅓ cup/10 grams minced fresh tarragon
To make the reduction, heat the peppercorns in a small sauté pan over medium heat until you can smell them. Add the vinegar, tarragon, and shallot to the pan and simmer until almost all of the liquid is gone. Add the water, return to a simmer, then remove the pan from the heat.
To make the sauce, combine the yolks, water, lemon juice, and salt in a saucepan (a 3-quart/2.8-liter saucier is best for whisking). Strain the reduction through a fine-mesh sieve into the saucepan, pressing on the ingredients to squeeze out more liquid. Whisk to combine, then hold over medium heat, whisking continuously and occasionally removing the pan from the heat to avoid cooking the eggs. You're looking for a fluffy, ribbony mixture, warm but not hot. Reduce the heat to low.
Using a spoon, let a few drops of melted butter fall into the yolks while whisking vigorously to establish the emulsion, then whisk in the remaining butter. The butter solids and water at the bottom can be added or not. (If the sauce is very thick, or if you've cooked a lot of liquid out of it, endangering its stability, the water will stabilize it.) If the sauce is too thin, continue cooking it, carefully, until it falls thickly from the whisk. (You can make the blender version by adding the hot reduction to the yolks in the blender and continuing to blend as you pour in the hot butter.)
Add the fresh tarragon and whisk to distribute it.
Serve on freshly grilled rare hamburgers (or other grilled beef or lean fish).
#
# Crème Anglaise
MAKES ABOUT 3½ CUPS/840 MILLILITERS
_This is the workhorse sauce of the dessert kitchen, the all-purpose vanilla sauce, which can be transformed into numerous other preparations. It scarcely differs from the crème brûlée on the next page except in the way it's cooked. Indeed, you can enrich a crème anglaise by replacing half the milk with cream, a strategy I've seen in many kitchens. Feel free to do this if you wish, though you may face the wrath of traditionalists. Vanilla sauce can be manipulated in various ways to different ends: cooked in a water bath, it's a custard; frozen, it's ice cream; thickened with flour or cornstarch, it becomes pastry cream; and as is, of course, it's a basic sauce to eat with cakes, soufflés, and other desserts. But at its heart it's simply egg yolks cooked with milk and sugar, flavored with vanilla._
3 cups/720 milliliters milk
Pinch of salt
1 vanilla bean, split lengthwise
¾ cup/150 grams sugar
9 egg yolks
Combine the milk, salt, and vanilla bean in a small saucepan and bring to a simmer over medium-high heat. Remove the pan from the heat and let the bean steep for 15 minutes. With a paring knife, scrape the seeds from the pod into the milk. (Put the empty pod in your sugar bowl or bag to gently infuse the sugar.)
Combine the sugar and yolks in a medium bowl and whisk vigorously for 30 seconds or so (this will help the sugar to begin dissolving and will also help the egg cook more evenly).
Fill a large bowl with half ice and half water, and float a second bowl in the ice bath. Set a fine-mesh strainer in the bowl.
Over medium heat, bring the milk back to a simmer, then pour it slowly into the yolks while whisking continuously. Pour the mixture back into the pan and continue stirring over medium heat until the mixture is slightly thick, or _nappé_ —it should be pourable, but if you dip a spoon in it, it should be thick enough on the spoon to draw a line through. This will take 2 to 4 minutes.
Pour the sauce through the strainer into the bowl set in the ice bath. Stir the sauce with a rubber spatula until it is cold. Use right away or cover with plastic wrap, pressing down on the plastic so that it lies on the surface of the sauce, and refrigerate for up to 1 week.
# Crème Brûlée
SERVES 4
_This is as simple and perfect as a dessert gets. Because there are so few ingredients, their care is critical. Egg cooked fast and at high heat turns hard. Or it curdles, crosses its fleshy arms in anger at having been ill used. So you must treat the yolk with thoughtfulness, authority, and finesse. It's the yolk's richness and capacity to set as it heats the fluid dairy that make it the queen of this dessert and so many sauces._
_I'm offering here a traditional crème brûlée, nothing but milk and cream, vanilla, and sugar brought together by the enveloping egg yolk. The yolk gathers these ingredients and works them in concert to create a sweet pudding with a crunchy burnt-sugar top. I always tap the top with my spoon to hear the click on the burnt, candied surface, and then tap a little harder to break through to the cream, a pleasure like a child's in stepping on the frozen surface of a puddle. I then scoop up the vanilla custard below, to combine on the palate smooth luxury and brittle melting sweetness._
_A custard such as this requires gentle heat, thus a water bath, the self-regulating thermometer of H 2O, which will never go higher than 200°F/95°C in an oven set between 200° and 300°F/95° and 150°C. The water will remain below a boil if uncovered._
_Please don't desecrate this lovely dessert with vanilla extract or fancy aromatics. Appreciate it for what it is: one of the perfect dishes in the western repertoire._
1 cup/240 milliliters milk
1 cup/240 milliliters heavy cream
Pinch of salt
1 vanilla bean, split lengthwise
½ cup/100 grams plus ¼ cup/50 grams sugar
8 egg yolks
Preheat the oven to 300°F/150°C. Place four 4- to 5-ounce/120- to 150-milliliter ramekins in a large sauté pan or roasting pan and fill the pan so that the water comes three-quarters of the way up the sides of the ramekins. Remove the ramekins and place the pan of water in the oven.
Combine the milk, cream, salt, and vanilla bean in a small saucepan and bring to a simmer over medium-high heat. Remove the pan from the heat and let the bean steep for 15 minutes. With a paring knife, scrape the seeds from the pod into the milk-cream mixture. (Put the empty pod in your sugar bowl or bag to gently infuse the sugar.)
Combine ½ cup/100 grams of the sugar and the yolks in a medium bowl and whisk vigorously for 30 seconds or so (this will help the sugar begin to dissolve and will also help the egg cook more evenly). Slowly pour the cream mixture into the yolks while whisking continuously.
Pour the custard into the ramekins. Cover each with a piece of parchment paper followed by foil and put them in the water bath. Cook the custards until just set, about 30 minutes. Uncover them and allow to cool. (If you intend to serve them the following day, cover them again and refrigerate; remove them from the refrigerator several hours before serving to allow them to come to room temperature.)
Top each custard with enough of the remaining ¼ cup/50 grams sugar to coat the entire surface and pour off the excess. With a propane torch, heat the sugar till it melts, bubbles, and caramelizes—when it's cool, the browned sugar should create a delicate crust. Serve immediately.
# Pastry Cream
MAKES ABOUT 3½ CUPS/840 MILLILITERS
To make a great pastry cream with which to fill profiteroles (here) or undergird a berry tart or layer a cake, simply thicken our all-purpose vanilla sauce with a milk-and-cornstarch slurry. Follow the instructions for crème anglaise here, up through pouring the milk-egg mixture back into the pan. Before bringing it back up to heat, add 3 tablespoons cornstarch mixed with 3 tablespoons milk. Cook the sauce till it thickens, then, rather than straining it, simply submerge the pot in an ice bath to cool it and whip in 1 to 2 ounces/30 to 60 grams room-temperature butter while it's still warm enough to melt the butter. Keep stirring till the sauce has cooled. Refrigerate the pastry cream with plastic wrap pressed down on its surface so that it doesn't develop a skin.
# Vanilla Ice Cream
MAKES ABOUT 3½ CUPS/840 MILLILITERS
_This is, more or less, a crème brûlée cooked on the stovetop and then frozen, or a frozen crème anglaise, or a rich pastry cream that's thickened by freezing rather than with starch. The basic vanilla creams are all very similar, demonstrating the range of what can be done with the combination of dairy, sugar, and yolks. The yolks give the ice cream its beautiful color, richness, and depth of flavor._
1½ cups/360 milliliters milk
1½ cups/360 milliliters heavy cream
Pinch of salt
1 vanilla bean, split lengthwise
¾ cup/150 grams sugar
9 egg yolks
Combine the milk, cream, salt, and vanilla bean in a small saucepan and bring to a simmer over medium-high heat. Remove the pan from the heat and let the bean steep for 15 minutes. With a paring knife, scrape the seeds from the pod into the milk-cream mixture. (Put the empty pod in your sugar bowl or bag to gently infuse the sugar.)
Combine the sugar and yolks in a medium bowl and whisk vigorously for 30 seconds or so (this will help the sugar begin to dissolve and will also help the egg to cook more evenly).
Fill a large bowl with half ice and half water and float a second bowl in the ice bath. Set a fine-mesh strainer in the bowl.
Over medium heat, bring the milk-cream mixture back to a simmer, then pour it slowly into the yolks while whisking continuously. Pour the mixture back into the pan and continue stirring over medium heat until the mixture is slightly thick, or _nappé_ (it should be pourable, but if you dip a spoon in it, it should be thick enough on the spoon to draw a line through. This will take 2 to 4 minutes.
Pour the sauce through the strainer into the bowl set in the ice bath. Stir the sauce with a rubber spatula until it is cold. Cover and refrigerate till it's thoroughly chilled, preferably overnight. The colder it is before going into the ice cream machine, the better.
Freeze according to your machine's instructions.
# Poire Williams Sabayon
SERVES 4
_This yolk-based dessert sauce is typically made with a sweet wine such as a sauterne or, in Italy, where it's called_ zabaglione _, with an Italian sweet wine such as marsala. For this recipe I used a Michigan-made pear eau de vie; you can also use Grand Marnier or another favorite brandy or liqueur._
_The sauce is made by whipping yolks, sugar, and wine over a double boiler until the yolks are cooked and the sauce has become thick and ribbony. It can be served as is in small cups, or poured over fruit or cake. You could also whip the egg whites with sugar and make_ Île Flottante _(here)._
4 egg yolks
¼ cup/50 grams sugar
Pinch of salt
¼ cup/60 milliliters pear eau de vie (or other suitable spirit)
2 tablespoons water
1 teaspoon fresh lemon juice (optional)
Grated lemon zest, for garnish
In the top of a double boiler or a large metal bowl set over simmering water, combine all of the ingredients except the lemon zest and whisk continuously until the mixture is warm, has quadrupled in volume, and becomes a satiny, ribbony sauce, about 10 minutes. Remove from the heat and serve warm, garnished with grated lemon zest.
The egg white is an astonishing culinary lever. A mixture of a dozen or so proteins, water, and a few trace elements, it is designed to protect the yolk from anything dangerous, whether virus, bacterium, or predator. But the feats those proteins can achieve in the kitchen are as diverse as they are extraordinary.
Egg whites act as a binder for the Seafood Roulade (here) or any mousseline; they set cream for the Orange-Ginger Panna Cotta (here); they clarify stock for Turkey Consommé (here); and they lend body—and nutrition!—to cocktails, such as the Clover Club Cocktail (here). They can also be used as a garnish, and are a powerful leavening device.
The yolk is a kind of diva in the kitchen. The white is more akin to an Olympic gymnast—its range and power are nothing short of astonishing.
#
# Seafood Roulade with Scallops and Crab
MAKES 8 (3-OUNCE) PORTIONS
_The foundation of this dish, a mousseline, is a fundamental preparation with a broad reach and infinite variations. A mousseline is simply a white meat or fish pureed with cream and bound by egg white—about two parts meat to one part cream, all of it bound with about 10 percent egg. Whole eggs can be used for richness, but for a very light, delicate mousseline I prefer egg whites only, which bind the meat and fat into a delicate but sliceable whole. The mousseline can be turned into pike quenelles or a ravioli stuffing of chicken with roasted garlic. It's most commonly used with fish because the lightness of the fat (cream) and texture (from the egg white) enhances the flavor of the fish without overpowering it._
_I'm offering here a shrimp mousseline because shrimp reliably binds with the cream and egg for a great texture. Salmon makes a terrific mousseline, too. Scallops, by contrast, carry inconsistent levels of water based on how they've been stored, so the texture of the cooked mousseline can vary. But I love the flavor of scallops and their impeccable whiteness as a garnish, so I've included them in the roulade, along with lump crabmeat, which provides an excellent flavor parallel. For an even more dramatic preparation, steep ½ teaspoon crumbled saffron in the cream, then chill it, for a color that will make the interior crab and scallops especially vivid._
_The roulade can be sliced and served cold with Lemon-Shallot Mayonnaise (here), which is how I prefer it. Or it can be sliced and gently sautéed in butter to warm through, then served with a light butter-and-lemon sauce (heat some lemon juice and shallot with a tablespoon of water or white wine and swirl in pats of butter) or a shellfish sauce made from the shrimp shells (as in the photo here). This mixture can also be stuffed into sheep casings (chop the scallops) and poached, then sautéed for outstanding seafood sausages._
1 tablespoon butter
1 leek, white part only, finely chopped
1 pound/450 grams peeled, deveined shrimp
2 egg whites
1 teaspoon salt
1 cup/240 milliliters heavy cream
4 ounces/120 grams scallops, cut into chunks if large or whole if small
4 ounces/120 grams lump crabmeat
2 tablespoons minced fresh chives
Heat the butter in a small sauté pan over medium heat. Add the leek and sauté until tender but not brown. Transfer to a bowl, cover, and refrigerate until chilled.
Puree the shrimp with the egg whites and salt in a food processor. With the machine running, slowly add half of the cream through the feed tube. The mixture should be stiff enough to shape. Continue adding the rest of the cream with the machine running.
Transfer the shrimp mousseline to a mixing bowl and add the chilled leeks, scallops, crab, and chives, gently folding to distribute everything evenly.
Wet your counter slightly and lay out a sheet of plastic wrap (use Glad wrap if you're concerned about cooking in plastic), at least 2 feet/60 centimeters long. Spoon the seafood mixture along the center of the plastic wrap. Fold the plastic wrap over the mousseline and roll it into a tube about 2½ inches/6 centimeters in diameter. Twist each end of the plastic wrap to form a tight roulade as you roll it on the counter. If it gets out of shape on you, unroll it onto a new sheet of plastic and start again.
Bring a large pot of water to 180°F/82°C. Drop the roulade into the water and weigh it down with an appropriately sized plate to keep it submerged. Cook the roulade, maintaining a water temperature of between 170° and 185°F/77° and 85°C, until an instant-read thermometer reads between 140° and 150°F/60° and 65°C when inserted into the center of the roulade, 45 to 50 minutes.
While the roulade is cooking, fill a large bowl with half ice and half water. When the roulade is done, submerge it in the ice bath until thoroughly chilled, 15 minutes or so. Remove the plastic wrap and serve (see the headnote for suggestions).
#### SEAFOOD ROULADE
1. / _Mousseline_ mise en place _(clockwise from top left): cream, herbs, interior garnish (crabmeat, scallops, and leeks), salt, shrimp, and egg white._
2. / _First, puree shrimp and egg white._
3. / _After you've slowly added the cream, with the blade running, the result will be a stiff shrimp paste, or mousseline._
4. / _Remove the mousseline to a bowl and fold in the garnishes, here crabmeat and scallops._
5. / _Last, add the leeks and stir until the garnishes are uniformly distributed._
6. / _Spread the mousseline onto plastic wrap to shape it._
7. / _Form the roulade by shaping it into a cylinder._
8. / _Press your hand or a flat edge such as a baking sheet into the cylinder to make it as tight as possible._
9. / _Roll the roulade, holding both ends of the plastic wrap, until it's a tight cylinder. It's now ready to poach._
10. / _The roulade, cooked and removed from the plastic wrap._
11. / _Slice the roulade with a single forward stroke, if possible._
12. / _These slices of roulade can be eaten cold, with some Lemon-Shallot Mayonnaise (here), or they can be sautéed if you'd like to serve them hot._
#
# Turkey Consommé
MAKES 8 (4-OUNCE) PORTIONS
_One of the most amazing things that egg whites can do is clarify a cloudy stock into a crystal-clear elixir for what we call consommé. In culinary school we were taught to make a chicken consommé because, I suppose, chicken stock was so plentiful. We were taught that it should be_ crystal _clear. Here was how the school stated the standard of clarity: you should be able to read the date on a dime at the bottom of a gallon of it. I loved that. I wrote about it. Some people have emailed me photos of dimes in their soup._
_I've noted earlier that certain comments from chefs stick with you. I learned how to make chicken consommé in my first kitchen at the Culinary Institute of America from Michael Pardus. In my last kitchen, at the American Bounty restaurant, the chef-instructor, Dan Turgeon, put a soup sampler on the menu, one sample of which was turkey consommé. It was so incredibly rich and tasty that Turgeon said, "I will never make chicken consommé again." Ever. Period. I loved how emphatic he was. And there's a great logic to this statement. Turkey_ is _a lot tastier than chicken, and more interesting. And turkey is abundant, available, and inexpensive._
_So turkey consommé it is. And here's how it works. The tightly bundled strands of protein in the egg whites unfold in the heat and create a kind of mesh that traps all of the fine fragments responsible for making a clear soup cloudy and dull looking as opposed to bright and clear as tea. It's also an excellent example of craftsmanship in the kitchen, using your knowledge and skill to perfect a common preparation._
_This is a fabulous soup to make if you prepare stock from the turkey carcass after Thanksgiving (there's a recipe for turkey stock at ruhlman.com). The key point to remember is that the egg whites also trap some flavor, so it's important to include flavorful ingredients in the egg-white mixture. The soup is fun to eat on its own, but more impressive if you add some garnish so that the clarity shines through._
4 egg whites, lightly whipped
½ Spanish onion, chopped
1 carrot, peeled and chopped
1 celery rib, chopped
12 ounces/340 grams ground turkey
1½ quarts/1.5 liters turkey stock
Optional: chopped plum tomato, fresh thyme, chopped fresh parsley, cracked black peppercorns, bay leaf
Combine all of the ingredients in a pot, preferably one that is taller than it is wide (too wide a pot spreads out the clarification and allows too much reduction during cooking). Stir the ingredients to distribute the egg white.
Place the pot over high heat and stir vigorously with a whisk. Then switch to a flat-edged wooden spoon, dragging it along the bottom to prevent the egg white from sticking and scorching. As the liquid gets hot, the protein will begin to coagulate and rise to the top. Continue to stir gently to make sure nothing is sticking to the bottom. As the liquid reaches a simmer, the solid ingredients will come together in a mass at the top, known as a raft. As the raft is forming, stop stirring and allow it to come together. Lower the heat before it boils, letting it get hot enough just to simmer over the raft and sink down.
You should be able to see how clear the stock is at this point. Continue to simmer like this for at least 45 minutes and up to an hour. Don't let it boil or the raft will disintegrate. After it's cooked, ladle the consommé through a strainer lined with a coffee filter. Your liquid should be perfectly clear. Taste and add salt if necessary. Serve immediately in warm serving bowls, into which you've divided your garnishes (see below), or cover the consommé and store in the refrigerator until you're ready to reheat and serve.
# SUGGESTIONS FOR GARNISH:
2 tablespoons small-diced carrot
2 tablespoons small-diced celery
1 teaspoon vegetable oil
4 shiitake mushrooms, stemmed
1½ tablespoons shallot, finely minced
Blanch the carrot and celery together in boiling water for 20 seconds, then strain under cold running water until thoroughly chilled.
Heat the oil in a small sauté pan over high heat, and sauté the mushrooms for about 1 minute per side. Let them drain on paper towels. Cut them into fine julienne. Combine the mushrooms, shallot, carrot, and celery.
#### TURKEY CONSOMMÉ
1. / _Whisk in the egg whites, then stir continuously with a flat-edged spoon to prevent sticking._
2. / _Once the stock is hot, the egg whites will cook and rise to the surface._
3. / _As the stock comes to a simmer, the egg whites will act as a filter._
4. / _After the stock has cooked for 45 to 60 minutes, strain it through a coffee filter._
5. / _Place your garnishes in a bowl. Both bowl and garnishes should be warm, if not hot._
6. / _Add your piping-hot consommé to your soup setup. Pouring the soup is best done at table, for the visual pleasure of your family or guests, and to ensure hot soup!_
#
# Orange-Ginger Panna Cotta
SERVES 4
_The luxuriously creamy panna cotta (Italian for cooked cream) became a popular dessert in the 1990s when pastry chef Claudia Fleming's buttermilk panna cotta became renowned and much imitated—and for good reason. Like nearly all panna cottas, that one was cream set with gelatin. For a purer flavor, you can instead set the cream with egg whites. It's a great way to make use of extra whites and yields a vegetarian dessert (since most gelatin is made from cow or pig parts). Panna cotta is often served unmolded. But because egg whites are sticky (and don't melt when warmed as gelatin does), serving this panna cotta unmolded is a little trickier, so I recommend serving it in the ramekins in which it was cooked. (If you do want to try to serve it unmolded, use a round ramekin or mold, and oil it first.)_
1 cup/240 milliliters heavy cream
½ cup/120 milliliters whole milk
¼ cup/50 grams sugar
Pinch of salt
1 piece of ginger, roughly 2 inches/5 centimeters long, peeled and thinly sliced
1 whole clove
Peel from ½ orange, pith removed
½ vanilla bean, split lengthwise
4 egg whites
Preheat the oven to 300°F/150°C. Place four 4- to 5-ounce/120- to 150-milliliter ramekins in a large sauté pan or roasting pan and fill the pan so that the water comes three-quarters of the way up the sides of the ramekins. Remove the ramekins and place the pan of water in the oven.
In a small saucepan combine the cream, milk, sugar, salt, ginger, clove, orange peel, and vanilla bean and bring to a soft boil over medium-high heat. Be sure not to burn or scorch the cream, and keep an eye on it as it heats—if it boils over, you'll have a mess and a less tasty panna cotta. Once a boil has been reached, remove the pot from the heat, cover it, and allow it to steep for 1 hour.
In a medium bowl, beat the whites together so they are just combined. Strain the cream base into the egg whites and whisk together. Scrape the seeds from the vanilla bean into the cream mixture. (Put the empty pod in your sugar bowl or bag to gently infuse the sugar.)
Fill the ramekins with the cream mixture. Using sturdy tongs or a spatula or both, place the ramekins in the water bath. Cover the pan with foil—try to do this without moving the pan or the rack, risking water sloshing into the ramekins—and make several slices in the foil to allow the steam to escape. Cook until the panna cotta is set (there should be little to no jiggle when shaken), 35 to 40 minutes.
Remove the ramekins from the water bath, allow to cool, then cover and refrigerate. Serve chilled.
#
# Clover Club Cocktail
SERVES 4
_Without question, one of the finest uses of the egg white is in a cocktail, giving the libation's flavors lovely hang time in the mouth. When we take care to make a great cocktail, we want some lingering, and the body that the protein adds to a cocktail gives the loose fluids this quality. It is also a handy excuse: now you can call your cocktail a protein snack. Many drinks use an egg white—the gin fizz, the whiskey sour, and the following, a cocktail that I learned about via a nifty e-book called_ Twenty-Five Classic Cocktails _. As with many cocktails, a slight variation changes its name. In this case, if you add ¾ ounce apple brandy it becomes a Pink Lady, also a fabulous drink but not something I'm inclined to order in public. So if this is what you're after, I recommend that you add some Key lime juice instead of the lemon juice and call it a Key Sunrise._
_The white needs to be broken up from its initial viscous state, something bartenders accomplish by giving it a dry shake, that is, a shake without ice; when the egg white has begun to be mixed, ice is added and the drink is shaken further to mix and chill it. I find that it's easier to mix such drinks in a blender, then add the ice, then strain it. This drink calls for grenadine for color and sweetness. It's worth seeking out a true pomegranate syrup rather than a mass-produced grenadine, which is little more than colored sugar water. This cocktail serves four but could easily be doubled in a blender. You could also use a hand blender or a whisk and make a batch in a large glass measuring cup._
6 ounces/180 milliliters gin
2 ounces/60 milliliters fresh lemon juice
2 ounces/60 milliliters simple syrup (equal parts sugar and water, heated until the sugar is dissolved)
1 ounce/30 milliliters grenadine (pomegranate syrup)
2 egg whites
Freeze four martini or coupe glasses.
Combine all of the ingredients in a blender or, if using a hand blender, in an appropriately sized vessel. Blend till frothy. Fill the container with ice and stir for 1 to 2 minutes, then pour through a fine-mesh strainer into the icy glasses.
# The Amazing Meringue
OF ALL THE CULINARY MAGIC MADE POSSIBLE by the egg, perhaps the most extraordinary feats are accomplished when we take the white away from the yolk and whip it into a foam. The white is 90 percent water, but the rest of it is almost entirely composed of protein molecules, tight coils of amino acids. When the proteins are vigorously knocked about with a whisk, they unwind and form webs that capture air bubbles. The egg white's ability to trap infinitesimal bubbles of air, creating what we call a meringue—egg white whipped to many times its initial volume—allows for all kinds of delectable creations in the kitchen.
Meringue leavens cakes, makes chocolate sauce airy for mousse, lightens pastry cream (called Chiboust), and makes soufflés go pouf! Cook a meringue in moist heat, directly in liquid, in a water bath, or in steam, and it becomes a pillowy dessert. Simply bake a common meringue and it becomes a delicious, crunchy confection—a protein-and-sugar cookie—or a shell in which to put other ingredients. Add a little flour and bake the meringue and it becomes angel food cake.
Meringue mixed with hot sugar and honey is the delectable mortar holding together nuts and fruit in a nougat. Mixed with gelatin, meringue becomes marshmallow. Mixed with almond flour it becomes the cookie sandwiching a creamy filling for _macarons_. Mixed with confectioners' sugar it becomes royal icing, a satiny white cream used for decoration. Mixed with sugar and butter it becomes a tasty Italian buttercream icing.
Sugar or some sort of sweetener is always added to meringue for sweet preparations, but how you add the sugar determines the type of meringue you're making. If you simply add the sugar to the meringue while whipping it, it's called a common, or French, meringue. If you cook the egg whites and sugar over a water bath to dissolve the sugar, and then whip it, it's called a Swiss meringue, and if you cook the sugar separately (typically to 250°F/120°C or higher) and add it hot to the whipping egg whites, it's an Italian meringue. The amount of sugar varies from chef to chef. A traditional ratio, and what is commonly taught in cooking schools, is two parts sugar to one part egg white by weight. That's a very sweet meringue and about all the sugar the egg-white foam can carry. Many chefs reduce the amount of sugar by as much as half, to as little as equal parts sugar and egg white, which is my preference. Other chefs use confectioners' sugar, which contains some cornstarch and so absorbs water.
Meringues are not stable. They can be overwhipped and become dry and clumpy. If they are left to sit for too long, the water leaks out of them. And they can be difficult to work with if a drop of yolk, or oil, or dish soap somehow gets into your mixing bowl. Acid, in the form of lemon juice or powdered cream of tartar, helps to stabilize egg whites and make a sturdier meringue. As this also balances sweetness, acid is always good to add to meringues. Some chefs, when making a meringue with cooked sugar added late in the mixing, recommend adding a little raw sugar when they start the meringue for stability as well.
#### WHIPPING EGG WHITES
1. / _Egg whites whipped until foamy. This is the stage when you can begin to add the sugar._
2. / _Whip the egg whites on high, adding the sugar slowly as you do._
3. / _These egg whites are whipped to stiff, glossy peaks._
4. / _These egg whites have been overwhipped. Notice the grainy, lumpy texture, compared with the creamy, glossy look of the egg whites in illustration 3._
# WHAT MERINGUE CAN BECOME AND DO
#### Cooked crisp:
Crispy cookie or shell ( _vacherin_ and _pavlova_ )
_Macarons_
#### Set with sugar:
Nougat
#### Set with gelatin:
Marshmallow
#### Cooked in moist heat:
_Île flottante_
#### To lighten texture:
Mousse
Chiboust
Semifreddo
#### As an icing base or component:
A range of icings
#### Raw (or browned) garnish:
Pie/tart topping
Baked Alaska
#### As a leavener:
Soufflé
Cake
#### Cooked with flour:
Angel food cake
#
# Crunchy French Meringue
MAKES ABOUT 32 (2-INCH/5-CENTIMETER) COOKIES
_This is the most basic, easiest form of meringue you can make, and it creates delicious cookies._ Macarons _, shells of meringue mixed with almond flour sandwiching a creamy filling, have all but avalanched this country. It's no wonder, since done well, they're ethereal, and for decades America has known only the coconut ball version. Meringue can also be piped into varying shapes to create a dessert base, such as a shell in which to serve a mousse. My kids simply eat them as they would any crunchy cookie._
_A basic ratio for crispy meringue is one and a half parts sugar to one part egg white. An egg white weighs about 1.4 ounces/40 grams, so you would measure by weight 2.1 ounces/60 grams of sugar (seriously, if you're weighing, use your scale's gram button—grams are so much easier to work with). There's plenty of leeway when making meringue—you can reduce the ratio of sugar to egg white to as little as one to one or increase it to as much as two to one._
_I always use a standing mixer, but these can be whisked by hand or using a hand blender fitted with the whisk attachment._
3 egg whites
¼ teaspoon cream of tartar or 1 teaspoon fresh lemon juice
1 teaspoon pure vanilla extract
¾ cup/150 grams sugar
Preheat the oven to 180°F/85°C, or its lowest setting. Line two baking sheets with parchment paper or Silpats and set aside.
Put the egg whites, cream of tartar or lemon juice, and vanilla in a standing mixer fitted with a whisk attachment. Mix on high for 30 seconds, then gradually pour in the sugar. Continue to mix until the meringue is satiny and white and holds gentle peaks.
Pipe or spoon the meringue in whatever shape you wish onto the prepared baking sheets. Put the baking sheets in the oven with the door ajar and bake until the meringues are dry all the way through, 4 to 8 hours (they can't be overbaked if your oven temperature is low; you're really just dehydrating them).
#
NOUGAT AND MARSHMALLOWS ARE TWO preparations that are just too cool not to write about. They're similar in construction but very different when finished, largely due to the sugar content and the temperature that sugar reaches. As with most confections, you'll need a candy thermometer to do the job properly. Arguably these are more sugar concoctions than egg preparations, but each relies on meringue to hang the cooked sugar on, as it were. The nougat is hard but chewy, set by the high volume of heavily cooked sugar, and marshmallows are of course soft, meringue set with gelatin.
# Nougat
MAKES ABOUT 24 STRIPS
_I learned this nougat while working on the_ Bouchon Bakery _cookbook, wherein executive pastry chef Sebastien Rouxel makes a version for the bakery. I've simplified his recipe for the home kitchen (corn syrup instead of glucose and no cocoa butter, which he uses to get a cleaner cut). But it all basically comes down to the temperature you cook the sugar to; because it's added hot to the egg whites, it ultimately cooks the whites and becomes a foamy white candy into which nuts and fruit are stirred. I love the combination of pistachio, almond, cashew, and cherry, but virtually any toasted nuts or dried fruits can be used._
3 cups/600 grams plus 1 tablespoon sugar
⅓ cup/100 grams light corn syrup
½ cup/120 milliliters water
1 cup/300 grams high-quality honey, preferably local to you
3 egg whites
Pinch of salt
3 to 4 cups toasted unsalted nuts and dried fruit (a mixture of almonds, cashews, pistachios, hazelnuts, walnuts, dried cherries, cranberries, or anything else you choose)
Combine the 3 cups/600 grams sugar, corn syrup, and water in a high-sided saucepan, bring to a simmer, and cook till it reaches 298°F/148°C, about 10 minutes. Remove the pan from the heat.
In a small saucepan, bring the honey to a simmer and cook it till it reaches 257°F/125°C. Remove the pan from the heat.
Place the egg whites and salt in the clean bowl of a standing mixer fitted with a whisk attachment and beat on high speed. Once a foam has begun to form, add the remaining 1 tablespoon sugar. Continue to beat the eggs until firm, glossy peaks form, 3 to 5 minutes.
Reduce the mixer speed to medium-high and slowly pour in the warm honey. Whip the whites for a few more minutes. Slowly pour in the cooked sugar mixture and continue mixing until the mixture cools to the point that you can hold your hand against the bowl, 15 to 20 minutes.
Replace the whisk attachment with a paddle attachment and paddle in the nuts and fruit.
Line a baking sheet with parchment paper and grease it generously with butter. Lay another sheet of parchment on top and rub it well, so that both sheets are equally well buttered, then set aside the top sheet. Butter a rubber spatula or coat it with nonstick baking spray, and scrape the nougat into the center of the buttered parchment on the baking sheet. Cover it with the second buttered sheet of parchment. Use a rolling pin to roll the nougat out to about 1 inch/2.5 centimeters thick. Allow it to cool completely, several hours or overnight, then slice into strips (about 1 by ½ by 6 inches/2.5 by 1 by 15 centimeters). This will keep for a week in an airtight container.
#### NOUGAT
1. / _Egg whites heavily whipped with cooked honey and sugar will nearly fill your mixing bowl._
2. / _Paddle in the nuts and fruit._
3. / _Spread the nougat out on buttered parchment paper._
4. / _Cover it with a second piece of buttered parchment and use a rolling pin to flatten the nougat to an even thickness._
5. / _Slice the uneven edges off to achieve uniform strips._
6. / _Slice the strips into individual pieces._
# Marshmallows
MAKES ABOUT 100 (1-INCH/2.5-CENTIMETER) SQUARE MARSHMALLOWS
_For me, marshmallows are an example of how processed food has distanced us from real food. I had a childhood filled with marshmallows out of a bag, whether roasted on a stick over a fire, sandwiched between graham crackers and chocolate, melted and mixed with Rice Krispies, or dropped into steaming mugs of hot chocolate after sledding on dark Cleveland winter afternoons. The realization that one could actually_ make _marshmallows came only when I was an adult and began exploring and writing about the world of the professional kitchen. As restaurants began to search for more interesting and varied sweet treats and confections, they began offering house-made marshmallows as components of desserts. Marshmallows can be made with gelatin alone or, as here, with a combination of gelatin and egg white. Marshmallows can be flavored with citrus, with jams added to the warm sugar, or, in treating them as a variation on nougat, here with honey. The flavor of the honey is dominant in the finished marshmallow, so use the best-quality honey you can find._
½ cup/64 grams cornstarch
½ cup/50 grams confectioners' sugar
2 tablespoons unflavored gelatin
¾ cup/180 milliliters water
1 cup/200 grams plus 1 tablespoon granulated sugar
½ cup/170 grams high-quality honey, preferably local to you
2 tablespoons light corn syrup
3 egg whites, at room temperature
¼ teaspoon salt
1 tablespoon pure vanilla extract
Mix the cornstarch and confectioners' sugar together till uniformly combined. Line a rimmed baking sheet with parchment paper and dust it generously with some of the cornstarch mixture.
In a microwavable ramekin, sprinkle the gelatin over ¼ cup/60 milliliters of the water and allow it to absorb the water, or "bloom."
Combine the 1 cup/200 grams granulated sugar, honey, corn syrup, and the remaining ½ cup/120 milliliters water in a medium saucepan over medium-high heat and bring to a simmer. Cook the mixture till it reaches between 265°F/130°C and 285°F/140°C and remove from the heat. Allow the sugar syrup to cool to 210°F/100°C.
While the sugar is cooling, heat the bloomed gelatin in a microwave for 20 or 30 seconds to melt the gelatin.
Place the egg whites and salt in the clean bowl of a standing mixer fitted with a whisk attachment and beat on high speed. Once a foam has begun to form, add the remaining 1 tablespoon sugar. Continue to beat the eggs until firm, glossy peaks form, 3 to 5 minutes.
If the sugar is not cool yet, turn the mixer to low until it is. When the sugar syrup has cooled, turn the mixer to medium and begin to pour the syrup in between the spinning whisk; pour near the side of the bowl.
Once all of the syrup is in the egg whites, add the liquid gelatin and beat on medium-high speed for another 3 to 5 minutes, and then add the vanilla. The marshmallow base will be fluffy and white.
Using a spatula, spread the marshmallow onto the baking sheet with the cornstarch. Spread it toward the corners and continue to spread it out evenly (it should fill about half the baking sheet). Alternatively, you can put the marshmallow base into a piping bag and pipe shapes onto the baking sheet.
Dust the top of the marshmallow with more of the cornstarch mixture and cover it with another layer of parchment. Using a rolling pin or anything with a straight edge, such as a long wooden spoon handle, spread the marshmallow out so that it's smooth and level with the edges of the baking sheet, filling roughly half the sheet. Let the marshmallow set in a cool, dry place for 3 to 5 hours.
Once it is cool and set, cut the marshmallow into squares or other shapes with a pair of scissors or trim with a knife. Keep your scissors or knife clean. Once you cut your shapes, toss them with the remaining cornstarch mixture to coat. They're best eaten that day but can be kept for several days in an airtight container.
#
# Île Flottante
SERVES 4
_I first had_ île flottante _, floating island, at the French Laundry restaurant in the Napa Valley and fell in love with its texture. Traditionally, meringue is dropped off spoons into sweet simmering milk (filled with air, it floats), cooked for a couple minutes, flipped, and served. The French Laundry bakes tiny ones in small foil cups in a water bath, unmolding for an easy and clean presentation (also filling them with chocolate mousse, serving on a crème anglaise with mint oil, and garnishing with chocolate). I do mine in small ramekins, and I find that steaming them is an easy, fast, and excellent way to cook the meringues._
_Serve with Poire Williams Sabayon made from the yolks and garnish with shaved chocolate or sliced fruit and berries._
4 egg whites
¼ teaspoon cream of tartar or ½ teaspoon fresh lemon juice
1 teaspoon pure vanilla extract
⅔ cup/130 grams sugar
1 recipe Poire Williams Sabayon (here)
Butter four 3- to 4-ounce/85- to 115-milliliter ramekins. Place a rack or steamer basket in a pot large enough to hold all four ramekins. Pour in enough water to just reach the rack and bring it to a simmer over high heat.
Combine the egg whites, cream of tartar or lemon juice, and vanilla in the bowl of a standing mixer fitted with a whisk attachment. Begin mixing on high. Slowly pour in the sugar. Continue mixing until you have stiff but still soft, glossy peaks.
Fill each ramekin with the meringue and level off the top. Put them on the rack in the pot of simmering water, cover, and steam until the meringues are puffy and firm, 3 to 4 minutes. These can be unmolded and served warm, left to cool to room temperature, or refrigerated for up to 3 hours and served cold or at room temperature.
To serve with Poire Williams Sabayon, spoon about ¼ cup/60 milliliters of the sauce into the center of each plate and top with an Île Flottante.
#
# Angel Food Cake with Berry Compote
MAKES 1 CAKE, SERVES 12
_Angel food is one of my favorite cakes and it's probably the easiest to be successful with—a meringue into which a little flour has been folded. The flour gives it a bit more structure, a chewiness from the gluten, as compared to a floating island or marshmallow. Like most cakes, this freezes well. It can be made 3 or 4 weeks in advance of using it, if you wish; double-wrap it in plastic and freeze, then let it defrost at room temperature at least an hour before you want to serve it._
_I loved angel food cake so much as a boy that my mom created a special version for me that was iced with whipped cream and topped with chopped Heath bars. You can top this cake with some if you like—it will go great with the berry compote as well. It's an easy, lovely dessert for early summertime, when ripe berries are plentiful._
# FOR THE CAKE:
12 egg whites
Pinch of salt
1 tablespoon fresh lemon juice
2 cups/400 grams sugar
1 teaspoon pure vanilla extract
½ teaspoon pure almond extract
1 cup/140 grams flour
# FOR THE BERRY COMPOTE:
1 cup/140 grams raspberries
1½ cups/210 grams strawberries, sliced
1 cup/140 grams blueberries
½ cup/70 grams blackberries, halved
1 tablespoon fresh lemon juice
⅓ cup/65 grams sugar
Pinch of salt
2 tablespoons Grand Marnier
Preheat the oven to 350°F/180°C.
Whisk the egg whites in a standing mixer with the salt and the lemon juice. Begin to whip at high speed; once the whites are foamy, rain in the sugar, then add the extracts and whip until soft peaks are formed, 5 to 7 minutes in all.
Whisk the flour to break up any clumps. When the egg whites have formed soft peaks, remove the mixing bowl and fold in the flour, one-third at a time.
Scoop the batter into an angel food cake pan or a tube springform pan (a tube pan provides for the biggest rise, but you can use a non-tube pan if you have a way of cooling the cake upside down, which inhibits the cake from sinking as it cools).
Bake until the top of the cake is golden and springs back to the touch, 35 to 40 minutes. Remove the cake from the oven and invert it, still in its pan, on a baking sheet lined with parchment. Allow the cake to cool, upside down, for 1 hour.
While the cake is cooling, make the berry compote. Combine all of the ingredients in a medium saucepan and place over medium heat. Cook until the fruit becomes soft, about 15 minutes. You can use it right away or refrigerate it to serve cool.
Once the cake is cooled, run a knife along the sides to remove the cake from the pan. You may have to give it a good thump to release it from the pan if you are not using a springform pan. Cut into slices and serve with the warm or chilled berry compote.
Preparations that take advantage of the unique nature of the richness of the yolk by itself and the white by itself are some of the most interesting and dramatic in the kitchen. A cake leavened by egg white is unlike any other kind of cake; a soufflé base, enriched with yolks, leaps out of its ramekin when egg whites are folded into it and baked.
But there are also times when you want to cook the white and the yolk separately and then recombine them as a garnish or as a central preparation. That same soufflé, not baked but rather chilled and eaten, is a mousse. Sometimes their separateness is obscured, the yolk setting the Key lime pie filling, and the white creating the meringue garnish.
But even when separated, they make a dynamic duo.
#
# Leeks Vinaigrette
SERVES 4
_In a classic leeks vinaigrette, the leeks are cooked and served in long halves. I'm changing it up here for easier consumption, because even when they're thoroughly cooked they can be tough. And because the egg garnish is so fundamental to the satisfying nature of this dish, slicing the leeks makes the combination of the leek and egg more felicitous and easier to eat. This is a delicious first-course salad that can be made a day ahead, vinaigrette and all, and assembled at the last minute. And it's a great lesson in how a cooked yolk has a smoothing, mellowing, enriching effect on the acidic components of a dish._
8 fat leeks
2 tablespoons/30 grams butter
Salt
⅓ cup/75 milliliters good red wine vinegar
1 tablespoon Dijon mustard
1 raw egg yolk (optional)
Pinch of cayenne pepper
1 cup/240 milliliters vegetable oil
6 hard-cooked eggs, peeled, yolks and whites separated and finely chopped or passed through a sieve or basket strainer
Salt and freshly ground black pepper
Cut the root ends off the leeks, then cut them in half lengthwise. Rinse them thoroughly. Remove the dark green tops and save them for the next time you make stock. Cut each leek half widthwise into ½-inch strips. (If they are still dirty, soak them in cold water to remove the dirt, then strain.)
Melt the butter in a large sauté pan over medium heat, and add the leeks and an aggressive four-finger pinch of salt. Cook gently until tender but not brown, 10 to 15 minutes. When they taste good and are tender, remove them to a plate lined with a paper towel and allow them to cool. Discard any pieces that are brown and papery, as they are likely to be tough.
Make the vinaigrette by combining the vinegar, Dijon, ½ teaspoon salt, raw yolk (if using), and cayenne. Drizzle in the oil, whisking continuously as for a mayonnaise. The yolk will help make it a thick, creamy vinaigrette, which is best for the leeks, but a thin one without the yolk is fine as well. Alternatively, you can use a strong blender with or without the yolk for the vinaigrette, pouring the oil in a thin stream over the other ingredients with the blade running on high until it's too thick to blend; remove the emulsified vinaigrette to a bowl and finish whipping in the oil by hand.
When the leeks are at room temperature, put them in a mixing bowl and add as much of the vinaigrette as suits you; any leftover vinaigrette will keep, covered and refrigerated, for up to 3 days. Either finish the dish now or refrigerate the dressed leeks in a covered container for up to 1 day.
To finish the dish, divide the leeks among four plates, spreading the leeks into a disc shape (use a large ring mold if you have one). Season with salt. Spoon some chopped egg white in a smaller circle on top of the leeks (or fill a ramekin with the egg white and upend it on the leeks), and top with a still smaller circle of chopped yolk (about a tablespoon). Grind pepper over the egg and serve.
#
# Curried Deviled Egg Canapés
MAKES 48 CANAPÉS
_These are easy, quick, inexpensive crowd pleasers. The drawback to deviled eggs, though, is that they're too good—you want to eat more than you should. Four halves, with scrumptious yolk and flavorful, spicy mayonnaise in the semi-spherical space left by the yolk, is a meal in itself. So, I've scaled these back to quarter eggs, and in true canapé fashion, I serve them on a crouton to make them easier to eat. Of course you could simply make Curried Egg Salad (here)—or any egg salad—and spread it on a crouton for a great canapé, but these look prettier._
_If you have a pressure cooker, use it here, as the peeling is guaranteed to be easy (seehere). Each egg will be sliced lengthwise into quarters; for perfectly smooth and even cuts, a wire cheese cutter is ideal. But if you don't have one, a piece of plain dental floss or thread slips right through a hard-cooked egg. The croutons can be made up to a day ahead and kept in a plastic bag or airtight container at room temperature. The deviled eggs should be made as close to serving time as is convenient; if necessary, they can be made several hours in advance, covered, and refrigerated, but the flavor is best just after mixing._
48 canapé-size bread slices or baguette slices
Salt and freshly ground black pepper
1 dozen eggs, hard-cooked and thoroughly chilled in an ice bath, then peeled
1 recipe curried mayonnaise (here)
Cayenne pepper or minced fresh chives, for garnish
Preheat the oven to 300°F/150°C. Toast the bread slices on a baking sheet or rack until crisp, about 10 minutes. Sprinkle them with salt and a grind of pepper.
Quarter each egg lengthwise. Put the egg yolks in a large bowl (for perfectly smooth deviled eggs, pass the yolks through a sieve) and mash them with the curried mayonnaise until the mixture is evenly distributed. Season with salt and pepper to taste.
Place an egg-white quarter on each crouton. Fill a piping bag fitted with a star tip (or a sturdy plastic bag with a ½-inch/12-millimeter hole cut from one corner) with the curried yolk mixture and pipe about a tablespoon onto each egg-white quarter. Garnish with a sprinkling of cayenne or chives and serve immediately.
# Mango-Lime Semifreddo
SERVES 12
_I love the effect of freezing a meringue, giving what is essentially an ice cream base a lightness it wouldn't otherwise have. I usually see chocolate or mocha or coffee semifreddos, but when my assistant suggested making a citrus semifreddo, it made me think of the orange Creamsicles of my youth, so we went that route. Still it remains a semifreddo: a yolk-sugar base, a flavoring component, whipped cream, and meringue. This fills a big springform pan that serves many, for a good make-ahead warm-weather dessert._
2 ripe mangos
⅓ cup/75 milliliters fresh lime juice (from about 3 large limes)
6 egg yolks
1¼ cups/250 grams sugar
1 tablespoon water
3 egg whites
Pinch of salt
2 teaspoons grated lime zest (from about 3 large limes)
2 cups/480 milliliters heavy cream
Peel and dice the mango and place in a blender. Add the lime juice and buzz until the mango is puréed. Refrigerate it.
In the top of a double boiler (or a metal bowl or saucepan) set over simmering water, combine the egg yolks and ¾ cup/150 grams of the sugar. Whip the eggs and the sugar for several minutes, adding the water midway through, until the mixture is very warm, has quadrupled in volume, and becomes a satiny, ribbony sauce, about 10 minutes in all. Set aside.
In a medium bowl, whip the egg whites with a pinch of salt on high speed. Once foamy, add the remaining ½ cup/100 grams sugar. Whip until firm, moist peaks are reached.
Remove the mango-lime puree from the fridge and begin to fold it into the yolk base. Fold in the lime zest and transfer to a large bowl. Fold the meringue into the puree mixture, one-third at a time, until uniformly combined.
In another large bowl, whip the cream to soft peaks; then fold the cream into the mango-lime base, one-third at a time.
Line the inside of a 12-inch/30.5-centimeter springform pan with plastic wrap. Scoop the semifreddo into the mold, cover with plastic wrap, and place in the freezer overnight. (You can also freeze in individual cups if you prefer.)
Unmold and slice to serve (or serve straight from the freezer if serving in cups).
# ON SERVING RAW YOLK
A raw egg yolk is a marvelous thing in and of itself. It can be used as a garnish or as a ready-made sauce of the perfect consistency—rich and luxurious. It can be put on virtually any food and thereby make that food better. Make a dimple in a hamburger and put a yolk there. Top a lemony salad with it. Drop it into a soup. Make it the bull's eye in a potato pancake. It seems to me that the only time it doesn't make the food better is when it's already part of the food. It would be a little ridiculous to put an egg yolk on a custard, or ice cream, or mayonnaise, for example. But simply recognizing that an egg yolk on just about any food makes that food tastier, more visually appealing, and more nutritious turns you into a better cook.
If you're serving a whole unbroken yolk, you may want to remove (or hide) the two white chalazae at either pole of the yolk; these protein coils keep the egg suspended in the white. Pinch them off if you wish, for presentation, but be warned that they may break open the yolk.
Some people are concerned about the safety of raw eggs, and there can be an issue of their carrying the bacterium salmonella, which can give you some GI distress; if you're otherwise compromised by age or health issues, it can land you in the hospital. I have never worried about it and have never been ill from salmonella that I know of in a lifetime of eating raw eggs (my mom used to drop one into my Carnation Instant Breakfast in the mid-1970s).
What you need to know about bacteria in food generally is this: if the bacteria exists in or on the food, it multiplies at an astonishing rate at room temperature, a rate that increases even more quickly if you make it warmer, say 100°F/38°C. If you've got three yolks in a pot for a hollandaise sauce with a small amount of the bacteria in it, and you keep it warm for several hours, you've got a small bacteria bomb with which to sicken friends and family. Therefore, when working with eggs, keep them cold, or keep them hot for as long as you wish and serve freely, or keep them at room temperature but not for longer than an hour. Do that and you should never have to be concerned about consuming egg in any of its forms.
One of my favorite dishes to serve is beef tartare with a raw yolk as a central garnish. Because we've so abused our land and our animals, this dish presents a double whammy, bacteria-wise—salmonella and _E. coli._ I thumb my nose at the danger and eat away because I treat the situation with care.
I buy eggs from local farms, beg them from friends who raise chickens (yes, even here in Cleveland Heights—thanks, Amelia!), or buy organic eggs from the grocery store. I grind my own meat after rinsing it and salting it. You can never be 100 percent certain that your eggs are free of bacteria. But as long as you keep them out of danger-zone temperatures for more than an hour, you shouldn't have any problems with local eggs (or even factory-farmed eggs) because you haven't let the bacteria multiply to harmful levels.
#
# Steak Tartare with Egg Yolk, Capers, and Red Onion
SERVES 4
_When I realized that I had cooked yolk/raw white recipes but no cooked white/raw yolk recipes, I asked Twitter for suggestions. Hilmi Ahmad, @hilmiahmad77, who cooks at the Sparrow Bar + Cookshop in Houston, suggested adding the white to steak tartare, which I think is a splendid idea._
_This makes a delicious and easy hors d'oeuvre or canapé presented on a platter, with the red onion, capers, and anchovy on the side so people can pick and choose their own garnish. The recipe that follows serves up to eight as hors d'oeuvres._
_But here I'm serving the dish in individual portions as a first course or lunch entrée. I'm going Craig Claiborne 1970s-style, forming patties with the garnish and seasoning mixed in, accompanied by a toasted baguette and arugula salad (just before serving, toss four handfuls of arugula with olive oil, give the greens a squeeze of lemon, several grinds of pepper, and a few pinches of salt)._
_As for those concerned about bad bugs, bacteria don't exist within the muscle of raw beef, only on its surface. So if I'm planning to serve a large portion of raw beef at room temperature, I run cold tap water over it heavily to wash off any surface bacteria. I then pat it dry, coat it in salt, wrap it, and refrigerate it for 1 to 3 days. The salt should—I repeat,_ should _, as this isn't a guarantee—take care of any remaining harmful bacteria, in effect curing the meat. (Never eat large quantities of any kind of raw meat ground at a grocery store; you can get really sick.) Then I cut and grind; I prefer a coarse grind, but a fine grind is good, too; you can chop the meat in a food processor or mince by hand._
1 pound/450 grams beef eye of round
Kosher salt
¼ cup/25 grams minced red onion
1 tablespoon fresh lemon juice (or more to taste)
1 tablespoon red wine vinegar
4 eggs, separated
2 tablespoons capers, chopped
2 to 4 tablespoons extra-virgin olive oil (or to taste)
1 teaspoon fish sauce or 1 anchovy, mashed
1 tablespoon minced fresh parsley
1 tablespoon minced fresh chives
Freshly ground black pepper
16 baguette slices, toasted
Thoroughly rinse the meat under cold running water, then pat dry. Rain kosher salt over all surfaces so that it's uniformly salted, pressing it into the surface. It should be completely coated in salt. Wrap in plastic and put it in the refrigerator for at least 2 hours or up to 72 hours.
Put the red onion in a small bowl, sprinkle with ½ teaspoon salt, and add the lemon juice and red wine vinegar; set aside to rest for at least 10 minutes.
Remove the meat from the refrigerator. Rinse off the salt and pat the meat dry. Cut the meat into 1-inch/2.5-centimeter dice and grind it through the largest die of your grinder (or chop it in a food processor or by hand). The grinding can be done up to 1 day before serving; store the ground meat in a tightly covered container in the refrigerator.
Bring a medium pot of water to a boil over high heat. Strain the egg whites through a large perforated spoon or mesh strainer to remove the less viscous part of the white, and pour the egg whites into the water. Reduce the heat to low and cook them till they're set, a few minutes. Drain the hot water and run cold water over the egg whites until they're completely chilled. Pat them dry and press them through a sieve, ricer, or basket strainer.
In a large bowl combine the ground beef, egg whites, red onion mixture, capers, olive oil, fish sauce or anchovy, and the herbs. Mix it thoroughly with a wooden spoon until the garnishes are uniformly distributed. Taste a teaspoonful and adjust the seasoning with more oil, salt, or lemon juice or vinegar and plenty of black pepper. The mixture can be covered and refrigerated for an hour or two before serving.
To serve, form into four patties and set them each on a plate; make a yolk-sized divot in each patty. Place a yolk in each divot and grind more pepper onto the yolks. Serve with toasted baguette slices and arugula salad (see headnote).
#
# Chocolate Espresso Kahlúa Soufflé
SERVES 8
_Soufflés have a reputation for being difficult, but in fact they're easy to make, and you can even mix and freeze them ahead of time. The tricky part is timing, because you've got to eat them straight from the oven. The meringue that's folded into the flavorful yolky base (here a simple chocolate pastry cream) is filled with countless little air bubbles that expand and lift that base right out of the cup—but as soon as they cool, they'll shrink, and your little miracle in a ramekin will fall, humiliated by the lack of attention._
_So just pay attention to timing and you can wow your family and friends—I've yet to meet someone unimpressed by a soufflé. This one's flavored with espresso powder and Kahlúa, but you can mix and match as you wish. Use Grand Marnier instead of the Kahlúa, for instance._
_There are all manner of soufflé recipes out there, but I favor the streamlined efficiency of this one. It doesn't result in an over-the-top rise but rather a gentle one, so that the finished soufflé is powerfully rich and flavorful._
¾ cup/180 milliliters milk
1 tablespoon espresso powder
1 tablespoon unsweetened cocoa powder
1 teaspoon pure vanilla extract
Pinch of salt
6 eggs, separated
¾ cup/150 grams granulated sugar, plus more for coating the ramekins
2 tablespoons flour mixed with 2 tablespoons Kahlúa
¾ cup/150 grams semisweet chocolate chips
1 teaspoon fresh lemon juice
Confectioners' sugar, for dusting
Preheat the oven to 350°F/180°C.
Coat the insides of eight 4-ounce/120-milliliter ramekins with butter. Toss a little sugar around in the cups to coat the insides; pour out the excess.
Combine the milk, espresso powder, cocoa powder, vanilla, and salt in a small saucepan and bring to a simmer over medium-high heat.
Whisk together the yolks and ¼ cup/50 grams of the granulated sugar in a medium bowl till the mixture is uniform and the flour evenly distributed.
Add about ¼ cup of the simmering milk mixture to the yolks while whisking, then pour the yolk mixture back into the pot and whisk over the heat until just before it reaches a simmer. Add the flour-Kahlúa mixture, and whisk until the sauce thickens just as it returns to a simmer, 30 to 60 seconds. Remove the pan from the heat and add the chocolate, stirring just to cover the chips with the sauce so that they'll melt.
While the sauce sits, whip the whites on high speed in a standing mixer fitted with a whisk attachment. Once they're frothy, add the lemon juice, then drizzle in the remaining ½ cup/100 grams sugar while continuing to whip. Continue whipping the egg whites until they hold peaks.
Transfer the chocolate mixture to a large bowl. Fold in the egg whites, one-third at a time, folding slowly and gently until the whites are incorporated. Spoon the mixture into the prepared ramekins, filling them about three-quarters full.
Bake for 30 minutes. Dust with confectioners' sugar and serve immediately.
# Chocolate Mousse
SERVES 6
_Raw egg white lightens and sweetens a chocolate ganache for a classic chocolate mousse, an easy, make-ahead dessert that is great for entertaining._
4 ounces/120 grams chocolate (at least 60% cacao)
1 tablespoon unsalted butter
Salt
2 eggs, separated
1 cup/240 milliliters heavy cream
¼ cup/60 grams sugar
In the top of a double boiler (or a metal bowl or saucepan) set over simmering water, combine the chocolate, butter, and a pinch of salt. When about 70 percent of the chocolate has melted, turn off the heat, stir the mixture, and allow it to cool somewhat, 5 minutes or so. Stir in the yolks.
In a large bowl, begin to whip the heavy cream with a whisk. Once soft peaks are reached, rain in 2 tablespoons/30 grams of the sugar. Keep whipping until firm peaks are reached. Transfer the whipped cream to the refrigerator while you prepare the rest of the chocolate mousse.
Be sure to clean your whisk and bowl thoroughly before starting the meringue. Add the egg whites and a pinch of salt. Begin whipping until foamy. Then rain in the remaining 2 tablespoons/30 grams sugar. Keep whipping the meringue until stiff, moist peaks are formed. Set aside.
By now the chocolate ganache should be completely cooled. Remove the whipped cream from the refrigerator and fold one-third of the cream into the chocolate ganache. Once it is mixed in, fold in another one-third of the cream. Once that is mixed in, fold the entire chocolate-cream mixture into the larger bowl with the rest of the whipped cream. Fold until the cream is uniform in color.
Begin to fold the meringue into the chocolate cream, half at a time. Be sure that the chocolate cream and meringue are well combined.
Fill six serving vessels—coupes, large wine glasses, or regular old bowls or ramekins—with the mousse. Cover with plastic wrap and place in the refrigerator for at least 3 hours and as many as 8 hours to set. Serve chilled.
#
# Cooked Eggnog
SERVES 4 OR 5
_Cooking eggs to make eggnog results in a truly delicious elixir—in effect it's a thin crème anglaise. It has the added benefit of taking care of bacteria for those who have an uncommon fear of salmonella poisoning or who, for whatever reason, must avoid raw-egg preparations (in which case, sadly, you'll also need to omit the meringue in this recipe). When I wrote about this technique on ruhlman.com, one of the commenters said she was grateful for it as she needed to serve eggnog to a lot of seniors (who can be badly affected by salmonella poisoning) at a nursing home, and this was the perfect solution._
_I include the recipe because it is a delicious example of raw egg white manipulated into an ethereal garnish. A perfect holiday treat._
1½ cups/360 milliliters milk
1 cup/240 milliliters cream
1 vanilla bean, split lengthwise
Freshly grated nutmeg
4 egg yolks
¼ cup/50 grams plus 2 tablespoons sugar
2 egg whites
1 cup/240 milliliters rum, brandy, or bourbon
Combine 1 cup/240 milliliters of the milk, the cream, and the vanilla bean in a small saucepan, bring it to a simmer over medium-high heat, and remove it from the heat. Add plenty of nutmeg shavings and let the mixture steep for 10 minutes. With a paring knife, scrape the seeds from the vanilla bean into the milk-cream mixture. (Put the empty pod in your sugar bowl or bag to gently infuse the sugar.)
Put the yolks and ¼ cup/50 grams of the sugar in a medium bowl and whisk to combine. Whisking continuously, add the milk-cream mixture.
Fill a large bowl with half ice and half water, and float a second bowl in the ice bath. Set a fine-mesh strainer in the bowl.
Return the yolk-cream mixture to the saucepan and stir it with a flat-edged spoon or heatproof spatula over medium heat until the mixture thickens, a few minutes. It should coat the back of a spoon (you can take it as high as 165°F/75°C if you want to measure). Pour it through the strainer into the bowl set in ice. Add the remaining ½ cup/120 milliliters milk and stir to combine and fully cool the mixture. Cover and refrigerate the eggnog until ready to serve.
To make the meringue, whip the egg whites till frothy, then add the remaining 2 tablespoons sugar and whip to very soft peaks (do this shortly before serving for the best consistency).
For each serving, combine 4 ounces eggnog with 2 ounces alcohol in a tumbler, then add ice, top with a dollop of meringue, and garnish with gratings of nutmeg.
# Key Lime Tart with Almond Crust and Meringue Topping
MAKES 1 (9-INCH) TART
_I was planning to do a yolk-based lemon tart but had recently been in Key West, so I decided to do a lime version. I love this preparation because it uses the egg in three different ways. The yolks enrich and help set the custard, while the white both helps bind the crust and is the basis for the meringue garnish. You can use a standard pie plate if you don't have a tart mold._
1¼ cups/175 grams almond flour (almond meal)
¼ cup/35 grams all-purpose flour
5 eggs, separated
½ cup/100 grams plus 3 tablespoons sugar
3 tablespoons butter, melted
1 (14-ounce/396-gram) can sweetened condensed milk
½ cup/120 milliliters fresh Key lime juice (or Nellie & Joe's Famous Key West Lime Juice)
1 tablespoon grated lime zest
Preheat the oven to 350°F/180°C. Position a rack in the center of the oven.
To make the crust, combine the almond flour and all-purpose flour in a medium bowl. In a small bowl, lightly whip 2 of the egg whites with 3 tablespoons of the sugar to dissolve the sugar. Add the egg white mixture and the melted butter to the flour mixture. Stir till it all comes together. Press the mixture into the bottom and up the sides of a 9-inch/23-centimeter tart pan.
Bake the crust till it looks appealingly golden brown, 10 to 12 minutes. Set aside to cool.
For the filling, whisk together all 5 egg yolks, the sweetened condensed milk, lime juice, and lime zest in a medium bowl. Pour the mixture into the cooled tart crust.
Bake until the center is set but still moves a bit when the pan is nudged, 20 to 30 minutes. Remove the tart from the oven and allow it to cool completely.
Before serving, make an Italian meringue with the remaining 3 egg whites (equal parts egg white and sugar by weight, cooking the sugar to 250°F/120°C). Pipe or spread the meringue onto the cooled pie and broil the top to brown it lightly or hit it with a blowtorch for color. Serve.
#### MERINGUE TOPPING
1. / _Use a large star tip for a decorative meringue topping, moving it up and down to form waves._
2. / _Take care while the meringue is under the broiler: turn the tart pan a few times to make sure it ends up evenly browned._
#
# Traditional Eggnog
SERVES 2
_This is a great last-minute treat for two people. You could make a big batch, but I prefer to make a traditional eggnog spontaneously when, say, Donna and I skipped dessert and want something festive, sweet, and rich. Use organic eggs and you shouldn't need to worry about salmonella, though, as noted before, the elderly or anyone with a compromised immune system should probably try the cooked version above or the aged eggnog that follows._
2 eggs, separated
4 teaspoons sugar
½ cup/120 milliliters half-and-half
½ cup/120 milliliters bourbon, rye, or brandy
Freshly grated nutmeg
Combine the yolks and 2 teaspoons of the sugar in a large glass measuring cup. Whisking, pour in the half-and-half, then the alcohol, and whisk to combine.
In a small bowl, whisk the whites till frothy, then add the remaining 2 teaspoons sugar. Continue whisking till it reaches your desired thickness, loose if you want to incorporate it into the drink, stiffer for a more festive-looking drink.
Pour the eggnog into two tumblers filled with ice, garnish with the egg white, and grate nutmeg over the top to finish. (Alternatively, fold the whites into the eggnog, then pour over ice and garnish with nutmeg.)
#### TRADITIONAL EGGNOG
1. / _Homemade eggnog takes all of about 2 minutes. Combine half the sugar and the egg yolks (1 per person)._
2. / _Whisk in the half-and-half and the bourbon, rye, or brandy._
3. / _Begin whipping the egg whites._
4. / _About midway through, add the remaining sugar. Whip until the egg whites are as frothy as you wish._
5. / _Pour the half-and-half mixture over ice._
6. / _Garnish with the meringue and some gratings of nutmeg._
#
# Aged Eggnog
MAKES ABOUT 3 QUARTS/3 LITERS
_Here is a recipe dear to my heart. I first read about it on chow.com and was so intrigued that I tried it myself and wrote about it onruhlman.com. My friend chef-restaurateur Jonathon Sawyer started making it on a regular basis at his restaurant, trying different combinations of spirits (one favorite is scotch aged in a sherry cask; another uses a coffee-infused rum—you can see the leeway you have here). A little research online revealed that after several weeks the alcohol disables both harmful bacteria and spoilage bacteria, so this recipe will last at least 3 years in the fridge (that's as long as I've managed to keep it without consuming it; you could probably age it for 5 years before it started to taste more funky than is pleasurable—if you try it, email me!). It's well worth making even 3 weeks before serving but takes on a distinctive and pleasing funkiness and color over time._
_Serve over ice with grated nutmeg or garnished with a froth of freshly whipped egg whites, as with the Traditional Eggnog (above)._
12 to 14 egg yolks
1½ cups/300 grams sugar
1 quart/1 liter half-and-half
1 quart/1 liter bourbon
1 cup/240 milliliters Cognac
2 teaspoons pure vanilla extract
Cream the yolks and sugar by whisking them together in a large bowl. Add the remaining ingredients and whisk to dissolve the sugar. Pour the mixture into a plastic or glass container, cover, and refrigerate for at least 3 months.
Serve over ice or straight up with some grated nutmeg and, if you wish, garnished with egg white, whipped to a meringue with sugar.
# Acknowledgments
First, thanks go to my wife and collaborator, Donna, whose photography makes these books so valuable to use. Donna employs her camera more as a photojournalist than food photographer. Our goal with photography is to convey information, to show home cooks what they should be able to achieve in their own kitchens. All of the shots in this book were taken in our kitchen or in the makeshift studio we created in our dining room. I am very grateful to those who have written to thank me for the beauty of Donna's photographs, in the books and on the blog, but our main goal is not to induce aesthetic pleasure; rather, we just want to help cooks feel more comfortable in the kitchen. (The aesthetic pleasure is just a happy by-product.) What people don't know is how much Donna pushes me to get things right; where the work is concerned, that is what I thank my partner for most.
I would not be able to get done all that I do were it not for my Michigan-based assistant, Emilia Juocys. In addition to helping create the "Nutritional Differences in Eggs" sidebar in the introduction, she helped develop and test many of the recipes in this book and kept my crazy life somewhat organized. Trained under my colleague on _Charcuterie_ and _Salumi,_ chef and teacher Brian Polcyn, Emilia has always had my back, and I am grateful to be able to acknowledge her publicly.
This is the fourth book whose recipe selection and testing have been overseen by my Calgary-based colleague, Marlene Newell. In addition to running a fine cooking site called cookskorner.com, a wonderfully warm virtual cooking community, she has tested or overseen the testing of every recipe in this book. If I've gotten anything wrong in a recipe, it's all but certainly my negligence and not hers, as she is tenacious with details. I am also grateful to Matthew Kayahara, an Ontario-based translator whose love of cooking has led to stages (internships) at many fine restaurants, who was also a lead tester.
I would like to thank the many chefs I've worked with over the years. I've learned from all of you. I'm particularly grateful that, where this book is concerned, I was able to reach out via email to the pastry chefs Michael Laiskonis, Cory Barrett, Shuna Fish Lydon, and David Lebovitz, who helped when sugar and egg whites were making my head ache.
The cake recipes in this book would not have been possible were it not for Donna's sister, Regina Simmons, a professional baker in New York's Hudson Valley, who, as I note in the text, flew in for tutorials on cake basics and decorating (at which I'm particularly lame).
I would like to thank my daughter, Addison, and my son, James, for their forbearance when Donna and I turned the kitchen and dining room, the most trafficked areas of the house, into a photo studio, and for generally putting up with their odd parents, who never seem to leave the house to go to work (quoting one: "Like, what am I supposed to tell my friends you _do?_ ").
Finally, this book is dedicated to the writer Blake Bailey. I have no friend dearer than he. During the past thirty years, we have shared many apprentice-writer travails, many meals together, and many, many cocktails. I feel it somewhat inadequate, though, as he is a seriously brainy chap who writes massive (and massively praised) biographies of famous writers (John Cheever and Richard Yates, to name two), and here I am dedicating a book on the egg to this august intellect. My ode to the humble egg will have to suffice, Blake. For thee, I conclude with this fine egg anecdote from P. G. Wodehouse, because I know it will please you.
A perilously hungover Bertie Wooster recalls meeting a prospective gentleman's gentleman named Jeeves:
"If you would drink this, sir," he said, and with a kind of bedside manner rather like the royal doctor shooting the bracer into the sick prince. "It is a little preparation of my own invention. It is the Worcester Sauce that gives it its color. The raw egg makes it nutritious. The red pepper gives it its bite. Gentlemen have told me they have found it extremely invigorating after a late evening."
I would have clutched at anything that looked like a lifeline that morning. I swallowed the stuff. For a moment I felt as if somebody had touched off a bomb inside the old bean and was strolling down my throat with a lighted torch, and then everything seemed suddenly to get right. The sun shone in through the window; the birds twittered in the tree-tops; and, generally speaking, hope dawned once more.
"You're engaged!" I said, as soon as I could say anything.
We may not be able to have our Jeeves, but eggs, and dawning hope, indeed, yes.
# About the Author
**Michael Ruhlman's** innovative and successful food reference books include _Ratio, The Elements of Cooking, Ruhlman's Twenty,_ and _Charcuterie_. He has appeared as a judge on _Iron Chef America_ and as a featured guest on Anthony Bourdain's _No Reservations_. He has also coauthored books with Thomas Keller, Eric Ripert, and Michael Symon. He lives in Cleveland with his wife, photographer Donna Turner Ruhlman, and their two children.
**Donna Turner Ruhlman** has photographed many of her husband's books, including _Ratio_ and _Ruhlman's Twenty,_ and is the sole photographer for his blog, ruhlman.com.
# ALSO BY MICHAEL RUHLMAN
_The Book of Schmaltz: Love Song to a Forgotten Fat_
_Ruhlman's Twenty: 20 Techniques, 100 Recipes, A Cook's Manifesto_
_Ratio: The Simple Codes Behind the Craft of Everyday Cooking_
_The Elements of Cooking: Translating the Chef 's Craft for Every Kitchen_
_The Reach of a Chef: Professional Cooks in the Age of Celebrity_
_The Soul of a Chef: The Journey Toward Perfection_
_The Making of a Chef: Mastering Heat at the Culinary Institute of America_
_Salumi: The Craft of Italian Dry Curing_ (with Brian Polcyn)
_Charcuterie: The Craft of Salting, Smoking, and Curing_ (with Brian Polcyn)
_Bouchon Bakery_ (with Thomas Keller and Sebastien Rouxel)
_Ad Hoc at Home_ (with Thomas Keller)
_Under Pressure: Cooking Sous Vide_ (with Thomas Keller)
_Bouchon_ (with Thomas Keller)
_The French Laundry Cookbook_ (with Thomas Keller)
_A Return to Cooking_ (with Eric Ripert)
_Michael Symon's Live to Cook_ (with Michael Symon)
# Recipe Index by Technique
EGG/WHOLE/COOKED IN SHELL
hard-cooked eggs
egg salad
Egg Salad with Homemade Lemon-Shallot Mayonnaise
Egg Salad with Tarragon and Chives
Curried Egg Salad
Warm Hard-Cooked, Creamy-Yolk Egg with Ham and Cheddar
_mollet_ eggs
soft-cooked eggs
Soft-Cooked Eggs on Artichoke Hearts with Creamy Lemon-Shallot Vinaigrette
Pork Ramen with Soft-Cooked Egg and Scallions
Soft-Cooked Egg with Buttered Toast for Two
Crispy Mollet Egg with Asparagus
Ramen with Soft-Cooked Eggs
_sous vide_ eggs
Ramen with Miso-kombu Broth, Shiitake Mushrooms, and Sous Vide Egg
EGG/WHOLE/COOKED OUT OF SHELL
Weekend Broken-Yolk Fried-Egg Sandwich
Gently Fried Eggs on Grits with Bacon and Toast
Quail Eggs Croque Madame
Michael Pardus's Bibimbap
Frisée Salad with Bacon Vinaigrette and Deep-Fried Egg
coddled eggs
Coddled Eggs with Truffle Butter
shirred eggs
Shirred Eggs Florentine for My Mom
poached eggs
Eggs Benedict
Poached Duck Egg on Duck Confit Hash
eggs poached in broth
Oeufs en Meurette (Eggs Poached in Red Wine Sauce)
Eggs in Puttanesca Sauce with Angel Hair Pasta
Poached-in-a-Bag Egg Sandwich with Caramelized Onion and Roasted Red Pepper
EGG/WHOLE/COOKED OUT OF SHELL/BLENDED
perfect scrambled eggs
Scrambled Eggs with Fines Herbes
perfect omelet
Omelet with Creamy Morel Mushrooms
Potato, Onion, and Cheese Frittata
Daniel Patterson's Poached Omelet
Traditional Egg Flower Soup
Marlene's Bacon and Sausage Breakfast Strata
Breaded Chicken Cutlets with Dijon and Thyme
Picadillo Meatballs
Classic French Quiche with Chorizo and Roasted Peppers
Chawanmushi
Crème Caramel
Basic Bread Pudding
Bourbon Brioche Bread Pudding
Almond Challah French Toast
Nineteenth-Century Ale and Rum Flip
EGG/AS INGREDIENT/THE DOUGH-BATTER CONTINUUM
Brioche
Challah
All-Purpose Pasta
Yolk-Only Pasta
Italian Drop Cookies with Lemon Glaze
Norwegian Berlinerkranser
Gougères
Profiteroles with Vanilla Ice Cream and Chocolate
Crisp and Fluffy Potato Pancakes
Regina's Lemon Cream Cake
Basic Sponge Cake
Lemon Curd
French Buttercream
Coconut Cream Cake
Vanilla Pastry Cream
Chocolate Mocha Cake
Chocolate Sponge Cake
Mocha Pastry Cream
Chocolate Buttercream
Rum-Soaked Cherry Bread (or Muffins) with Almond-Sugar Topping
Emilia's Carrot Cake
Make-Ahead Brownies
Pancakes
Corn and Sweet Pepper Fritters with Chipotle-Lime Mayonnaise
Shrimp Tempura with Tentsuyu Dipping Sauce
Popovers with Raspberry Jam and Powdered Sugar
Crêpes
Crêpes Suzette
EGG/SEPARATED/THE YOLK
Egg Ravioli with Chèvre, Brown Butter, and Thyme
Caesar Dressing for Crisp Romaine Lettuce and Croutons
Blanquette de Veau
Chicken Fricassee
mayonnaise
aioli
Basic Mayonnaise
Lemon-Shallot Mayonnaise
Grilled Chicken Salad with Chipotle-Lime Mayonnaise
Pasta Carbonara
hollandaise and béarnaise
Blender Hollandaise Sauce
Traditional Béarnaise Sauce
Crème Anglaise
Crème Brûlée
Pastry Cream
Vanilla Ice Cream
Poire Williams Sabayon
EGG/SEPARATED/THE WHITE
Seafood Roulade with Scallops and Crab
Turkey Consommé
Orange-Ginger Panna Cotta
Clover Club Cocktail
meringue
Crunchy French Meringue
Nougat
Marshmallows
Île Flottante
Angel Food Cake with Berry Compote
EGG/SEPARATED BUT USED TOGETHER
Leeks Vinaigrette
Curried Deviled Egg Canapés
Mango-Lime Semifreddo
Steak Tartare with Egg Yolk, Capers, and Red Onion
Chocolate Espresso kahlúa Soufflé
Chocolate Mousse
Cooked Eggnog
key Lime Tart with Almond Crust and Meringue Topping
Traditional Eggnog
Aged Eggnog
# James Beard Award–winning author Michael Ruhlman explains why the egg is the key to the craft of cooking.
For culinary wizard Michael Ruhlman, the question is not whether the chicken or the egg came first; it's how anything could be accomplished in the kitchen without the magic of the everyday egg. In this groundbreaking book, Ruhlman explains how to make perfect poached and scrambled eggs and builds up to recipes for brioche and soufflés. Readers learn to make their own mayonnaise, custards, quiches, and cakes, mastering foods from sweet to savory, from light-as-air meringues to hearty bread and homemade pasta.
More than 100 recipes are grouped by technique, and range from simple (Egg Salad with Tarragon and Chives) to sophisticated (Seafood Roulade with Scallops and Crab). Multiple photographs guide the reader through this extraordinary journey, which unlocks the secrets of the egg for the home cook.
Michael Ruhlman started writing about the lives of chefs twenty years ago, and he soon became interested in training as a chef himself. His groundbreaking and successful food reference books include _The Book of Schmaltz, Ruhlman's Twenty, Ratio, The Elements of Cooking,_ and _Charcuterie_ (with Brian Polcyn), and he cowrote Thomas Keller's _The French Laundry Cookbook, Ad Hoc at Home,_ and _Bouchon_ cookbooks. He lives in Cleveland with his wife, Donna Turner Ruhlman, who has done the photography for many of his books and is the sole photographer for his blog, **ruhlman.com**.
**littlebrown.com**
@littlebrown
littlebrownandcompany
Author photograph by Donna Turner Ruhlman
For more about this book and author, visit Bookish.com.
* The quality of dry yeast is now very good and most bakers rely on it. I prefer Red Star or SAF instant dried yeast, which is just slightly stronger than active dried yeast; use either, as they work more or less the same.
### Thank you for buying this ebook, published by Hachette Digital.
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Sign Up
Or visit us at hachettebookgroup.com/newsletters
For more about this book and author, visit Bookish.com.
# Contents
1. Cover
2. Title Page
3. Welcome
4. Dedication
5. Introduction
6. PART ONE: Egg / Whole / Cooked in Shell
7. PART TWO: Egg / Whole / Cooked out of Shell
8. PART THREE: Egg / Whole / Cooked out of Shell / Blended
9. PART FOUR: Egg / As Ingredient / The Dough-Batter Continuum
10. PART FIVE: Egg / Separated / The Yolk
11. PART SIX: Egg / Separated / The White
12. PART SEVEN: Egg / Separated but Used Together
13. Acknowledgments
14. About the Author
15. Also by Michael Ruhlman
16. Recipe Index by Technique
17. Newsletters
18. Copyright
# Navigation
1. Begin Reading
2. Table of Contents
# Copyright
Copyright © 2014 by Ruhlman Enterprises, Inc.
Cover design by Gary Tooth/Empire Design Studio
Cover photograph by Donna Turner Ruhlman
Cover copyright © 2014 by Hachette Book Group, Inc.
Book design by Gary Tooth/Empire Design Studio
All rights reserved. In accordance with the U.S. Copyright Act of 1976, the scanning, uploading, and electronic sharing of any part of this book without the permission of the publisher constitute unlawful piracy and theft of the author's intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained by contacting the publisher at permissions@hbgusa.com. Thank you for your support of the author's rights.
Little, Brown and Company
Hachette Book Group
237 Park Avenue, New York, NY 10017
littlebrown.com
twitter.com/littlebrown
facebook.com/littlebrownandcompany
First ebook edition: April 2014
The publisher is not responsible for websites (or their content) that are not owned by the publisher.
The Hachette Speakers Bureau provides a wide range of authors for speaking events. To find out more, go to hachettespeakersbureau.com or call (866) 376-6591.
ISBN 978-0-316-25407-6
E3
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{
"redpajama_set_name": "RedPajamaBook"
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Q: Ansible role task delegation Given the below inventory/role/playbook examples, how can I delegate a task within a role to localhost and have it print the ip address of the node the role is executing on?
To be clear, my desired output is the IP address of app1 in this scenario. However, I want localhost to print that IP Address. I don't want the print task to run on the app1 node.
## inventory
[app_nodes]
app1
app2
## Role: foobar/main.yml
task:
debug:
msg: {{ansible_default_ipv4.address}}
delegate_to: localhost
## playbook
hosts: app_nodes
roles:
- { role: foobar }
A: The answer was to use the host_vars magic variable combined with the inventory_hostname
## inventory
[app_nodes]
app1
app2
## Role: foobar/main.yml
task:
debug:
msg: {{host_vars[inventory_hostname]['ansible_facts']['default_ipv4_address']}}
delegate_to: localhost
## playbook
hosts: app_nodes
roles:
- { role: foobar }
On run, this outputs the IP address of app1 and app2.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"http:\/\/library.kiwix.org\/stats.stackexchange.com_eng_all_2018-08\/A\/question\/4551\/what-are-common-statistical-sins.html","text":"## What are common statistical sins?\n\n213\n\n217\n\nI'm a grad student in psychology, and as I pursue more and more independent studies in statistics, I am increasingly amazed by the inadequacy of my formal training. Both personal and second hand experience suggests that the paucity of statistical rigor in undergraduate and graduate training is rather ubiquitous within psychology. As such, I thought it would be useful for independent learners like myself to create a list of \"Statistical Sins\", tabulating statistical practices taught to grad students as standard practice that are in fact either superseded by superior (more powerful, or flexible, or robust, etc.) modern methods or shown to be frankly invalid. Anticipating that other fields might also experience a similar state of affairs, I propose a community wiki where we can collect a list of statistical sins across disciplines. Please, submit one \"sin\" per answer.\n\n1@whuber There was some good answers, so I've merged them both. \u2013 mbq \u2013 2011-02-06T11:02:41.973\n\nI just gave a talk on this subject... A link to the video follows if you are interested. http:\/\/www.youtube.com\/watch?v=1SNQQvY1ESo&feature=g-upl\n\n\u2013 None \u2013 2012-10-21T01:13:53.057\n\n1Hi @Amanda, could you give some indication here of what's in the talk? No-one likes the possibility of being rick-rolled. \u2013 naught101 \u2013 2012-10-21T02:17:50.833\n\nApplying statistics where it doesn't belong is the main sin. \u2013 Aksakal \u2013 2016-08-11T19:19:29.643\n\n5I'm aware that \"sin\" is possibly inflammatory and that that some aspects of statistical analysis are not black-and-white. My intention is to solicit cases where a given commonly-taught practice is pretty clearly inappropriate. \u2013 Mike Lawrence \u2013 2010-11-15T18:53:14.270\n\n5You can also add biology\/life sciences students to the mix if you like ;) \u2013 nico \u2013 2010-11-15T19:03:17.187\n\n1maybe retitle it life science statistical sins?... or something else more specific... \u2013 John \u2013 2010-11-15T19:27:28.377\n\n111\n\nFailing to look at (plot) the data.\n\n1Very, very important! \u2013 deps_stats \u2013 2011-02-04T17:18:08.080\n\n1Probably the most common one. \u2013 Carlos Cinelli \u2013 2014-03-16T00:28:04.160\n\n+1 Well done! I'm shocked this hasn't been mentioned yet. \u2013 whuber \u2013 2010-12-16T23:28:00.023\n\n108\n\nMost interpretations of p-values are sinful! The conventional usage of p-values is badly flawed; a fact that, in my opinion, calls into question the standard approaches to the teaching of hypothesis tests and tests of significance.\n\nHaller and Krause have found that statistical instructors are almost as likely as students to misinterpret p-values. (Take the test in their paper and see how you do.) Steve Goodman makes a good case for discarding the conventional (mis-)use of the p -value in favor of likelihoods. The Hubbard paper is also worth a look.\n\nHaller and Krauss. Misinterpretations of significance: A problem students share with their teachers. Methods of Psychological Research (2002) vol. 7 (1) pp. 1-20 (PDF)\n\nHubbard and Bayarri. Confusion over Measures of Evidence (p's) versus Errors (\u03b1's) in Classical Statistical Testing. The American Statistician (2003) vol. 57 (3)\n\nGoodman. Toward evidence-based medical statistics. 1: The P value fallacy. Ann Intern Med (1999) vol. 130 (12) pp. 995-1004 (PDF)\n\nAlso see:\n\nWagenmakers, E-J. A practical solution to the pervasive problems of p values. Psychonomic Bulletin & Review, 14(5), 779-804.\n\nfor some clear cut cases where even the nominally \"correct\" interpretation of a p-value has been made incorrect due to the choices made by the experimenter.\n\nUpdate (2016): In 2016, American Statistical Association issued a statement on p-values, see here. This was, in a way, a response to the \"ban on p-values\" issued by a psychology journal about a year earlier.\n\nSee also: Gigerenzer, G., Krauss, S., & Vitouch, O. (2004). The null ritual. The Sage book of quantitative methodology for the social sciences, 391-408. http:\/\/library.mpib-berlin.mpg.de\/ft\/gg\/GG_Null_2004.pdf\n\n\u2013 None \u2013 2014-12-09T13:05:26.973\n\n1Sadly, my medical boards had a question which encouraged misinterpretation of p-values. And they refused to correct it even after I contested in a letter. Insanity. \u2013 Ari B. Friedman \u2013 2011-08-10T20:29:35.630\n\n2@Michael (+1) I added links to abstracts and ungated PDFs. Hope you don't mind. \u2013 chl \u2013 2010-11-16T10:24:17.297\n\n@Chi Thanks! I should have done that myself. Next time... \u2013 Michael Lew \u2013 2010-11-16T19:44:01.083\n\n7+1, but I would like to make some critical comments. Regarding the opening line, one could just as well say that \"almost all\" (in the measure theoretic sense) interpretations of any well-defined concept are incorrect, because only one is correct. Second, to what do you refer when you say \"the conventional usage\" and \"standard approaches\"? These vague references sound like a straw man. They do not accord with what one can find in the literature on statistics education, for example. \u2013 whuber \u2013 2010-11-17T13:43:58.040\n\n4@Whuber Have a look at the Goodman paper. It accords pretty well with my experience in the field of pharmacology. Methods say \"Results where P<0.05 were taken as statistical significant\" and then results are presented with + for p<0.05, ++ for p<0.01 and +++ for p<0.0001. The statement implies the control of error rates a la Neyman and Pearson, but the use of different levels of p suggest Fisher's approach where the p value is an index of the strength of evidence against the null hypothesis. As Goodman points out, you cannot simultaneously control error rates and assess strength of evidence. \u2013 Michael Lew \u2013 2010-11-18T02:41:07.073\n\n8@Michael There are alternative, more generous interpretations of that kind of reporting. For example, the author might be aware that readers might want to apply their own thresholds of significance and therefore do the flagging of p-values to help them out. Alternatively, the author might be aware of possible multiple-comparisons problems and use the differing levels in a Bonferroni-like adjustment. Perhaps some portion of the blame for misuse of p-values should be laid at the feet of the reader, not the author. \u2013 whuber \u2013 2010-11-18T14:04:04.577\n\n4@Whuber I agree entirely, but only that what you suggest is true in some small fraction of cases (a restricted version of 'entirely'). There are some journals that specify that p values should be reported at one, two or three star levels rather than exact values, so those journals share some responsibility for the outcome. However, both that ill-considered requirement and the apparently naive use of p values might be a result of the lack of a clear explanation of the differences between error rates and evidence in the several introductory statistics texts that are on my shelves. \u2013 Michael Lew \u2013 2010-11-19T03:40:25.297\n\n@Michael All good points; thank you. And thank you for sharing the references: they make good reading (even though the papers addressed to medical practitioners belabor their arguments). \u2013 whuber \u2013 2010-11-19T13:49:32.273\n\n@Michael: (+1). The Goodman article was especially insightful in supporting your argument. \u2013 Christopher Aden \u2013 2010-11-19T16:36:04.347\n\n67\n\nThe most dangerous trap I encountered when working on a predictive model is not to reserve a test dataset early on so as to dedicate it to the \"final\" performance evaluation.\n\nIt's really easy to overestimate the predictive accuracy of your model if you have a chance to somehow use the testing data when tweaking the parameters, selecting the prior, selecting the learning algorithm stopping criterion...\n\nTo avoid this issue, before starting your work on a new dataset you should split your data as:\n\n\u2022 development set\n\u2022 evaluation set\n\nThen split your development set as a \"training development set\" and \"testing development set\" where you use the training development set to train various models with different parameters and select the bests according to there performance on the testing development set. You can also do grid search with cross validation but only on the development set. Never use the evaluation set while model selection is not 100% done.\n\nOnce you are confident with the model selection and parameters, perform a 10 folds cross-validation on the evaluation set to have an idea of the \"real\" predictive accuracy of the selected model.\n\nAlso if your data is temporal, it is best to choose the development \/ evaluation split on a time code: \"It's hard to make predictions - especially about the future.\"\n\n9In general it takes an enormous dataset for data splitting to be reliable. That's why stringent internal validation with the bootstrap is so attractive. \u2013 Frank Harrell \u2013 2013-06-15T15:53:57.770\n\nEspecially when the development set is past data and the evaluation set future data. Why not, after all model tuning, train the final model with it's fixed parameters on the entire development set and predict the entire evaluation set with it. In a real scenario, you couldn't cross validate through future data the way you describe anyway, so you would use all relevant past data. \u2013 David Ernst \u2013 2017-09-15T14:50:14.787\n\n5I agree with this in principle but in the case of a small data set (I often have only 20-40 cases) use of a separate evaluation set is not practical. Nested cross-validation can get around this but may lead to pessimistic estimates on small data sets \u2013 BGreene \u2013 2012-07-24T16:39:46.160\n\n61\n\nReporting p-values when you did data-mining (hypothesis discovery) instead of statistics (hypothesis testing).\n\n2Can you (or someone) elaborate? \u2013 antoine-sac \u2013 2015-09-22T22:05:24.020\n\n1\n\n\u2013 Neil McGuigan \u2013 2015-11-17T19:17:54.250\n\nWhat about p-values corrected for multiple hypothesis testing (with some flavour of Bonferroni method or a more advanced correction)? I would tend to think it is fine, even in the context of data mining? \u2013 antoine-sac \u2013 2015-11-19T15:46:06.330\n\nI like the general idea, but it's a distortion to equate statistics with hypothesis testing when the latter is a subset of the former. \u2013 rolando2 \u2013 2017-01-25T16:34:03.700\n\n44\n\nTesting the hypotheses $H_0: \\mu=0$ versus $H_1: \\mu\\neq 0$ (for example in a Gaussian setting)\n\nto justify that $\\mu=0$ in a model (i.e mix \"$H_0$ is not rejected\" and \"$H_0$ is true\").\n\nA very good example of that type of (very bad) reasoning is when you test whether the variances of two Gaussians are equal (or not) before testing if their mean are equal or not with the assumption of equal variance.\n\nAnother example occurs when you test normality (versus non normality) to justify normality. Every statistician has done that in is life ? it is baaad :) (and should push people to check robustness to non Gaussianity)\n\n3I try to be statistically literate and still fall for this one from time to time.\n\nWhat are the alternatives? Change your model so the old null becomes $H_1$? The only other option I can think of is power your study enough that a failure to reject the null is in practice close enough to confirming the null. E.g. if you want to make sure that adding a reagent to your cells won't kill off more than 2% of them, power to a satisfactory false negative rate. \u2013 DocBuckets \u2013 2013-05-15T17:38:37.830\n\nGreat!! Yes, this drives me crazy.. \u2013 jpillow \u2013 2011-07-11T06:32:45.423\n\n@DocBuckets equivalence testing with two one sided tests is more rigorous than the power based approach. But you need to set a minimum relevant effect size below which you can speak of practical equivalence. \u2013 David Ernst \u2013 2017-09-15T14:57:10.763\n\n6The same logic (taking \"absence of evidence in favor H1\" as \"evidence of absence of H1\") essentially underlies all goodness-of-fit tests. The reasoning also often crops up when people state \"the test was non significant, we can therefore conclude there is no effect of factor X \/ no influence of variable Y\". I guess the sin is less severe if accompanied by reasoning about the test's power (e.g., a-priori estimation of sample size to reach a certain power given a certain relevant effect size). \u2013 caracal \u2013 2010-11-16T23:07:06.833\n\nIf you do not make any concideration about the power, I would say claming $H_0$ is true when it is not rejected is very very bad while claming $H_1$ is true while $H_0$ is rejected is just a little wrong :). \u2013 robin girard \u2013 2010-11-17T07:22:53.877\n\n43\n\nA few mistakes that bother me:\n\n1. Assuming unbiased estimators are always better than biased estimators.\n\n2. Assuming that a high $R^2$ implies a good model, low $R^2$ implies a bad model.\n\n3. Incorrectly interpreting\/applying correlation.\n\n4. Reporting point estimates without standard error.\n\n5. Using methods which assume some sort of Multivariate Normality (such as Linear Discriminant Analysis) when more robust, better performing, non\/semiparametric methods are available.\n\n6. Using p-value as a measure of strength between a predictor and the response, rather than as a measure of how much evidence there is of some relationship.\n\n5Would you break these out into separate options? \u2013 russellpierce \u2013 2010-12-12T21:27:42.730\n\n40\n\nRitualized Statistics.\n\nThis \"sin\" is when you apply whatever thing you were taught, regardless of its appropriateness, because it's how things are done. It's statistics by rote, one level above letting the machine choose your statistics for you.\n\nExamples are Intro to Statistics-level students trying to make everything fit into their modest t-test and ANOVA toolkit, or any time one finds oneself going \"Oh, I have categorical data, I should use X\" without ever stopping to look at the data, or consider the question being asked.\n\nA variation on this sin involves using code you don't understand to produce output you only kind of understand, but know \"the fifth column, about 8 rows down\" or whatever is the answer you're supposed to be looking for.\n\nTo me Epigrad's description is of someone who cares inordinately about inference and neglects things such as reflection, discovery, and consideration of causality. \u2013 rolando2 \u2013 2013-11-26T03:03:27.110\n\n6Unfortunately, if you aren't interested in statistical inference, or are scarce on time and\/or resources, the ritual does seem very appealling... \u2013 probabilityislogic \u2013 2012-03-09T05:32:49.393\n\n40\n\nNot really answering the question, but there's an entire book on this subject:\n\nPhillip I. Good, James William Hardin (2003). Common errors in statistics (and how to avoid them). Wiley. ISBN 9780471460688\n\n6+1 I made sure to read this book shortly after it came out. I get plenty of opportunities to make statistical mistakes so I'm always grateful to have them pointed out before I make them! \u2013 whuber \u2013 2010-12-12T23:02:01.107\n\n39\n\nDichotomization of a continuous predictor variable to either \"simplify\" analysis or to solve for the \"problem\" of non-linearity in the effect of the continuous predictor.\n\nThis one really bugs me. My clients do this all the time. And when I point out the error, they understand, but insist on doing it anyway \u2013 Peter Flom \u2013 2011-02-06T15:54:50.273\n\n2+1 and it becomes a serious sin when they start choosing the dichotomization cutoff so that it optimizes some sort of difference which is then tested. \u2013 Erik \u2013 2013-09-16T10:36:26.687\n\n@Iterator I for one would rather use lat\/long if I had them, and the patience to interpret the results carefully. It could be quite interesting to find fine-grained \/ dimensional differences within countries or even cities, and it could help reduce residuals if analyzing something that changes dramatically across a border. E.g., even analyses of pro-Palestinian liberation attitudes might be more or less polarized closer to the border. Just calling it Palestine vs. Israel is a rather clumsy way to dumb down results, and I don't see what else is achieved beyond that...Less time learning stats? \u2013 Nick Stauner \u2013 2014-03-02T21:41:45.713\n\n4@Iterator you start to get at the real reason to aggregate (to two or more categories), which is because one has a priori theoretical reasons to believe that variance is meaningfully compartmentalized into those categories. For example, we do this all the time by assuming that collections of a trillion or so cells comprise an individual, or that a contiguous 24-hour period here on Earth is meaningfully interpreted as a unit. But arbitrarily aggregation does not just \"throw out\" information (e.g. statistical power), but can lead to (serious) biases about relationships between phenomena. \u2013 Alexis \u2013 2014-06-25T19:44:44.203\n\n2This isn't even a sin if there are two or more distinct populations. Suppose you have separable classes or sub-populations, then it can make sense to discretize. A very trivial example: Would I rather use indicators for site\/location\/city\/country or lat\/long? \u2013 Iterator \u2013 2011-08-09T23:38:34.560\n\n17I don't think this is really a \"sin\" as the results obtained are not wrong. However, it does throw away a lot of useful information so is not good practice. \u2013 Rob Hyndman \u2013 2010-11-15T22:51:35.173\n\n2Along these lines, using extreme groups designs over-estimates effect sizes whereas the use of a mean or median split under-estimates effect sizes. \u2013 russellpierce \u2013 2010-11-16T02:47:46.777\n\n39\n\ninterpreting Probability(data | hypothesis) as Probability(hypothesis | data) without the application of Bayes' theorem.\n\n5http:\/\/en.wikipedia.org\/wiki\/Prosecutor%27s_fallacy \u2013 finnw \u2013 2011-12-19T23:38:24.327\n\n30\n\nMaybe stepwise regression and other forms of testing after model selection.\n\nSelecting independent variables for modelling without having any a priori hypothesis behind the existing relationships can lead to logical fallacies or spurious correlations, among other mistakes.\n\nUseful references (from a biological\/biostatistical perspective):\n\n1. Kozak, M., & Azevedo, R. (2011). Does using stepwise variable selection to build sequential path analysis models make sense? Physiologia plantarum, 141(3), 197\u2013200. doi:10.1111\/j.1399-3054.2010.01431.x\n\n2. Whittingham, M. J., Stephens, P., Bradbury, R. B., & Freckleton, R. P. (2006). Why do we still use stepwise modelling in ecology and behaviour? The Journal of animal ecology, 75(5), 1182\u20139. doi:10.1111\/j.1365-2656.2006.01141.x\n\n3. Frank Harrell, Regression Modeling Strategies, Springer 2001.\n\n29\n\nSomething I see a surprising amount in conference papers and even journals is making multiple comparisons (e.g. of bivariate correlations) and then reporting all the p<.05s as \"significant\" (ignoring the rightness or wrongness of that for the moment).\n\nI know what you mean about psychology graduates, as well- I've finished a PhD in psychology and I'm still only just learning really. It's quite bad, I think psychology needs to take quantitative data analysis more seriously if we're going to use it (which, clearly, we should)\n\n8This is particularly important. I remember reading a study about whether Ramadan was bad for babies whose mothers were fasting. It looked plausible (less food, lower birth weight), but then I looked at the appendix. Thousands of hypotheses, and a few percent of them were in the \"significant\" range. You get weird \"conclusions\" like \"it's bad for the kid if Ramadan is the 2nd, 4th or 6th month\". \u2013 Carlos \u2013 2011-01-31T13:06:21.333\n\n27\n\nBeing exploratory but pretending to be confirmatory. This can happen when one is modifying the analysis strategy (i.e. model fitting, variable selection and so on) data driven or result driven but not stating this openly and then only reporting the \"best\" (i.e. with smallest p-values) results as if it had been the only analysis. This also pertains to the point if multiple testing that Chris Beeley made and results in a high false positive rate in scientific reports.\n\n25\n\nThe one that I see quite often and always grinds my gears is the assumption that a statistically significant main effect in one group and a non-statistically significant main effect in another group implies a significant effect x group interaction.\n\n23\n\nEspecially in epidemiology and public health - using arithmetic instead of logarithmic scale when reporting graphs of relative measures of association (hazard ratio, odds ratio or risk ratio).\n\n5\n\nNot to mention not labeling them at all http:\/\/xkcd.com\/833\/\n\n22\n\nAnalysis of rate data (accuracy, etc) using ANOVA, thereby assuming that rate data has Gaussian distributed error when it's actually binomially distributed. Dixon (2008) provides a discussion of the consequences of this sin and exploration of more appropriate analysis approaches.\n\nThis is only as bad as the normal approximation to the binomial - should be fine, provided that each case is weighted by the denominator used in calculating the rate. Would expect it to perform poorly for rates below 10% and above 90%. \u2013 probabilityislogic \u2013 2014-07-02T05:24:48.890\n\n4How much does this decrease the power of the analysis? In what conditions is it most problematic? In many cases deviations from the assumptions of ANOVA do not substantially affect the outcomes to an important extent. \u2013 Michael Lew \u2013 2010-11-16T07:48:44.663\n\nWhat is the alternative do the ANOVA procedure? \u2013 Henrik \u2013 2010-11-16T12:55:24.730\n\n@Michael Lew & Henrik: I just updated this entry to include a link to Dixon (2008) \u2013 Mike Lawrence \u2013 2010-11-16T15:48:33.303\n\n2But in short, it is most problematic when probabilities observed are low or high as the range of values are constricted and unable to meet Gaussian assumptions. \u2013 russellpierce \u2013 2010-11-17T15:14:31.483\n\n21\n\nCorrelation implies causation, which is not as bad as accepting the Null Hypothesis.\n\n3Google makes all that money not caring about causation at all. Indeed, why would it? Prediction is the thing... \u2013 conjugateprior \u2013 2012-03-30T14:56:11.567\n\nbut sometimes... sometimes the potential directions of causation have highly disparate probabilities. I'm certainly not going to think that a correlation between age and height could be caused by the height... or some intervening variable either. Also, I think that this is one that behavioural science training is generally quite sensitive to. \u2013 John \u2013 2010-11-15T19:21:49.960\n\nindeed, inferring something from A and B are correlated usually only see A causes B but not B causes A...(and forget about C which causes A and B) \u2013 Andre Holzner \u2013 2010-12-01T20:13:59.237\n\n12google makes $65B a year not caring about the difference... \u2013 Neil McGuigan \u2013 2010-12-03T07:41:55.810 5I agree with your points and they all are valid. But does Google's profit imply: correlation => causation? \u2013 suncoolsu \u2013 2010-12-03T07:53:58.967 17 A current popular one is plotting 95% confidence intervals around the raw performance values in repeated measures designs when they only relate to the variance of an effect. For example, a plot of reaction times in a repeated measures design with confidence intervals where the error term is derived from the MSE of a repeated measures ANOVA. These confidence intervals don't represent anything sensible. They certainly don't represent anything about the absolute reaction time. You could use the error term to generate confidence intervals around the effect but that is rarely done. Is there a standard article that can be cited to dissuade reviewers from demanding this all-too-common practice? \u2013 russellpierce \u2013 2010-11-26T03:56:18.223 The only critique I know is Blouin & Riopelle (2005) but they don't get to the heart of the matter. I generally don't insist on not showing them but doing something correct as in the effect graphs of Masson & Loftus (2003, see figure 4, right panel... if they were removed from the left one you'd have done it right). \u2013 John \u2013 2010-11-26T13:41:01.367 Just to be clear, the problem with those CI's is that they're purely used for inferential reasons with respect to differences among conditions and therefore are worse even than PLSD... in fact I prefer them. At least they're honest. \u2013 John \u2013 2010-11-26T13:42:01.457 16 While I can relate to much of what Michael Lew says, abandoning p-values in favor of likelihood ratios still misses a more general problem--that of overemphasizing probabilistic results over effect sizes, which are required to give a result substantive meaning. This type of error comes in all shapes and sizes and I find it to be the most insidious statistical mistake. Drawing on J. Cohen and M. Oakes and others, I've written a piece on this at http:\/\/integrativestatistics.com\/insidious.htm . @rolando2 - You still want a likelihood ratio, but when there is a substantive value that has meaning, you should really be testing on this value rather than on$0$. So if$|\\beta|=1$has physical meaning in terms of the coefficient, then you should really be testing for$|\\beta|>1$vs$|\\beta|\\leq 1$. And not$\\beta=0$vs$\\beta\\neq 0$\u2013 probabilityislogic \u2013 2012-03-09T05:31:11.800 3I'm actually unclear as to how a likelihood ratio (LR) does not achieve everything that an effect size achieves, while also employing an easily interpretable scale (the data contains X times more evidence for Y than for Z). An effect size is usually just some form of ratio of explained to unexplained variability, and (in the nested case) the LR is the ratio of unexplained variability between a model that has an effect and one that doesn't. Shouldn't there at least be a strong correlation between effect size and LR, and if so, what is lost by moving to the likelihood ratio scale? \u2013 Mike Lawrence \u2013 2011-01-06T19:54:51.660 Mike - You've got me interested, but do your points extend to effect sizes as simple as mean differences between groups? These can be easily interpreted by a lay person and can also be assigned confidence intervals. \u2013 rolando2 \u2013 2011-01-06T20:22:53.617 Ah, so by effect size, you mean absolute effect size, a value that is meaningless unto itself, but that can be made meaningful by transformation into relative effect size (by dividing by some measure of variability, as I mentioned), or by computing a confidence interval for the absolute effect size. My argument above applies to the merits of LRs vs relative effect sizes. There may utility to computing effect CIs in cases where the actual value of the effect is of interest (eg. prediction), but I still stand by the LR as a more intuitive scale for talking about evidence for\/against effects. \u2013 Mike Lawrence \u2013 2011-01-06T23:11:56.483 I guess the use of LRs vs CIs will likely vary according to the context, which may be usefully summarized as follows: More exploratory stages of science, where theories are roughly characterized by the existence\/absence of phenomena, may prefer LRs to quantify evidence. On the other hand, CIs may be preferred in more advanced stages of science, where theories are sufficiently refined to permit nuanced prediction including ranges of expected effects or, conversely, when different ranges of effect magnitudes support different theories. Finally, predictions generated from any model need CIs. \u2013 Mike Lawrence \u2013 2011-01-06T23:18:49.550 14 Failing to test the assumption that error is normally distributed and has constant variance between treatments. These assumptions aren't always tested, thus least-squares model fitting is probably often used when it is actually inappropriate. 9What's inappropriate about least squares estimation when the data are non-normal or heteroskedastic? It is not fully efficient, but it is still unbiased and consistent. \u2013 Rob Hyndman \u2013 2010-11-16T03:18:20.200 3If the data are heteroscedastic you can end up with very innacurate out of sample predictions because the regression model will try too hard to minimise the error on samples in areas with high variance and not hard enough on samples from areas of low variance. This means you can end up with a very badly biased model. It also means that the error bars on the predictions will be wrong. \u2013 Dikran Marsupial \u2013 2010-11-16T09:31:44.380 6No, it is unbiased, but the variance is larger than if you used a more efficient method for the reasons you explain. Yes, the prediction intervals are wrong. \u2013 Rob Hyndman \u2013 2010-11-16T12:39:49.377 4Yes (I was using biased in a colloquial rather than a statistical sense to mean the model was systematically biased towards observations in high-variance regions of the feature space - mea culpa!) - it would be more accurate to say that the higher variance means there is an increased chance of getting a poor model using a finite dataset. That seems a reasonable answer to your question. I don't really view unbiasedness as being that much of a comfort - what is important is that the model should give good predictions on the data I actually have and often the variance is more important. \u2013 Dikran Marsupial \u2013 2010-11-16T22:18:23.977 12 My intro psychometrics course in undergrad spent at least two weeks teaching how to perform a stepwise regression. Is there any situation where stepwise regression is a good idea? I'm working on a project that uses stepwise regression. The reason is because I have D >> N, where D is dimensionality and N is sample size (thus ruling out using one model with all the variables), subsets of the features are highly correlated with each other, I want a statistically principled way of selecting maybe 2-3 \"best\" features, and I don't intend to report the P-values, at least without some kind of fairly conservative correction. \u2013 dsimcha \u2013 2011-01-29T04:16:14.197 6\"Good idea\" depends on the situation. When you want to maximize prediction it isn't a horrible idea - though it may lead to over fitting. There are some rare cases where it is inevitable - where there is no theory to guide the model selection. I wouldn't count stepwise regression as a \"sin\" but using it when theory is sufficient to drive model selection is. \u2013 russellpierce \u2013 2010-11-16T02:52:32.267 19Perhaps the sin is doing statistical tests on a model obtained via stepwise regression. \u2013 Rob Hyndman \u2013 2010-11-16T03:19:25.860 3It's fine if you use cross-validation, and don't extrapolate. Don't publish the p-values though, as they are meaningless. \u2013 Neil McGuigan \u2013 2010-11-30T06:16:54.457 11 This may be more of a pop-stats answer than what you're looking for, but: Using the mean as an indicator of location when data is highly skewed. This isn't necessarily a problem, if you and your audience knows what you're talking about, but this generally isn't the case, and the median is often likely to give a better idea of what's going on. My favourite example is mean wages, which are usually reported as \"average wages\". Depending on the income\/wealth inequality in a country, this can be vastly different from the median wage, which gives a much better indicator for where people are at in real life. For example, in Australia, where we have relatively low inequality, the median is 10-15% lower than the mean. In the US the difference is much starker, the median is less than 70% of the mean, and the gap is increasing. Reporting on the \"average\" (mean) wage results in a rosier picture than is warranted, and could also give a large number of people the false impression that they aren't earning as much as \"normal\" people. 1This is not just related to skewness, but is a general problem that the mean, or any other measure of central tendency is not enough without considering dispersion. For example, if the medians of two groups were equal, but the inter quartile range was 100 times as big for one population. Just looking at the median, you would say they're the \"same population distribution\", when in reality they would be very different. Not to mention multiple modes creating problems... \u2013 probabilityislogic \u2013 2014-07-02T04:48:41.930 But, for some purposes mean is relevant: wage is an extensive variable, meaning that sums of wages are meaningful. For questions where total wage income of some (sub)group is relevant, means are the right thing: The total can be recovered from the mean, not from the median. \u2013 kjetil b halvorsen \u2013 2015-08-16T12:19:30.050 @kjetilbhalvorsen: Why not just use the total then? \u2013 naught101 \u2013 2015-08-17T00:45:00.350 @naught101: Because diffwrent (sub)groups generaly will have diferent$n$s. \u2013 kjetil b halvorsen \u2013 2015-08-17T09:02:55.530 There's a semi-related discussion of this as it applies to trend analysis here: https:\/\/tamino.wordpress.com\/2012\/03\/29\/to-robust-or-not-to-robust-that-is-the-question\/ \u2013 naught101 \u2013 2012-04-11T01:05:40.757 10 My old stats prof had a \"rule of thumb\" for dealing with outliers: If you see an outlier on your scatterplot, cover it up with your thumb :) This is akin to Winsorization which isn't too terrible. \u2013 Ari B. Friedman \u2013 2011-08-10T20:28:23.353 9 That the p-value is the probability that the null hypothesis is true and (1-p) is the probability that the alternative hypothesis is true, of that failing to reject the null hypothesis means the alternative hypothesis is false etc. 1Interestingly, Atkin shows thhe pvalue is the posterior probability that the likelihood ratio is less than$1$(for the fixed data that was observed) \u2013 probabilityislogic \u2013 2012-05-05T01:36:24.833 Interesting, can you give me a reference to read up about it? \u2013 Dikran Marsupial \u2013 2012-05-05T14:04:38.707 2 (here you go)[http:\/\/www.ece.uvic.ca\/~bctill\/papers\/mocap\/Aitkin_1997.pdf] personally, while I do find it interesting, I struggle with the question of why the posterior distribution of the likelihood ratio is the quantity of interest. \u2013 probabilityislogic \u2013 2012-05-06T00:08:44.527 8 Repeating the same or similar experiment over 20 times on the same data and then reporting a statistically significant result with$\\alpha = 0.05$. Incidentally there is a comic about this one. And similarly to (or almost the same as) @ogrisel's answer, performing a Grid search and reporting only the best result. I think you meant to link to a different comic, though that's an immortal one. \u2013 rolando2 \u2013 2017-01-25T16:39:47.133 Possibly, if I remember well enough what I had in mind back then: https:\/\/xkcd.com\/882\/ \u2013 Andrew \u2013 2017-01-25T20:27:47.753 8 (With a bit of luck this will be controversial.) Using a Neyman-Pearson approach to statistical analysis of scientific experiments. Or, worse, using an ill-defined hybrid of Neyman-Pearson and Fisher. sorry to be ignorant, but what's wrong with a Neyman-Pearson construction for the analysis of (the outcome of) scientific experiments ? \u2013 Andre Holzner \u2013 2010-12-01T20:15:43.133 @Andre I think this remark may be closely related to another one offered by @Michael Lew elsewhere in this thread (http:\/\/stats.stackexchange.com\/questions\/4551\/what-are-common-statistical-sins\/4567#4567 ). \u2013 whuber \u2013 2010-12-12T23:04:48.967 8 Using pie charts to illustrate relative frequencies. More here. 2Would be good to include some reasoning on-site. \u2013 naught101 \u2013 2012-04-10T12:32:24.497 8 In similar vein to @dirkan - The use of p-values as a formal measure of evidence of the null hypothesis being true. It does have some good heuristic and intuitively good features, but is essentially an incomplete measure of evidence because it makes no reference to the alternative hypothesis. While the data may be unlikely under the null (leading to a small p-value), the data may be even more unlikely under the alternative hypothesis. The other problem with p-values, which also relates to some styles of hypothesis testing, is there is no principle telling you which statistic you should choose, apart from the very vague \"large value\"$\\rightarrow$\"unlikely if null hypothesis is true\". Once again, you can see the incompleteness showing up, for you should also have \"large value\"$\\rightarrow$\"likely if alternative hypothesis is true\" as an additional heuristic feature of the test statistic. I am not answering because I don't want to go to the trouble of thinking one up and for that matter wading through all the ones already given to make sure I don't repeat one! But I think i can be helpful. There is a book by Good and Hardin called \"Common Errors in Statistics and How to Avoid Them.\" You can find a lot of great examples there. It is a popular book that is already going into its fourth edition. \u2013 Michael Chernick \u2013 2012-05-04T18:04:35.533 Also Altman's book with Chapman & Hall\/CRC \"Practical Statistics in Medical Research\" has a chapter on the medical literature where many statistical sins are revealed that occurred in published papers. \u2013 Michael Chernick \u2013 2012-05-04T18:04:42.553 8 Using statistics\/probability in hypothesis testing to measure the \"absolute truth\". Statistics simply cannot do this, they can only be of use in deciding between alternatives, which must be specified from \"outside\" the statistical paradigm. Statements such as \"the null hypothesis is proved true by the statistics\" are just incorrect; statistics can only tell you \"the null hypothesis is favoured by the data, compared to the alternative hypothesis\". If you then assume that either the null hypothesis or the alternative must be true, you can say \"the null proved true\", but this is only a trivial consequence of your assumption, not anything demonstrated by the data. 7 Requesting, and perhaps obtaining The Flow Chart: That graphical thing where you say what the level of your variables are and what sort of relationship you're looking for, and you follow the arrows down to get a Brand Name Test or a Brand Name Statistic. Sometimes offered with mysterious 'parametric' and 'non-parametric' paths. 7 Perhaps the poor teaching of statistics to end consumers. The fact is that most courses have given a medieval menu, not including new theoretical developments, computational and best practices, insufficient teaching of modern and complete analysis of real data sets, at least in poor and developing countries, what is the situation in developed countries? 3The situation in developed countries is exactly the same. \u2013 Flounderer \u2013 2014-08-17T01:41:40.493 7 Temptation to use advanced statistical methods without understanding them, just because they sound impressive or because they happen to better support researcher's initial hypothesis. When one uses an advanced method he or she should have solid reasons as to why the method is appropriate. 6 In psychology, the cardinal sin (for me) is the use of principal components analysis to examine the hypothesised latent structure underlying a psychometric test. Not testing for normality before using tests which assume this. https:\/\/stats.stackexchange.com\/questions\/2492\/is-normality-testing-essentially-useless ... I especially recommend Harvey's answer there. \u2013 Glen_b \u2013 2017-10-18T04:41:45.243 1Take a look at the sin I have already proposed about testing. If you do the test and do not reject normality, it does not mean that you have a normal sample... it only means that you cannot say the sample is not normal. sin ! \u2013 robin girard \u2013 2010-12-01T16:04:56.537 6 Probably not as applicable to psych stats (or is it? I'm not sure) but failing to account for a split plot design in an analysis of an experiment. I've seem way too many people do this. A pre-post, experimental-control group design is extremely common in psychology. I agree that few people seem to be aware of the comparatively complicated and strict assumptions. Mixed models often seem to be beyond the statistical horizon. \u2013 caracal \u2013 2010-12-01T21:36:20.573 5 I would say, doing tests and regressions on a small set of data. Edit: Without looking at the confidence intervals, or when the confidence intervals\/error bars are not easy to calculate. 4Perhaps I don't see why this is such a problem. Hypothesis testing a small sample size using a normal distribution, sure, but using a more conservative\/nonparametric test, is this so bad? \u2013 Christopher Aden \u2013 2010-11-16T10:18:30.767 I agree that using a more conservative model to fit the data is the best we can do. But in any case you will have to trust this model. It will be a fitting, not a model. A model requires a representative set of data otherwise it may not work in the future. \u2013 RockScience \u2013 2010-11-17T02:17:33.240 If you use Bayesian regression then the error bars also indicate the uncertainty due to the finite nature of the dataset (given the prior), you only trust the model as far as the error bars suggest you ought to trust it. If you don't have enough data to make a useful inference it will generally be evident in the posterior distributions for the parameters and\/or predictions. The usual frequentist error bars will probably say pretty much the same thing. At the end of the day, sometimes only a small dataset is available, it just limits the confidence in your conlcusions. \u2013 Dikran Marsupial \u2013 2010-11-19T09:14:11.553 Agreed for the Bayesian regression. Thanks for pointing that out. But if you have two points that form a straight line, how do you calculate the frequentist error bars? And let's say that you have enough points to calculate the frequentist error bars, from how many points can you trust them (should we use the error bars of the error bars?) \u2013 RockScience \u2013 2010-11-19T09:26:57.503 Sorry, I am not that familiar with the limitations of frequentist error bars (I only mentioned them so as not to appear bigoted ;o). IIRC frequentist error bars for ridge regression are very similar to the Bayesian ones for a fixed value of the ridge parameter (a full Bayesian analysis would marginalise over the ridge parameter as well). \u2013 Dikran Marsupial \u2013 2010-11-19T09:50:49.527 1Isn't performing a test implicitly looking at the confidence intervals? Perhaps it would be that the sin test-wise is ignoring the power of the test? \u2013 Dikran Marsupial \u2013 2010-11-19T13:11:43.460 5 Using Analysis of Covariance (ANCOVA) to try to \"control for\" or \"regress out\" the influence of a covariate that is known to be correlated with, or affect the influence of, other predictor variables. More discussion at this question: Checking factor\/covariate independence in ANCOVA 4 This isn't generally considered a sin but I hope it will be one day: using a bad model that doesn't describe reality just because it's \"interpretable.\" 4 Rush into modeling before spending enough time on understanding and preprocessing the data. This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review \u2013 kjetil b halvorsen \u2013 2016-11-28T08:26:19.143 This is probably adequate, IMO, given the context of the thread to which it was posted. I think it's OK. @Aliweb, you might want to develop \/ elaborate the idea a little, though. \u2013 gung \u2013 2016-11-28T13:06:57.247 @kjetilbhalvorsen This exactly answers the question. The author is asking for some statistical sins and if you read the question, my answer couldn't be seen as a critic to the author. \u2013 Aliweb \u2013 2016-11-28T17:42:15.177 1@gung I tried to answer the question the same way many others have done and considering what the asker is asking for. I think elaborating the idea would be kind of repeating myself or I should start talking about \"how to understand\/preprocess the data\" which is not what the question is about. \u2013 Aliweb \u2013 2016-11-28T17:44:36.117 1@Aliweb, that's fair. I only said \"might\"; if you can't go further w\/o repeating yourself or moving off topic, then you're best staying put. \u2013 gung \u2013 2016-11-28T18:31:42.700 I upvoted the answer though I bet many of us would benefit from seeing an example or 2. \u2013 rolando2 \u2013 2017-01-25T16:44:44.697 4 Interpreting a$100\\alpha \\%$Confidence Interval$I$as the probability of finding the \"real\" parameter inside the interval. The most common case is when someone calculates this C.I. ($I$) and interprets the number$\\alpha$as the probability of finding the \"true mean\" say,$\\mu$, inside the interval, i.e., interpreting the C.I. as$P(\\mu \\in I)=\\alpha$. 3Is this such a bad thing? I understand the usual argument of \"the mean is either in the interval or it isn't\", or at least I think I do. But the endpoints of the confidence interval are random variables, so why is it wrong to talk about the probability that they take values above and below the true mean? \u2013 mark999 \u2013 2012-04-08T22:43:34.040 It IS bad because you are giving a probability statement that doesn't exist. The phrase \"the true population mean$\\mu$is either in the interval or isn't\" states a beautiful fact:$P(\\mu \\in I)$is either 1 or 0. On the other hand, recall that what you calculate when creating confidence intervals is$P(L_1<\\bar{X}-\\mu<L_2)=\\alpha$, where$\\bar{X}$is a STATISTIC (i.e. a function of random variables) and is, therefore, the random variable on which you are calculating the probability. The 100$\\alpha$% confidence interval that you calculate is one of many other (random) intervals that... \u2013 N\u00e9stor \u2013 2012-04-09T03:58:15.360 ...might \"appear\" as you sample from the population. Simply put,$I$is one realization of many random intervals (say,$I_r$) that are generated as you sample from the population. You can only say that$I_r$is a range in which the mean will occur 95% of the time, but that says nothing about your particular realization (or estimation) of interval$I$that one usually calculates, and, therefore, says nothing about the probability of$I$containing$\\mu\\$. \u2013 N\u00e9stor \u2013 2012-04-09T04:06:57.843\n\n2Thanks for the explanation, I think I understand now. I wasn't suggesting that a particular realisation of a 95% confidence interval has 95% chance of containing the true value, although I didn't make that clear. What I meant was that saying \"the probability that the (generic) interval contains the true value is 0.95\" seems to me to be equivalent to saying \"if repeated many times, 95% of the intervals will contain the true value\". \u2013 mark999 \u2013 2012-04-09T04:55:03.887\n\nOh, yes! That is in fact true :-). Just a misunderstanding then. \u2013 N\u00e9stor \u2013 2012-04-09T04:57:47.943\n\n4\n\nOver-interpreting OLS regression in the presence of known outliers. If you know that there are particular data in your dataset which are generated by a different process to the process that generates most of the data, and this different process generates wildly different results which show up as outliers, then you have to be very careful in interpreting the model output because the outliers often do substantially move the OLS results. That's not to say OLS is bad, just that you need to think about the data when interpreting the results.\n\nWhat's worse is that we often have \"never throw away outliers\" as common advice to early students. Sometimes it translates into an attitude of keeping the data, warts and all, without really discussing anomalies and outliers.\n\nBetter advice might be: \"use a mixture model\" or \"use Huber\/quantile-based\/other robust techniques\" or \"go Bayes and use a hierarchical model\". But everyone should at least learn to just \"reanalyse without the suspect outliers and print both analyses and show us a plot\" or even \"talk qualitatively for a bit about outliers in the conclusion of your paper and suggest it might be a good idea to redo the experiment with fewer foul-ups\".\n\nHah. I already added this as a comment to my answer, but it's more relevant to this answer: https:\/\/tamino.wordpress.com\/2012\/03\/29\/to-robust-or-not-to-robust-that-is-the-question\/ discusses situations where OLS may be better than robust regression, even in cases with (apparent) outliers.\n\n\u2013 naught101 \u2013 2012-04-11T01:28:54.183\n\n4\n\nApplication of least-squares minimization when maximum-likelihood procedures exist.\n\n3@MikeLawrence - \"maximum likelihood\" = \"weighted least squares\" in many cases. even maximum entropy is approximately least squares (when the initial measure isn't too far from the optimum measure). You could do a lot worse than least squares... \u2013 probabilityislogic \u2013 2012-03-09T05:43:06.650\n\n3Would you please explain the consequences of this sin? \u2013 russellpierce \u2013 2010-11-16T02:50:28.617\n\n1If the data are generated by a process with a heteroscedastic noise process, the regression model is likely to give very inacurate out-of-sample predictions. \u2013 Dikran Marsupial \u2013 2010-11-16T09:26:46.770\n\nThis entry was originally inspired by the observation that some folks estimate non-linear psychometric functions by minimizing least squares. For example, Murd et al (2009, http:\/\/www.perceptionweb.com\/abstract.cgi?id=p6145 , free pdf available by googling the title) fit a probit function through data by minimizing least-squares.\n\n\u2013 Mike Lawrence \u2013 2010-11-16T16:04:32.697\n\nSo, should the \"answer\" be amended to \"application of least-squares minimization on heteroscedatic data\"? \u2013 russellpierce \u2013 2010-11-17T15:13:00.360\n\nif the noise process is skewed but homoscedastic then it is still inappropriate if you need the error bars. \u2013 Dikran Marsupial \u2013 2010-11-19T13:10:33.167\n\n3\n\nInterpreting a statistically significant result as \"meaningfully large\".\n\n3\n\nSpecifically in psychology, and even more so in marketing, the technique of partial least squares (PLS) is used to \"fit\" structural equation models and path models, despite being deficient on almost any imaginable performance metric. See McIntosh, Edwards and Antonakis (2014 ORM), R\u00f6nkk\u00f6, McIntosh and Antonakis (2015 PID) and\/or R\u00f6nkk\u00f6, McIntosh, Antonakis and Edwards (2016 JOM) for detailed treatment, including spelling out some natural requirements like bias and consistency, and demonstration of how PLS fails them (compared to other regression-type methods such as Bollen's model-implied instrumental variables). (I don't know how substitutable the papers are for one another, though; they must cover very similar topics, but may be aimed at somewhat different audiences.)\n\nDidn't you just list a different paper from the chemometrics literature somewhere? It might be worth adding to the list. \u2013 gung \u2013 2016-08-11T20:00:31.303\n\nGood point -- it was @amoeba who was familiar with the chemometrics literature; I am not qualified to speak on that. My understanding is that it is used with much less fanfare there than in psych and marketing, just to honestly reduce dimensionality rather than to claim that you found some underlying factors and quantified them perfectly. \u2013 StasK \u2013 2016-08-11T21:08:25.157\n\n3\n\nNot paying attention to levels of measurement, and treating polytomous nominal scales as though they were ordinal, interval, or ratio scales (Ouch).\n\n2\n\nUsing technical replication instead of true replication, and similarly, using MSE as the denominator in a nested ANOVA F-statistic.\n\n2\n\nCompletely forgetting about checking calibration or normalization, when datasets come for different sensors, different times, different observers.","date":"2019-02-16 00:08:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.609007716178894, \"perplexity\": 1953.0801140372332}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550247479627.17\/warc\/CC-MAIN-20190215224408-20190216010408-00370.warc.gz\"}"}
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<pre>public class <span class="strong">RSAKeyPairGenerator</span>
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<div class="block">A simple utility class that generates a RSA PGPPublicKey/PGPSecretKey pair.
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usage: RSAKeyPairGenerator [-a] identity passPhrase
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Where identity is the name to be associated with the public key. The keys are placed
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| 6,435
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Gossys Reef is a small granite island, with an area of 1.5 ha, in south-eastern Australia. It is part of Tasmania's Sentinel Island Group, lying in eastern Bass Strait off the north-west coast of Flinders Island in the Furneaux Group.
Fauna
Seabirds and waders recorded as breeding on the island include silver gull, Caspian tern and sooty oystercatcher. The metallic skink is also present.
References
Furneaux Group
|
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\section{Introduction}
Nonlinear wave phenomena are of wide physical and mathematical interest as they arise in diverse areas of science such as nonlinear optics, fluid dynamics,
lattice dynamics, plasma physics, bio-physics and Bose-Einstein condensates (BEC) \cite{fluids,ablowitz,wiley,pana}.
In the context of BECs several kinds of nonlinear coherent structures,
such as, solitons \cite{dj}, discrete breathers \cite{dbrea}, rouge waves \cite{bludov}, Faraday waves \cite{engels}, gap solitons \cite{kivshar}
and vortices \cite{vortex} have been observed. Particularly, the multicomponent BECs host rich dynamical behaviors because of the vector order parameters. These systems display interesting effects
such as Josephson effect \cite{joseph}, Efimov effect \cite{efimov}, spin textures \cite{spintext},
spin-orbit coupling \cite{spinorbit}, just to cite a few of them. The fundamental multicomponent BEC is the two component BEC. Such two-component BEC has been realized by using a mixture of atoms with two hyperfine states of
$^{87}Rb$ \cite{rabi11}. The theoretical studies on spin dipole oscillations \cite{rabi1}, vortex pairs \cite{rabi2}, phase transition \cite{abad}, localized waves \cite{pair} and bosons in optical lattices \cite{mole} emphasize the need for investigation of BECs with coherent (phase-dependent) coupling.\\
Observing solitons in multicomponent condensates is one of the current research topics. The formation and dynamical properties of solitons in BECs are determined by the nature of their two-body atomic interactions, i.e., the sign of the $s$-wave scattering length which may be positive (negative) for repulsive (attractive) interatomic interactions. The $s$-wave scattering length and hence the nonlinearity co-efficient can be controlled by means of Feshbach resonance \cite{fesh}. If we use a time-dependent magnetic field, the strength of the nonlinear interaction can be tuned by time-dependent Feshbach resonance. In real experiments, various forms of time-dependence of nonlinearity have been explored \cite{greiner}.
In theoretical studies, many authors have thoroughly studied the dynamics of BECs for several forms of time (or) spatial dependent nonlinearity co-efficients
(see, e.g., \cite{boaris,boaris1,abdul} and references therein). In the present work, we will be interested in a system of two coupled Gross-Pitaevskii equations with a coherent coupling between both components \cite{rabi,abad} along with time-varying scattering lengths and
external potential. \\
On the other hand, study of nonlinear waves like, protean rogue waves and breathers in BECs is also one of the main objectives of modern day research \cite{pana,bludov,bec_brea}. A formal mathematical description of single rogue wave is provided by the nonlinear Schr\"odinger (NLS) equation in the self-focusing regime \cite{solli}. Here the mechanism which leads to the generation of rogue waves is nonlinear wave mixing, that generates modulation instability (MI) of the continuous wave (CW) background~\cite{mi}. The nonlinear development of MI is described by families of exact breathers. A special member of this solution family is the Peregrine soliton \cite{peregrine}, which represents a wave that is localized both in space and time dimensions. In recent years, rogue waves and breathers have attracted much attention \cite{akmbec,spinrogue,bec_brea,degas}. A number of recent reviews have attempted to summarize the study of rogue waves in different contexts \cite{review}. The space-periodic breather type solution of NLS system has been obtained by Akhmediev et al., \cite{ussr} and the time-periodic solution of NLS was derived by Ma \cite{ma}.
\\
In this paper, we carry out a thorough analysis of the dynamics of superposed nonlinear waves
in autonomous and non-autonomous coherently coupled Gross-Pitaevskii (CCGP) system. The autonomous CCGP equations considered in this paper are shown to be integrable in Ref.~\cite{Park} and Kanna et al., have constructed the explicit soliton solutions of the underlying equation by applying a non-standard Hirota's bilinearization procedure in Ref. \cite{kannajpa}. Very recently, in Ref.~\cite{zhao}
by applying the Darboux transformation method, the soliton and rogue wave solutions of CCGP equations
have been constructed. In the present work, we decouple the autonomous CCGP equations into two uncoupled NLS equations and make use of their different nonlinear wave solutions profitably to construct the superposed nonlinear wave solution of the autonomous CCGP equations and study their subsequent dynamics. Then we make use of these superposed nonlinear waves to obtain the nonlinear wave solutions of the non-autonomous CCGP
equations with the aid of a similarity transformation. We have observed an interesting soliton phenomenon namely, soliton compression and also shown the existence of creeping soliton. The obtained solutions are new and display rich dynamical features that can find applications in atom interferometry and matter wave switches.\\
The remaining part of this paper is arranged as follows: We present the model equation and its physical significance in Sec. II. In Sec.
III, the set of autonomous CCGP equations is converted into two independent NLS systems.
We review the solutions of scalar (decoupled) NLS system and discuss their properties in Sec. III A.
Sec. III B, deals with the interesting coherent structures of superposed nonlinear waves,
namely colliding Ma breathers, coexisting rogue wave and Ma breathers, and coexisting
Ma and Akhmediev breathers. In Sec. IV, we convert the non-autonomous CCGP system into an integrable autonomous
CCGP system by using a similarity transformation. Also we clearly bring out the role of variable
nonlinearity by considering two types of time-varying nonlinearities namely, kink-like and
periodic modulated (Mathieu function) nonlinearity.
Finally, the results are summarized in Sec.~V.\\
\section{The Model}
Here we consider the quasi-one-dimensional (cigar-shaped) BEC. For this case, the three dimensional CCGP system can be reduced to an one dimensional system and we can write the governing equations in dimensionless form as \cite{rabi,abad}
\begin{subequations}\begin{eqnarray}
i\frac{\partial\psi_1}{\partial t}&=&L_1\psi_1-\alpha_1(|\psi_1|^2+2|\psi_2|^2)\psi_1-\alpha_2 \psi_2^2\psi_1^*,\\
i\frac{\partial\psi_2}{\partial t}&=&L_2\psi_2-\alpha_1(2|\psi_1|^2+|\psi_2|^2)\psi_2-\alpha_2 \psi_1^2\psi_2^*,
\end{eqnarray}\label{nccnls}\end{subequations}
where $L_i$ = $-\partial_x^2+U_i(x,t)$ with $i=1,2$. Here, we measure
the length and energy in units of
$a_{ho} = \sqrt{\hbar/m\omega_\bot}$, where $a_{ho}$ is the characteristic length of the
condensate and $\hbar\omega_\bot$, in which $\omega_\bot$ is the transversal frequency.
In Eqs.~(\ref{nccnls}), $\psi_{j}$~($j=1,2$) are the condensate wave functions;
the coefficients $\alpha_1$ and $\alpha_2$ introduce incoherent and coherent coupling between the two components and
they can be tuned using Feshbach resonance mechanism and $U_i(x,t)(i=1,2)$ are the external potentials. For the homogeneous system the external potentials $U_i(x,t)$= 0. \\
The physical significance of the above proposed model can be realized in the following two contexts. (i) In spin-1 BECs, the governing equation is three-component GP equations with the
components [i.e., $\Psi$=($\psi_{+1}/\sqrt{2}$, $\psi_0$, $\psi_{-1})^T$] corresponding to the three values of the vertical
spin projection, $m_F$ = $-1, 0,+1$ (see Eqs. (4) and (5) in ref. \cite{spincomplex}). For the special case ($\psi_{+1}=\psi_{-1}$ and $c_0=c_2=-c$), the set of three-component GP equations is reduced to two component GP system (\ref{nccnls}) with $\alpha_1$=$\alpha_2$=$c$. (ii) In the context of nonlinear optics, the model equation exactly same as that of (\ref{nccnls}) governs the pico-second pulse propagation in nonlinear Kerr media with low birefringence or beam propagation in weakly anisotropic media and is usually referred as coherently coupled nonlinear Schr\"odinger equations \cite{book,kannajpa}.
Apart from these, Eqs.~(\ref{nccnls}) can also be viewed as the continuum limit of the model for the discrete coupled atomic condensates describing the four-wave mixing effects discussed in ref. \cite{discrete}
\section{Autonomous CCGP system}\label{trans}
First, we consider the homogeneous binary condensates ($U_i(x,t)=0$) with constant nonlinearity co-efficient $\alpha_1 = \alpha_2= \alpha$ in Eq.~(\ref{nccnls}). Here
$\alpha$ is a real integer and is independent of time. The resulting equations can be expressed as
\begin{subequations}\begin{eqnarray}
i\frac{\partial\psi_1}{\partial t}&=&-\frac{\partial^2\psi_1}{\partial x^2 }-\alpha(|\psi_1|^2+2|\psi_2|^2)\psi_1-\alpha\psi_2^2\psi_1^*,\\
i\frac{\partial\psi_2}{\partial t}&=&-\frac{\partial^2\psi_2}{\partial x^2 }-\alpha(2|\psi_1|^2+|\psi_2|^2)\psi_2- \alpha\psi_1^2\psi_2^*.
\end{eqnarray}\label{ccnls}\end{subequations}
In this paper, we refer to Eqs.~(\ref{ccnls}) as autonomous CCGP equations.
These autonomous CCGP equations are integrable \cite{Park} and several interesting soliton solutions
have been obtained in Ref.\cite{kannajpa} by applying a non-standard Hirota's bilinearization technique.
We note that Eqs.~(\ref{ccnls})
with $\alpha=2$, can be decoupled into two independent NLS equations of the form
\begin{eqnarray}
~~~~~~~~~~iu_{j,t}+u_{j,xx} + 2u_j^*u_j^2=0,\qquad j=1,2,
\label{singlenls}
\end{eqnarray}
by applying a linear transformation
\begin{subequations}\begin{eqnarray}
&&\psi_1(x,t) = \frac{1}{2}(u_1+u_2),\\ &&\psi_2(x,t) = \frac{1}{2}(u_1-u_2),
\end{eqnarray}\label{q1}
\end{subequations}
where $u_1$ and $u_2$ are arbitrary analytic functions of $x$ and $t$ and satisfy the NLS equation (\ref{singlenls}).
The above NLS system (\ref{singlenls}) is completely integrable
and has been investigated extensively by using various analytical methods.
Interestingly, it admits many kinds of nonlinear wave solutions such as solitons,
rogue waves and breathers (including Ma and Akhmediev breathers) \cite{Chab,tkopt}
and they have been experimentally observed under different contexts. Particularly the Peregrine soliton of
the NLS system has been recently observed experimentally in nonlinear optics \cite{kibler}.
In this paper, we address the dynamical features of
superposition of these various structures which are also possible solutions of Eqs.~(\ref{ccnls}).
As a prelude we briefly revisit some of the interesting nonlinear wave solutions of the scalar NLS system (\ref{singlenls}) in the following sub-section.
\subsection{Revisit of nonlinear waves in the NLS system }\label{trans}
\subsection*{(i)~ Bright one-soliton}
The bright one-soliton solution $u_1$ of the NLS equation (\ref{singlenls}) with $j=1$
is obtained as \cite{bookml}
\begin{eqnarray}
u_1(x,t) = k^{(1)}_{R}~\mbox{sech}(z^{(1)}_{2})e^{iz^{(1)}_{1}},
\label{soliton}
\end{eqnarray}
where $z^{(1)}_{1} = k^{(1)}_{I}x+((k^{(1)}_{R})^2-(k^{(1)}_{I})^2)t+\eta_{I}^{(1)}$, $z^{(1)}_{2} =
k^{(1)}_{R}(x-2k^{(1)}_{I}t)+\eta_{R}^{(1)}+R^{(1)}$, and
$R^{(1)} = \ln \left[\frac{1}{2k^{(1)}_{R}}\right]$.
The above one-soliton solution $u_1$ is characterized by four real parameters $k^{(1)}_{R}$, $k^{(1)}_{I}$,
$\eta_{R}^{(1)}$ and $\eta_{I}^{(1)}$. Similar solution for $u_2$ can be obtained by replacing the superscript (1) by (2). Here and in the following $R$ and $I$ appearing in the suffices of the soliton parameters denote the real and imaginary parts, respectively. The amplitude and velocity of the NLS soliton can be controlled by tuning $k^{(1)}_{R}$ and $k^{(1)}_{I}$, respectively,
while the soliton position depends on the parameter $\eta_{R}^{(1)}$.
\subsection*{(ii)~ Bright two-soliton}
The bright two-soliton solutions of the above two different NLS systems (\ref{singlenls}) ($u_j$ with $j=1$ corresponds to the
solution of the first NLS system and $j=2$ corresponds to the
solution of the second NLS system) are given by \cite{bookml}
\begin{subequations}\begin{eqnarray}
&&u_j(x,t) =\frac{G^{(j)}}{F^{(j)}}, \quad j=1,2,
\label{twosoliton}
\end{eqnarray}
where
\begin{eqnarray} &&G^{(j)}=e^{\eta^{(j)}_{1}}+e^{\eta^{(j)}_{2}}+e^{\eta^{(j)}_{1}+\eta_{1}^{(j)*}+\eta^{(j)}_{2}+\delta^{(j)}_{1}}+e^{\eta^{(j)}_{1}+\eta_{2}^{(j)*}+
\eta^{(j)}_{2}+\delta^{(j)}_{2}},\\
&&F^{(j)}=1+e^{\eta^{(j)}_{1}+\eta_{1}^{(j)*}+R^{(j)}_{1}}+e^{\eta^{(j)}_{1}+\eta_{2}^{(j)*}+\delta^{(j)}_{0}}+
e^{\eta^{(j)}_{2}+\eta_{1}^{(j)*}+\delta_{0}^{(j)*}}+e^{\eta^{(j)}_{2}+\eta_{2}^{(j)*}+R^{(j)}_{2}}+
e^{\eta^{(j)}_{1}+\eta_{1}^{(j)*}+\eta^{(j)}_{2}+\eta_{2}^{(j)*}+R^{(j)}_{3}}.
\end{eqnarray} \end{subequations}
In the above solution, $\eta^{(j)}_{1}=k^{(j)}_{1} x+i (k_{1}^{(j)})^2t+\eta_{1}^{(j)(0)}$, $\eta^{(j)}_{2}=
k^{(j)}_{2} x+i (k_{2}^{(j)})^2t+\eta_{2}^{(j)(0)}$, $e^{R^{(j)}_{1}}=\frac{1}{(k^{(j)}_{1} + k_{1}^{(j)*})^2}$,
$e^{R^{(j)}_{2}}=\frac{1}{(k^{(j)}_{2} + k_{2}^{(j)*})^2}$, $e^{\delta^{(j)}_{0}}=\frac{1}{(k^{(j)}_{1}+k_{2}^{(j)*})^2}$,
$e^{\delta_{0}^{(j)*}}=\frac{1}{(k_{1}^{(j)*}+k^{(j)}_{2})^2}$, $e^{\delta^{(j)}_{1}}=\frac{(k^{(j)}_{1}-k^{(j)}_{2})^2}{(k^{(j)}_{1}+k_{1}^{(j)*})^2(k_{1}^{(j)*}+k^{(j)}_{2})^2}$,
$e^{\delta^{(j)}_{2}}=\frac{(k^{(j)}_{2}-k^{(j)}_{1})^2}{(k^{(j)}_{1}+k_{2}^{(j)*})^2(k_{2}^{(j)*}+k^{(j)}_{2})^2}$ and
$e^{R^{(j)}_{3}}=\frac{(k^{(j)}_{1}-k^{(j)}_{2})^2(k_{1}^{(j)*}-k_{2}^{(j)*})^2}{(k^{(j)}_{1}+k_{1}^{(j)*})^2(k_{1}^{(j)*}+k^{(j)}_{2})^2(k^{(j)}_{1}+k_{2}^{(j)*})^2(k^{(j)}_{2}+k_{2}^{(j)*})^2}$.
The bright two-soliton solution of the NLS equation is characterized by four complex parameters
$k^{(j)}_{1}$, $k^{(j)}_{2}$, $\eta_{1}^{(j)(0)}$ and $\eta_{2}^{(j)(0)}$.
Here the real parts of $k^{(j)}_{1}$ and $k^{(j)}_{2}$ control the amplitude of the respective soliton,
while their imaginary parts influence the velocity. One can observe two kinds of dynamics from the
above two-soliton solution: two-soliton bound state with beating for same soliton velocities ($k^{(1)}_{1I}=k^{(1)}_{2I}$)
and a head-on elastic soliton collision for equal but opposite soliton velocities ($k^{(1)}_{1I}=-k^{(1)}_{2I}$).
\subsection*{(iii)~ Rogue wave}
Rogue waves are nothing but a sudden peak (hump) localized both in space and time with maximum amplitude on
a continuous wave background.
The rogue wave (rational) solutions ($u_j,~j=1,2$) of the two independent NLS systems (\ref{singlenls})
obtained in Ref.~\cite{rogue}
can be written as
\begin{eqnarray}
u_j(x,t) = \left[1-\frac{2(1+4i|\delta^{(j)}|^2t)}{2|\delta^{(j)}|^2(x^2+4|\delta^{(j)}|^2t^2+\frac{1}{4|\delta^{(j)}|^2})}\right]\times \delta^{(j)} e^{2i|\delta^{(j)}|^2 t}, \quad j=1,2.
\label{rogue}
\end{eqnarray}
Here the arbitrary complex parameter $\delta^{(j)}$ influences the
amplitude as well as the width of the rogue wave in the $j^{th}$ NLS system. In general, the amplitude
of rogue wave is three-times (or more) higher than the background carrier wave.
\subsection*{(iv)~ Akhmediev breather}
Akhmediev breather (AB) solution ($u_j$, where $j=1$ corresponds to the
solution of the first NLS system and $j=2$ corresponds to the
solution of the second NLS system) of the two independent NLS equations (\ref{singlenls}) is given by
\begin{eqnarray}
u_j(x,t) = \left[\frac{\cosh(\Omega t-2ia^{(j)})-\cos(a^{(j)})\cos(b^{(j)}x)}{\cosh(\Omega t)-\cos(a^{(j)})\cos(b^{(j)}x)}\right]\times e^{2it}, \quad j=1,2,
\label{akh}
\end{eqnarray}
where $\Omega = 2 \sin(2a^{(j)})$, $b^{(j)} = 2 \sin(a^{(j)})$ and $a^{(j)}$ $\in$ $\mathbb{R}$.
The role of parameter $a^{(j)}$ in AB is directly related to the
number of breathers and inversely related to its maximum amplitude.
To be more precise, for increasing value of $a^{(j)}$ the number of breathers over a definite region
increases but their amplitude decreases, whereas the reverse scenario takes place for decreasing value of $a^{(j)}$ \cite{Chab}.
\subsection*{(v)~ Ma breather}
The Ma breather (MB) solutions of the two independent NLS equations (\ref{singlenls}) can be expressed as
\begin{eqnarray}
u_j(x,t) = \left[\frac{\cos(\Omega t-2ia^{(j)})-\cosh(a^{(j)})\cosh(b^{(j)}x)}{\cos(\Omega t)-\cosh(a^{(j)})\cosh(b^{(j)}x)}\right]\times e^{2it}, \quad j=1,2,
\label{Ma}
\end{eqnarray}
where $\Omega = 2 \sinh(2a^{(j)})$, $b^{(j)} = 2 \sinh(a^{(j)})$ and $a^{(j)}$ $\in$ $\mathbb{R}$.
These Ma breathers are localized in spatial~($x$) coordinate and periodic in time~($t$) coordinate.
The effect of parameter $a^{(j)}$ is straightforward in the case of MB, where the number of
breathers as well as their maximum amplitude increases (decreases) when $a^{(j)}$ is increased (decreased) \cite{Chab}.
\subsection{Dynamics of superposed nonlinear waves}
In the starting of this section, we have shown that the solution of the autonomous CCGP system (\ref{ccnls}) can be obtained by superposing the solutions of two different NLS systems. Following that, we revisit several nonlinear wave solutions of NLS system.
Now we look into the various combinations of these solutions discussed in Sec.
IIA and construct several superposed solutions of the autonomous CCGP system. This will give
rise to the possibility of non-trivial localized nonlinear coherent structures in the autonomous CCGP system.\\
\noindent{\textbf{Case~(i)} Superposition of two different one-soliton solutions}
To start with, we consider two different one-soliton solution forms obtained from (\ref{soliton}) with
$j=1$ and $2$ and write down the solutions $\psi_j,(j=1,2)$ of the autonomous CCGP system using (\ref{q1}).
First we superimpose two one-solitons travelling with different amplitudes and opposite velocities.
The resultant structure is a head on collision of two soliton in the $\psi_1$ and $\psi_2$ components. This is depicted in Fig.~\ref{s-s}.
\begin{figure}[!pht]
\centering\includegraphics[width=0.8\linewidth]{Fig_1}
\caption{Superposition of two different one-soliton. Top panel: Soliton collision with parameters
$k^{(1)}_R=1.3$, $k^{(1)}_{I}=0.25$, $k^{(2)}_{R}=1$, $k^{(2)}_{I}=-0.25$, $\eta_{R}^{(1)(0)}=\eta_{R}^{(2)(0)}=1$ and $\eta_{I}^{(1)(0)}=\eta_{I}^{(2)(0)}=0.2$;
Bottom panel: Bound soliton for the parametric choice $k^{(1)}_{R}=1.3$, $k^{(1)}_{I}=0$, $k^{(2)}_{R}=1$,
$\eta_{R}^{(1)(0)}=\eta_{R}^{(2)(0)}=1$ and $\eta_{I}^{(1)(0)}=\eta_{I}^{(2)(0)}=0.2$.}
\label{s-s}
\end{figure}
The expression for $|\psi_1|^2 = \frac{1}{4} \left[(k^{(1)}_{R})^2 \mbox{sech}^2(z^{(1)}_{2})+(k^{(2)}_{R})^2
\mbox{sech}^2(z^{(2)}_{2})+2 k^{(1)}_{R}k^{(2)}_{R}\mbox{sech}(z^{(1)}_{2})\mbox{sech}(z^{(2)}_{2})
\cos(z^{(1)}_{1}-z^{(2)}_{1})\right]$ and $|\psi_2|^2 = \frac{1}{4} \left[(k^{(1)}_{R})^2
\mbox{sech}^2(z^{(1)}_{2})+(k^{(2)}_{R})^2 \mbox{sech}^2(z^{(2)}_{2})-2 k^{(1)}_{R}k^{(2)}_{R}
\mbox{sech}(z^{(1)}_{2})\mbox{sech}(z^{(2)}_{2})\cos(z^{(1)}_{1}-z^{(2)}_{1})\right]$.
At the origin [$(x,t)$=0], the intensity of the resulting superposed coherent structure attains a maximum value in the $\psi_1$ component while it reaches a minimum value in the $\psi_2$ component. This is a consequence of conservation of total energy $(\int (|\psi_|^2+|\psi_2|^2) dt = const.)$. If we choose $k^{(1)}_{I}$ and $k^{(2)}_{I}$ to be zero or to be equal with $z^{(1)}_{1} \neq z^{(2)}_{1}$,
we obtain soliton bound state with oscillations along $t$ axis. This is clearly shown in the bottom panel
of Fig.~\ref{s-s}. Another interesting possibility arises for $k^{(1)}$=$k^{(2)}$, where the superposed
soliton exists only in the $\psi_1$ component and is absent in the $\psi_2$ component. Thus we can engineer the nature of soliton propagation by tuning the $k^{(j)}$ parameters appropriately.\\
\noindent{\textbf{Case~(ii)} Superposition of one-soliton ($u_1$) with two-soliton ($u_2$)}
Next we construct $\psi_1$ and $\psi_2$ (see Eq.~(\ref{q1})) by superposing
one-soliton solution ($u_1$) with two-soliton solution ($u_2$).
Fig.~\ref{fig7} depicts the density plot of such superposed nonlinear coherent structures.
Here we discuss two possible interesting coherent structures in the $\psi_1$ and $\psi_2$ components:
(i)~superposition of bound soliton (which can be obtained by setting same values for the imaginary parts
of $k^{(2)}_{1}$ and $k^{(2)}_{2}$ parameters in the two-soliton solution) with one soliton.
This is shown in the top panel of Fig.~\ref{fig7}. There is another possibility which is shown in the middle panel of figure (\ref{fig7}). Here the coherent structure displays a collision of bound soliton with a single soliton. Specifically a double-hump breather is converted into a single-hump breather with increased intensity after collision. (ii)~superposition of two-soliton with one-soliton in which all the three solitons are travelling at different velocities.
To obtain this we set different values for $k^{(1)}$, $k^{(2)}_{1}$ and $k^{(2)}_{2}$ (see bottom panel of Fig.~\ref{fig7}). The resulting coherent structure is an interaction of three independent solitons.
\begin{figure}[!pht]
\vspace{-0.5cm}
\centering\includegraphics[width=0.8\linewidth]{Fig_2}
\caption{Superposition of single- and two- solitons. Top Panel:
Collision of bound state and soliton with parameters $k^{(1)}=1.2-0.35i$, $k^{(2)}_{1}=1.2+0.2i$, $k^{(2)}_{2}=1+0.2i$,
$\eta^{(2)(0)}_{1}=1$, $\eta^{(2)(0)}_{2}=1$, $\eta^{(1)}_R=1$, and $\eta^{(1)}_I=0.2$;
Middle Panel: Collision of one- soliton with bound soliton with parameters $k^{(1)}=1-0.4i$, $k^{(2)}_{1}=1.25-0.4i$, $k^{(2)}_{2}=1+0.3i$,
$\eta^{(2)(0)}_{1}=1$, $\eta^{(2)(0)}_{2}=1$, $\eta^{(1)}_R=1$, and $\eta^{(1)}_I=0.1$; Bottom Panel: Three soliton collision with parameters $k^{(1)}=1.2+0.2i$, $k^{(2)}_{1}=1.3-0.25i$,
$k^{(2)}_{2}=1+0.35i$ and all other parameters are same as given in the top panel.}
\label{fig7}
\end{figure}
\noindent{\textbf{Case~(iii)} Superposition of one-soliton ($u_1$) with Akhmediev breather ($u_2$)}
In this case, we choose the one-soliton solution for $u_1$ as given by Eq.~(\ref{soliton}) and
Akhmediev breather (spatially periodic) form for $u_2$ in (\ref{q1}). After superposing a stationary soliton ($k^{(1)}_I=0$) with Akhmediev breather, we obtain special nonlinear coherent structures in which the Ma and
Akhmediev breathers coexist in both $\psi_1$ and $\psi_2$ components.
\begin{figure}[!pht]
\centering\includegraphics[width=0.8\linewidth]{Fig_3}
\caption{Superposition of one-soliton with Akhmediev breather. The parameters $k^{(1)}_{R}=0.9$,
$k^{(1)}_{I}=0$, $a^{(2)}=1$, $\eta_{R}^{(1)}$=1, and $\eta_{I}^{(1)}$=0.2.}
\label{fig9}
\end{figure}
Such coherent structures are shown in Fig.~\ref{fig9}. For solitons with non-zero velocity, also similar co-existence of Ma and Akhmediev breathers exists.\\
\noindent{\textbf{Case~(iv)} Superposition of one-soliton ($u_1$) with Ma breather ($u_2$)}
In this case, we choose the one-soliton solution as given by Eq.~(\ref{soliton}) and
MB (time periodic) form for $u_2$ in Eq.~(\ref{q1}). In $\psi_1$ and $\psi_2$ components,
we get nonlinear coherent structures akin to the collision of two MBs.
This is shown in Fig.~\ref{fig10} for illustrative purpose. If we choose the soliton velocity to be zero, the two MBs overlap with each other.
\begin{figure}[!pht]
\centering\includegraphics[width=0.8\linewidth]{Fig_4}
\caption{Superposition of one-soliton with Ma breather. The parameters are $k^{(1)}_{R}=0.9$, $k^{(1)}_{I}=0.3$, $a^{(2)}=0.2$,
$\eta_{R}^{(1)}=-1$, and $\eta_{I}^{(1)}=0.2$.}
\label{fig10}
\end{figure}
\noindent{\textbf{Case~(v)}~Superposition of one-soliton ($u_1$) with a Rogue wave ($u_2$)}\\
In this case, for $u_1$ and $u_2$ in (\ref{q1}), we choose one soliton and rogue wave solution, respectively.
When we superpose these two types of nonlinear waves using the transformation (\ref{q1}),
the resultant structure is the coexistence of MB and rogue wave in the $\psi_1$ and $\psi_2$ components.
This is depicted in figure \ref{fig8}.
\begin{figure}[!pht]
\centering\includegraphics[width=0.8\linewidth]{Fig_5}
\caption{Superposition of one-soliton with rogue wave with parameters $k^{(1)}_{R}=1$, $k^{(1)}_{I}=0$, $\eta_{R}^{(1)}=-4$, $\eta_{I}^{(2)}=0.2$ and $\delta^{(2)}$=$\sqrt{1.3}$.}
\label{fig8}
\end{figure}
Here we note that the soliton is converted into a MB.
This can be understood by writing down the detailed expression for $|\psi_1|^2$ and $|\psi_2|^2$. We find $|\psi_1|^2$= $\frac{1}{4}\left(|\lambda|^2+(k_{R}^{(1)})^2\mbox{sech}^2(z^{(1)}_2)+k_{R}^{(1)}\mbox{sech}(z^{(1)}_2)[2\lambda_R\cos(\chi)-2\lambda_I\sin(\chi)]\right)$ and $|\psi_2|^2$=$\frac{1}{4}\left(|\lambda|^2+(k_{R}^{(1)})^2\mbox{sech}^2(z^{(1)}_2)+k_{R}^{(1)}\mbox{sech}(z^{(1)}_2)[-2\lambda_R\cos(\chi)+2\lambda_I\sin(\chi)]\right)$,
where $\chi$ = $2|\delta^{(2)}|^2t-z_1^{(1)}$ and
$\lambda$=$\delta^{(2)} \left[1-\frac{2(1+4i|\delta^{(2)}|^2t)}{2|\delta^{(2)}|^2(x^2+4|\delta^{(2)}|^2t^2+\frac{1}{4|\delta^{(2)}|^2})}\right]$.
We observe that the interference terms for $|\psi_1|^2$ and $|\psi_1|^2$ involve $cosine$ and $sine$ functions.
Ultimately, this results in breathing oscillations in the soliton intensity along the $t$ axis thereby converting
it into a MB.\\
\noindent{\textbf{Case~(vi)} Superposition of two-soliton solution ($u_1$) with Rogue wave ($u_2$)}
Here we choose two-soliton solution and rogue wave for $u_1$ and $u_2$, respectively, in Eq.~(\ref{q1}).
The two colliding solitons are now converted into interacting MBs. As a result of this we get a
nonlinear coherent structure comprising of a rogue wave and two colliding MBs.
This is shown in the top panel of Fig. \ref{fig11} for the choice of parameters
$k^{(1)}_{1}$ = $1 + 0.2i$, $k^{(1)}_{2}$ = $1- 0.2i$ and $\delta^{(2)}=\sqrt{1.5}$.
If we choose the imaginary part of $k^{(2)}_{1}$ and $k^{(2)}_{2}$ parameters to be zero,
which means the velocities of respective solitons are zero, then we obtain bound state with breathing oscillations (see bottom panel).
\begin{figure}[!pht]
\centering\includegraphics[width=0.8\linewidth]{Fig_6}
\caption{Superposition of two-soliton with rogue wave.
Top Panel: Coexistence of interactions between two breathers and rogue wave with parameters are
$k^{(1)}_{1}=1.25+0.2i$, $k^{(1)}_{2}=1-0.2i$,
$\delta^{(2)}=\sqrt{1.5}$, and $\eta^{(1)(0)}_1$=$\eta^{(1)(0)}_2$=$2.5+i$.
Bottom panel: Breather bound states with parameters $k^{(1)}_{1}=1.2$, $k^{(1)}_{2}=1$, $\delta^{(2)}=\sqrt{1.5}$ and
$\eta^{(1)(0)}_1$=$\eta^{(1)(0)}_2$=$2.5+i$.}
\label{fig11}
\end{figure}
\noindent{\textbf{Case~(vii)} Superposition of Rogue waves}
We consider two distinct rogue wave solutions for $u_1$ and $u_2$, and insert them in Eq.~(\ref{q1}).
We plot the resulting intensities $|\psi_1|^2$ and $|\psi_2|^2$ in Fig.~\ref{fig12}.
\begin{figure}[!pht]
\centering\includegraphics[width=0.8\linewidth]{Fig_7}
\caption{Superposition of two different rogue waves. The parameters are
$\delta^{(1)}=\sqrt{1.5}$ and $\delta^{(2)}$=$\sqrt{0.3}-0.1i$.}
\label{fig12}
\end{figure}
In the $\psi_1$ component the rogue waves merge together along with a periodic oscillation of the background.
In the $\psi_2$ component we get a non-trivial twin peak rogue wave. Here too the background shows significant oscillations.
To facilitate the understanding of the oscillations we explicitly present the expressions
for $|\psi_1|^2$ and $|\psi_2|^2$.
$|\psi_1|^2=\frac{1}{4}\left(|\lambda_1|^2+|\lambda_2|^2+2A \cos[2(|\delta^{(1)}|^2-|\delta^{(2)}|^2)t]-2B\sin[2(|\delta^{(1)}|^2-|\delta^{(2)}|^2)t]\right)$
and \\
$|\psi_2|^2 = \frac{1}{4} \left( |\lambda_1|^2+|\lambda_2|^2-2A \cos[2(|\delta^{(1)}|^2-|\delta^{(2)}|^2)t]+
2B\sin[2(|\delta^{(1)}|^2-|\delta^{(2)}|^2)t]\right)$,
where $\lambda_1 = \delta^{(1)} \left[1-\frac{2(1+4i|\delta^{(1)}|^2t)}{2|\delta^{(1)}|^2(x^2+4|\delta^{(1)}|^2t^2+\frac{1}{4|\delta^{(1)}|^2})}\right]$,
$\lambda_2 = \delta^{(2)} \left[1-\frac{2(1+4i|\delta^{(2)}|^2t)}{2|\delta^{(2)}|^2(x^2+4|\delta^{(2)}|^2t^2+\frac{1}{4|\delta^{(2)}|^2})}\right]$,
$A$ and $B$ are real and imaginary parts of $\lambda_1\lambda_2^*$.
From this expression we find that the oscillations originate from the circular functions ($cosine$ and $sine$)
appearing in the cross terms and depend upon the difference $|\delta^{(1)}|^2-|\delta^{(2)}|^2$. Especially, for $\delta^{(1)}=\delta^{(2)}$,
the background oscillations are completely suppressed and the rogue wave appears only in the $\psi_1$ component while it disappears in the $\psi_2$ component.\\
\noindent{\textbf{Case~(viii)} Superposition of Rogue wave ($u_1$) with Akhmediev breather ($u_2$)}
We choose the solution $u_1$ to admit rogue wave and $u_2$ to possess the form of Akhmediev breather.
This type of superposition allows the possibility of interesting distinct nonlinear coherent
structures in the $\psi_1$ and $\psi_2$ components.
In the $\psi_1$ component the dominant behaviour is displayed by the rogue wave whose amplitude
is increased significantly while that of the AB gets suppressed.
The reverse scenario takes place in the $\psi_2$ component, that is,
here the amplitude of the rogue wave is suppressed whereas that of the AB is enhanced significantly.
This shows that there is an energy redistribution among the components.
In both the components the background executes oscillations.
Particularly in the $\psi_2$ component the background oscillates vibrantly. Such coherent structures are shown in Fig.~\ref{fig13}.
\begin{figure}[!pht]
\centering\includegraphics[width=0.8\linewidth]{Fig_8}
\caption{Superposition of rogue wave with Akhmediev breather for $\delta^{(1)}=\sqrt{1.5}$ and $a^{(2)}=0.9$.}
\label{fig13}
\end{figure}
\noindent{\textbf{Case~(ix)} Superposition of Rogue wave ($u_1$) with Ma breather ($u_2$)}
Finally, we consider the rogue wave solution for $u_1$ and Ma breather solution for $u_2$ in Eq.~(\ref{q1}).
The resulting intensity plots for the $\psi_1$ and $\psi_2$ components are shown in Fig.~\ref{fig14}.
This type of superposition results in nonlinear coherent structures with special dynamical properties.
In the $\psi_1$ component the rogue wave appears predominantly. But it disappears completely in the $\psi_2$ component.
In both the components the MB periodically reaches maximum and minimum intensities.
Especially in the $\psi_2$ component the breather is dominant.
Thus here too an energy redistribution takes place among the components $\psi_1$ and $\psi_2$. Also,
the background displays periodic oscillations in both the components.
\begin{figure}[!pht]
\centering\includegraphics[width=0.8\linewidth]{Fig_9}
\caption{Superposition of rogue wave with Ma breather.
The parameters are $\delta^{(1)}=\sqrt{1.2}$ and $a^{(2)}=0.3$}
\label{fig14}
\end{figure}
In the following table, we have summarized all the above interesting nonlinear superposed coherent structures.\\
\begin{table}[H]
\caption {Superposition of several kinds of nonlinear waves}
\centering
\begin{tabular}{|c|c|p{2.1cm}|p{5.5cm}|p{5.5cm}|c|c|}
\hline
Case & Form of & ~~Form of & \multicolumn{2}{c|}{Nature of nonlinear coherent structures} \\
\cline{4-5}
&$~~u_1$&~~~~$u_2$& ~~~~~~~$\psi_1$ component & ~~~~~~~$\psi_2$ component \\
\hline
\hline
(i) & One-soliton & One-soliton & (a) Two soliton collision with higher intensity, (b) Oscillating bound soliton with larger period.& (a) Two soliton collision with lower intensity, (b) Oscillating bright soliton with smaller period.\\
\hline
(ii) & One-soliton & Two-soliton & (a) Collision of bound soliton with one soliton, (b) Three soliton collision with higher amplitude, and (c) Three soliton bound state. & (a) Similar behavior as in $\psi_1$ component, (b) Three soliton collision with lower amplitude, and (c) Three soliton bound state. \\
\hline
(iii) & One-soliton & Rogue wave & \multicolumn{2}{c|}{\multirow{1}{*}{Coexistence of Ma breather with rogue wave.}}\\
\hline
(iv) & One-soliton & Akhmediev breather & Coexistence of Ma and Akhmediev breathers with lower amplitude. & Coexistence of Ma and Akhmediev breathers with higher amplitude.\\
\hline
(v) & One-soliton & Ma breather & \multicolumn{2}{c|}{\multirow{1}{*}{Colliding Ma breathers}}.\\
\hline
(vi) & Two-soliton & Rogue wave & \multicolumn{2}{c|}{\multirow{1}{*}{(a) Coexistence of colliding Ma breathers with rouge wave}}.\\
&&& \multicolumn{2}{c|}{\multirow{1}{*}{\hspace{-1.3cm}(b) Bound state of Ma breathers with rouge wave}}.\\
\hline
(vii) & Rogue wave & Rogue wave & Higher amplitude rogue wave with oscillating background. & Lower amplitude twin peak rogue waves with oscillating background.\\
\hline
(viii) & Rogue wave & Ma breather & Rogue wave is dominant and Ma breather periodically reaches minimum and maximum intensities. & Ma breather is dominant.\\
\hline
(ix) & Rogue wave& Akhmediev breather & Rogue wave is dominant with higher amplitude and Akhmediev breather gets suppressed. & Akhmediev breather is dominant and rogue wave gets suppressed.\\[1ex]
\hline
\end{tabular}
\end{table}
Apart from this we can have other possible superpositions such as (i)~Two-soliton with Akhmediev breather, (ii)
Two-soliton with Ma breather, (iii)~Two-soliton with Two-soliton, (iv)~Ma breather with Ma breather and (v)~ Akhmediev breather with Akhmediev breather. For brevity we don't discuss these cases here and the interested readers can explore these possibilities in a straightforward way.
\section{Non-autonomous CCGP system}
Next we turn our focus to the following non-autonomous CCGP system:
\begin{subequations}\begin{eqnarray}
i\frac{\partial\psi_1}{\partial t}&=&\left[-\frac{\partial^2}{\partial x^2 }+U_1(x,t)\right]\psi_1-\alpha_1(t)(|\psi_1|^2+2|\psi_2|^2)\psi_1-\alpha_2(t)\psi_2^2\psi_1^*,\\
i\frac{\partial\psi_2}{\partial t}&=&\left[-\frac{\partial^2}{\partial x^2 }+U_2(x,t)\right]\psi_2-\alpha_1(t)(2|\psi_1|^2+|\psi_2|^2)\psi_2- \alpha_2(t)\psi_1^2\psi_2^*,
\end{eqnarray}\label{hccnls}\end{subequations}
with time-dependent nonlinearity coefficient $\alpha_j(t)$ and external potential $U(x,t)$.
Here the external potentials $U_1=U_2=U(x,t)$ = $\frac{1}{2}\Omega^2(t)x^2$.
\subsection{Similarity transformation}\label{trans}
First, we look for a similarity transformation that
transforms the above non-autonomous CCGP system (\ref{hccnls}) into an integrable autonomous CCGP system. This will be of use in identifying the explicit
forms of the variable nonlinearity coefficient and the nature of
corresponding trapping potential which can support different nonlinear
coherent structures in atomic systems.
\indent For this purpose we introduce the following transformation for the dependent and independent variables in Eqs.~(\ref{hccnls}) with $\alpha_1(t)$=$\alpha_2(t)$=$\alpha(t)$:
\begin{subequations}\begin{eqnarray}
&&\psi_j(x,t) = \xi_1\sqrt{{2\alpha}(t)}~q_j(X(x,t),T(t))~e^{i\varphi(x,t)},\quad j=1,2,
\end{eqnarray}
where
\begin{eqnarray}
&&\varphi(x,t) = \left[-\frac{1}{4}\frac{d}{dt}(\ln \alpha)\right]x^2 + \xi_1^2 \xi_2 \left({\alpha}x - \xi_2 \xi_1^2\int {\alpha}^2 dt\right)
\end{eqnarray}
and the new coordinates $X$ and $T$ are defined as
\begin{eqnarray}
X = ~\xi_1 \left({\alpha} x - 2\xi_2 \xi_1^2\int {\alpha}^2 dt\right),~~~~~~~~
T = \xi_1^2 \int {\alpha}^2 dt.
\end{eqnarray}\label{str}\end{subequations}
Here $\xi_j,~j=1,2,$ are real arbitrary constants.
Then Eq.(\ref{hccnls}) reduces to the following set of
integrable equations \cite{kannajpa}:
\begin{subequations}\begin{eqnarray}
&iq_{1,T}+ q_{1,XX}+2(|q_1|^2+2|q_2|^2)q_1 + 2q_2^2q_1^*=0,\\
&iq_{2,T}+ q_{2,XX}+2(|q_2|^2+2|q_1|^2)q_2+ 2q_1^2q_2^*=0,
\end{eqnarray}\label{ccnls1}
along with a constraint condition
\begin{eqnarray}
\Omega^2(t)=\frac{2\dot{\alpha}^2-\ddot{\alpha}\alpha}{2\alpha^2},
\label{ricatti}
\end{eqnarray}
\end{subequations}
where the overdot denotes differentiation with respect to time $t$.
Such type of transformation is possible mainly due to the temporal dependence of the nonlinearity
and that of external harmonic potential; in the absence of such dependence these transformations are not at all possible. Now we can examine the dynamics of different nonlinear waves for a given type of
nonlinearity and potential by expressing Eqs.~(\ref{ccnls1}) as two decoupled NLS equations
using the transformation (\ref{q1}) and then by reconstructing $\psi$ in terms of the original co-ordinates $x$ and $t$ by making use of the similarity transformation (\ref{str}).
\subsection{Forms of time- dependent nonlinear coefficient}
Here we choose two types of nonlinearities namely, periodically
and kink-like modulated nonlinearities, which are of physical interest in BECs.\\
\noindent{(a) {\bf Periodically modulated nonlinearity}} \\
First we choose a periodic form for the nonlinearity coefficient, namely,
\begin{subequations}\begin{eqnarray}
\alpha(t) =1+\epsilon\cos(t),
\label{cos}\end{eqnarray}
where $\epsilon$ is a real arbitrary parameter. Such interesting periodic variations
of the nonlinearity coefficient is considered in BECs \cite{periodic}. This interesting nonlinearity is plotted in Fig.~(\ref{fig15}). The corresponding expression for $\Omega^2(t)$ is given by:\\
\begin{eqnarray}
\Omega^2(t) =\frac{\epsilon[\cos(t)+\epsilon\cos^2(t)+2\epsilon \sin^2(t)]}{2(1+\epsilon\cos(t))^2}.
\end{eqnarray}
\end{subequations}
\noindent{(b) {\bf Kink-like nonlinearity}} \\
We also choose another form for variable nonlinearity co-efficient, given by
\begin{subequations}\begin{eqnarray}
\alpha(t) =2+\tanh(\epsilon t),
\label{kink}\end{eqnarray}
where $\epsilon$ is an arbitrary real function. In this case, we envision a nonlinearity
that is rapidly varied from one value to another. The evolution of this nonlinearity is shown in Fig.~(\ref{fig15}). Such variation of nonlinearity has been observed in atomic
BECs \cite{pollack}. The corresponding form of $\Omega^2(t)$ is obtained as
\begin{eqnarray}
\Omega^2(t) =\frac{\epsilon^2 sech^2(\epsilon t)[1+2 \tanh(\epsilon t)]}{(2+\tanh(\epsilon t))^2}.
\end{eqnarray}
\end{subequations}
\begin{figure}[H]
\vspace{-0.5cm}
\centering\includegraphics[width=0.5\linewidth]{Fig_10}
\caption{Typical form of periodically modulated
and kink-like nonlinearity $\alpha(t)$ for $\epsilon=0.4$.}
\label{fig15}
\end{figure}
\subsection{Dynamics of superposed nonlinear waves in non-autonomous CCGP system}
For brevity, here we consider three types of superposed non-autonomous nonlinear waves namely,
superposition of two different non-autonomous solitons, superposition of two-soliton with a non-autonomous rogue wave and superposition of two different non-autonomous rogue waves. The general solution of the non-autonomous CCGP equations (\ref{hccnls}) can be constructed by
using the transformations (\ref{str}) and (\ref{q1}). It can be written as
\begin{subequations}\begin{eqnarray}
\psi_1(x,t) = \xi_1\sqrt{{2\alpha}(t)}~e^{i\varphi(x,t)}(u_1(X,T)+u_2(X,T))/2,\\
\psi_2(x,t) = \xi_1\sqrt{{2\alpha}(t)}~e^{i\varphi(x,t)}(u_1(X,T)-u_2(X,T))/2,
\end{eqnarray}\end{subequations}\\
where $u_1(X,T)$ and $u_2(X,T)$ can be any of the five solutions of NLS system (\ref{singlenls}) discussed in Sec. III A, with the replacement of the old-coordinates $x$ and $t$ by the new co-ordinates $X$ and $T$.\\
\noindent{{\bf Case(i)}~ Superposition of two different non-autonomous one solitons}
Here, we have chosen $u_1$ and $ u_2$ to be of the form given by Eq.~(\ref{soliton}) with $x$ and $t$ replaced by $X$ and $T$, respectively.
The periodically modulated nonlinearity affects both soliton velocity and its shape.
This kind of oscillatory nonlinearity induces oscillations in the soliton profile and results in snake like propagation.
Such non-autonomous solitons can be viewed as ``creeping solitons".
This is shown in the top panel of Fig.~\ref{fig16}.
\begin{figure}[!pht]
\centering\includegraphics[width=0.8\linewidth]{Fig_11}
\caption{Superposition of two different one solitons. Top panel: Periodically modulated nonlinearity for $\epsilon=0.4$, $\xi_1=0.8$ and $\xi_2=0$
; Bottom panel: Kink-like nonlinearity for $\epsilon=0.2$, $\xi_1=0.6$ and $\xi_2=0$.
All other parameters are same as given in the top panel of Fig.~\ref{s-s}.}
\label{fig16}
\end{figure}
Next, after introducing the kink nonlinearity, the direction of propagation is also affected (see Eq.~(\ref{kink})), the shape and amplitude of the soliton are altered significantly by the inhomogeneity.
Another interesting mechanism due to kink-like nonlinearity is ``soliton compression".
This type of superposition looks similar to the collision of two one solitons.
Here the amplitudes of the two solitons before collision are lower than that of after collision.
After collision the width of the solitons (for large positive $t$) get compressed and their amplitudes are enhanced.
Also the separation distance between two solitons before and after
collision is different. In addition to this there is a bending in the path of
the solitons during propagation due to the nature of nonlinearity.
Such a dynamical behaviour is shown in the bottom panel of Fig.~\ref{fig16}.\\
\noindent{{\bf Case(ii)}~ Superposition of two-soliton with rogue wave}
In this case, we examine the superposition of two-soliton of the form (\ref{twosoliton}) with a rogue
wave (\ref{rogue}) in the presence of periodically modulated nonlinearity. Here, interestingly, the two-solitons are converted into breathers and the rogue wave remains as it is. A distinct feature of this nonlinearity
is the wings of the breathers are stretched due to oscillation. Note that, here the background is oscillating in
both the components in a periodic manner as compared with Fig.~\ref{fig11}, where the periodically modulated nonlinearity is absent. This is shown in the top panel of Fig.~\ref{fig17} for illustrative purpose.\\
\begin{figure}[!pht]
\centering\includegraphics[width=0.8\linewidth]{Fig_12}
\caption{Superposition of two-solitons with a rogue wave. Top panel: Periodically modulated nonlinearity
with parameters $\epsilon=0.5$, $\xi_1=0.8$ and $\xi_2=0$. Bottom panel:
Kink-like nonlinearity with parameters $\epsilon=2.5$, $\xi_1=0.6$ and $\xi_2=0$.
All other parameters are same as given in Fig.~\ref{fig11}.}
\label{fig17}
\end{figure}
Next we consider the superposition of two non-autonomous solitons with a non-autonomous rogue wave in the presence of kink-like nonlinearity (see Eq.~(\ref{kink})). This shows several dynamical features.
First of all the background does not oscillate instead it displays a step-like profile.
Here too the two-solitons are converted into breathers. Before collision the
time period of breathers is greater whereas after collision it decreases and as a result of this
the period of the breathers is narrowed down. Hence the number of
breathers has enhanced after collision ($t>0$).
This is shown in the bottom panel of Fig.~\ref{fig17}.\\
\noindent{{\bf Case(iii)}~ Superposition of non-autonomous rogue waves}\\
By considering two different rogue waves in the presence of periodically modulated nonlinearity as given by Eq. (\ref{cos}), we find that
the period of oscillation in the background is well diminished as compared to that of Fig. \ref{fig12} (autonomous case). In the second component we observe a non-trivial asymmetric twin peak rogue wave.
\begin{figure}[!pht]
\centering\includegraphics[width=0.8\linewidth]{Fig_13}
\caption{Superposition of rogue wave with another rogue wave. Top panel:
Periodically modulated nonlinearity with parameters $\epsilon=0.4$, $\xi_1=0.6$ and $\xi_2=0.1$. Bottom panel:
Kink-like nonlinearity with parameters $\epsilon=0.5$, $\xi_1=0.6$ and $\xi_2=0$.
All other parameters are same as given in Fig.~\ref{fig12}.}
\label{fig17}
\end{figure}
Next we consider the kink-like nonlinearity (see Eq.~(\ref{kink})). Here
the period of oscillating background is greater (almost suppressed) from -$\infty$ to $0$ (see Fig.~\ref{fig12}).
For $t>0$, the period of oscillation of background is decreased significantly
with step-like enhancement in its background.
This is shown in the bottom panel of Fig.~\ref{fig17}. Also the intensity of the rogue wave is increased significantly in the $\psi_1$ and $\psi_2$ components as compared with that of Fig.~\ref{fig12}.
\section{Conclusions}
In conclusion, we have studied the superposed nonlinear waves of coherently coupled GP system. Based on a set of linear transformations,
the coherently coupled GP equations are converted into two decoupled NLS equations. We briefly discuss the available nonlinear wave solutions of scalar NLS equation, namely
solitons (one- and two-), rogue waves and breathers (Ma- and Akhmediev).
We show that the existence of various interesting coherent nonlinear structures such as,
collision of bound soliton with soliton, three soliton collision, coexistence of
rogue wave and Ma breather, coexistence of Ma and Akhmediev breathers, collision of Ma breathers,
bound state of Ma breather by superimposing different types of nonlinear waves. We have also shown that
by such superposition the conventional rogue waves and breathers (Ma/Akhmediev) can be engineered.
Finally, we consider the non-autonomous CCGP system with physically relevant time-dependent nonlinearity coefficients and external potential. With the aid of a similarity transformation, the non-autonomous CCGP system is converted into an integrable autonomous CCGP system with a constraint condition. We examine the two forms
of nonlinearity coefficient, namely kink-like and periodically modulated function and
investigate their effects on the novel coherent structures of superposed nonlinear waves. Especially, we demonstrate the possibility of manipulating the nonlinear waves in autonomous as well as non-autonomous settings of the CCGP systems.\\
{\bf Acknowledgments}. The work of T.K. is supported by Department of Science and Technology , Government of India, in the form of a major research project. The authors acknowledge the Principal and Management of Bishop Heber College (Autonomous), Tiruchirappalli for the constant support and encouragement. The authors also thank K. Sakkaravarthi for useful discussions.
\section*{\bf References}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,660
|
\section{Introduction}\label{se:one}
\subsection{Background and context}
If $T_1,\ldots,T_N \colon \mathbb{R}^d \to \mathbb{R}^d$ are contractions it is well-known that there exists a unique nonempty compact set $X \subset \mathbb{R}^d$ such that $X=\bigcup_{i=1}^NT_iX$. In this case $(T_1,\ldots,T_N)$ is called an \emph{iterated function system} and the set $X$ its \emph{attractor}. When each transformation $T_i$ is a similitude with contraction ratio $r_i \in (0,1)$ and the distinct images $T_iX \cap T_jX$ do not overlap too strongly it is classical that the box dimension and Hausdorff dimension of the attractor are both equal to the unique real number $s>0$ such that $\sum_{i=1}^Nr_i^s=1$ (see for example \cite[Theorem 9.3]{Fa14} or the original article \cite{Hu81}). In the case where each $T_i$ is instead an affine map $T_ix=A_ix+v_i$ the Hausdorff dimension and box dimension of the attractor $X$ -- which in this context we call a \emph{self-affine set} -- are more challenging to calculate. The problem of determining the Hausdorff dimension of such sets, even implicitly, has been an active topic of research since the 1980s and has received particularly intense research interest within the last decade (see for example the classic articles \cite{Be84,Ed92,Fa88,Fa92,HuLa95,Mc84} and more recent contributions such as \cite{Ba08,BaHoRa18,BoMo18,DaSi17,FaKe18,FeSh14,Fr12,KaSh09,MoSh17}). In the landmark article \cite{Fa88} K. Falconer defined an implicit formula which is known to give the correct value for the Hausdorff dimension of a wide variety of self-affine sets. The subject of this article is the numerical estimation of the value predicted by Falconer's formula.
In order to define Falconer's formula we require a few preliminary definitions. Let $M_d(\mathbb{R})$ denote the set of all real $d \times d$ matrices. If $A \in M_d(\mathbb{R})$ we recall that the \emph{singular values} of $A$ are defined to be the square roots of the eigenvalues of the positive semidefinite matrix $A^\top A$. We denote the singular values of $A \in M_d(\mathbb{R})$ by $\sigma_1(A),\ldots,\sigma_d(A)$ in decreasing order of absolute value. For each $A \in M_d(\mathbb{R})$ and $s\geq 0$ let us define
\[\varphi^s(A):=\left\{\begin{array}{cl}\sigma_1(A)\cdots \sigma_{\lfloor s \rfloor}(A) \sigma_{\lceil s \rceil}(A)^{s-\lfloor s \rfloor}&\text{if }0 \leq s \leq d,\\
|\det A|^{\frac{s}{d}}&\text{if }s \geq d.\end{array}\right.\]
It was shown in \cite{Fa88} that for each $s \geq 0$ we have $\varphi^s(AB) \leq \varphi^s(A)\varphi^s(B)$ for all $A,B \in M_d(\mathbb{R})$. The \emph{affinity dimension} of the iterated function system $T_ix:=A_ix=v_i$, where $1 \leq i \leq N$, is then defined to be the quantity
\[\dimaff (T_1,\ldots,T_N) := \inf \left\{s>0 \colon\sum_{n=1}^\infty \sum_{i_1,\ldots,i_n=1}^N \varphi^s(A_{i_1}\cdots A_{i_n})<\infty\right\}.\]
Since $\dimaff (T_1,\ldots,T_N)$ depends only on $A_1,\ldots,A_N$ and not on the additive part of the transformations $T_i$ we will also denote it by $\dimaff (A_1,\ldots,A_N)$.
If the matrices $A_1,\ldots,A_N$ are assumed to be invertible and contracting with respect to some norm on $\mathbb{R}^d$ then the affinity dimension is the unique $s>0$ such that the quantity
\[ P(A_1,\ldots,A_N;s):= \lim_{n \to \infty }\frac{1}{n}\log \sum_{i_1,\ldots,i_n=1}^N \varphi^s(A_{i_1}\cdots A_{i_n})\]
is equal to zero.
Let $\|\cdot\|$ denote the Euclidean norm on $\mathbb{R}^d$. It was shown in \cite{Fa88} that when $\max_{1 \leq i \leq N}\|A_i\|<1$ the affinity dimension $\dimaff (A_1,\ldots,A_N)$ is well-defined and is an upper bound for the box dimension of the attractor. (This argument may easily be adapted to the case where $\max_{1 \leq i \leq N}\vertle{A_i}<1$ in the operator norm induced by some norm $\vertle{\cdot}$ on $\mathbb{R}^d$.) It was additionally shown that when matrices $A_1,\ldots,A_N$ satisfying $\max_{1 \leq i \leq N}\|A_i\|<\frac{1}{3}$ are fixed, then for Lebesgue-a.e. choice of $(v_1,\ldots,v_N) \in (\mathbb{R}^d)^N$ the attractor of the affine transformations $T_1,\ldots,T_N$ given by $T_ix:=A_ix+v_i$ has Hausdorff dimension equal to $\min\{d,\dimaff(A_1,\ldots,A_N)\}$. Subsequent research focused on providing explicit examples for which the Hausdorff dimension of the attractor equals the affinity dimension of the defining iterated function system, with explicit special cases being given in articles such as \cite{FaKe18,Fr12,HuLa95,MoSh17}. Very recently, B. Bara\'ny, M. Hochman and A. Rapaport have shown that the Hausdorff dimension of a planar self-affine set is always equal to the affinity dimension of the defining iterated function system as long as the matrices $A_i$ are invertible, the affine transformations satisfy the strong open set condition, and the matrices $|\det A_i|^{-1/2}A_i$ neither belong to a compact subgroup of $GL_2(\mathbb{R})$ nor preserve a finite subset of $\mathbb{RP}^1$. At the present time, however, results on higher-dimensional self-affine sets additional to that of Falconer are essentially unavailable.
Despite its prominent r\^ole in the dimension theory of self-affine sets, the properties of the affinity dimension itself have been investigated only very recently. In the 2014 article \cite{FeSh14} D.-J. Feng and P. Shmerkin showed for the first time that the affinity dimension $\dimaff (A_1,\ldots,A_N)$ depends continuously on the entries of the matrices $A_1,\ldots,A_N$, and in \cite{Mo16} it was shown that the affinity dimension is computable in principle in the sense that for any given $\varepsilon>0$ we may algorithmically compute an explicit approximation to $\dimaff(A_1,\ldots,A_N)$ which is guaranteed to be accurate to within the prescribed error $\varepsilon$ and which requires only finitely many arithmetical operations to calculate. However, the method of \cite{Mo16} does not result in an algorithm which is fast enough to be useful in practical computations. Further general properties of the affinity dimension were investigated in \cite{BoMo18,KaMo16}.
At the present time there are very few practical techniques available for the computation of the affinity dimension. In the article \cite{Mo18} the author gave a simple closed-form expression for the affinity dimension in the very special case where the matrices $A_i$ are generalised permutation matrices, that is, matrices having exactly one nonzero entry in every row and column. Closed-form expressions are also available in the case of diagonal and upper-triangular matrices \cite{FaMi07,KaMo16}. To the best of the author's knowledge there so far exists only one result in the literature which is powerful enough to be able to estimate the affinity dimension for a nonempty open set of examples in a practicable time frame. The following result was proved by M. Pollicott and P. Vytnova in \cite{PoVy16}. Here and throughout this article $\rho(A)$ denotes the spectral radius of the matrix or linear operator $A$.
\begin{theorem}\label{th:one}
Let $A_1,\ldots,A_N$ be $2\times 2$ matrices which satisfy the following conditions:
\begin{enumerate}[(i)]
\item
We have $\sigma_1(A_i)^2< \sigma_2(A_i)<1$ for all $i=1,\ldots,N$.
\item
If $\mathcal{Q}_2$ is defined to be the open second quadrant $\{(x,y)\in \mathbb{R}^2 \colon x<0<y\}$, then the sets $A_1^{-1}\mathcal{Q}_2,\ldots,A_N^{-1}\mathcal{Q}_2$ are subsets of $\mathcal{Q}_2$ and have pairwise disjoint closures in $\mathcal{Q}_2$.
\item
All entries of the matrices $A_i$ are strictly positive\footnote{This hypothesis is invoked in Pollicott and Vytnova's section 3 but is not explicitly stated in their introduction. It does not follow automatically from the other hypotheses unless the determinants are assumed positive.}.
\end{enumerate}
For each $n \geq 1$ and $s \in \mathbb{C}$ define
\[t_n(s)=\sum_{i_1,\ldots,i_n=1}^N \frac{\rho(A_{i_1}\cdots A_{i_n})^{2+s}}{\rho(A_{i_1}\cdots A_{i_n})^2 - \det A_{i_1}\cdots A_{i_n}},\]
\[a_n(s):=\sum_{k=1}^n \frac{(-1)^k}{k!} \sum_{\substack{(n_1,\ldots,n_k) \in \mathbb{N}^k \\ \sum_{i=1}^k n_i=n}} \prod_{i=1}^k \frac{t_{n_i}(s)}{n_i}\]
and $a_0(s):=1$, and for each $n\geq 1$ let $s_n \in \mathbb{R}$ denote the smallest positive real number $s$ such that $\sum_{i=0}^n a_i(s)=0$. Then $\dimaff(A_1,\ldots,A_N) \in (0,1)$, $s_n$ is well-defined for all sufficiently large $n$, and there exists $\gamma>0$ such that
\[\left|\dimaff (A_1,\ldots,A_N)-s_n\right|=O\left(\exp(-\gamma n^2)\right).\]
\end{theorem}
\emph{Remark.} The quantity $a_n(s)$ may be alternatively characterised as
\[ \frac{(-1)^n}{n!}\det\begin{pmatrix}
t_1(s)& n -1& 0&\cdots &0 &0\\
t_2(s)&t_1(s)&n-2 &\cdots &0 &0\\
t_3(s)&t_2(s)&t_1(s) &\ddots &0 &0\\
\vdots & \vdots & \vdots & \ddots &\ddots& \vdots\\
t_{n-1}(s) &t_{n-2}(s)&t_{n-3}(s)&\cdots &t_1(s) &1\\
t_n(s) &t_{n-1}(s)&t_{n-2}(s)&\cdots &t_2(s) &t_1(s)
\end{pmatrix},\]
and we will prefer this format in our exposition.
The methods underlying the proof of Theorem \ref{th:one} will be described in more detail in the following section. We remark that condition (i) above implies that the matrices are invertible, and the combination of the three conditions implies $0<\dimaff(A_1,\ldots,A_N)<1$ (see \cite{HuLa95} for details).
In fact the only condition which is really essential to Pollicott and Vytnova's argument is that the matrix entries are positive, although in cases where we have $\dimaff(A_1,\ldots,A_N)\in (1,2)$ the formula for $t_n(s)$ instead becomes
\[t_n(s):=\sum_{i_1,\ldots,i_n=1}^N \frac{\rho(A_{i_1}\cdots A_{i_n})^{4-s}|\det A_{i_1}\cdots A_{i_n}|^{s-1}}{\rho(A_{i_1}\cdots A_{i_n})^2 - \det A_{i_1}\cdots A_{i_n}}.\]
In this article we aim to prove as comprehensive as possible an extension of Theorem \ref{th:one}. In particular, as well as removing hypotheses (i)--(ii) from Theorem \ref{th:one} we will establish a version of that theorem which is valid for affine iterated function systems in dimensions higher than two, in which $\dimaff(A_1,\ldots,A_N)$ may take any value in the range $(0,d)$, and in which the hypothesis of positivity is weakened to one of domination. In order to state our results in full we will require a number of definitions, which relate to multilinear algebra, to positivity and to domination.
\subsection{Multilinear algebra}
In extending Theorem \ref{th:one} one of our concerns will be to allow matrices of arbitrary dimension. Whereas in two dimensions the function $\varphi^s(A)$ admits the simple characterisation
\[\varphi^s(A)=\left\{\begin{array}{cl} \|A\|^s&\text{if }0 \leq s \leq 1,\\
|\det A|^{s-1}\|A\|^{2-s}&\text{if }1 \leq s \leq 2,\end{array}\right.\]
when $s>1$ and $d>2$ the analogous formula involves exterior powers of the matrix $A$. In order to study the singular value function $\varphi^s$ in dimensions higher than two we therefore need to recall some concepts and notation from multilinear algebra.
Recall that when $1 \leq k \leq d$ the real vector space $\wedge^k \mathbb{R}^d$ is the vector space spanned by the formal expressions $\{v_1\wedge v_2 \wedge \cdots \wedge v_k \colon v_1,\ldots v_k \in \mathbb{R}^d\}$ subject to the identifications
\[\lambda(v_1 \wedge v_2\wedge \cdots \wedge v_k) = (\lambda v_1)\wedge v_2\wedge \cdots \wedge v_k,\]
\[(u_1\wedge v_2\wedge \cdots \wedge v_k)+(v_1\wedge v_2\wedge \cdots \wedge v_k)=(u_1+v_1)\wedge v_2\wedge \cdots \wedge v_k,\]
\[v_1 \wedge v_2 \wedge \cdots \wedge v_k = (-1)^{\sign (\pi)} v_{\pi(1)}\wedge v_{\pi(2)}\wedge \cdots v_{\pi(k)}\]
for all $v_1,\ldots,v_k,u_1 \in \mathbb{R}^d$, $\lambda \in \mathbb{R}$ and permutations $\pi \colon \{1,\ldots,k\}\to \{1,\ldots,k\}$. The vector space $\wedge^k \mathbb{R}^d$ is ${d\choose k}$-dimensional and if $v_1,\ldots,v_d$ is any basis for $\mathbb{R}^d$ then $\{v_{i_1}\wedge \cdots \wedge v_{i_k} \colon 1 \leq i_1<i_2<\cdots<i_k\leq d\}$ is a basis for $\wedge^k\mathbb{R}^d$. The ${d \choose k}$-dimensional vector space $\wedge^k\mathbb{C}^d$ may be constructed analogously.
The space $\wedge^k\mathbb{R}^d$ inherits an inner product $\langle\cdot,\cdot\rangle_{\wedge^k\mathbb{R}^d}$ from the standard inner product $\langle\cdot,\cdot\rangle$ on $\mathbb{R}^d$ which satisfies
\[\langle u_1\wedge \cdots \wedge u_k,v_1\wedge \cdots \wedge v_k\rangle_{\wedge^k\mathbb{R}^d} = \det \left(\left[\langle u_i,v_j\rangle\right]_{i,j=1}^k\right).\]
If $A \in M_d(\mathbb{R})$ then we may define a linear map $A^{\wedge k} \colon \wedge^k \mathbb{R}^d \to \wedge^k\mathbb{R}^d$ by $A^{\wedge k}(v_1 \wedge \cdots \wedge v_k)=Av_1 \wedge \cdots \wedge Av_k$. If $v_1,\ldots,v_d$ is a basis for $\mathbb{C}^d$ consisting of eigenvectors and generalised eigenvectors for $A$ then the vectors $v_{i_1}\wedge \cdots \wedge v_{i_k}$ form a basis for $\wedge^k\mathbb{C}^d$ and it is not hard to see that if $\lambda_1,\ldots,\lambda_d$ are the eigenvalues of $A$ then the eigenvalues of $A^{\wedge k}$ are precisely the ${d \choose k}$ different products $\lambda_{i_1}\cdots \lambda_{i_k}$ with $1 \leq i_1<\cdots<i_k\leq d$. It is clear from the definition of the inner product on $\wedge^k\mathbb{R}^d$ that $(A^{\wedge k})^\top =(A^\top )^{\wedge k}$. Combining these observations we may easily see that
\[\left\|A^{\wedge k}\right\|_{\wedge^k\mathbb{R}^d}=\rho\left(\left({A^\top}A\right)^{\wedge k}\right)^{\frac{1}{2}}=\sigma_1(A)\cdots \sigma_k(A)\]
for all $A \in M_d(\mathbb{R})$. By convention we also define $\wedge^0 \mathbb{R}^{d}=\mathbb{R}$ and $A^{\wedge 0}=1$. It follows easily that we may write
\[\varphi^s(A)=\left\|A^{\wedge\lfloor s\rfloor}\right\|^{1+s-\lfloor s\rfloor}\left\|A^{\wedge \lceil s \rceil}\right\|^{\lceil s \rceil-s}\]
for all $A \in M_d(\mathbb{R})$ and $s \in [0,d]$.
\subsection{Positivity and domination}
As well as increasing the dimension of the matrices to be considered in our extension of Theorem \ref{th:one} we would like to weaken as much as possible the hypothesis that the matrices have positive entries. To this end we introduce the following definition:
\begin{definition}\label{de:mc}
Let $\mathsf{A} \subset M_d(\mathbb{R})$ be nonempty. We say that $(\mathcal{K}_1,\ldots,\mathcal{K}_m)$ is a \emph{multicone} for $\mathsf{A}$ if the following properties hold:
\begin{enumerate}[(i)]
\item
Each $\mathcal{K}_j$ is a closed, convex subset of $\mathbb{R}^d$ with nonempty interior such that $\lambda \mathcal{K}_j \subseteq \mathcal{K}_j$ for every non-negative real number $\lambda$.
\item
There exists a unit vector $w \in \mathbb{R}^d$ such that $\langle u,w\rangle> 0$ for all nonzero vectors $u \in \bigcup_{j=1}^m \mathcal{K}_j$.
\item
For every $A \in \mathsf{A}$ and $j \in \{1,\ldots,m\}$ there exists $\ell =\ell(j,A) \in \{1,\ldots,m\}$ such that $A(\mathcal{K}_j\setminus \{0\})\subset (\Int \mathcal{K}_\ell) \cup (-\Int \mathcal{K}_\ell)$.
\item
For all distinct $j_1,j_2 \in \{1,\ldots,m\}$ we have $\mathcal{K}_{j_1}\cap \mathcal{K}_{j_2}=\{0\}$.
\end{enumerate}
When (ii) holds we say that $w$ is a \emph{transverse-defining vector} for $(\mathcal{K}_1,\ldots,\mathcal{K}_m)$ since the hyperplane normal to $w$ is transverse to $\bigcup_{j=1}^m \mathcal{K}_j$. If a multicone for $\mathsf{A}$ exists then we say that $\mathsf{A}$ is \emph{multipositive}.
\end{definition}
We will say that $(A_1,\ldots,A_N) \in M_d(\mathbb{R})^N$ is \emph{$k$-multipositive} if and only if the set $\{A_1^{\wedge k},\ldots,A_N^{\wedge k}\}$ is multipositive in the sense defined above. (By abuse of notation we shall say that a tuple of matrices is $k$-multipositive if and only if the corresponding set is.) We observe that a tuple of $d \times d$ matrices with all entries positive is multipositive since we may take $m=1$ and $\mathcal{K}_1$ to be the closed positive orthant in $\mathbb{R}^d$. We also observe that every tuple of $d \times d$ matrices is $0$-multipositive, and every tuple of $d \times d$ invertible matrices is $d$-multipositive.
If $1 \leq k < d$ then a tuple of invertible matrices $(A_1,\ldots,A_N) \in GL_d(\mathbb{R})^N$ is called \emph{$k$-dominated} if there exist $C,\gamma>0$ such that
\[\sigma_{k+1}(A_{i_1}\cdots A_{i_n}) \leq Ce^{-\gamma n}\sigma_k(A_{i_n}\cdots A_{i_1})\]
for all $i_1,\ldots,i_n \in \{1,\ldots,n\}$ and $n \geq 1$. By convention we will say that every $(A_1,\ldots,A_N) \in GL_d(\mathbb{R})^N$ is both $0$- and $d$-dominated. It is not difficult to show using the previous observations that $(A_1,\ldots,A_N)$ is $k$-dominated if and only if $(A_1^{\wedge k},\ldots,A_N^{\wedge k})$ is $1$-dominated. For $0< k < d$ let $\mathrm{Gr}(k,d)$ denote the Grassmannian manifold of $k$-dimensional linear subspaces of $\mathbb{R}^d$. In the case $k=1$ we will prefer the notation $\mathrm{Gr}(1,d)=\mathbb{RP}^{d-1}$. If $E \subset \mathbb{R}^d$ then we shall write $PE:=\{V \in \mathbb{RP}^{d-1} \colon E \cap (V \setminus \{0\})\neq \emptyset\}$. The property of $k$-domination (which is based on the earlier concept of dominated splittings in smooth ergodic theory) has numerous equivalent formulations which were explored in \cite{BoGo09}:
\begin{theorem}[Bochi-Gourmelon]\label{th:bogo}
Let $\mathsf{A} \subset M_d(\mathbb{R})$ be a nonempty compact set of invertible matrices and let $k \in \{1,\ldots,d-1\}$. Then the following statements are equivalent:
\begin{enumerate}[(i)]
\item
There exist $C,\gamma>0$ such that
\begin{equation}\label{eq:horse}\sup_{A_1,\ldots,A_n \in \mathsf{A}} \frac{\sigma_{k+1}(A_n\cdots A_1)}{\sigma_k(A_n\cdots A_1)}\leq Ce^{-\gamma n}\end{equation}
for every integer $n \geq 1$.
\item
There exists a nonempty set $\mathcal{C}\subset \mathrm{Gr}(k,d)$ such that the closure of $\bigcup_{A
\in \mathsf{A}}A\mathcal{C}$ is a subset of the interior of $\mathcal{C}$, and such that there exists a $(d-k)$-dimensional linear subspace of $\mathbb{R}^d$ which is transverse to every element of $\mathcal{C}$.
\item
There exists a nonempty subset $\mathcal{C}$ of real projective space $\mathbb{RP}^{d-1}$ such that $PE \subset \mathcal{C}$ for some $k$-dimensional subspace $E$ of $\mathbb{R}^d$, such that $PF \cap \mathcal{C}=\emptyset$ for some $(d-k)$-dimensional subspace $F$ of $\mathbb{R}^d$, and such that the closure of $\bigcup_{A
\in \mathsf{A}}A\mathcal{C}$ is a subset of the interior of $\mathcal{C}$.
\item
For every nonempty compact metric space $X$, homeomorphism $T \colon X \to X$ and continuous function $\mathcal{A} \colon X \to \mathsf{A}$ there exist continuous functions $\mathcal{U} \colon X \to \mathrm{Gr}(k,d)$ and $\mathcal{V} \colon X \to \mathrm{Gr}(d-k,d)$ such that $\mathcal{A}(x)\mathcal{U}(x)=\mathcal{U}(Tx)$, $\mathcal{A}(x)\mathcal{V}(x)=\mathcal{V}(Tx)$ and $\mathbb{R}^d = \mathcal{U}(x)\oplus \mathcal{V}(x)$ for all $x \in X$, and such that for some constants $C,\gamma>0$ depending on $\mathcal{A}$ we have for all $x \in X$ and $n \geq 1$
\[\|\mathcal{A}(T^{n-1}x)\cdots \mathcal{A}(x)u\| \geq Ce^{\gamma n}\|\mathcal{A}(T^{n-1}x)\cdots \mathcal{A}(x)v\|\]
for all unit vectors $u\in \mathcal{U}(x)$ and $v \in \mathcal{V}(x)$.
\end{enumerate}
Moreover the sets $\mathcal{C}$ in (ii) and (iii) may without loss of generality be taken to be closed and to have finitely many connected components.
\end{theorem}
If $(A_1,\ldots,A_N)\in M_d(\mathbb{R})^N$ is a tuple of invertible matrices then $(A_1,\ldots,A_N)$ is $k$-dominated if and only if it is $k$-multipositive. This result was obtained in \cite{BaVi12} where it is expressed in quite different language; to save the reader the labour of translating that argument into the present article's terminology we provide a proof of this result in the appendix, where it is stated as Proposition \ref{pr:app}.
\subsection{The main theorem}
In order to state our main theorem we require just a few more items of notation. For each $N \geq 1$ let us define
\[\Sigma_N^*:=\bigcup_{n=1}^\infty \{1,\ldots,N\}^n.\]
If $\mathtt{i}=(i_k)_{k=1}^n \in \Sigma_N^*$ we write $|\mathtt{i}|=n$ and refer to $|\mathtt{i}|$ as the \emph{length} of $\mathtt{i}$. If $\mathtt{i},\mathtt{j} \in \Sigma_N^*$ we let $\mathtt{i}\mathtt{j} \in \Sigma_N^*$ denote the sequence of length $|\mathtt{i}|+|\mathtt{j}|$ obtained by running first through the symbols of $\mathtt{i}$ and then through those of $\mathtt{j}$ in the obvious fashion. Clearly $\Sigma_N^*$ is a semigroup with respect to the operation $(\mathtt{i},\mathtt{j}) \mapsto \mathtt{i}\mathtt{j}$. If $A_1,\ldots,A_N \in M_d(\mathbb{R})$ and $\mathtt{i}=(i_k)_{k=1}^n \in \Sigma_N^*$ then we write $A_\mathtt{i}:=A_{i_n}\cdots A_{i_1}$. We observe that $A_\mathtt{i} A_\mathtt{j} =A_{\mathtt{j} \mathtt{i}}$ for all $\mathtt{i},\mathtt{j} \in \Sigma_N^*$.
If $B$ is a linear transformation of a finite-dimensional real vector space we let $\lambda_1(B),\ldots,\lambda_d(B)$ denote the eigenvalues of $B$ listed with repetition according to multiplicity and listed in decreasing order of absolute value. While this notation \emph{a priori} introduces ambiguities when distinct eigenvalues of the same modulus exist, we will see that this consideration does not affect the statements of our results.
We may now present the following generalisation of Pollicott and Vytnova's result:
\begin{theorem}\label{th:main}
Let $d,N \geq 2$, let $(A_1,\ldots,A_N) \in M_d(\mathbb{R})^N$ and let $0 \leq k <d$. Suppose that $(A_1,\ldots,A_N)$ is both $k$-multipositive and $(k+1)$-multipositive. For each integer $n \geq 1$ and $s \in \mathbb{R}$ define
\[t_n(s):=\sum_{|\mathtt{i}|=n} \frac{\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)^{{d \choose k} -1} \lambda_1\left(A_\mathtt{i}^{\wedge\left(k+1\right)}\right)^{{d \choose k+1} -1} \rho\left(A_\mathtt{i}^{\wedge k}\right)^{k+1-s} \rho\left(A_\mathtt{i}^{\wedge \left(k+1\right)}\right)^{s-k} }{ p'_{A_\mathtt{i}^{\wedge k}}\left(\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)\right)p'_{A_\mathtt{i}^{\wedge \left(k+1\right)}}\left(\lambda_1\left(A_\mathtt{i}^{\wedge \left(k+1\right)}\right)\right) }
\]
where $p_B'(x_0)$ denotes the first derivative of the characteristic polynomial $p_B(x):=\det (xI-B)$ evaluated at the point $x_0$. Define also
\[a_n(s):= \frac{(-1)^n}{n!}\det\begin{pmatrix}
t_1(s)& n -1& 0&\cdots &0 &0\\
t_2(s)&t_1(s)&n-2 &\cdots &0 &0\\
t_3(s)&t_2(s)&t_1(s) &\ddots &0 &0\\
\vdots & \vdots & \vdots & \ddots &\ddots& \vdots\\
t_{n-1}(s) &t_{n-2}(s)&t_{n-3}(s)&\cdots &t_1(s) &1\\
t_n(s) &t_{n-1}(s)&t_{n-2}(s)&\cdots &t_2(s) &t_1(s)
\end{pmatrix}\]
for all $n \geq 1$, and $a_0(s):=1$.
For each $s \in [k,k+1]$ let $r_n(s)$ denote the smallest positive real root of the polynomial $p_{n,s}(x):=\sum_{i=0}^n a_n(s)x^i$. Then there exists $n_0 \in \mathbb{N}$ such that $r_n(s)$ is well-defined for all $s \in [k,k+1]$ and $n\geq n_0$, and we have
\[\left|e^{P(A_1,\ldots,A_N;s)}-\frac{1}{r_n(s)}\right| \leq K\exp\left(-\gamma n^\alpha\right)\]
for some constants $K,\gamma>0$ not depending on $s \in [k,k+1]$, where
\[\alpha:=\frac{{d+1\choose k+1}-1}{{d+1\choose k+1}-2}>1.\]
Suppose additionally that there is a norm $\vertle{\cdot}$ on $\mathbb{R}^d$ such that $\max_{1 \leq i \leq N} \vertle{A_i}<1$, and that $\dimaff (A_1,\ldots,A_N) \in (k,k+1)$. Then for all for all sufficiently large $n$ the function $s \mapsto 1/r_n(s)$ is a strictly decreasing convex function $[k,k+1]\to \mathbb{R}$ and there exists a unique $s_n \in [k,k+1]$ such that $r_n(s_n)=1$.
There exist constants $K',\gamma'>0$ depending on $A_1,\ldots,A_N$ such that for all such $n$ we have
\[\left|\dim_{\mathsf{aff}} (A_1,\ldots,A_N)- s_n\right| \leq K'\exp\left(-\gamma' n^\alpha\right).\]
\end{theorem}
Since every matrix tuple is $0$-multipositive, in the case $k=0$ the hypothesis of Theorem \ref{th:main} reduces to the requirement that $\dimaff (A_1,\ldots,A_N)$ is $1$-multipositive and $\dimaff(A_1,\ldots,A_N) \in (0,1)$. Since $B^{\wedge 0}$ is the identity map on $\mathbb{R}$ the expressions involving $A_\mathtt{i}^{\wedge k}$ reduce to $1$ in the case $k=0$, resulting in the formula
\[t_n(s):=\sum_{|\mathtt{i}|=n} \frac{\lambda_1\left(A_\mathtt{i}\right)^{d-1} \rho\left(A_\mathtt{i}\right)^s}{p'_{A_\mathtt{i}}\left(\lambda_1\left(A_\mathtt{i}\right)\right) }.
\]
In particular when $d=2$, $k=0$ and the matrices $A_i$ have positive entries we may recover the conclusion of Theorem \ref{th:one}. Similarly, since every tuple in $M_d(\mathbb{R})^N$ is $d$-multipositive and $B^{\wedge d}=\det B$, the expressions involving $A_\mathtt{i}^{\wedge(k+1)}$ simplify when $k=d-1$ yielding
\[t_n(s):=\sum_{|\mathtt{i}|=n} \frac{\lambda_1\left(A_\mathtt{i}^{\wedge (d-1)}\right)^{d-1} \rho\left(A_\mathtt{i}^{\wedge(d-1)}\right)^{d-s} |\det A_\mathtt{i}|^{s+1-d}}{p'_{A_\mathtt{i}^{\wedge(d-1)}}\left(\lambda_1\left(A_\mathtt{i}^{\wedge(d-1)}\right)\right) }.
\]
and the hypotheses are reduced to the requirement that $(A_1,\ldots,A_N)$ is $(d-1)$-multipositive and $\dimaff(A_1,\ldots,A_N)\in (d-1,d)$. We remark that hypotheses of domination and positivity analogous to those in Theorem \ref{th:main} have been a feature of numerous recent works on affine iterated function systems such as \cite{BaKa17,BaKaKo17,BaRa18,FaKe17,FaKe18} as well as the older article \cite{HuLa95}.
If it is known that the tuple $(A_1^{\wedge k},\ldots,A_N^{\wedge k})$ preserves a single cone in $\wedge^k\mathbb{R}^d$ and similarly $(A_1^{\wedge (k+1)},\ldots,A_N^{\wedge (k+1)})$ preserves a single cone in $\wedge^{k+1}\mathbb{R}^d$ then the condition $\dimaff (A_1,\ldots,A_N) \in (k,k+1)$ may be easily checked. A theorem of V. Yu. Protasov \cite{Pr10} implies that if $B_1,\ldots,B_N$ preserve a cone then
\[\lim_{n \to \infty} \left(\sum_{i_1,\ldots,i_n=1}^N \left\|B_{i_1}\cdots B_{i_n}\right\|\right)^{\frac{1}{n}}=\rho\left(\sum_{i=1}^N B_i\right),\]
and so in this case
\[\lim_{n \to \infty} \left(\sum_{i_1,\ldots,i_n=1}^N \varphi^k\left(A_{i_1}\cdots A_{i_n}\right)\right)^{\frac{1}{n}}=\rho\left(\sum_{i=1}^N A_i^{\wedge k}\right),\]
\[\lim_{n \to \infty} \left(\sum_{i_1,\ldots,i_n=1}^N \varphi^{k+1}\left(A_{i_1}\cdots A_{i_n}\right)\right)^{\frac{1}{n}}=\rho\left(\sum_{i=1}^N A_i^{\wedge (k+1)}\right)\]
using the identity $\varphi^\ell(B)=\|B^{\wedge \ell}\|$ for $\ell=0,\ldots,d$.
It follows that in this situation Theorem \ref{th:main} is applicable if
\[\rho\left(\sum_{i=1}^N A_i^{\wedge (k+1)}\right)<1<\rho\left(\sum_{i=1}^N A_i^{\wedge k}\right).\]
An example of this situation is presented in \S\ref{se:apps} below.
In the situation where $(A_1,\ldots,A_N)$ fails to be both $k$- and $(k+1)$-multipositive we believe it to be unlikely that any analogue of Theorem \ref{th:main} can be proved. The precise role of the multipositive hypothesis is discussed in more detail in the following section, and in the final section \S\ref{se:nondom}.
\section{Overview of the method and statement of the main technical theorem}
The method underlying Theorem \ref{th:main} is, like Theorem \ref{th:one}, based on Fredholm determinants of transfer operators, and in broad terms resembles many other arguments of this type such as \cite{JePo02,JePo05,JePo16,Po10,PoJe00,PoVy16,PoWe08}. Both in order to give a sense of the organisation of this article and to indicate the difficulties present in the proof of Theorem \ref{th:main} which do not occur in the context of Theorem \ref{th:one} let us briefly describe this strategy. We recall that an operator $\mathscr{L}$ on an infinite-dimensional Hilbert space is called trace-class if the sequence of approximation numbers
\[\mathfrak{s}_n(\mathscr{L}):=\inf\left\{\|\mathscr{L}-\mathscr{F}\| \colon \text{rank }\mathscr{F}<n\right\}\]
is summable; we observe in particular that such an operator is compact (being a limit in the norm topology of a sequence of finite-rank operators) and cannot be invertible. We also observe that clearly $\mathfrak{s}_n(\mathscr{L}^\ell) \leq \|\mathscr{L}^{\ell-1}\| \mathfrak{s}_n(\mathscr{L})$ for every $n,\ell \geq 1$ and consequently every power of a trace-class operator is also trace-class. The notion of trace-class operator is reviewed in detail for the reader's convenience in \S\ref{se:prelims}.
Suppose then that $\mathscr{H}$ is a separable complex Hilbert space and $\mathscr{L} \colon \mathscr{H} \to \mathscr{H}$ a trace-class linear operator, and let $(\lambda_\ell)_{\ell=1}^\infty$ be the sequence of nonzero eigenvalues of $\mathscr{L}$ listed with repetition according to their algebraic multiplicity. (If only $M<\infty$ nonzero eigenvalues exist then define $\lambda_\ell=0$ for $\ell>M$.) It is a classical fact that the function $z \mapsto \det(I-z\mathscr{L})$ defined by
\[\det(I-z\mathscr{L}):=\prod_{\ell=1}^M \left(1-z\lambda_\ell\right)\]
is an entire function from $\mathbb{C}$ to $\mathbb{C}$, and moreover one may show that in the power series $\det(I-z\mathscr{L})=\sum_{n=0}^\infty a_\ell z^\ell$ the coefficients are given by $a_0=1$ and
\[a_\ell= \frac{(-1)^\ell}{\ell!}\det\begin{pmatrix}
\tr \mathscr{L} & \ell -1& 0&\cdots &0 &0\\
\tr \mathscr{L}^2&\tr \mathscr{L} &\ell-2 &\cdots &0 &0\\
\tr \mathscr{L}^3&\tr \mathscr{L}^2&\tr \mathscr{L} &\ddots &0 &0\\
\vdots & \vdots & \vdots & \ddots &\ddots& \vdots\\
\tr \mathscr{L}^{\ell-1} &\tr \mathscr{L}^{\ell-2}&\tr \mathscr{L}^{\ell-3}&\cdots &\tr \mathscr{L} &1\\
\tr \mathscr{L}^{\ell} &\tr \mathscr{L}^{\ell-1}&\tr \mathscr{L}^{\ell-2}&\cdots &\tr \mathscr{L}^2 &\tr \mathscr{L}
\end{pmatrix}\]
for $\ell \geq 1$. If we write
\[\sum_{\ell=0}^\infty a_\ell z^\ell=\det(I-z\mathscr{L})=\prod_{\ell=1}^M \left(1-z\lambda_\ell\right)\]
then by equating coefficients of $z^n$ we find (at least informally) that also
\begin{equation}\label{eq:fast}a_n= (-1)^n\sum_{i_1<i_2<\cdots <i_n} \lambda_{i_1}\cdots \lambda_{i_n}. \end{equation}
for each $n \geq 1$. Suppose now that we wished to calculate the spectral radius $\rho(\mathscr{L})$, knowing the values of the traces $\mathscr{L}^\ell$ for $\ell=1,\ldots,n$, say, and knowing also that the spectral radius is an eigenvalue of $\mathscr{L}$. The roots of $\det(I-z\mathscr{L})$ are precisely the reciprocals of the eigenvalues of $\mathscr{L}$ and therefore $\rho(\mathscr{L})^{-1}$ is the smallest positive root of $\sum_{\ell=0}^\infty a_\ell z^\ell$. In particular, the smallest positive root of $\sum_{\ell=0}^n a_\ell z^\ell$ should be a good approximation to $\rho(\mathscr{L})^{-1}$ as long as $\sum_{\ell=n+1}^\infty |a_\ell|$ is small. But if we are able to show that the eigenvalues $(\lambda_n)$ decay exponentially (or even just stretched-exponentially) in $n$, then the expression \eqref{eq:fast} implies a super-exponential decay estimate for the coefficients $a_n$. Such an estimate will hold in particular if the approximation numbers of $\mathscr{L}$ decay stretched-exponentially. In such a situation we may therefore reasonably hope that the approximation procedure just outlined provides an estimate which becomes super-exponentially more accurate as $n$ increases.
In order to implement this line of reasoning we need therefore to construct, for each $s \in [k,k+1]$, a trace-class operator $\mathscr{L}_s$ on a Hilbert space $\mathscr{H}$ such that $e^{P(A_1,\ldots,A_N;s)}$ is an eigenvalue of $\mathscr{L}_s$ and is equal to the spectral radius of $\mathscr{L}_s$, such that $\mathscr{L}_s$ is trace-class, such that the sequence of approximation numbers of $\mathscr{L}_s$ decays rapidly to zero, and such that the sequence of traces $\tr \mathscr{L}_s^n$ is easy to compute. Once such a family of operators has been constructed the result follows by relatively straightforward manipulations which, while they do not correspond precisely to any prior work, share a degree of familial resemblance with calculations occurring in numerous earlier articles such as \cite{BaJePo14,JePo01,JePo02,JePo05,JePo07,JePo16,JePoVy17,KaPo15,Po11,PoFe14,PoJe00,PoVy15,PoVy16,PoWe08}.
If $V$ is a finite-dimensional real vector space let $PV$ denote the real projective space of lines through the origin in $V$. Intuitively, in order to construct an operator $\mathscr{L}_s$ with spectral radius $e^{P(A_1,\ldots,A_N;s)}$, we might consider an operator acting on some space of continuous functions $P(\wedge^k\mathbb{R}^d)\times P(\wedge^{k+1}\mathbb{R}^d) \to \mathbb{C}$ defined by
\[\left(\mathscr{L}_sf\right)(\overline{u},\overline{v}) = \sum_{i=1}^N \left(\frac{\left\|A^{\wedge k}_iu\right\|}{\|u\|}\right)^{k+1-s}\left(\frac{\left\|A^{\wedge(k+1)}_iv\right\|}{\|v\|}\right)^{s-k} f\left(\overline{A^{\wedge k}_iu},\overline{A^{\wedge (k+1)}_iv}\right)\]
where for $v \in V$ the notation $\overline{v}$ represents the one-dimensional subspace spanned by the vector $v$. Since we would then have
\[\left(\mathscr{L}_s^nf\right)(\overline{u},\overline{v}) = \sum_{|\mathtt{i}|=n} \left(\frac{\left\|A^{\wedge k}_\mathtt{i} u\right\|}{\|u\|}\right)^{k+1-s}\left(\frac{\left\|A^{\wedge(k+1)}_\mathtt{i} v\right\|}{\|v\|}\right)^{s-k} f\left(\overline{A^{\wedge k}_\mathtt{i} u},\overline{A^{\wedge (k+1)}_\mathtt{i} v}\right)\]
for each $n\geq 1$ we might then reasonably expect that
\[\lim_{n \to \infty} \left\|\mathscr{L}_s^n\right\|^{\frac{1}{n}}= \lim_{n\to \infty}\left(\sum_{|\mathtt{i}|=n} \left\|A^{\wedge k}_\mathtt{i}\right\|^{k+1-s}\left\|A^{\wedge(k+1)}_\mathtt{i}\right\|^{s-k}\right)^{\frac{1}{n}}= \lim_{n\to \infty}\left(\sum_{|\mathtt{i}|=n}\varphi^s(A_\mathtt{i})\right)^{\frac{1}{n}}\]
so that $e^{P(A_1,\ldots,A_N;s)}$ is equal to the spectral radius of $\mathscr{L}_s$. Indeed, such operators were successfully constructed by Guivarc'h and Le Page on spaces of H\"older continuous functions $P(\wedge^k\mathbb{R}^d)\times P(\wedge^{k+1}\mathbb{R}^d) \to \mathbb{C}$ in the article \cite{GuLe04}.
However, notwithstanding the minor additional complications posed by the fact that the spaces defined above are not Hilbert, there is no reason to believe that $\mathscr{L}_s$ acting on such a space should have a summable sequence of approximation numbers $\mathfrak{s}_n(\mathscr{L}_s)$. Indeed, $\mathscr{L}_s$ as constructed is equal to a sum of weighted composition operators $f \mapsto g\cdot f \circ T$ where $T$ is an invertible transformation of $P(\wedge^k\mathbb{R}^d)\times P(\wedge^{k+1}\mathbb{R}^d)$ and $g$ is nowhere zero. Such an operator might reasonably be expected to be invertible, and there is certainly no reason to believe that $\mathscr{L}_s$ should be trace-class.
The problem is thus to define $\mathscr{L}_s$ approximately as above in such a way that it is a sum of trace-class, non-invertible operators. It is here that the hypothesis of $k$- and $(k+1)$-multipositivity becomes relevant: this hypothesis implies that for $\ell=k,k+1$ the matrices $A_1^{\wedge \ell},\ldots,A_N^{\wedge \ell}$ map a finite union of patches of $P(\wedge^\ell \mathbb{R}^d)$ strictly inside itself. By taking $\mathscr{H}$ to be a suitable Hilbert space of functions defined only on the patches, composition with the projective action of the matrices should then induce an operator which is non-invertible and hopefully trace-class. It transpires that composition operators on spaces of holomorphic functions are reliably trace-class subject to moderate geometrical conditions, and as such our strategy will involve passing to a space of holomorphic functions defined on complex extensions of the patches in real projective space. Once we have verified that such an extension can be constructed in such a way that the operator $\mathscr{L}_s$ is well-defined on the patches we may proceed to prove Theorem \ref{th:main} along the lines outlined above. The principal task arising in this article is therefore to construct the complex patches in such a way that that the operator $\mathscr{L}_s$ is well-defined and has the aforementioned necessary properties.
In the two-dimensional context of Theorem \ref{th:one} the construction of these patches is straightforward. Since Theorem \ref{th:one} is restricted to affine transformations whose linear parts contract the positive cone in $\mathbb{R}^2$, it is sufficient to consider the projective action of those linear maps on the interval $\{(x,1-x)\colon x \in [0,1]\}$, which is an action by linear fractional transformations. A finite collection of linear fractional transformations each of which maps an interval strictly inside itself can easily be shown to also map a corresponding complex disc inside itself, and this complex disc can be used as the domain of the holomorphic functions on which the operator $\mathscr{L}_s$ acts. In the two-dimensional case the construction of $\mathscr{L}_s$ and the space on which it acts is thus rather trivial. In higher dimensions and using multicones instead of cones, the corresponding problem is to understand (in place of one-dimensional intervals) a family of $(d-1)$-dimensional sections of cones in $\mathbb{R}^d$ -- in effect, a finite collection of arbitrary $(d-1)$-dimensional convex bodies -- and a collection of linear fractional transformations between them, and to contrive a system of extensions of those convex bodies into $\mathbb{C}^{d-1}$ which is also preserved by the same family of linear fractional transformations. This rather technical procedure, undertaken in \S\ref{se:complex-cones}, is responsible for much of the length of the present article.
The outcome of the constructions outlined above is summarised in the following technical theorem:
\begin{theorem}\label{th:opter}
Let $d,N \geq 2$ and let $A_1,\ldots,A_N$ be real $d \times d$ matrices and suppose that $(A_1,\ldots,A_N)$ is both $k$-multipositive and $(k+1)$-multipositive, where $0\leq k<d$. There exist a separable complex Hilbert space $\mathscr{H}$ and a family of bounded linear operators $\mathscr{L}_s \colon \mathscr{H} \to \mathscr{H}$ defined for all $s \in \mathbb{C}$ with the following properties:
\begin{enumerate}[(i)]
\item
There exist $C,\kappa,\gamma>0$ such that for all $s \in \mathbb{C}$ and $n \geq 1$ we have $\mathfrak{s}_n(\mathscr{L}) \leq C\exp\left(\kappa|s|-\gamma n^\beta\right)$, where
\[\beta:=\frac{1}{{d+1 \choose k+1}-2} \in (0,1].\]
In particular each $\mathscr{L}_s$ is trace-class.
\item
For every $s \in \mathbb{C}$ and $n \geq 1$ we have
\[\tr \mathscr{L}_s^n=\sum_{|\mathtt{i}|=n} \frac{\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)^{{d \choose k} -1} \lambda_1\left(A_\mathtt{i}^{\wedge\left(k+1\right)}\right)^{{d \choose k+1} -1} \rho\left(A_\mathtt{i}^{\wedge k}\right)^{k+1-s} \rho\left(A_\mathtt{i}^{\wedge \left(k+1\right)}\right)^{s-k} }{ p'_{A_\mathtt{i}^{\wedge k}}\left(\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)\right)p'_{A_\mathtt{i}^{\wedge \left(k+1\right)}}\left(\lambda_1\left(A_\mathtt{i}^{\wedge \left(k+1\right)}\right)\right) }
\]
where $p_B(x):=\det(xI-B)$ denotes the characteristic polynomial of $B$ and $p'_B(x_0)$ its derivative evaluated at $x_0$.
\item
For every $s \in \mathbb{R}$ the spectral radius of $\mathscr{L}_s$ is equal to
\[\lim_{n \to \infty} \frac{1}{n}\log \sum_{|\mathtt{i}|=n} \left\|A_{\mathtt{i}}^{\wedge k}\right\|^{k+1-s} \left\|A_{\mathtt{i}}^{\wedge (k+1)}\right\|^{s-k}.\]
In particular the above limit exists for all $s \in \mathbb{R}$, and for every $s \in [k,k+1]$ the spectral radius of $\mathscr{L}_s$ is equal to $e^{P(A_1,\ldots,A_N;s)}$. For all $s \in \mathbb{R}$ the spectral radius of $\mathscr{L}_s$ is a simple eigenvalue of $\mathscr{L}_s$ and there are no other eigenvalues of the same modulus.
\end{enumerate}
\end{theorem}
Theorem \ref{th:opter} is a special case of a slightly more general result, Theorem \ref{th:topaff}, which will be proved later.
The remainder of this article is structured as follows. In \S\ref{se:complex-cones} we undertake the construction of the complex extensions of the patches in real projective space. It is essentially immediate from the hypotheses of Theorem \ref{th:main} that the projective action of each $A_\mathtt{i}^{\wedge k}$ and $A_\mathtt{i}^{\wedge(k+1)}$ maps an open region of the appropriate real projective space (corresponding to the projectivisation of the real cones witnessing the multipositivity of the matrices $A_i$) inside a compact subset of that region, but it does not automatically follow that there exists a complex region with the same property, which \emph{a priori} is a much stronger requirement. (For example, the real map $f(x):=\frac{1}{2}\sin 100x$ maps the real interval $(-1,1)$ analytically inside a compact subset of $(-1,1)$, but there can be no bounded open subset $U$ of $\mathbb{C}$ which contains $(-1,1)$ and is mapped inside a compact subset of itself by $f$ in the same manner: if such a set $U$ existed then by the Earle-Hamilton fixed point theorem $f$ would have a unique fixed point in $U$, which is manifestly false.) In order to obtain this stronger property we apply the theory of complex cones and gauges introduced by H. H. Rugh in \cite{Ru10} and extended by L. Dubois in \cite{Du09}.
Once the system of complex neighbourhoods underlying the domain of $\mathscr{L}_s$ has been constructed, we review in \S\ref{se:prelims} the properties of trace-class operators which will be needed in this article, and prove suitable extensions of some standard results in view of the fact that we will be working with space of holomorphic functions defined on a non-connected region. We then proceed in \S\ref{se:operators} to establish the properties of the operator $\mathscr{L}_s$ and deduce Theorem \ref{th:opter}. In \S\ref{se:proofs} we derive Theorem \ref{th:main} from Theorem \ref{th:opter} above. Some examples of the application of Theorem \ref{th:main} are presented in \S\ref{se:apps}. In \S\ref{se:nondom} we consider the problem of calculating the affinity dimension in situations where the hypotheses of Theorem \ref{th:main} do not apply. We remark that sections \ref{se:proofs}--\ref{se:nondom} depend only on the statement of Theorem \ref{th:opter} and the material presented in sections 1 and 2 and as such may be read independently of sections \ref{se:complex-cones}--\ref{se:operators} in which the proof of Theorem \ref{th:opter} is prepared for and presented.
\section{The projective action on complex cones}\label{se:complex-cones}
Our first task in proving Theorem \ref{th:main} is to translate the matter from the context of linear maps between cones to the context of holomorphic maps between complex domains.
For this task we will use the machinery of complex cones and gauges introduced by H. H. Rugh and extended by L. Dubois \cite{Du09,Ru10}.
Given a nonempty compact set $\mathsf{A}\subset M_d(\mathbb{R})$ we let $\mathcal{S}(\mathsf{A})$ denote the semigroup generated by $\mathsf{A}$. In the applications considered in this article we will consider only the finitely-generated semigroup $\mathcal{S}(\{A_1^{\wedge \ell},\ldots,A_N^{\wedge \ell}\})=\{A_\mathtt{i}^{\wedge \ell} \colon \mathtt{i} \in \Sigma_N^*\}$ for some $\ell \in \{1,\ldots,d-1\}$, but it is almost as easy to study the compactly-generated case and we include this case for possible use in future research.
Here and throughout this article we shall say that if $U_1,U_2$ are subsets of a metric space $X$ then $U_1$ is \emph{compactly contained in } $U_2$ if the closure of $U_1$ is a compact subset of the interior of $U_2$. We express this relation by writing $U_1 \Subset U_2$.
The results to be proved in this section are summarised in the following theorem:
\begin{theorem}\label{th:multicones}
Let $d \geq 1$, let $\mathsf{A}\subset M_d(\mathbb{R})$ be compact and nonempty and suppose that $(\mathcal{K}_1,\ldots,\mathcal{K}_m)$ is a multicone for $\mathsf{A}$ with transverse-defining vector $w$. For each $j=1,\ldots,m$ define
\[\mathcal{K}_j^{\mathbb{C}}:=\left\{\lambda((u+v)+i(u-v))\colon \lambda \in \mathbb{C}\text{ and }u,v \in \mathcal{K}_j\right\},\]
and let
\[\Omega:=\left\{z \in \mathbb{C}^d \colon z \in \bigcup_{j=1}^m \Int \mathcal{K}_j^{\mathbb{C}} \text{ and }\langle z,w\rangle=1\right\}.\]
For each $A \in \mathcal{S}(\mathsf{A})$ and $z \in \Omega$ let us write $\overline{A} z:=\langle A z,w\rangle^{-1}Az$. Then:
\begin{enumerate}[(i)]
\item
Every $A \in \mathcal{S}(\mathsf{A})$ has a simple leading eigenvalue $\lambda_1(A)$ which is real and is strictly larger in modulus than all of the other eigenvalues of $A$.
\item
There is a constant $\tau>0$ such that $\|A_1 A_2 \| \geq \tau \|A_1\|\cdot\|A_2\|$ for every $A_1,A_2 \in \mathcal{S}(\mathsf{A})$.
\item
For every $z \in \Omega$ and $A \in \mathcal{S}(\mathsf{A})$ we have $\Re(\langle Az,w\rangle)\neq 0$.
\item
$\Omega$ is a nonempty, open, bounded subset of the complex hyperplane $\{z \in \mathbb{C}^d \colon \langle z,w\rangle=1\}$, and for every $A \in \mathcal{S}(\mathsf{A})$ the map $\overline{A} \colon \Omega \to \Omega$ is well-defined.
\item
There exist constants $C,\gamma>0$ such that
\[\sup_{A_1,\ldots,A_n \in \mathsf{A}} \diam \overline{A_1\cdots A_n}(\Omega)\leq Ce^{-\gamma n}\]
for every $n \geq 1$.
\item
For each $A \in \mathcal{S}(\mathsf{A})$ the map $\overline{A} \colon \Omega \to \Omega$ has a unique fixed point $z_A \in \Omega$. We have $z_A \in \Omega \cap \mathbb{R}^{d}$ and $\langle A z_A,w\rangle= \lambda_1(A)$. The eigenvalues of the derivative $D_{z_A}\overline{A}$ are precisely the numbers $\lambda_j(A)/\lambda_1(A)$ for $j=2,\ldots,d$, and in particular
\[\det (I-D_{z_A}\overline{A})= \frac{p_A'(\lambda_1(A))}{\lambda_1(A)^{d-1}} \neq 0\]
where $p_A(x):=\det (xI-A)$ denotes the characteristic polynomial of $A$ and $p_A'$ its first derivative.
\item
There is a constant $C>0$ such that for each $A \in \mathcal{S}(\mathsf{A})$ and $z \in \Omega$ we have $C^{-1}\|A\|\leq |\langle Az,w\rangle| \leq C \|A\|$.
\item
The set $\bigcup_{A \in \mathcal{S}(\mathsf{A})} \overline{A}(\Omega)$ is compactly contained in $\Omega$.
\end{enumerate}
\end{theorem}
Theorem \ref{th:multicones} is trivial in the case $d=1$ and for the remainder of this section we shall ignore this case, assuming at all times that $d \geq 2$. (When $d=1$ the determinant in (vi) above will be interpreted as being equal to $1$.) Here and throughout the remainder of this article we use the notation $z^*$ to denote the complex conjugate of $z \in \mathbb{C}$ and reserve the notation $\overline{z}$ and $\overline{A}$ for projective quantities.
We will establish some preliminary results towards the proof of Theorem \ref{th:multicones} in the following three subsections and then combine them to give the proof itself at the end of this section.
\subsection{Elementary lemmas and growth rates in real cones}
Let us define an \emph{$\mathbb{R}$-cone} to be a closed, convex set $\mathcal{K}\subseteq \mathbb{R}^d$ with nonempty interior such that $\lambda \mathcal{K}=\mathcal{K}$ for all $\lambda>0$, and such that $\mathcal{K} \cap -\mathcal{K}=\{0\}$. (This definition is slightly more restrictive than that used in Rugh's article \cite{Ru10} in that we require $\mathbb{R}$-cones to have interior and Rugh does not.) If $\mathcal{K}\subseteq \mathbb{R}^d$ is an $\mathbb{R}$-cone, we recall that the \emph{dual cone} is defined to be the set
\[\mathcal{K}':=\left\{v \in \mathbb{R}^d \colon \langle u,v\rangle \geq 0\text{ for all }u \in \mathcal{K}\right\}.\]
Obviously $\mathcal{K}'$ is closed and convex and satisfies $\lambda\mathcal{K}'=\mathcal{K}'$ for all real $\lambda>0$. We remark that the dual cone of an $\mathbb{R}$-cone (in our sense of an $\mathbb{R}$-cone) is also an $\mathbb{R}$-cone, but this fact will not be required.
It is easier to provide a proof of the following elementary separation lemma than to find a crisp reference:
\begin{lemma}\label{le:sep}
Let $\mathcal{K}\subset \mathbb{R}^d$ be an $\mathbb{R}$-cone and $w \in \mathbb{R}^d \setminus\mathcal{K}$. Then there exists $v \in \mathcal{K}'$ such that $ \langle w,v\rangle<0$.
\end{lemma}
\begin{proof}
By a trivial compactness argument there exists $u_0 \in \mathcal{K}$ which minimises the distance $\|u_0-w\|$. We claim that $\langle u_0,u_0-w\rangle=0$. The definitions of $u_0$ and of $\mathcal{K}$ together imply that the function $\lambda \mapsto \|\lambda u_0 -w\|^2$ defined for $\lambda \geq 0$ has a local minimum at $\lambda=1$. Differentiating $\|\lambda u_0 -w\|^2 = \lambda^2\|u_0\|^2 -2\lambda \langle u_0,w\rangle + \|w\|^2$ at $\lambda=1$ yields $2\|u_0\|^2-2\langle u_0,w\rangle=0$ since the derivative must be zero at a local minimum. It follows that $\langle u_0,u_0-w\rangle = \|u_0\|^2-\langle u_0,w\rangle=0$ as claimed. Since $\langle u_0-w,u_0-w\rangle>0$ we deduce that $\langle w,u_0-w\rangle<0$.
If $u \in \mathcal{K}$, $\lambda \in (0,1)$ are arbitrary then $u_0+\lambda (u-u_0) \in \mathcal{K}$ and therefore $\|u_0-w\|^2 \leq \|u_0+\lambda (u-u_0)-w\|^2 =\|u_0-w\|^2+\lambda \langle u-u_0,u_0-w\rangle + \lambda^2\|u-u_0\|^2$. Hence $\langle u-u_0,u_0-w\rangle +\lambda \|u-u_0\|^2 \geq 0$ for all $\lambda \in (0,1)$ and it follows that $\langle u-u_0,u_0-w\rangle \geq 0$ for all $u \in \mathcal{K}$. Applying $\langle u_0,u_0-w\rangle=0$ yields $\langle u,u_0-w\rangle \geq 0 >\langle w,u_0-w\rangle$ for all $u \in \mathcal{K}$ and the result follows by taking $v:=u_0-w$.
\end{proof}
The following result is elementary, but its proof illuminates the version for complex cones which will shortly follow.
\begin{lemma}\label{le:intreal}
Let $\mathcal{K}\subseteq \mathbb{R}^d$ be a closed convex cone. Then
\[\Int \mathcal{K}=\left\{u \in \mathbb{R}^d \colon \langle u,v\rangle >0\text{ for all nonzero }v \in \mathcal{K}'\right\}.\]
\end{lemma}
\begin{proof}
If $\langle u,v\rangle>0$ for all nonzero $v \in \mathcal{K}'$ then $u \in \mathcal{K}$ since otherwise Lemma \ref{le:sep} would be contradicted. Clearly $\langle u,v\rangle>0$ for all nonzero $v \in \mathcal{K}'$ if and only if $\langle u,v\rangle>0$ for all $v \in \mathcal{K}'$ such that $\|v\|=1$, if and only if
\[ \inf_{\substack{v \in \mathcal{K}'\\ \|v\|=1}} \langle u,v\rangle>0;\]
but this quantity is locally Lipschitz continuous in $u$, so the set of all $u \in \mathbb{R}^d$ such that $\langle u,v\rangle>0$ for all nonzero $v \in \mathcal{K}'$ is an open subset of $\mathcal{K}$ and in particular is contained in $\Int \mathcal{K}$. We now claim that every point \emph{not} in this set does not belong to $\Int \mathcal{K}$. Indeed, suppose $u \in \mathbb{R}^d$ and $\langle u,v\rangle \leq 0$ for some nonzero $v \in \mathcal{K}'$. We have $\langle u-\varepsilon v,v\rangle=\langle u,v\rangle -\varepsilon\|v\|^2<0$ for every $\varepsilon>0$, so $u$ is an accumulation point of the complement of $\mathcal{K}$ and hence is not in $\Int \mathcal{K}$.
\end{proof}
\begin{proposition}\label{pr:bob}
Let $\mathsf{A}\subset M_d(\mathbb{R})^N$ be compact and nonempty and suppose that $(\mathcal{K}_1,\ldots,\mathcal{K}_m)$ is a multicone for $\mathsf{A}$ with transverse-defining vector $w$. Then there exist constants $\tau_1,\tau_2,\tau_3>0$ such that:
\begin{enumerate}[(i)]
\item
For every $u \in \bigcup_{j=1}^m \mathcal{K}_j$ and every $A \in \mathcal{S}(\mathsf{A})$ we have $\|A u\|\geq \tau_1 \|A\|\cdot \|u\|$.
\item
For every $u \in \bigcup_{j=1}^m \mathcal{K}_j$ and every $A \in \mathcal{S}(\mathsf{A})$ we have $|\langle A u,w\rangle|\geq \tau_2 \|A\|\cdot \|u\|$.
\item
For every $A_1,A_2 \in \mathcal{S}(\mathsf{A})$ we have $\|A_1 A_2\|\geq \tau_3 \|A_1\|\cdot\|A_2\|$.
\end{enumerate}
\end{proposition}
\begin{proof}
We will prove (i) first and then deduce (ii) and (iii). Since $\mathsf{A}$ is compact and each $A \in \mathsf{A}$ maps $\bigcup_{j=1}^m \mathcal{K}_j\setminus\{0\}$ into $\bigcup_{j=1}^m (\Int \mathcal{K}_j \cup -\Int \mathcal{K}_j)$, by slightly enlarging each $\mathcal{K}_j$ we may find new $\mathbb{R}$-cones $\mathcal{K}_1^+,\ldots,\mathcal{K}_m^+$ such that we still have $A\left(\bigcup_{j=1}^m \mathcal{K}_j^+\right)\subseteq \bigcup_{j=1}^m (\mathcal{K}_j \cup - \mathcal{K}_j)$ for all $A \in \mathsf{A}$, but such that additionally $\bigcup_{j=1}^m \mathcal{K}_j \setminus \{0\} \subseteq \bigcup_{j=1}^m \Int \mathcal{K}_j^+$. Define
\[\mathfrak{S}:=\left\{B \in M_d(\mathbb{R}) \colon B\left(\bigcup_{j=1}^m \mathcal{K}_j^+\right) \subseteq \bigcup_{j=1}^m( \mathcal{K}_j\cup -\mathcal{K}_j)\right\}.\]
We note that $\mathfrak{S}$ is a closed subsemigroup of $M_d(\mathbb{R})$ which includes $\mathsf{A}$, so in particular it contains $\mathcal{S}(\mathsf{A})$.
We claim that there exists $\tau_1>0$ such that $\|Bu\|\geq \tau_1 \|B\|\cdot \|u\|$ for every $B \in \mathfrak{S}$ and $u \in \bigcup_{k=1}^m \mathcal{K}_j$. By homogeneity it is sufficient to consider only the case where $\|B\|=\|u\|=1$. By compactness it is in turn sufficient to show that $\|Bu\|>0$ whenever $B \in \mathfrak{S}$, $u \in \bigcup_{k=1}^m \mathcal{K}_j$ and $\|B\|=\|u\|=1$. Suppose for a contradiction that there exist $B \in \mathfrak{S}$ and $u \in \bigcup_{j=1}^m \mathcal{K}_j$ such that $\|B\|=\|u\|=1$ and $Bu=0$. Let $u \in \mathcal{K}_{j_1}$, say. Since $\|B\|\neq 0$ there exists $v \in \mathbb{R}^d$ such that $Bv \neq 0$. Note that since $u$ is nonzero we have $u \in \Int \mathcal{K}_{j_1}^+$, and in particular there exists $\varepsilon>0$ such that $u+\varepsilon v$ and $u-\varepsilon v$ both belong to $\mathcal{K}_{j_1}^+$.
Since $B\mathcal{K}_{j_1}^+ \subseteq \bigcup_{j=1}^m (\mathcal{K}_j\cup -\mathcal{K}_j)$
and the sets $(\mathcal{K}_j \cup -\mathcal{K}_j)\setminus \{0\}$ are disjoint for distinct values of $j$, the nonzero vector $Bv$ belongs to $\mathcal{K}_j \cup -\mathcal{K}_j$ for some unique value of $j$, say $j_2$. In particular $B\mathcal{K}_{j_1}^+ \cap ( (\mathcal{K}_{j_2}\cup -\mathcal{K}_{j_2})\setminus\{0\})$ is nonempty since it contains $B(u+\varepsilon v)=\varepsilon Bv$. Since $B\mathcal{K}_{j_1}^+$ is connected it can intersect at most one of the pairwise disjoint sets $ (\mathcal{K}_{j}\cup -\mathcal{K}_{j})\setminus\{0\}$, so we have either $B\mathcal{K}_{j_1}^+ \subseteq \mathcal{K}_{j_2}$ or $B\mathcal{K}_{j_1}^+ \subseteq -\mathcal{K}_{j_2}$. If the former, we find that
\[\varepsilon Bv = B(u+\varepsilon v)\in B\mathcal{K}_{j_1}^+ \subseteq \mathcal{K}_{j_2}\]
and
\[-\varepsilon Bv = B(u-\varepsilon v)\in B\mathcal{K}_{j_1}^+ \subseteq \mathcal{K}_{j_2}\]
which implies that $\mathcal{K}_{j_2}\cap -\mathcal{K}_{j_2} \ni \varepsilon Bv \neq 0$, contradicting the statement that $\mathcal{K}_{j_2}$ is an $\mathbb{R}$-cone; if the latter, the same result holds by the same argument modulo some appropriate changes of sign. We conclude that there must exist
$\tau_1>0$ such that $\|Bu\|\geq \tau_1 \|B\|\cdot \|u\|$ for every $B \in \mathfrak{S}$ and $u \in \bigcup_{k=1}^m \mathcal{K}_j$, and since $\mathcal{S}(\mathsf{A})\subseteq \mathfrak{S}$ this completes the proof of (i).
To deduce (ii), we note that by (i) it is sufficient to find a constant $\tilde \tau>0$ such that $|\langle A u,w\rangle| \geq \tilde\tau \|A u\|$ for every $A\in\mathcal{S}(\mathsf{A})$ and $u \in\bigcup_{j=1}^m \mathcal{K}_j$. To obtain this inequality it clearly in turn suffices to show that $|\langle v,w\rangle| \geq \tilde\tau \|v\|$ for all $v \in \bigcup_{j=1}^m \mathcal{K}_j$. By homogeneity it is sufficient to consider only unit vectors $v \in \bigcup_{j=1}^m \mathcal{K}_j$. Since the intersection of the unit sphere of $\mathbb{R}^d$ with $\bigcup_{j=1}^m \mathcal{K}_j$ is compact we may achieve this by showing that $|\langle v,w\rangle| \neq 0$ for every unit vector $v \in \bigcup_{j=1}^m \mathcal{K}_j$, but this is true by Definition \ref{de:mc}(ii). We deduce (ii). To see that (iii) also follows, let $u_0 \in \mathcal{K}_1$ be an arbitrary unit vector. We have
\[\|A_1 A_2\|\geq \|A_1 A_2 u_0\|\geq \tau_1 \|A_1\|\cdot\| A_2 u_0\| \geq \tau^2_1 \|A_1\|\cdot\| A_2\|\cdot\| u_0\| =\tau^2_1 \|A_1\|\cdot\| A_2\| \]
which yields (iii).
\end{proof}
\subsection{The complexification of a real cone}
Following H. H. Rugh \cite{Ru10} we shall say that a set $\mathsf{K}\subseteq \mathbb{C}^d$ is a \emph{$\mathbb{C}$-cone} if it is nonempty and closed, satisfies $\lambda \mathsf{K}= \mathsf{K}$ for all nonzero $\lambda \in \mathbb{C}$, and contains no $2$-dimensional subspace of $\mathbb{C}^d$. Following \cite[\S5]{Ru10} we define the \emph{complexification} of an $\mathbb{R}$-cone $\mathcal{K}\subseteq \mathbb{R}^d$ to be the set
\[\mathcal{K}^{\mathbb{C}}:=\left\{z \in \mathbb{C}^d \colon \Re\left(\langle z,w_1\rangle \langle z,w_2\rangle^*\right)\geq 0\text{ for all }w_1,w_2 \in \mathcal{K}'\right\} \]
where $\omega^*$ denotes the complex conjugate of $\omega$. The set $\mathcal{K}^{\mathbb{C}}$ admits the alternative characterisation
\[\mathcal{K}^{\mathbb{C}}= \left\{\lambda ((u+v)+i(u-v)) \colon \lambda \in \mathbb{C}\text{ and }u,v \in \mathcal{K}\right\}\]
which is proved in \cite[Proposition 5.2]{Ru10}. By replacing $u$ and $v$ with $|\lambda|u$ and $|\lambda|v$ if necessary we may clearly assume if desired that $\lambda=e^{i\theta}$ for some $\theta \in \mathbb{R}$. The complexification $\mathcal{K}^{\mathbb{C}}$ of an $\mathbb{R}$-cone is always a $\mathbb{C}$-cone \cite[Theorem 5.5]{Ru10}.
In this subsection we will prove some elementary geometric and topological properties of complexifications which will be useful in the following two subsections.
\begin{lemma}\label{le:w}
Let $\mathcal{K} \subseteq \mathbb{R}^d$ be an $\mathbb{R}$-cone and suppose that $w \in \mathbb{R}^d$ satisfies $\langle u,w\rangle> 0$ for all nonzero $u \in \mathcal{K}$. Then $\langle z,w\rangle\neq 0$ for all nonzero $z \in \mathcal{K}^{\mathbb{C}}$.
\end{lemma}
\begin{proof}
For a contradiction take $z =\lambda((u+v)+i(u-v)) \in \mathcal{K}^{\mathbb{C}}$ where $u,v \in \mathcal{K}$, $z \neq 0$ and $\langle z,w\rangle=0$. We have $\langle \lambda^{-1}z,w\rangle=0$ which is to say $\langle (u+v)+i(u-v),w\rangle=0$, but then $\langle u+v,w\rangle=0$ which implies that $\langle u,w\rangle=-\langle v,w\rangle$ with $u,v \in \mathcal{K}$. If $u \neq 0$ then $\langle u,w\rangle > 0$ which implies that $\langle v,w\rangle<0$ with $v \in \mathcal{K}$, a contradiction. It follows that $u=0$ and therefore $\langle v,w\rangle=0$ which implies $v=0$ and therefore $z=0$, which is also a contradiction. The proof is complete.
\end{proof}
\begin{lemma}\label{le:bdd}
Let $\mathsf{K}\subseteq \mathbb{C}^d$ be a $\mathbb{C}$-cone and let $w \in \mathbb{C}^d$ be a unit vector such that $\langle z,w\rangle \neq 0$ for all nonzero $z \in \mathsf{K}$. Then there exists $C>0$ such that $\|z\|\leq C |\langle z,w\rangle|$ for all $z \in \mathsf{K}$.
\end{lemma}
\begin{proof}
We will show that there exists $\tau>0$ such that $ |\langle z,w\rangle|\geq \tau \|z\|$ for all $z \in \mathsf{K}$. By homogeneity it is clearly sufficient to consider only those cases where $\|z\|=1$. By compactness it in turn suffices to show that we cannot simultaneously have $\|z\|=1$, $z \in \mathsf{K}$ and $|\langle z,w\rangle|=0$, but this is true by hypothesis. The result follows.\end{proof}
The following complex version of Lemma \ref{le:intreal} seems not to have been previously remarked.
\begin{lemma}\label{le:intcomp}
Let $\mathcal{K}\subseteq \mathbb{R}^d$ be an $\mathbb{R}$-cone. Then
\[\Int \mathcal{K}^\mathbb{C}=\left\{z \in \mathbb{C}^d \colon \Re(\langle z,w_1\rangle \langle z,w_2\rangle^*)>0\text{ for all nonzero }w_1,w_2 \in \mathcal{K}'\right\}.\]
\end{lemma}
\begin{proof}
It is clear that if $z$ belongs to the set
\begin{equation}\label{eq:int}\left\{z \in \mathbb{C}^d \colon \Re(\langle z,w_1\rangle \langle z,w_2\rangle^*)>0\text{ for all nonzero }w_1,w_2 \in \mathcal{K}'\right\}\end{equation}
then
\[\inf_{\substack{w_1,w_2\in \mathcal{K}'\\\|w_1\|=\|w_2\|=1}}\Re(\langle z,w_1\rangle \langle z,w_2\rangle^*) >0\]
by continuity and compactness. It follows easily that every sufficiently small perturbation of $z$ belongs to the same set and in particular belongs to $\mathcal{K}^{\mathbb{C}}$, so the set \eqref{eq:int} is contained in the interior of $\mathcal{K}^{\mathbb{C}}$.
We now wish to show that if $z \in \mathcal{K}^{\mathbb{C}}$ and $\Re(\langle z,w_1\rangle \langle z,w_2\rangle^*)=0$ for some nonzero $w_1,w_2 \in \mathcal{K}'$ then $z$ is not an interior point of $\mathcal{K}^{\mathbb{C}}$. We consider three cases. Suppose firstly that $z \in \mathcal{K}^{\mathbb{C}}$ is equal to $0$. If $z=0 \in \Int \mathcal{K}^{\mathbb{C}}$ then by homogeneity $\mathbb{C}^d\subseteq \mathcal{K}^{\mathbb{C}}$ and therefore $\mathcal{K}^{\mathbb{C}}$ contains a two-dimensional subspace of $\mathbb{C}^d$, which is impossible since $\mathcal{K}^{\mathbb{C}}$ is a $\mathbb{C}$-cone. It follows that $0 \notin \Int \mathcal{K}^{\mathbb{C}}$.
Suppose secondly that $z \in \mathcal{K}^{\mathbb{C}}$ and $\langle z,w_1\rangle=0$ for some nonzero $w_1 \in \mathcal{K}'$, but $z \neq 0$. Without loss of generality we assume $\|w_1\|=1$. We claim that there exists nonzero $w_2 \in \mathcal{K}'$ such that $\langle z,w_2\rangle\neq 0$. Let us write $z_2=\lambda((u+v)+i(u-v))$ where $\lambda \in \mathbb{C}$ and $u,v \in \mathcal{K}$. If we have $u+v=0$ then $u=-v \in \mathcal{K} \cap -\mathcal{K}=\{0\}$ which implies $z=0$, so $u+v \in \mathcal{K}$ must be nonzero. It follows that $-u-v \notin \mathcal{K}$ and therefore $\langle -u-v,w_2\rangle<0$ for some $w_2\in\mathcal{K}'$ by Lemma \ref{le:sep}, where we may freely assume $\|w_2\|=1$. In particular $\Re(\langle \lambda^{-1}z,w_2\rangle)=\langle u+v,w_2\rangle \neq 0$ which suffices to prove the claim. Now choose $\theta \in \mathbb{R}$ such that $e^{i\theta}\langle z,w_2\rangle^*=-|\langle z,w_2\rangle|$. For every $\varepsilon>0$ we have
\[\Re(\langle z+\varepsilon e^{i\theta}w_1,w_1\rangle \langle z+\varepsilon e^{i\theta}w_1,w_2\rangle^*)= -\varepsilon|\langle z,w_2\rangle| + \varepsilon^2\langle w_1,w_2\rangle \]
and this implies that $z+\varepsilon e^{i\theta}w_1 \notin \mathcal{K}^{\mathbb{C}}$ for all sufficiently small $\varepsilon>0$, so in particular $z \notin \Int \mathcal{K}^{\mathbb{C}}$.
Consider finally the case in which $z\in \mathcal{K}^{\mathbb{C}}$, $\langle z,w\rangle \neq 0$ for all nonzero $w \in \mathcal{K}'$, and $\Re(\langle z,w_1\rangle \langle z,w_2\rangle^*)=0$ for some $w_1,w_2 \in \mathcal{K}'$ where we assume that $\|w_1\|=\|w_2\|=1$. We in particular have $\langle z,w_1\rangle \langle z,w_2\rangle^*\neq 0$. By interchanging $w_1$ with $w_2$ if necessary we assume that $|\langle z,w_2\rangle| \geq |\langle z,w_1\rangle|>0$. If $|\langle w_1,w_2\rangle|=1$ then $w_1=\pm w_2$ so that $\Re(\langle z,w_1\rangle \langle z,w_2\rangle^*)=\pm|\langle z,w_1\rangle|^2\neq 0$, a contradiction, so necessarily $|\langle w_1,w_2\rangle|<1$. Choose $\theta \in \mathbb{R}$ such that $e^{i\theta}\langle z,w_2\rangle^*=-|\langle z,w_2\rangle|$. We have
\begin{eqnarray*}\lefteqn{\Re\left(\left\langle z+\varepsilon e^{i\theta}w_1,w_1\right\rangle \left\langle z+\varepsilon e^{i\theta}w_1,w_2\right\rangle^*\right)}& &\\
&=& \Re\left(\left\langle \varepsilon e^{i\theta}w_1,w_1\right\rangle \left\langle z,w_2\right\rangle^*\right)+\Re\left(\left\langle z,w_1\right\rangle \left\langle \varepsilon e^{i\theta}w_1,w_2\right\rangle^*\right)\\
& &+ \Re\left(\left\langle \varepsilon e^{i\theta}w_1,w_1\right\rangle \left\langle \varepsilon e^{i\theta}w_1,w_2\right\rangle^*\right)\\
& =&-\varepsilon\left|\left\langle z,w_2\right\rangle\right| +\varepsilon \Re\left(e^{-i\theta}\langle z,w_1\rangle\right) \langle w_1,w_2\rangle + \varepsilon^2\langle w_1,w_2\rangle.\end{eqnarray*}
Since $|\langle z,w_2\rangle| \geq |\langle z,w_1\rangle|>|e^{i\theta}\langle z,w_1\rangle \langle w_1,w_2\rangle|$ this quantity is negative for all sufficiently small $\varepsilon>0$. It follows that $z$ is an accumulation point of the complement of $\mathcal{K}^{\mathbb{C}}$ and as claimed is not interior to $\mathcal{K}^{\mathbb{C}}$. The proof is complete.
\end{proof}
\begin{corollary}\label{co:cks}
Let $\mathcal{K}_1,\mathcal{K}_2 \subseteq \mathbb{R}^d$ be $\mathbb{R}$-cones and let $A \in M_d(\mathbb{R})$ such that $A(\mathcal{K}_1\setminus \{0\}) \subseteq \Int \mathcal{K}_2 \cup -\Int \mathcal{K}_2$. Then $A(\mathcal{K}_1^{\mathbb{C}}\setminus \{0\})\subseteq \Int \mathcal{K}_2^{\mathbb{C}}$.
\end{corollary}
\begin{proof}
By connectedness of $\mathcal{K}_1\setminus \{0\}$ we either have $A(\mathcal{K}_1\setminus \{0\}) \subseteq \Int \mathcal{K}_2$ or $A(\mathcal{K}_1\setminus \{0\}) \subseteq - \Int \mathcal{K}_2$. In the former case we have $\langle Au,w\rangle>0$ for every nonzero $u \in \mathcal{K}_1$ and $w \in \mathcal{K}_2'$ by Lemma \ref{le:intreal}, and in the latter case we have $\langle Au,w\rangle<0$ for every nonzero $u \in \mathcal{K}_1$ and $w \in \mathcal{K}_2'$. In either case the sign of $\langle Au,w\rangle$ is constant with respect to the choice of nonzero $u \in \mathcal{K}_1$ and $w \in \mathcal{K}_2'$, and is never zero for any such $u$ and $w$.
If $z \in \mathcal{K}^{\mathbb{C}}_1$ is nonzero and $w_1,w_2 \in \mathcal{K}_2'$ are arbitrary nonzero vectors, we must by Lemma \ref{le:intcomp} show that $\Re(\langle Az,w_1\rangle \langle Az,w_2\rangle^*)>0$. Since this condition is unaffected by the substitution $z \mapsto e^{i\theta}z$ we may assume that $z \in \mathcal{K}_1^{\mathbb{C}}$ has the form $z=u+v+i(u-v)$ where $u,v \in \mathcal{K}_1$. We then have
\begin{align*}\Re(\langle Az,w_1\rangle \langle Az,w_2\rangle^*)=&\langle Au+Av,w_1\rangle \langle Au+Av,w_2\rangle \\
&+\langle Au-Av,w_1\rangle \langle Au-Av,w_2\rangle\\
=&2\langle Au,w_1\rangle\langle Au,w_2\rangle + 2\langle Av,w_1\rangle\langle Av,w_2\rangle >0\end{align*}
since at least one of $u$ and $v$ is nonzero and since $\langle Au,w_1\rangle$, $\langle Au,w_2\rangle$, $\langle Av,w_1\rangle$ and $\langle Av,w_2\rangle$ cannot have mixed positive and negative signs. It follows that $Az \in \Int \mathcal{K}_2^{\mathbb{C}}$ as required.
\end{proof}
\begin{lemma}\label{le:disjt}
Let $\mathcal{K}_1,\mathcal{K}_2 \subseteq \mathbb{R}^d$ be $\mathbb{R}$-cones such that $\mathcal{K}_1\cap\mathcal{K}_2=\{0\}$, and suppose that there exists $w \in \mathbb{R}^d$ such that $\langle u,w\rangle > 0$ for all nonzero $u \in \mathcal{K}_1\cup\mathcal{K}_2$. Then $\mathcal{K}_1^{\mathbb{C}} \cap \mathcal{K}_2^{\mathbb{C}}=\{0\}$.\end{lemma}
\begin{proof}
We observe that the hypothesis directly implies $\mathcal{K}_2 \cap (\mathcal{K}_1\cup -\mathcal{K}_1)=\{0\}$. Suppose that $z \in \mathcal{K}_1^{\mathbb{C}} \cap \mathcal{K}_2^{\mathbb{C}}$; we wish to show that $z=0$. Write $z=\lambda_1((u_1+v_1)+i(u_1-v_1))=\lambda_2((u_2+v_2)+i(u_2-v_2))$ where $\lambda_1,\lambda_2 \in \mathbb{C}$ and $u_1,v_1 \in \mathcal{K}_1$, $u_2,v_2 \in \mathcal{K}_2$. If $\lambda_2=0$ or $\lambda_1=0$ then we are done; otherwise, let $\lambda_2^{-1}\lambda_1=a+ib\neq 0$ with $a,b\in\mathbb{R}$. We have
\[(a+ib)((u_1+v_1)+i(u_1-v_1)) = u_2+v_2+i(u_2-v_2)\]
which is to say
\[(a-b)u_1 + (a+b)v_1 = u_2+v_2,\qquad (a+b)u_1 - (a-b)v_1 = u_2-v_2\]
by separating real and imaginary parts, and therefore
\[u_2=au_1+bv_1,\qquad v_2=-bu_1+av_1.\]
Consider first the case in which one of $a$ and $b$ is strictly positive and the other strictly negative. In this case $v_2 =-bu_1+av_1 \in \mathcal{K}_2\cap (\mathcal{K}_1\cup -\mathcal{K}_1)=\{0\}$ so that $v_2=0$. It follows that $bu_1=av_1$, but since $a$ and $b$ have opposite signs this implies $bu_1,av_1 \in \mathcal{K}_1 \cap -\mathcal{K}_1=\{0\}$. Since neither $a$ nor $b$ is zero it follows that $u_1=v_1=0$ and therefore $z=0$ as required.
On the other hand, if $a$ and $b$ are both non-negative then $au_1+bv_1 \in \mathcal{K}_1$, and if $a$ and $b$ are both non-positive then $au_1+bv_1\in -\mathcal{K}_1$. In either case $u_2=au_1+bv_1 \in \mathcal{K}_2 \cap (\mathcal{K}_1\cup -\mathcal{K}_1)=\{0\}$ and so $u_2=0$. This implies $au_1+bv_1=0$ so $au_1=-bv_1 \in \mathcal{K}_1\cap-\mathcal{K}_1=\{0\}$ and therefore $au_1=bv_1=0$.
If $a=0$ then $b \neq 0$ which implies $v_1=0$, so $v_2=-bu_1 \in \mathcal{K}_2 \cap (\mathcal{K}_1\cup -\mathcal{K}_1)=\{0\}$ and we have $u_2=v_2=0$ so that $z=0$. If $a \neq 0$ then $u_1=0$ and therefore $v_2=av_1 \in \mathcal{K}_2 \cap (\mathcal{K}_1\cup -\mathcal{K}_1)=\{0\}$. We again have $u_2=v_2=0$ so that $z=0$. The proof is complete.
\end{proof}
\subsection{The projective contraction}
In this subsection we will apply a projective version of Hilbert's metric to obtain contraction results for the complexified cones $\mathcal{K}_j^{\mathbb{C}}$. We will prefer the technique of L. Dubois \cite{Du09} to that of H. H. Rugh \cite{Ru10} since Dubois' projective distance is more easily defined.
If $\mathsf{K}\subseteq \mathbb{C}^d$ is a $\mathbb{C}$-cone then for all $z_1,z_2 \in \mathsf{K}$ we define
\[E_{\mathsf{K}}(z_1,z_2):=\left\{\lambda \in \mathbb{C} \colon \lambda z_1 -z_2 \notin \mathsf{K}\right\}.\]
It is easy to see that $E_{\mathsf{K}}(\lambda z_1,z_2)=\lambda^{-1}E_{\mathsf{K}}(z_1,z_2)$ and $E_{\mathsf{K}}(z_1,\lambda z_2)=\lambda E_{\mathsf{K}}(z_1,z_2)$ for every nonzero $\lambda \in \mathbb{C}$, and that $\lambda \in E_{\mathsf{K}}(z_1,z_2)$ if and only if $\lambda^{-1} \in E_{\mathsf{K}}(z_2,z_1)$. Clearly we always have $0 \notin E_{\mathsf{K}}(z_1,z_2)$. Moreover, $\Int E_{\mathsf{K}}(z_1,z_2)$ can never be empty when $\{z_1,z_2\}$ is linearly independent: if this were possible for some $z_1,z_2$ then $\lambda z_1- z_2$ would belong to $\mathsf{K}$ for a dense set of $\lambda \in \mathbb{C}$, which since $\mathsf{K}$ is closed would imply that the span of $z_1$ and $z_2$ is contained in $\mathsf{K}$, contradicting the definition of a $\mathbb{C}$-cone. In particular $E_{\mathsf{K}}(z_1,z_2)$ itself is never empty when $z_1$ and $z_2$ are not colinear.
If $z \in \mathbb{C}^d$ is nonzero, let $\overline{z}$ denote the (complex) one-dimensional subspace spanned by $z$ (which should not be confused with the complex conjugate of $z$, denoted in this article by $z^*$). Following \cite{Du09} we define a notion of distance between two one-dimensional subspaces of a $\mathbb{C}$-cone $\mathsf{K}\subseteq \mathbb{C}^d$ by
\[d_{\mathsf{K}}(\overline{z_1},\overline{z_2}) :=\left\{\begin{array}{cl}0&\text{if }\overline{z_1}=\overline{z_2},\\
\log \left(\frac{\sup \left\{|\lambda|\colon \lambda \in E_{\mathsf{K}}(z_1,z_2)\right\}}{ \inf \left\{|\lambda|\colon \lambda \in E_{\mathsf{K}}(z_1,z_2)\right\}}\right)&\text{otherwise}.\end{array}
\right.\]
Note that $d_{\mathsf{K}}(\overline{z_1},\overline{z_2})$ may take the value $+\infty$. It is easy to see that $d_{\mathsf{K}}(\overline{z_1},\overline{z_2})$ depends only on the subspaces $\overline{z_1},\overline{z_2}$ and not on the choice of spanning vector $z_1$ or $z_2$ within those subspaces. The function $d_{\mathsf{K}}$ is easily seen to satisfy $d_{\mathsf{K}}(\overline{z_1},\overline{z_2})=d_{\mathsf{K}}(\overline{z_2},\overline{z_1})$, and $d_{\mathsf{K}}(\overline{z_1},\overline{z_2})\geq 0$ with equality if and only if $\overline{z_1}=\overline{z_2}$. (If we had $d_{\mathsf{K}}(\overline{z_1},\overline{z_2})=0$ with $\overline{z_1}\neq \overline{z_2}$ then $E_{\mathsf{K}}(z_1,z_2)$ would be a subset of a circle in $\mathbb{C}$ and hence have empty interior, which we noted previously to be impossible.) However, even where it takes finite values $d_{\mathsf{K}}$ is not in general a metric, since the triangle inequality is not satisfied for certain $\mathbb{C}$-cones $\mathsf{K}$ (see \cite[Remark 6]{Du09}). We shall nonetheless write $\diam_{\mathsf{K}} X:= \sup\{d_{\mathsf{K}}(\overline{z_1},\overline{z_2}) \colon \overline{z_1},\overline{z_2}\in X\}$ in the same manner as we would if $d_{\mathsf{K}}$ were a \emph{bona fide} metric. By abuse of notation if $X \subseteq \mathbb{C}^d$ is an arbitrary set then we shall also write $\diam_{\mathsf{K}} X$ for the diameter of the set of all one-dimensional subspaces generated by nonzero elements of $X$.
The following is a modification of part of \cite[Lemma 2.2]{Du09}:
\begin{lemma}\label{le:apert}
Let $\mathsf{K}\subseteq \mathbb{C}^d$ be a $\mathbb{C}$-cone and $w \in \mathbb{C}^d$ a unit vector such that $\langle z,w\rangle \neq 0$ for all nonzero $z \in \mathsf{K}$. Then there exists $C>0$ such that for all nonzero $z_1,z_2 \in \mathsf{K}$
\[\left\|\frac{z_1}{\langle z_1,w\rangle} - \frac{z_2}{\langle z_2,w\rangle} \right\| \leq Cd_{\mathsf{K}}(\overline{z_1},\overline{z_2}).\]
\end{lemma}
\begin{proof}
Let $z_1,z_2\in \mathsf{K}$ be nonzero. If $z_1$ and $z_2$ are colinear or if $d_{\mathsf{K}}(\overline{z_1},\overline{z_2})=+\infty$ then the result is trivial, so we assume otherwise. By homogeneity it is clearly sufficient to consider the case in which $\langle z_1,w\rangle =\langle z_2,w\rangle =1$. Choose $a \in (0,\inf E_{\mathsf{K}}(z_1,z_2))$ and $b\in (\sup E_{\mathsf{K}}(z_1,z_2),+\infty)$ which is possible since $d_{\mathsf{K}}(\overline{z_1},\overline{z_2})<+\infty$. Since $z_1-z_2\neq 0$ and $\langle z_1-z_2,w\rangle=0$ we have $z_1-z_2\notin \mathsf{K}$ so in particular $1 \in E_{\mathsf{K}}(z_1,z_2)$ and therefore $a<1<b$. Let $C_1>0$ be the constant given by Lemma \ref{le:bdd}. Since $bz_1-z_2, az_1-z_2 \in \mathsf{K}$ we may estimate
\begin{align*}\left\|z_1-z_2\right\|&=\left\|\frac{1-a}{b-a}(bz_1-z_2)-\frac{1-b}{b-a}(az_1-z_2) \right\|\\
&\leq 2C_1\frac{(1-a)(b-1)}{b-a}=2C_1\frac{b+a-1-ab}{b-a}\\
&\leq 2C_1\frac{b+a-2\sqrt{ab}}{b-a}=2C_1\frac{\sqrt{b}-\sqrt{a}}{\sqrt{b}+\sqrt{a}}\\
&=2C_1\frac{\sqrt{b/a}-1}{\sqrt{b/a}+1} =2C_1\tanh \left(\frac{\log b/a}{4}\right)< \frac{C_1}{2}\log b/a\end{align*}
using the arithmetic-geometric mean inequality and the elementary estimate $\tanh t < t$ for real $t> 0$. Since $a<\inf E_{\mathsf{K}}(z_1,z_2)$ and $b>\sup E_{\mathsf{K}}(z_1,z_2)$ were arbitrary the result follows.
\end{proof}
The following result is a special case of \cite[Theorem 2.3]{Du09}:
\begin{proposition}[\cite{Du09}]\label{pr:dubois}
Let $\mathsf{K}_1,\mathsf{K}_2 \subseteq \mathbb{C}^d$ be $\mathbb{C}$-cones, let $A\in M_d(\mathbb{C})$ and suppose that $A(\mathsf{K}_1\setminus \{0\})\subseteq \mathsf{K}_2\setminus \{0\}$ and that $\Delta:=\diam_{\mathsf{K}_2} A\mathsf{K}_1$ is finite. Then
\[d_{\mathsf{K}_2}(\overline{Az_1},\overline{Az_2}) \leq \left(\tanh \frac{\Delta}{4}\right)d_{\mathsf{K}_1}(\overline{z_1},\overline{z_2})\]
for all nonzero $z_1,z_2 \in \mathsf{K}_2$.
\end{proposition}
The key result of this subsection is the following:
\begin{proposition}\label{pr:dam}
Let $\mathsf{A}\subset M_d(\mathbb{R})$ be compact and nonempty and suppose that $(\mathcal{K}_1,\ldots,\mathcal{K}_m)$ is a multicone for $\mathsf{A}$ with transverse-defining vector $w$. Then there exist $C,\gamma>0$ such that for every $n \geq 1$
\[\left\|\frac{A_n\cdots A_1 z_1}{\langle A_n\cdots A_1 z_1,w\rangle} - \frac{A_n\cdots A_1z_2}{\langle A_n\cdots A_1z_2,w\rangle} \right\| \leq Ce^{-\gamma n}\]
for every nonzero $z_1,z_2 \in \bigcup_{j=1}^m \mathcal{K}_j^{\mathbb{C}}$ and every $A_1,\ldots,A_n \in \mathsf{A}$.
\end{proposition}
\begin{proof}
For each $A \in \mathsf{A}$ and $j=1,\ldots,m$ there exists $\ell=\ell(A,j)$ such that $A(\mathcal{K}_j\setminus \{0\}) \subseteq (\Int \mathcal{K}_{\ell})\cup(-\Int \mathcal{K}_\ell)$, and in particular by Corollary \ref{co:cks} we have $A(\mathcal{K}_j^{\mathbb{C}}\setminus\{0\}) \subset \Int \mathcal{K}_\ell^{\mathbb{C}}$ for this value of $\ell$. We claim that there exists $\Delta>0$ such that
\[\diam_{\mathcal{K}_{\ell(A,j)}^{\mathbb{C}}} A\mathcal{K}_j^{\mathbb{C}}\leq \Delta<\infty\]
independently of $A \in \mathsf{A}$ and $j \in \{1,\ldots,m\}$.
To this end fix $j,\ell \in \{1,\ldots,m\}$ and define
\[X_{j,\ell}:=\left\{Az \colon z \in \mathcal{K}_j^{\mathbb{C}},\text{ }Az \in \mathcal{K}_\ell^{\mathbb{C}}\text{, }A \in \mathsf{A}\text{ and }\langle Az,w\rangle=1\right\}.\]
We assert that $X_{j,\ell}$ is compact. If $X_{j,\ell}=\emptyset$ then this is trivial. Otherwise, let $(A_n)$ be a sequence of elements of $\mathsf{A}$ and $(z_n)$ a sequence of elements of $\mathcal{K}_j$ such that $A_nz_n \in X_{j,\ell}$ for every $n\geq 1$, and let us show that $(A_nz_n)$ has a limit point in $X_{j,\ell}$. By passing to a subsequence we may clearly assume that $(A_n)$ converges to some $A \in \mathsf{A}$, say, and it is clear that $A\mathcal{K}_j^{\mathbb{C}}\subseteq \mathcal{K}_\ell^{\mathbb{C}}$ since every $A_n$ has this property. By passing to a further subsequence we assume that $\|z_n\|^{-1}z_n$ converges to a nonzero limit $z \in \mathcal{K}_j^{\mathbb{C}}$. It is clear that $\lim_{n \to \infty} \|z_n\|^{-1}A_nz_n=Az$ and therefore $ \lim_{n\to \infty} \|z_n\|^{-1}=\langle Az,w\rangle $. If $\langle Az,w\rangle=0$ then by Lemma \ref{le:w} we have $Az=0$ contradicting Corollary \ref{co:cks}, so we have $\lim_{n\to \infty} \|z_n\|^{-1}>0$. In particular $(z_n)$ is bounded, and by passing to a subsequence we assume it to converge to a limit $z' \in \mathcal{K}_j^{\mathbb{C}}$. We note that $\lim_{n \to \infty} A_nz_n= Az'$ and consequently $\langle Az',w\rangle=\lim_{n\to \infty} \langle A_nz_n,w\rangle=1$. In particular $z' \in X_{j,\ell}$ as required to prove the assertion.
By Corollary \ref{co:cks} the compact set $X_{j,\ell}$ is a subset of $\Int \mathcal{K}_\ell^{\mathbb{C}}$. By Lemma \ref{le:bdd} there exists $C_1>0$ such that $\|z\|\leq C_1$ for every $z \in X_{j,\ell}$. Choose $r=r(j,\ell)>0$ such that for every $z \in X_{j,\ell}$ the open $r$-ball centred at $z$ is a subset of $\mathcal{K}_\ell^{\mathbb{C}}$. If $z_1,z_2 \in X_{j,\ell}$ it follows that $z_1+\lambda z_2 \in \mathcal{K}_\ell^{\mathbb{C}}$ whenever $\|\lambda z_2\|<r$, which is true in particular whenever $|\lambda| < C_1^{-1}r$. Equally if $|\lambda|>C_1r^{-1}$ then $ z_2 +\lambda^{-1}z_1 \in \mathcal{K}_\ell^{\mathbb{C}}$ and therefore $z_1+\lambda z_2 = \lambda(z_2+\lambda^{-1}z_1) \in \lambda\mathcal{K}_\ell^{\mathbb{C}}=\mathcal{K}_\ell^{\mathbb{C}}$. It follows that $E_{\mathcal{K}_\ell^{\mathbb{C}}}(z_1,z_2) \subseteq [C_1^{-1}r,C_1r^{-1}]$ and therefore $d_{\mathcal{K}_\ell^{\mathbb{C}}}(\overline{z_1},\overline{z_2})\leq \log C_1^2/r^2$. Hence $\diam_{\mathcal{K}_\ell^{\mathbb{C}}} X_{j,\ell} \leq \log C_1^2/r^2<\infty$. To prove the claim we define
\[\Delta:=\max_{1 \leq j,\ell \leq m} \diam_{\mathcal{K}_{\ell}^{\mathbb{C}}} X_{j,\ell}<\infty.\]
By Lemma \ref{le:apert} there exists $C_2>0$ such that if $j \in \{1,\ldots,m\}$ and $z_1,z_2 \in \mathcal{K}_j^{\mathbb{C}}\setminus \{0\}$ then
\[\left\|\frac{z_1}{\langle z_1,w\rangle} - \frac{z_2}{\langle z_2,w\rangle} \right\| \leq C_2d_{\mathcal{K}_j^{\mathbb{C}}}(\overline{z_1},\overline{z_2}).\]
We claim next that if $j \in \{1,\ldots,m\}$, $n \geq 1$, $A_1,\ldots,A_n \in \mathsf{A}$ and $z_1,z_2 \in \mathcal{K}_j^{\mathbb{C}}\setminus \{0\}$ then
\[\left\|\frac{A_n\cdots A_1 z_1}{\langle A_n\cdots A_1 z_1,w\rangle} - \frac{A_n\cdots A_1 z_2}{\langle A_n\cdots A_1 z_2,w\rangle} \right\| \leq C_2\Delta\left(\tanh \frac{\Delta}{4}\right)^{n-1}.\]
To see this let $A_1,\ldots,A_n \in \mathsf{A}$ and $z_1,z_2 \in \mathcal{K}_j^{\mathbb{C}}\setminus \{0\}$ and choose integers $\ell_1,\ldots,\ell_n$ such that $A_k\cdots A_1(\mathcal{K}_j\setminus \{0\})\subseteq \Int \mathcal{K}_{\ell_k}$ for each $k=1,\ldots,n$. We have $A_{k}\cdots A_{1}(\mathcal{K}_j^{\mathbb{C}}\setminus \{0\})\subseteq \Int \mathcal{K}_{\ell_k}^{\mathbb{C}}$ for each $k=1,\ldots,n$ by Corollary \ref{co:cks}. By inductive application of Proposition \ref{pr:dubois} we have for each $k=1,\ldots,n$
\begin{align*}d_{\mathcal{K}_{\ell_k}^{\mathbb{C}}}\left(\overline{A_{k}\cdots A_{1}z_1},\overline{ A_{k}\cdots A_{1}z_2}\right) &\leq \left(\tanh\frac{\Delta}{4}\right)^{k-1}d_{\mathcal{K}_{\ell_1}^{\mathbb{C}}}\left(\overline{A_{1}z_1},\overline{A_{1}z_2}\right)\\
& \leq \Delta\left(\tanh\frac{\Delta}{4}\right)^{k-1}\end{align*}
using the definition of $\Delta$, so in particular by Lemma \ref{le:apert}
\begin{align*}\left\|\frac{A_n \cdots A_1 z_1}{\langle A_n\cdots A_1 z_1,w\rangle} - \frac{A_n \cdots A_1 z_2}{\langle A_n \cdots A_1 z_2,w\rangle} \right\| &\leq C_2 d_{\mathcal{K}_{\ell_n}^{\mathbb{C}}}\left(\overline{A_n \cdots A_1 z_1},\overline{ A_n \cdots A_1 z_2}\right)\\
&\leq C_2 \Delta\left(\tanh\frac{\Delta}{4}\right)^{n-1}\end{align*}
as required to prove the claim.
We claim lastly that there exists $N\geq 1$ such that for every $A_1,\ldots,A_N \in \mathsf{A}$ there exists $\ell=\ell(A_1,\ldots,A_N)$ such that $A_N\cdots A_1\left( \bigcup_{j=1}^m \mathcal{K}_j^{\mathbb{C}} \right)\subseteq \mathcal{K}_\ell^{\mathbb{C}}$. If this is not the case then there exist a strictly increasing sequence of natural numbers $(n_k)$, a sequence of matrices $(B_k)$ of the form $B_k=A_{n_k}\cdots A_1$ for some $A_1,\ldots,A_{n_k} \in \mathsf{A}$ depending on $k$, and four sequences $(j_{1,k})$, $(j_{2,k})$, $(\ell_{1,k})$, $(\ell_{2,k})$ of integers in $\{1,\ldots,m\}$ such that $B_{k}\mathcal{K}_{j_{1,k}}^{\mathbb{C}} \subseteq \mathcal{K}_{\ell_{1,k}}^{\mathbb{C}}$ and $B_{k}\mathcal{K}_{j_{2,k}}^{\mathbb{C}} \subseteq \mathcal{K}_{\ell_{2,k}}^{\mathbb{C}}$ for each $k \geq 1$, and such that $\ell_{1,k}\neq \ell_{2,k}$ for every $k \geq 1$. By passing to a subsequence if necessary we may assume that $\|B_k\|^{-1}B_k$ converges to a limit matrix $B$ with $\|B\|=1$, that $(j_{1,n})$ and $(j_{2,n})$ take constant values $j_1$ and $j_2$, and that $(\ell_{1,n})$ and $(\ell_{2,n})$ take constant values $\ell_1$ and $\ell_2$ with $\ell_1 \neq \ell_2$. Let $z \in \mathcal{K}_{j_1}^{\mathbb{C}}$ be nonzero and write $z=e^{i\theta}((u+v)+i(u-v))$ with $u,v \in \mathcal{K}_{j_1}$ and $\theta \in \mathbb{R}$. We note that for each $k \geq 1$
\begin{align*}|\langle B_k z,w\rangle|^2 =|e^{-i\theta}\langle B_kz,w\rangle|^2 &=\langle B_k(u+v),w\rangle^2+\langle B_k(u-v),w\rangle^2\\
&=2\langle B_ku,w\rangle^2 + 2\langle B_kv,w\rangle^2\\
&\geq \tau_2\|B_k\|^2(2\|u\|^2+2\|v\|^2)\\
&= \tau_2 \|B_k\|^2 (\|u+v\|^2+\|u-v\|^2)\\
&= \tau_2 \|B_k\|^2\|e^{-i\theta} z\|^2=\tau_2 \|B_k\|^2\|z\|^2\end{align*}
using Proposition \ref{pr:bob}(ii) and the parallelogram law, and it follows in particular that the limit $\langle Bz,w\rangle = \lim_{k \to \infty}\langle \|B_k\|^{-1}B_kz,w\rangle$ cannot be zero. Hence
\[\lim_{k \to \infty} \langle B_kz,w\rangle^{-1}B_kz = \lim_{k \to \infty} \langle \|B_k\|^{-1}B_kz,w\rangle^{-1} \|B_k\|^{-1}B_kz = \langle Bz,w\rangle^{-1}Bz \in\mathcal{K}_{\ell_1}^{\mathbb{C}} \]
for every nonzero $z \in \mathcal{K}_{j_1}^{\mathbb{C}}$. Applying the previous claim we notice that the limit vector $\langle Bz,w\rangle^{-1}Bz$ does not depend on the choice of $z \in \mathcal{K}_{j_1}^{\mathbb{C}}$ and thus $B$ maps the whole $\mathbb{C}$-cone $\mathcal{K}_{j_1}^{\mathbb{C}}$ to a one-dimensional subspace of $\mathcal{K}_{\ell_1}^{\mathbb{C}}$. Similarly $B$ maps the $\mathbb{C}$-cone $\mathcal{K}_{j_2}^{\mathbb{C}}$ to a one-dimensional subspace of $\mathcal{K}_{\ell_2}^{\mathbb{C}}$. Since $\mathcal{K}_{\ell_1}^{\mathbb{C}} \cap \mathcal{K}_{\ell_2}^{\mathbb{C}} =\{0\}$ by Lemma \ref{le:disjt}, the image subspace $B\mathcal{K}_{j_1}^{\mathbb{C}}$ is distinct from the image subspace $B\mathcal{K}_{j_2}^{\mathbb{C}}$. But it is impossible for a linear map to take two sets with disjoint nonempty interior to distinct one-dimensional subspaces. This contradiction proves the existence of the desired constant $N$.
We may now prove the full statement of the proposition. Let $n \geq 1$ and $A \in \mathsf{A}_n$. If $n \leq N$ then we have
\[\left\|\frac{A z_1}{\langle A z_1,w\rangle} - \frac{A z_2}{\langle A z_2,w\rangle} \right\| \leq \sup_{z \in \bigcup_{j=1}^m \mathcal{K}_j^{\mathbb{C}}\setminus \{0\}} \frac{2\|z\|}{|\langle z,w\rangle|}\leq C_3,\]
say, by Lemma \ref{le:bdd}. Otherwise write $A=A_2A_1$ where $A_1 \in \mathsf{A}_{N}$ and $A_2 \in \mathsf{A}_{n-N}$. We have
\begin{align*}\left\|\frac{Az_1}{\langle A z_1,w\rangle} - \frac{A z_2}{\langle A z_2,w\rangle} \right\|&= \left\|\frac{A_2 A_1z_1}{\langle A_2 A_1 z_1,w\rangle} - \frac{A_2 A_1 z_2}{\langle A_2 A_1 z_2,w\rangle} \right\| \\
&\leq C_2\Delta \left(\tanh\frac{\Delta}{4}\right)^{n-N}\end{align*}
since $A_1 z_1$ and $A_1 z_2$ belong to the same cone $\mathcal{K}_\ell^{\mathbb{C}}$. Since $\tanh (\Delta/4)<1$ this suffices to complete the proof of the proposition.
\end{proof}
\subsection{Proof of Theorem \ref{th:multicones}}
(i). Let $A \in \mathcal{S}(\mathsf{A})$. For each $n \geq1$ let $\ell_n \in \{1,\ldots,m\}$ be the unique integer such that $A^n (\mathcal{K}_1 \setminus\{0\}) \subseteq \Int\mathcal{K}_{\ell_n}\cup -\Int\mathcal{K}_{\ell_n}$. By the pigeonhole principle there exist integers $n_1,n_2$ such that $n_2>n_1\geq 1$ and $\ell_{n_2}=\ell_{n_1}=\ell$, say, so we have $A^{n_2-n_1}(\mathcal{K}_{\ell}\setminus\{0\})\subseteq \Int \mathcal{K}_{\ell} \cup -\Int\mathcal{K}_{\ell}$. In particular $A^{2(n_2-n_1)}$ maps $\mathcal{K}_{\ell}\setminus \{0\}$ to its interior, so by the Perron-Frobenius Theorem we have $|\lambda_1(A^{2(n_2-n_1)})|>|\lambda_2(A^{2(n_2-n_1)})|$ and the eigenvector corresponding to this eigenvalue belongs to $\Int \mathcal{K}_{\ell}$. It follows that $|\lambda_1(A)|>|\lambda_2(A)|$ as required, and since the associated eigenvector belongs to $\Int \mathcal{K}_{\ell}$ and is in particular real, the associated eigenvalue of $\lambda_1(A)$ must be real also.
Part (ii) is given directly by Proposition \ref{pr:bob}(iii). Let us now prove (iii), let $z \in \Omega$ and $A \in \mathcal{S}(\mathsf{A})$ and write
\[z=(a+ib)((u+v)+ i(u-v)) = (a-b)u+(a+b)v+i((a+b)u-(a-b)v)\]
where $u,v \in \mathcal{K}_j$ and $a,b \in \mathbb{R}$. We have $\langle u,w\rangle \geq 0$ and $\langle v,w\rangle \geq 0$, and since for some $\ell \in \{1,\ldots,m\}$ we have either $A \mathcal{K}_j \subseteq \mathcal{K}_\ell$ or $A \mathcal{K}_j \subseteq -\mathcal{K}_\ell$, the real numbers $\langle A u,w\rangle$ and $\langle A v,w\rangle$ either are both greater than or equal to zero, or are both less than or equal to zero. Since $Az$ cannot be zero we note that $Au$ and $Av$ cannot both be zero, and consequently $\langle Au,w\rangle$ and $\langle Av,w\rangle$ cannot both be zero.
We observe that
\[\Re(\langle Az,w\rangle) = (a-b)\langle Au,w\rangle + (a+b)\langle Av,w\rangle\]
and the equation $\langle z,w\rangle=1$ implies
\[(a-b)\langle u,w\rangle + (a+b)\langle v,w\rangle = 1, \qquad (a+b)\langle u,w\rangle - (a-b)\langle v,w\rangle = 0 \]
by separating real and imaginary parts.
If $v=0$ then we have $(a-b)\langle u,w\rangle=1$ and it follows that $\Re(\langle Az,w\rangle) = (a-b)\langle Au,w\rangle \neq 0$ since otherwise $\langle Au,w\rangle$ and $\langle Av,w\rangle$ would both be zero. If $v \neq 0$ but $a-b=0$ then it follows that $Av \neq 0$ and $(a+b)\langle v,w\rangle = 1$, so $\langle Av,w\rangle \neq 0$ and $a+b \neq 0$ and we obtain $\Re(\langle Az,w\rangle) = (a+b)\langle Av,w\rangle \neq 0$ as desired. If $v$ and $a-b$ are both nonzero the equation $(a+b)\langle u,w\rangle - (a-b)\langle v,w\rangle = 0$ implies that $a+b$, $\langle u,w\rangle \neq 0$. Since necessarily $\langle u,w\rangle, \langle v,w\rangle>0$ it follows that the real numbers $a+b$ and $a-b$ have the same sign, and since $\langle Au,w\rangle $ and $\langle Av,w\rangle$ agree with one another in sign we deduce that $\Re(\langle Az,w\rangle) = (a-b)\langle Au,w\rangle + (a+b)\langle Av,w\rangle$ is the sum of two nonzero real numbers with matching signs, hence nonzero. In all cases we find that $\Re(\langle Az,w\rangle)\neq 0$ and this completes the proof of (iii).
To prove (iv), we first note that by hypothesis each of the real cones $\mathcal{K}_j$ has nonempty interior as a subset of $\mathbb{R}^d$. If $u \in \Int \mathcal{K}_j$ then $\langle u,v\rangle>0$ for every $v \in \mathcal{K}'_j$ by Lemma \ref{le:intreal}, so in particular $\Re(\langle u,\ell\rangle\langle u,m\rangle^*)>0$ for every $\ell,m \in \mathcal{K}_j'$ and hence by Lemma \ref{le:intcomp} $u$ belongs to the interior of the complexification $\mathcal{K}_j^{\mathbb{C}}$. In particular $\Int \mathcal{K}_j^{\mathbb{C}}$ is nonempty for each $j$ and it follows by homogeneity that $\Omega$ is nonempty. It is clear that $\Omega$ is open as a subset of the hyperplane $\{z \in \mathbb{C}^d \colon \langle z,w\rangle=1\}$. If $z \in \Omega$ and $A \in \mathcal{S}(A)$ then $z \in \Int \mathcal{K}_j^{\mathbb{C}}$ for some $j \in \{1,\ldots,m\}$ and it follows easily via Corollary \ref{co:cks} that $Az \in \Int \mathcal{K}_\ell^{\mathbb{C}}$ for some $\ell \in \{1,\ldots,m\}$. By (iii) we have $\langle Az,w\rangle \neq 0$ and therefore $\overline{A}z$ is well-defined, and it is clear that $\overline{A}z \in \Omega$. In particular $\overline{A}\colon \Omega \to \Omega$ is well-defined and is clearly holomorphic for every $A \in \mathcal{S}(\mathsf{A})$. To see that $\Omega$ is bounded we note that each of the sets
\[\left\{z \in \mathbb{C}^d \colon z \in \mathcal{K}_j^{\mathbb{C}}\text{ and }\langle z,w\rangle=1\right\}\]
is bounded by Lemma \ref{le:bdd}. This completes the proof of (iv).
Part (v) follows directly from Proposition \ref{pr:dam} together with the definition of $\Omega$. Let us now prove (vi). Fix $A \in \mathcal{S}(\mathsf{A})$. We showed when proving (i) that there exists an eigenvector $v_A \in \bigcup_{j=1}^m \Int \mathcal{K}_j$ such that $A v_A =\lambda_1(A)v_A$. In previous arguments we have seen that $\bigcup_{j=1}^m \Int \mathcal{K}_j\subseteq \bigcup_{j=1}^m \Int \mathcal{K}_j^{\mathbb{C}}$ and it follows that $z_A:=\langle v_A,w\rangle^{-1}v_A$ belongs to $\Omega$, and indeed to $\Omega\cap \mathbb{R}^d$. It is clear that $\overline{A}z_A = z_A$ and that $\langle A z_A,w\rangle = \lambda_1(A)$. It follows from (v) that $\lim_{n \to \infty} \diam \overline{A^n}(\Omega)=0$ and this implies that $\overline{A}$ cannot have a second fixed point in $\Omega$ which is distinct from $z_A$.
Let us now calculate the eigenvalues of the derivative $D_{z_A}\overline{A}$. Let $u_1,\ldots,u_d \in \mathbb{C}^d$ be a Jordan basis for $A$ with basis elements listed in descending order of the absolute value of the corresponding eigenvalue, and with $u_1=z_A$. Since $|\lambda_1(A)|>|\lambda_2(A)|$ we have $A u_1=\lambda_1(A)u_1$ and $A u_2=\lambda_2(A)u_2$. For each $j \in \{3,\ldots,d\}$, let $\delta_j \in \{0,1\}$ such that $A u_j=\lambda_j(A )u_j+\delta_ju_{j-1}$.
For every $v$ in the tangent space $\{v \in \mathbb{C}^d \colon \langle v,w\rangle=0\}$ to $\Omega$ at $z_A$ we have
\begin{align*}\left(D_{z_A}\overline{A}\right) v&:=\lim_{\varepsilon \to 0}\frac{1}{\varepsilon}\left( \overline{A}(u_1+\varepsilon v) -\overline{A}u_1\right)\\
&= \lim_{\varepsilon \to 0} \frac{1}{\varepsilon}\left(\frac{A(u_1+\varepsilon v)}{\langle A(u_1 + \varepsilon v),w\rangle } - \frac{Au_1}{\langle A u_1,w\rangle } \right)\\
&=\lim_{\varepsilon \to 0} \frac{1}{\varepsilon}\left(\frac{\langle A u_1,w\rangle\cdot A(u_1+\varepsilon v)- \langle A (u_1+\varepsilon v),w\rangle\cdot A u_1 }{\langle A(u_1 + \varepsilon v) ,w\rangle \langle A u_1,w\rangle } \right)\\
&=\lim_{\varepsilon \to 0} \frac{\langle A u_1,w\rangle\cdot A v- \langle A v,w\rangle\cdot A u_1 }{\langle A(u_1 + \varepsilon v) ,w\rangle \langle A u_1,w\rangle } \\
&=\frac{\langle A u_1,w\rangle\cdot A v- \langle A v,w\rangle\cdot A u_1 }{\langle Au_1 ,w\rangle \langle A u_1,w\rangle } \\
&=\frac{1}{\lambda_1(A)} \left(Av - \langle A v,w\rangle u_1 \right).\end{align*}
Clearly the vectors $v_j:=u_j-\langle u_j,w\rangle u_1$, where $j$ runs from $2$ to $d$, form a basis of the tangent space $\{z \in \mathbb{C}^d \colon \langle z,w\rangle=0\}$. We have
\begin{align*}\left(D_{z_A}\overline{A}\right)v_2
&=\frac{1}{\lambda_1(A)} \left(Av_2 - \langle Av_2,w \rangle u_1\right)\\
&=\frac{1}{\lambda_1(A)} \left(\lambda_2(A)u_2 - \lambda_1(A)\langle u_2,w \rangle u_1 - \lambda_2(A)\langle u_2,w \rangle u_1 + \lambda_1(A)\langle u_2,w \rangle u_1\right)\\
&=\frac{1}{\lambda_1(A)} \left(\lambda_2(A)u_2 - \lambda_2(A)\langle u_2,w \rangle u_1\right)\\
&=\frac{\lambda_2(A)}{\lambda_1(A)} v_2,\end{align*}
and for $j=3,\ldots,d$ we similarly have
\begin{align*}\left(D_{z_A}\overline{A}\right) v_j
=&\frac{1}{\lambda_1(A)} \left(Av_j - \langle Av_j,w \rangle u_1\right)\\
=& \frac{1}{\lambda_1(A)} \big(\lambda_j(A)u_j + \delta_ju_{j-1} - \lambda_1(A)\langle u_j,w\rangle u_1\\
& - \lambda_j(A)\langle u_j,w\rangle u_1 -\delta_j\langle u_{j-1},w \rangle u_1 + \lambda_1(A)\langle u_j,w\rangle u_1\big)\\
=& \frac{1}{\lambda_1(A)} \left(\lambda_j(A)u_j -\lambda_j(A)\langle u_j,w\rangle u_1 + \delta_ju_{j-1}- \delta_j\langle u_{j-1},w \rangle u_1\right)\\
=&\frac{\lambda_j(A)}{\lambda_1(A)} v_j + \frac{\delta_j}{\lambda_1(A)} v_{j-1}.\end{align*}
It follows that with respect to the basis $v_2,\ldots,v_d$ the matrix of $D_{z_A}\overline{A}$ is upper triangular with the values $\lambda_j(A)/\lambda_1(A)$ along the diagonal. In particular its eigenvalues are precisely the numbers $\lambda_j(A)/\lambda_1(A)$ for $j=2,\ldots,d$ as claimed. Since $p_A(x)=\det(xI-A)=\prod_{j=1}^d (x-\lambda_j(A))$ we have
\[p_A'(x)=\sum_{\ell=1}^d \prod_{\substack{1\leq j \leq d \\ j \neq \ell}} (x-\lambda_j(A))\]
and therefore
\[\frac{p_A'(\lambda_1(A))}{\lambda_1(A)^{d-1}} = \frac{\prod_{j=2}^d (\lambda_1(A)-\lambda_j(A))}{\lambda_1(A)^{d-1}} =\prod_{j=2}^d \left(1-\frac{\lambda_j(A)}{\lambda_1(A)}\right)=\det(I-D_{z_A}\overline{A}).\]
Since $1-\lambda_j(A)/\lambda_1(A)$ is nonzero for all $j=2,\ldots,d$ this quantity is nonzero. This completes the proof of (vi).
In order to prove (vii) we first claim that there exists $\tau>0$ such that $\|A z\|\geq \tau \|A\|\cdot\|z\|$ for all $z \in \bigcup_{j=1}^m \mathcal{K}_j^{\mathbb{C}}$ and $A \in \mathcal{S}(\mathsf{A})$. To this end let $z \in \bigcup_{j=1}^m \mathcal{K}_j^{\mathbb{C}}$ and $A \in \mathcal{S}(\mathsf{A})$ and write $z=e^{i\theta}((u+v)+i(u-v))$ where $u,v \in \mathcal{K}_j$, say. By Proposition \ref{pr:bob}(i) there exists $\tau>0$ not depending on $A$ or $z$ such that $\|Au\|\geq \tau \|A\|\cdot\|u\|$ and $\|A v\|\geq \tau\|A\|\cdot\|v\|$. We therefore have
\begin{align*}\|A z\|^2 &=\|e^{i\theta} A ((u+v)+i(u-v))\|^2\\
& =\|A(u+v)\|^2+\|A(u-v)\|^2\\
& = 2\|A u\|^2+2\|A v\|^2 \\
&\geq 2\tau^2 \|A\|^2\left(\|u\|^2+\|v\|^2\right)\\
&= \tau^2 \|A\|^2\left(\|u+v\|^2+\|u-v\|^2\right)=\tau^2\|A\|^2\|z\|^2\end{align*}
using the parallelogram law, which proves the claim. Since each $\mathcal{K}_j^{\mathbb{C}}$ is a $\mathbb{C}$-cone and $\langle z,w\rangle \neq 0$ for all nonzero $z \in \mathcal{K}_j^{\mathbb{C}}$, by Lemma \ref{le:bdd} there exists $\delta>0$ such that $|\langle z,w\rangle| \geq \delta \| z\|$ for every $z \in \bigcup_{j=1}^m \mathcal{K}_j^{\mathbb{C}}$. In particular
\[|\langle A z,w\rangle| \geq \delta \|A z\| \geq \delta \tau \|A\|\cdot\|z\| \geq \delta \tau \|A\|\cdot|\langle z,w\rangle|=\delta \tau \|A\|\]
for every $z \in \Omega$ and $A \in \mathcal{S}(\mathsf{A})$ as required. The opposing inequality $|\langle Az,w\rangle|\leq C\|A\|$ follows by the Cauchy-Schwarz inequality and the boundedness of $\Omega$. This completes the proof of (vii).
To prove (viii) we note that $\bigcup_{A \in \mathcal{S}(\mathsf{A})}\overline{A}(\Omega) \subseteq \bigcup_{A \in \mathsf{A}}\overline{A}(\Omega)$ since for each $A \in \mathcal{S}(\mathsf{A})$ we may write $\overline{A}=\overline{A_1}\circ \cdots\circ \overline{A_n}$ for some $A_1,\ldots,A_n \in \mathsf{A}$ and therefore $\overline{A}(\Omega)\subseteq \overline{A_1}(\Omega)$. It is therefore sufficient to show that the set $\bigcup_{A \in \mathsf{A}}\overline{A}(\Omega)$ is compactly contained in $\Omega$. By (iv) this set is a subset of $\Omega$, so it is sufficient to show that its closure in $\mathbb{C}^d$ -- which is a closed, bounded subset of $\mathbb{C}^d$, and therefore compact -- is a subset of $\Omega$.
Suppose that $z_0$ belongs to this closure. We may write $z_0 = \lim_{n \to \infty} \langle A_n z_n,w\rangle^{-1} A_n z_n$ for some sequence $(z_n)$ of elements of $\Omega$ and some sequence $(A_n)$ of elements of $\mathsf{A}$. By passing to subsequences if necessary we may assume that $(A_n)$ converges to $A \in \mathsf{A}$, that each $z_n$ belongs to $\Int \mathcal{K}_j^{\mathbb{C}}$ for a particular constant value of $j \in \{1,\ldots,m\}$, and that $(z_n)$ converges to $z \in \mathcal{K}_j^{\mathbb{C}}$, since $\mathsf{A}$ is compact by hypothesis and $\Omega$ is bounded by (iv). It follows that $\lim_{n \to \infty} A_nz_n=Az$ and therefore $\langle A_nz_n,w\rangle = \langle Az,w\rangle$. By (vii) we have $C^{-1}\|A_n\| \leq |\langle A_nz_n,w\rangle|\leq C\|A_n\|$ for all $n \geq 1$ and therefore $\langle Az,w\rangle \neq 0$.
Hence $\lim_{n \to \infty} \langle A_n z_n,w\rangle^{-1} A_n z_n = \langle Az,w\rangle^{-1}Az$.
Clearly there exists $\ell \in \{1,\ldots,m\}$ such that $Az \in \mathcal{K}_\ell^{\mathbb{C}}$. By Corollary \ref{co:cks} we have $A(\mathcal{K}_j^{\mathbb{C}}\setminus \{0\})\subseteq \Int \mathcal{K}_\ell^{\mathbb{C}}$ and therefore $z_0=\langle Az,w\rangle^{-1}Az \in \Int \mathcal{K}_\ell^{\mathbb{C}}$. Since $\langle z_0,w\rangle=1$ we find that $z_0 \in \Omega$ as required. This completes the proof of (viii) and hence of the theorem.
\section{Operator-theoretic preliminaries}\label{se:prelims}
In this section we collect some preliminary results which will underpin the construction of the operators $\mathscr{L}_s$ defined in Theorem \ref{th:opter}.
\subsection{Bergman spaces}
If $\Omega \subset \mathbb{C}^k$ is open and nonempty we define the Bergman space $\mathcal{A}^2(\Omega)$ to be the set of all holomorphic functions $f\colon \Omega \to \mathbb{C}$ such that the integral $\int_\Omega |f(z)|^2dV(z)$ is finite, where $V$ denotes $2k$-dimensional Lebesgue measure on $\mathbb{C}^k\simeq \mathbb{R}^{2k}$. The space $\mathcal{A}^2(\Omega)$ is a Hilbert space when equipped with the inner product $\langle f,g\rangle_{\mathcal{A}^2(\Omega)}:=\int_\Omega f(z)g(z)^*dV(z)$. In particular it is a closed subspace of the Hilbert space $L^2(\Omega)$ and is therefore separable. We note the following elementary estimate:
\begin{lemma}\label{le:supbound}
Let $\Omega \subseteq \mathbb{C}^k$ be a nonempty open set and let $K \subseteq \Omega$ be compact. Then there exists $C_K>0$ depending on $K$ such that $\sup_{z \in K}|f(z)| \leq C_K\|f\|_{\mathcal{A}^2(\Omega)}$ for every $f \in \mathcal{A}^2(\Omega)$.\end{lemma}
\begin{proof}
Choose $\varepsilon>0$ small enough that for every $z \in K$ the open ball $B_\varepsilon(z_0)$ is a subset of $\Omega$. By harmonicity we have
\begin{align*}|f(z_0)|^2 &=\left|\frac{1}{V(B_\varepsilon(z_0))} \int_{B_\varepsilon(z_0)} f(z)^2dV(z)\right|\\
&\leq \frac{1}{V(B_\varepsilon(z_0))}\int_{\Omega} |f(z)|^2dV(z)\\
&=\frac{1}{V(B_\varepsilon(z_0))} \|f\|_{\mathcal{A}(\Omega)}^2 = \frac{k!}{\pi^k\varepsilon^k} \|f\|_{\mathcal{A}(\Omega)}^2 \end{align*}
for all $f \in \mathcal{A}^2(\Omega)$ and $z_0 \in K$.
\end{proof}
We observe in particular that for every $z \in \Omega$ the evaluation map $f \mapsto f(z)$ is a continuous linear functional $\mathcal{A}^2(\Omega)\to \mathbb{C}$.
In practice we will be interested in the case where $\Omega$ is a bounded open subset of an affine subspace of $\mathbb{C}^d$ rather than of $\mathbb{C}^d$ itself. Clearly the results of this section will apply equally well in that context with $k$ being equal to the dimension of the affine subspace of $\mathbb{C}^d$ of which $\Omega$ is an open subset.
\subsection{Trace-class operators}
We define the \emph{singular values} or \emph{approximation numbers} $\mathfrak{s}_n(L)$ of a bounded linear operator $L \colon H\to H$ acting on a separable complex Hilbert space $H$ to be the quantities
\[\mathfrak{s}_n(L):=\inf\left\{\|L-F\|\colon F\colon H \to H \text{ is bounded with rank at most }n-1\right\},\]
where $n$ ranges over the positive integers. If $L$ is compact then the values $\mathfrak{s}_n(L)^2$ coincide with the sequence of eigenvalues of the positive self-adjoint operator $L^*L$ (see e.g. \cite[Theorem IV.2.5]{GoGoKr00}). If $L$ satisfies $\sum_{n=1}^\infty \mathfrak{s}_n(L)<\infty$ then $L$ is called \emph{trace-class}. Any trace-class operator is obviously the limit in the operator norm of a sequence of finite-rank operators and in particular is compact. It follows easily from the definition of $\mathfrak{s}_n$ that if $L_1$ and $L_2$ are bounded then $\mathfrak{s}_n(L_1L_2)$ and $\mathfrak{s}_n(L_2L_1)$ are both bounded by $\|L_1\|\mathfrak{s}_n(L_2)$ for every $n \geq 1$, and in particular the composition of a trace-class operator with a bounded operator is trace-class. In particular every power of a trace-class operator is trace-class.
The fundamental properties of the trace are summarised in the following result which combines several statements from \cite[\S3]{Si79}:
\begin{theorem}[Lidskii's theorem]\label{th:lid}
Let $L$ be a trace-class operator acting on a complex separable Hilbert space $H$ and let $(\lambda_n)_{n=1}^M$ be a complete enumeration of the nonzero eigenvalues of $L$, listed with repetition according to algebraic multiplicity, where $M \in \mathbb{N} \cup \{0,+\infty\}$. Then for every orthonormal basis $(e_n)_{n=1}^\infty$ of $H$ we have
\begin{equation}\label{eq:lid}\sum_{n=1}^\infty \langle Le_n,e_n\rangle = \sum_{n=1}^M \lambda_n\end{equation}
with both series being absolutely convergent.
The common value of \eqref{eq:lid} is defined to be the \emph{trace} of $L$ and is denoted $\tr L$.
\end{theorem}
It is clear from the definition that $\mathfrak{s}_{2n-1}(L_1+L_2)\leq \mathfrak{s}_n(L_1)+\mathfrak{s}_n(L_2)$ for every pair of bounded linear operators $L_1,L_2 \colon H \to H$ and every $n \geq 1$. It follows easily that if $L_1,\ldots,L_k$ are trace-class operators on $H$ then any finite linear combination $\sum_{i=1}^k a_i L_i$ is also trace-class and satisfies
\[\tr \sum_{i=1}^k a_i L_i = \sum_{i=1}^k a_i \tr L_i\]
as a consequence of \eqref{eq:lid}.
The following result also combines several statements from \cite[\S3]{Si79}, with the exception of the determinant formula for $a_n$ which may be found instead in, for example, \cite[Theorem 6.8]{Si77} or \cite[Theorem IV.5.2]{GoGoKr00}.
\begin{theorem}\label{th:ofundity}
Let $L$ be a trace-class operator on a separable complex Hilbert space $H$ and let $(\lambda_n)_{n=1}^\infty$ be an enumeration of the nonzero eigenvalues of $L$, repeated according to algebraic multiplicity. (If only $M<\infty$ nonzero eigenvalues exist then we define $\lambda_n:=0$ for all $n>M$.) For every $n \geq 1$ define
\[a_n:=(-1)^n \sum_{i_1<\cdots<i_n} \lambda_{i_1}(L)\cdots \lambda_{i_n}(L)\]
and define also $a_0:=1$. Then the function
\[\det(I-zL):=\sum_{n=0}^\infty a_nz^n\]
is well-defined and entire, and is equal to the absolutely convergent infinite product $\prod_{n=1}^\infty (1-z\lambda_n)$. The zeros of $z \mapsto \det(I-zL)$ are precisely the reciprocals of the nonzero eigenvalues of $L$ and the order of each zero is equal to the algebraic multiplicity of the corresponding eigenvalue. The coefficients $a_n$ satisfy
\[a_n=\frac{(-1)^n}{n!}\det\begin{pmatrix}
t_1(s)& n -1& 0&\cdots &0 &0\\
t_2(s)&t_1(s)&n-2 &\cdots &0 &0\\
t_3(s)&t_2(s)&t_1(s) &\ddots &0 &0\\
\vdots & \vdots & \vdots & \ddots &\ddots& \vdots\\
t_{n-1}(s) &t_{n-2}(s)&t_{n-3}(s)&\cdots &t_1(s) &1\\
t_n(s) &t_{n-1}(s)&t_{n-2}(s)&\cdots &t_2(s) &t_1(s)
\end{pmatrix},\]
and
\[\left|a_n\right| \leq \sum_{i_1<\dots<i_n}\mathfrak{s}_{i_1}(L)\cdots \mathfrak{s}_{i_n}(L)\]
for all $n\geq 1$.
\end{theorem}
\subsection{Weighted composition operators on Bergman spaces}
It has long been known that composition operators on Bergman spaces, and on other Banach spaces of holomorphic functions, are trace-class under mild conditions (see e.g. \cite{Gr55}). Historically most results in this context have assumed the set $\Omega \subset \mathbb{C}^k$ to be bounded and connected, but in this article we will need to work with sets having multiple connected components. The following is a special case of \cite[Theorem 5.9]{BaJe08a}.
\begin{theorem}\label{th:banjen}
Let $\Omega \subseteq \mathbb{C}^k$ be a nonempty open set, and let $\Omega_0\Subset \Omega$ be nonempty and open. Suppose that $\phi_1,\ldots,\phi_m \colon \Omega \to \Omega_0$ are holomorphic and $\psi_1,\ldots,\psi_m \colon \Omega \to \mathbb{C}$ are holomorphic and bounded. Then the operator $\mathscr{L} \colon \mathcal{A}^2(\Omega) \to \mathcal{A}^2(\Omega)$ given by
\[\left(\mathscr{L}f\right)(z):=\sum_{j=1}^m \psi_j(z) f(\phi_j(z))\]
is a well-defined bounded linear operator on $\mathcal{A}^2(\Omega)$, and there exist $C,\gamma>0$ depending only on $\Omega$ and $\Omega_0$ such that
\[\mathfrak{s}_n(\mathscr{L}) \leq C\left(\sum_{j=1}^m \sup_{z \in \Omega} |\psi_j(z)|\right)\exp\left(-\gamma n^{\frac{1}{k}}\right)\]
for every $n \geq 1$. In particular $\mathscr{L}$ is trace class.
\end{theorem}
In this article we will need to calculate explicitly the traces of a family of operators. The following result is a minor variation on a type of result appearing in work of D. Ruelle (\cite[Lemma 1]{Ru76}), D. Mayer (\cite[\S{III}]{Ma80} and remark following \cite[Corollary 7.11]{Ma91}), D. Fried (\cite[Lemma 5]{Fr86}) and other authors. The result may be proved easily by following the second, third and fourth paragraphs of the proof of \cite[Theorem 4.2]{BaJe08b}.
\begin{theorem}\label{pr:banjo}
Let $\Omega \subset \mathbb{C}^k$ be a bounded, connected, nonempty open set and suppose that $\phi \colon \Omega \to \Omega$ is a holomorphic function such that $\phi(\Omega)\Subset \Omega$. Let $\psi \colon \Omega \to \mathbb{C}$ be holomorphic and bounded. Then $\phi$ has a unique fixed point $z_0 \in \Omega$, the eigenvalues of the derivative $D_{z_0}\phi$ are all strictly less than $1$ in modulus, and the operator $\mathcal{L} \colon \mathcal{A}^2(\Omega) \to \mathcal{A}^2(\Omega)$ defined by $(\mathcal{L}f)(z):=\psi(z)f(\phi(z))$ is trace-class and has trace equal to $\psi(z_0)/ \det (I-D_{z_0}\phi)$.
\end{theorem}
Since we will in general need to study operators on Bergman spaces $\mathcal{A}^2(\Omega)$ for which $\Omega$ is not connected, we prove the following extension of Theorem \ref{pr:banjo} which does not seem to have been previously stated elsewhere:
\begin{theorem}\label{t:thrace}
Let $\Omega \subseteq \mathbb{C}^k$ be a bounded nonempty open set and suppose that $\phi \colon \Omega \to \Omega$ is a holomorphic function such that $\phi(\Omega)\Subset\Omega$. Let $\psi \colon \Omega \to \mathbb{C}$ be holomorphic and bounded. Then the set of fixed points $\Fix \phi := \{z \in \Omega \colon \phi(z)=z\}$ is either finite or empty, and each connected component of $\Omega$ contains at most one fixed point of $\phi$. At each fixed point $z \in \Fix \phi$ the eigenvalues of the derivative $D_z\phi$ are all strictly less than $1$ in modulus. The operator $\mathcal{L} \colon \mathcal{A}^2(\Omega) \to \mathcal{A}^2(\Omega)$ defined by $(\mathcal{L}f)(z):=\psi(z)f(\phi(z))$ is trace-class and satisfies
\begin{equation}\label{eq:slart}\tr \mathcal{L} = \sum_{z \in \Fix \phi} \frac{\psi(z)}{ \det (I-D_{z}\phi)}.\end{equation}
Additionally, if $\Omega$ is connected then $\Fix \phi$ is a singleton.
\end{theorem}
\begin{proof}
The case in which $\Omega$ is connected is precisely Theorem \ref{pr:banjo}, so we assume throughout that $\Omega$ is disconnected.
The operator $\mathcal{L}$ meets the hypotheses of Theorem \ref{th:banjen} with $m=1$ and hence in particular is trace-class. Note that since each connected component of $\Omega$ has positive Lebesgue measure, and their union $\Omega$, being bounded, has finite measure, $\Omega$ has at most countably many connected components. Let $M \in \mathbb{N}\cup\{\infty\}$ be the number of connected components of $\Omega$ and enumerate those components as $(\Omega_m)_{m=1}^M$.
For each integer $m$ such that $1 \leq m\leq M$ let $(f_{m,n})_{n=1}^\infty$ be an orthonormal basis for $\mathcal{A}^2(\Omega_m)$. Extend each $f_{m,n}$ to a function $\tilde{f}_{m,n}\colon \Omega \to \mathbb{C}$ by defining $\tilde{f}_{m,n}(z):=f_{m,n}(z)$ when $z\in \Omega_m$ and $\tilde{f}_{m,n}(z):=0$ otherwise. Clearly $(\tilde{f}_{m,n})$ is an orthonormal basis for $\mathcal{A}^2(\Omega)$, so by Theorem \ref{th:lid} we have
\[\tr \mathcal{L} = \sum_{m=1}^M\sum_{n=1}^\infty \langle \mathcal{L}\tilde{f}_{m,n},\tilde{f}_{m,n}\rangle_{\mathcal{A}^2(\Omega)}=\sum_{m=1}^M \sum_{n=1}^\infty \int_{\Omega} \psi(z)\tilde{f}_{m,n}(\phi(z))\tilde{f}_{m,n}(z)^*dV(z)\]
and this series is absolutely convergent.
Fix an integer $m$ such that $1 \leq m \leq M$ and consider the series
\begin{equation}\label{eq:buns}\sum_{n=1}^\infty \int_{\Omega} \psi(z)\tilde{f}_{m,n}(\phi(z))\tilde{f}_{m,n}(z)^*dV(z).\end{equation}
Since $\Omega_m$ is connected, either $\phi(\Omega_m)$ is a subset of $\Omega_m$ or it does not intersect $\Omega_m$. If the latter holds then by definition
$\tilde{f}_{m,n}(\phi(z))$ is zero for all $z \in \Omega_m$ and $n\geq 1$, and $\tilde{f}_{m,n}(z)^*$ is zero for all $z \in \Omega\setminus \Omega_m$ and $n \geq 1$. It follows that in the case where $\phi(\Omega_m) \not\subseteq \Omega_m$ all terms in the series \eqref{eq:buns} vanish and the series evaluates to zero.
In the case where $\phi(\Omega_m) \subseteq \Omega_m$, consider the operator $\mathcal{L}_m \colon \mathcal{A}^2(\Omega_m) \to \mathcal{A}^2(\Omega_m)$ defined by $(\mathcal{L}_mf)(z):=\psi(z)f(\phi(z))$. By Theorem \ref{pr:banjo} there is a unique fixed point $z_m$ of $\phi$ in $\Omega_m$, the operator $\mathcal{L}_m$ is trace-class, the eigenvalues of $D_{z_m}\phi$ are all less than one in absolute value, and we have
\[\tr \mathcal{L}_m = \sum_{n=1}^\infty \langle \mathcal{L}_mf_{m,n},f_{m,n}\rangle_{\mathcal{A}^2(\Omega_m)} = \frac{\psi(z_m)}{\det(I-D_{z_m}\phi)}\]
with this series being absolutely convergent. Since clearly
\begin{align*}\sum_{n=1}^\infty \int_{\Omega} \psi(z)\tilde{f}_{m,n}(\phi(z))\tilde{f}_{m,n}(z)^*dV(z)&=\sum_{n=1}^\infty \int_{\Omega_m} \psi(z)f_{m,n}(\phi(z))f_{m,n}(z)^*dV(z) \\
&=\sum_{n=1}^\infty \langle \mathcal{L}_mf_{m,n},f_{m,n}\rangle_{\mathcal{A}^2(\Omega_m)}\\
& = \frac{\psi(z_m)}{\det(I-D_{z_m}\phi)}\end{align*}
by the definition of $\tilde{f}_{m,n}$, we conclude that the series \eqref{eq:buns} evaluates to $\psi(z_m)/\det(I-D_{z_m}\phi)$. In each of the two cases we have obtained
\[\sum_{n=1}^\infty \int_{\Omega} \psi(z)\tilde{f}_{m,n}(\phi(z))\tilde{f}_{m,n}(z)^*dV(z)=\sum_{z \in \Omega_m \cap \Fix \phi} \frac{\psi(z)}{\det(I-D_z\phi)}\]
and we have also shown that each $\Omega_m$ can contain at most one fixed point, and that the derivative of $\phi$ at each fixed point has all eigenvalues less than one in absolute value. We deduce
\begin{align*}\tr \mathcal{L} &= \sum_{m=1}^M \sum_{n=1}^\infty \int_{\Omega} \psi(z)\tilde{f}_{m,n}(\phi(z))\tilde{f}_{m,n}(z)^*dV(z)\\
&=\sum_{m=1}^M\sum_{z \in \Omega_m \cap \Fix \phi} \frac{\psi(z)}{\det(I-D_z\phi)}\\
&=\sum_{z \in \Fix \phi} \frac{\psi(z)}{\det(I-D_z\phi)}\end{align*}
as required.
It remains only to show that $\Fix \phi$ is finite. If it is infinite then since $\Omega_m \cap \Fix \phi$ can contain at most one point for each $m$, there must exist an infinite sequence $(z_\ell)_{\ell=1}^\infty$ such that $\phi(z_\ell)=z_\ell$ for each $\ell$ and such that each $z_\ell$ belongs to a distinct component $\Omega_m$. In particular $(z_\ell)$ cannot have any accumulation points in $\Omega$ since if this were the case $(z_\ell)$ would have to return infinitely many times to the connected component containing the accumulation point; but $(z_\ell)$ necessarily has accumulation points in $\Omega$ because all its values belong to the compact set $\overline{\phi(\Omega)}\subset \Omega$. This contradiction shows that $\Fix \phi$ must be finite as claimed.
\end{proof}
\subsection{Operator Perron-Frobenius theory}
The last general functional-analytic result which we will require is the following:
\begin{theorem}[Kre{\u\i}n-Rutman]\label{th:kr}
Let $\mathcal{X}$ be a real Banach space and $\mathcal{C} \subseteq \mathcal{X}$ a subset such that:
\begin{enumerate}[(i)]
\item
$\mathcal{C}$ is closed and convex and satisfies $\lambda\mathcal{C}=\mathcal{C}$ for all real $\lambda>0$,
\item
$\mathcal{C} \cap -\mathcal{C}=\{0\}$,
\item
The span of $\mathcal{C}$ is dense in $\mathcal{X}$.
\end{enumerate}
If $\mathcal{L} \colon \mathcal{X} \to \mathcal{X}$ is a compact linear operator such that $\mathcal{L}\mathcal{C}\subseteq \mathcal{C}$ and $\rho(\mathcal{L})\neq 0$, then there exists nonzero $x \in \mathcal{C}$ such that $\mathcal{L}x=\rho(\mathcal{L})x$.
\end{theorem}
A proof of Theorem \ref{th:kr} may be found in \cite[Theorem 19.2]{De85}, or in a somewhat different form \cite[p.313]{Sc99}.
\section{Proof of Theorem \ref{th:opter}}\label{se:operators}
The following result, which will be proved in this section, clearly implies Theorem \ref{th:opter}:
\begin{theorem}\label{th:topaff}
Let $\mathsf{A}=(A_1,\ldots,A_N) \in M_d(\mathbb{R})^N$ and $k \in \{0,\ldots,d-1\}$. Suppose that $(A_1^{\wedge k},\ldots,A_N^{\wedge k})$ strictly preserves a multicone $( \mathcal{K}_{1}^{(1)},\ldots,\mathcal{K}_{m_1}^{(1)})$ in $\wedge^k \mathbb{R}^d$ with transverse-defining vector $w_1 \in \wedge^{k}\mathbb{R}^d$ and that $(A_1^{\wedge (k+1)},\ldots,A_N^{\wedge (k+1)})$ strictly preserves a multicone $( \mathcal{K}_{1}^{(2)},\ldots,\mathcal{K}_{m_2}^{(2)})$ in $ \wedge^{k+1}\mathbb{R}^d$ with transverse-defining vector $w_2 \in \wedge^{k+1}\mathbb{R}^d$. Define
\[\Omega_1:=\left\{z \in \wedge^k \mathbb{C}^d \colon z \in \bigcup_{j=1}^{m_1} \left(\Int \mathcal{K}_j^{(1)}\right)^{\mathbb{C}} \text{ and }\langle z,w_1\rangle=1\right\},\]
\[\Omega_2:=\left\{z \in \wedge^{k+1} \mathbb{C}^d \colon z \in \bigcup_{j=1}^{m_2} \left(\Int \mathcal{K}_j^{(2)}\right)^{\mathbb{C}} \text{ and }\langle z,w_2\rangle=1\right\}\]
and for every $f \in \mathcal{A}^2(\Omega_1 \times \Omega_2)$, $t=(t_1,t_2)\in \mathbb{C}^2$ and $(z_1,z_2) \in \Omega_1 \times \Omega_2$ define
\begin{align*}\left(\mathscr{L}_{t}f\right)(z):=\sum_{j=1}^N&\left(\frac{\langle A_j^{\wedge k} z_1,w_1\rangle}{\sign \Re(\langle A_j^{\wedge k} z_1,w_1\rangle)}\right)^{t_1} \left(\frac{\langle A_j^{\wedge (k+1)} z_2,w_2\rangle}{\sign \Re(\langle A_j^{\wedge (k+1)} z_2,w_2\rangle)}\right)^{t_2} \\
&\cdot f\left(\langle A_j^{\wedge k}z_1,w_1\rangle^{-1}A_j^{\wedge k}z_1,\langle A_j^{\wedge(k+1)}z_2,w_2\rangle^{-1}A_j^{\wedge(k+1)}z_2\right).\end{align*}
Then $\mathscr{L}_{t} \colon \mathcal{A}^2(\Omega_1 \times \Omega_2) \to \mathcal{A}^2(\Omega_1 \times \Omega_2)$ is a well-defined bounded linear operator, and:
\begin{enumerate}[(i)]
\item
There exist constants $C,\kappa,\gamma>0$ such that the approximation numbers $\mathfrak{s}_n(\mathscr{L}_t)$ satisfy $\mathfrak{s}_n(\mathscr{L}_t) \leq C\exp (\kappa\|t\|-\gamma n^{1/(\hat{d}-2)})$ for every $n \geq 1$ and $t=(t_1,t_2) \in \mathbb{C}^2$, where $\hat{d}:={d+1 \choose k+1}$.
\item
For each $n \geq 1$ the trace of the operator $\mathscr{L}_t^n$ is equal to
\begin{equation}\label{eq:trarse}\sum_{|\mathtt{i}|=n} \frac{\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)^{{d \choose k}-1}\lambda_1\left(A_\mathtt{i}^{\wedge (k+1)}\right)^{{d \choose k+1}-1} \rho\left(A_\mathtt{i}^{\wedge k}\right)^{t_1}\rho\left(A_\mathtt{i}^{\wedge (k+1)}\right)^{t_2}}{p_{A_\mathtt{i}^{\wedge k}}'\left(\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)\right)p_{A_\mathtt{i}^{\wedge (k+1)}}'\left(\lambda_1\left(A_\mathtt{i}^{\wedge (k+1)}\right)\right)}\end{equation}
where $p_B(x):=\det(xI-B)$ denotes the characteristic polynomial of $B$ and $p'_B(x_0)$ its derivative evaluated at $x_0$.
\item
If $t_1,t_2 \in \mathbb{R}$ then
\[\rho(\mathscr{L}_t)=\lim_{n \to \infty} \left(\sum_{|\mathtt{i}|=n} \left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1} \left\|A_\mathtt{i}^{\wedge(k+1)}\right\|^{t_2}\right)^{\frac{1}{n}}\]
and in particular this limit exists. Furthermore in this case $\rho(\mathscr{L}_t)$ is a simple eigenvalue of $\mathscr{L}_t$, and $\mathscr{L}_t$ has no other eigenvalues with modulus equal to $\rho(\mathscr{L}_t)$.
\end{enumerate}
\end{theorem}
If $k=0$ then $\left(\mathcal{K}_j^{(1)}\right)^{\mathbb{C}}$ is equal to $\mathbb{C}$ so that $\Omega_1$ is a single point and $\hat{d}-2=d-1$, and the expression \eqref{eq:trarse} simplifies considerably in the same manner as was noted subsequently to the statement of Theorem \ref{th:main}. Similarly if $k=d-1$ then $\left(\mathcal{K}_j^{(2)}\right)^{\mathbb{C}}=\wedge^{d}\mathbb{C} \simeq \mathbb{C}$, $\Omega_2$ is a point and the formula for $t_n(s)$ simplifies as described in \S\ref{se:one}.
We emphasise that the proof of Theorem \ref{th:topaff} will not in any respect use the fact that $A_\mathtt{i}^{\wedge k}$ and $A_\mathtt{i}^{\wedge(k+1)}$ are related by being different exterior powers of the same matrix. Indeed, one could as easily prove a more general result in which the two tuples $(A_1^{\wedge k},\ldots,A_N^{\wedge k})$ and $(A_1^{\wedge (k+1)},\ldots,A_N^{\wedge (k+1)})$ are replaced by $m$ unrelated tuples $(A_1^{(i)},\ldots,A_N^{(i)})\in M_{d_i}(\mathbb{R})^N$ for $i=1,\ldots,m$ with the property that each tuple individually is multipositive, resulting in a theorem describing an operator $\mathscr{L}_t$ which for $t=(t_1,\ldots,t_m) \in \mathbb{R}^m$ has spectral radius
\[\lim_{n \to \infty} \left(\sum_{|\mathtt{i}|=n} \prod_{i=1}^m \left\|A_\mathtt{i}^{(i)}\right\|^{t_i}\right)^{\frac{1}{n}}\]
and for $t \in \mathbb{C}^m$ has approximation numbers
\[\mathfrak{s}_n(\mathscr{L}_t)=O\left(\exp\left(\kappa \|t\|-\gamma n^\beta\right)\right)\]
and traces
\[\tr \mathscr{L}_t^n=\sum_{|\mathtt{i}|=n} \prod_{i=1}^m \frac{\lambda_1\left(A_\mathtt{i}^{(i)}\right)^{d_i-1} \rho\left(A_\mathtt{i}^{(i)}\right)^{t_i}} {p_{A_\mathtt{i}^{(i)}}'\left(\lambda_1\left(A_\mathtt{i}^{(i)}\right)\right)},\]
where $\beta:=\left(\sum_{i=1}^m (d_i-1)\right)^{-1}$. This underlying mathematical attitude to the singular value function $\varphi^s$ in Theorem \ref{th:topaff} -- namely, of treating it as a product of the powers of the norms of two essentially unrelated matrix products -- is identical to that used in \cite{BoMo18}, and we suspect that other results of a similar character such as \cite{FeSh14,GuLe04} could in principle be rewritten in those terms. We leave to the reader the labour of constructing in detail the more general version of Theorem \ref{th:topaff} just described.
The case $k=0$ of Theorem \ref{th:topaff} will be used in a subsequent article and we single out its statement for later convenience:
\begin{corollary}\label{co:toprad}
Let $\mathsf{A}=(A_1,\ldots,A_N) \in M_d(\mathbb{R})^N$ and suppose that $\mathsf{A}$ strictly preserves a multicone $(\mathcal{K}_1,\ldots,\mathcal{K}_m)$ in $\mathbb{R}^d$ with transverse-defining vector $w\in \mathbb{R}^d$. Define
\[\Omega:=\left\{z \in \mathbb{C}^d \colon z \in \bigcup_{j=1}^m \Int \mathcal{K}_j^{\mathbb{C}} \text{ and }\langle z,w\rangle=1\right\},\]
and for every $f \in \mathcal{A}^2(\Omega)$, $t\in \mathbb{C}$ and $z \in \Omega$ define
\[\left(\mathscr{L}_tf\right)(z):=\sum_{j=1}^N\left(\frac{\langle A_j z,w\rangle}{\sign \Re(\langle A_j z,w\rangle)}\right)^t f\left(\langle A_jz,w\rangle^{-1} A_jz\right).\]
Then $\mathscr{L}_t \colon \mathcal{A}^2(\Omega) \to \mathcal{A}^2(\Omega)$ is a well-defined bounded linear operator, and:
\begin{enumerate}[(i)]
\item
There exist constants $C,\kappa,\gamma>0$ such that the approximation numbers $\mathfrak{s}_n(\mathscr{L}_t)$ satisfy $\mathfrak{s}_n(\mathscr{L}_t) \leq C\exp (\kappa |t|-\gamma n^{1/(d-1)})$ for every $n \geq 1$ and $t \in \mathbb{C}$.
\item
For each $n \geq 1$ the trace of the operator $\mathscr{L}_t^n$ is equal to
\[\sum_{|\mathtt{i}|=n} \frac{\lambda_1(A_\mathtt{i})^{d-1}\rho\left(A_\mathtt{i} \right)^{t}}{p'_{A_\mathtt{i}}\left(\lambda_1\left(A_\mathtt{i}\right)\right)}\]
where $p_B(x):=\det(xI-B)$ denotes the characteristic polynomial of $B$ and $p'_B(x_0)$ its derivative evaluated at $x_0$.
\item
If $t \in \mathbb{R}$ then
\[\rho(\mathscr{L}_t)=\lim_{n \to \infty} \left(\sum_{|\mathtt{i}|=n} \left\|A_\mathtt{i}\right\|^t \right)^{\frac{1}{n}}.\]
Furthermore in this case $\rho(\mathscr{L}_t)$ is a simple eigenvalue of $\mathscr{L}_t$, and $\mathscr{L}_t$ has no other eigenvalues with modulus equal to $\rho(\mathscr{L}_t)$.
\end{enumerate}
\end{corollary}
\begin{proof}[Proof of Theorem \ref{th:topaff}]
We observe that Theorem \ref{th:multicones} applies to $(A_1^{\wedge k},\ldots,A_N^{\wedge k})$, $( \mathcal{K}_{1}^{(1)},\ldots,\mathcal{K}_{m_1}^{(1)})$ and $w_1$ and also applies to $(A_1^{\wedge (k+1)},\ldots,A_N^{\wedge (k+1)})$, $( \mathcal{K}_{1}^{(2)},\ldots,\mathcal{K}_{m_2}^{(2)})$ and $w_2$. Define $\Omega:=\Omega_1 \times \Omega_2$ which, via Theorem \ref{th:multicones}(iv), is a bounded, nonempty open subset of a $2$-codimensional affine subspace of $\wedge^k\mathbb{C}^d \oplus \wedge^{k+1}\mathbb{C}^d \simeq \mathbb{C}^{\hat{d}}$.
By Theorem \ref{th:multicones}(iii), for each $\mathtt{i} \in \Sigma_N^*$ the function $z_1 \mapsto \sign \Re(\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle)$ is constant and nowhere zero on each connected component of $\Omega_1$ and is therefore holomorphic on $\Omega_1$. For $\mathtt{i} \in \Sigma_N^*$ define
\[\phi_\mathtt{i}^{(1)}(z_1):=\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle^{-1} A_\mathtt{i}^{\wedge k}z_1\]
and
\[\psi_{\mathtt{i},t}^{(1)}(z_1):=\left(\frac{\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle}{\sign \Re(\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle)}\right)^{t_1}:= \exp\left(t_1 \log \left(\frac{\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle}{\sign \Re(\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle)}\right)\right)\]
for all $z_1 \in \Omega_1$. We observe that $\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle / \sign \Re(\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle)$ has positive real part for all $z_1 \in \Omega_1$ and therefore its logarithm is a well-defined holomorphic function of $z_1 \in \Omega_1$. These considerations also ensure that the definition of $\phi_\mathtt{i}^{(1)}$ does not constitute a division by zero. We also observe that the set $\Omega_1^0:=\bigcup_{i=1}^N \overline{\phi_i^{(1)}(\Omega_1)}$ is a compact subset of $\Omega_1$ and that $\phi^{(1)}_\mathtt{i}(\Omega_1)\subseteq \Omega_1^0$ for every $\mathtt{i} \in \Sigma_N^*$ by Theorem \ref{th:multicones}(iv) and (viii).
Since for all $z_1 \in \Omega_1$
\[ \left|\Re\left( \log\left(\frac{\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle}{\sign \Re(\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle)}\right)\right)-\log \left\| A_\mathtt{i}^{\wedge k}\right\|\right|\leq \log C_1\]
for some constant $C_1>1$ using Theorem \ref{th:multicones}(vii), and also
\[ \Im\left( \log\left(\frac{\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle}{\sign \Re(\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle)}\right) =\arg\left(\frac{\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle}{\sign \Re(\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle)}\right)\right) \in \left(-\frac{\pi}{2},\frac{\pi}{2}\right)\]
by the preceding observations, we have
\begin{align*}
\Re\left( t_1\log\left(\frac{\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle}{\sign \Re(\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle)}\right)\right)=&\,\,\Re(t_1)\Re\left( \log\left(\frac{\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle}{\sign \Re(\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle)}\right)\right)\\
& - \Im(t_1)\Im\left( \log\left(\frac{\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle}{\sign \Re(\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle)}\right)\right)\\
\leq&\,\,\Re(t_1) \log\left\|A_\mathtt{i}^{\wedge k}\right\| +|\Re(t_1)|\log C_1 + \frac{\pi}{2}\left|\Im(t_1)\right|\end{align*}
for all $z_1 \in \Omega_1$ and therefore
\[\sup_{z_1 \in \Omega_1} \left|\psi_{\mathtt{i},t}^{(1)}(z_1)\right| \leq \left(C_1e^{\pi/2}\right)^{|t_1|}\left\|A_\mathtt{i}^{\wedge k}\right\|^{\Re(t_1)}\]
for every $\mathtt{i} \in \Sigma_N^*$. In a similar manner we may define
\[\phi_\mathtt{i}^{(2)}(z_2):=\langle A_\mathtt{i}^{\wedge (k+1)}z_2,w_2\rangle^{-1} A_\mathtt{i}^{\wedge (k+1)}z_2\]
and
\[\psi_{\mathtt{i},t}^{(2)}(z_2):=\left(\frac{\langle A_\mathtt{i}^{\wedge (k+1)}z_2,w_2\rangle}{\sign \Re(\langle A_\mathtt{i}^{\wedge (k+1)}z_2,w_2\rangle)}\right)^{t_2}:=\exp\left(t_2 \log \left(\frac{\langle A_\mathtt{i}^{\wedge (k+1)}z_2,w_2\rangle}{\sign \Re(\langle A_\mathtt{i}^{\wedge (k+1)}z_2,w_2\rangle)}\right)\right)\]
for all $z_2 \in \Omega_2$ and obtain
\[\sup_{z_2 \in \Omega_2} \left|\psi_{\mathtt{i},t}^{(2)}(z_2)\right| \leq \left(C_1e^{\pi/2}\right)^{|t_2|}\left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{\Re(t_2)}\]
for every $\mathtt{i} \in \Sigma_N^*$, where the value of $C_1$ has been increased from its previous value if necessary. The analogously-defined set $\Omega_2^0:=\bigcup_{i=1}^N \overline{\phi_i^{(2)}(\Omega_2)}$ is a compact subset of $\Omega_2$ and similarly satisfies $\phi_\mathtt{i}^{(2)}(\Omega_2)\subseteq \Omega_2^0$ for every $\mathtt{i} \in \Sigma_N^*$.
Define $\phi_\mathtt{i} \colon \Omega \to \Omega$ by $\phi_\mathtt{i}(z_1,z_2):=(\phi_\mathtt{i}^{(1)}(z_1),\phi_\mathtt{i}^{(2)}(z_2))$ for all $z=(z_1,z_2)\in \Omega$ and $\mathtt{i} \in \Sigma_N^*$. Obviously each $\phi_\mathtt{i}$ is holomorphic and $\phi_\mathtt{i}(\Omega)$ is contained in the compact set $\Omega_0:=\Omega_1^0\times \Omega_2^0\Subset \Omega$. For every $t =(t_1,t_2) \in \mathbb{C}^d$, $\mathtt{i} \in \Sigma_N^*$ and $z=(z_1,z_2)\in \Omega$ define $\psi_{\mathtt{i},t}(z):=\psi_{\mathtt{i},t}^{(1)}(z_1)\psi_{\mathtt{i},t}^{(2)}(z_2)$. Clearly we have
\begin{equation}\label{eq:psibound}\sup_{z \in \Omega} \left|\psi_{\mathtt{i},t}(z)\right| \leq C_2^{\|t\|}\left\|A_\mathtt{i}^{\wedge k}\right\|^{\Re(t_1)}\left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{\Re(t_2)}\end{equation}
for every $\mathtt{i} \in \Sigma_N^*$ and $t \in \mathbb{C}^2$, where $C_2:=(C_1e^{\pi/2})^{\sqrt{2}}$. In particular
\begin{equation}\label{eq:weakpsibound}
\sum_{i=1}^N \sup_{z \in \Omega} \left|\psi_{i,t}(z)\right| \leq C_2^{\|t\|}\sum_{i=1}^N \left\|A_i^{\wedge k}\right\|^{\Re(t_1)}\left\|A_i^{\wedge (k+1)}\right\|^{\Re(t_2)} \leq NC_3^{\|t\|},
\end{equation}
say, for every $t \in \mathbb{C}^2$. We may now define the operator $\mathscr{L}_t$ by
\[\left(\mathscr{L}_tf\right)(z):=\sum_{i=1}^N \psi_{i,t}(z)f(\phi_i(z))\]
for all $f \in \mathcal{A}^2(\Omega)$ and $z \in \Omega$. The set $\Omega$ is a bounded, nonempty open subset of a $2$-codimensional affine subspace of $\mathbb{C}^{\hat{d}}$, so it follows by Theorem \ref{th:banjen} that $\mathscr{L}_t$ is a well-defined bounded linear operator acting on $\mathcal{A}^2(\Omega)$ and that there exist $C,\gamma>0$ depending only on $\Omega_0$ such that for all $t \in \mathbb{C}^2$ we have
\[\mathfrak{s}_n\left(\mathscr{L}_t\right) \leq C\left(\sum_{i=1}^N\sup_{z \in \Omega}\left|\psi_{i,t}(z)\right| \right)\exp\left(-\gamma n^{1/(\hat{d}-2)}\right) \leq CN\exp\left(\kappa\|t\|-\gamma n^{1/(\hat{d}-2)}\right)\]
as a consequence of \eqref{eq:weakpsibound}, where $\kappa:=\log C_3$. This proves (i).
It follows from (i) that $\mathscr{L}_t$ is a trace-class operator. For each $\mathtt{i} \in\Sigma_N^*$ and $t \in \mathbb{C}^2$ let us define an auxiliary operator $\mathscr{L}_{\mathtt{i},t}$ by
\[\left(\mathscr{L}_{\mathtt{i},t}f\right)(z):= \psi_{\mathtt{i},t}(z)f(\phi_\mathtt{i}(z)).\]
Theorem \ref{th:banjen} shows in the same manner as before that each $\mathscr{L}_{\mathtt{i},t}$ is a well-defined trace-class operator on $\mathcal{A}^2(\Omega)$. The reader may easily verify the equations
\[\psi_{\mathtt{j} \mathtt{i},t}(z)=\psi_{\mathtt{i},t}(\phi_\mathtt{j}(z)) \psi_{\mathtt{j},t}(z),\qquad \phi_{\mathtt{j}\mathtt{i}}(z)=\phi_\mathtt{i}(\phi_\mathtt{j}(z))\
and therefore $\mathscr{L}_{\mathtt{j}\mathtt{i},t}=\mathscr{L}_{\mathtt{i},t}\mathscr{L}_{\mathtt{j},t}$ for all $\mathtt{i},\mathtt{j} \in \Sigma_N^*$ and $t \in \mathbb{C}^2$. It follows by a simple inductive argument that $\mathscr{L}_t^n=\sum_{|\mathtt{i}|=n}\mathscr{L}_{\mathtt{i},t}$ for every $n\geq 1$ and $t \in \mathbb{C}^2$, so in particular we have
\begin{equation}\label{eq:trace-is-linear}\tr \mathscr{L}_t = \tr \sum_{|\mathtt{i}|=n}\mathscr{L}_{\mathtt{i},t}= \sum_{|\mathtt{i}|=n}\tr \mathscr{L}_{\mathtt{i},t}\end{equation}
for every $n\geq 1$ and $t \in \mathbb{C}^2$ by the linearity of the trace.
Let $\mathtt{i} \in \Sigma_N^*$. By Theorem \ref{th:multicones}(vi) the map $\phi^{(1)}_\mathtt{i}$ has a unique fixed point $z_\mathtt{i}^{(1)} \in \Omega_1$. This fixed point satisfies $z_\mathtt{i}^{(1)} \in \Omega_1\cap \wedge^k\mathbb{R}^d$ and $\langle A_\mathtt{i}^{\wedge k}z_\mathtt{i}^{(1)},w_1\rangle = \lambda_1(A_\mathtt{i}^{\wedge k})$ and it follows directly that $\psi_{\mathtt{i},t}^{(1)}(z_\mathtt{i}^{(1)})=\rho(A_\mathtt{i}^{\wedge k})^{t_1}$ by inspection of the definitions. Similarly $\phi^{(2)}_\mathtt{i}$ has a unique fixed point $z_\mathtt{i}^{(2)} \in \Omega_2$ which satisfies $z_\mathtt{i}^{(2)} \in \Omega_2\cap \wedge^{k+1}\mathbb{R}^d$ and $\psi_{\mathtt{i},t}^{(2)}(z_\mathtt{i}^{(2)})=\rho(A_\mathtt{i}^{\wedge (k+1)})^{t_2}$. If we define $z_\mathtt{i}:=(z_\mathtt{i}^{(1)},z_\mathtt{i}^{(2)}) \in \Omega \cap( \wedge^{k}\mathbb{R}^d \oplus \wedge^{k+1}\mathbb{R}^d)$ then $z_\mathtt{i}$ is clearly the unique fixed point of $\phi_\mathtt{i}$ in $\Omega$ and we have $\psi_{\mathtt{i},t}(z_\mathtt{i})=\rho(A_\mathtt{i}^{\wedge k})^{t_1}\rho(A_\mathtt{i}^{\wedge (k+1)})^{t_2}$. By Theorem \ref{th:multicones}(vi) the derivative $D_{z_\mathtt{i}^{(1)}}\phi_\mathtt{i}^{(1)}$ of $\phi_\mathtt{i}^{(1)}$ at $z_\mathtt{i}^{(1)}$ satisfies
\[\det\left(I-D_{z_\mathtt{i}^{(1)}}\phi_\mathtt{i}^{(1)}\right) = \frac{p'_{A_\mathtt{i}^{\wedge k}}\left(\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)\right)}{\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)^{{d \choose k}-1}}\]
and the derivative $D_{z_\mathtt{i}^{(2)}}\phi_\mathtt{i}^{(2)}$ of $\phi_\mathtt{i}^{(2)}$ at $z_\mathtt{i}^{(2)}$ satisfies
\[\det\left(I-D_{z_\mathtt{i}^{(2)}}\phi_\mathtt{i}^{(2)}\right) = \frac{p'_{A_\mathtt{i}^{\wedge (k+1)}}\left(\lambda_1\left(A_\mathtt{i}^{\wedge (k+1)}\right)\right)}{\lambda_1\left(A_\mathtt{i}^{\wedge (k+1)}\right)^{{d \choose k+1}-1}}.\]
Since clearly $D_{z_\mathtt{i}}\phi_\mathtt{i} = D_{z_\mathtt{i}^{(1)}}\phi_{\mathtt{i}}^{(1)} \oplus D_{z_\mathtt{i}^{(2)}}\phi_{\mathtt{i}}^{(2)}$ we easily obtain
\begin{align*}\det \left(I-D_{z_\mathtt{i}}\phi_\mathtt{i}\right) &= \det\left(I-D_{z_\mathtt{i}^{(1)}}\phi_\mathtt{i}^{(1)}\right)\det\left(I-D_{z_\mathtt{i}^{(2)}}\phi_\mathtt{i}^{(2)}\right)\\
&=\frac{p'_{A_\mathtt{i}^{\wedge k}}\left(\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)\right)p'_{A_\mathtt{i}^{\wedge (k+1)}}\left(\lambda_1\left(A_\mathtt{i}^{\wedge (k+1)}\right)\right)}{\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)^{{d \choose k}-1}\lambda_1\left(A_\mathtt{i}^{\wedge (k+1)}\right)^{{d \choose k+1}-1}}.\end{align*}
It follows by Theorem \ref{t:thrace} that
\begin{equation}\label{eq:trace-individual}\tr \mathscr{L}_{\mathtt{i},t} = \frac{\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)^{{d \choose k}-1}\lambda_1\left(A_\mathtt{i}^{\wedge (k+1)}\right)^{{d \choose k+1}-1} \rho\left(A_\mathtt{i}^{\wedge k}\right)^{t_1}\rho\left(A_\mathtt{i}^{\wedge (k+1)}\right)^{t_2}}{p_{A_\mathtt{i}^{\wedge k}}'\left(\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)\right)p_{A_\mathtt{i}^{\wedge (k+1)}}'\left(\lambda_1\left(A_\mathtt{i}^{\wedge (k+1)}\right)\right)}\end{equation}
for every $t =(t_1,t_2)\in \mathbb{C}^2$ and every $\mathtt{i} \in \Sigma_N^*$. We obtain \eqref{eq:trarse} by combining \eqref{eq:trace-is-linear} with \eqref{eq:trace-individual} which proves (ii).
The proof of (iii) is by far the longest part of Theorem \ref{th:topaff} and requires substantial groundwork. We begin this process by investigating the limit set $L:=\bigcap_{n=1}^\infty \bigcup_{|\mathtt{i}|=n}\overline{\phi_\mathtt{i}(\Omega)}$. (Subsequently, we will analyse the spectrum of $\mathscr{L}_t$ by studying the restrictions of certain eigenfunctions to this limit set.) We claim that every nonempty closed set $Z \subseteq L$ with the property $\bigcup_{i=1}^N \phi_i(Z)\subseteq Z$ is equal to the set
\[L':=\overline{\left\{z_\mathtt{i} \colon \mathtt{i} \in \Sigma_N^*\right\}}.\]
Since $L$ itself is such a set we in particular have $L'=L$, and therefore every nonempty closed set $Z \subseteq L$ with the property $\bigcup_{i=1}^N \phi_i(Z)\subseteq Z$ is in fact equal to $L$. The claim thus additionally yields $L \subseteq \Omega \cap (\wedge^{k}\mathbb{R}^d \oplus \wedge^{k+1}\mathbb{R}^d)$ since every $z_\mathtt{i}$ belongs to that set.
Let us prove the claim. Theorem \ref{th:multicones}(v) implies that
\[\lim_{n \to \infty}\max_{|\mathtt{i}|=n} \diam \phi_\mathtt{i}^{(1)}(\Omega_1) = \lim_{n \to \infty}\max_{|\mathtt{i}|=n} \diam \phi_\mathtt{i}^{(2)}(\Omega_2) =0\]
and it follows immediately that
\begin{equation}\label{eq:diambound}
\lim_{n\to \infty} \max_{|\mathtt{i}|=n} \diam \phi_\mathtt{i}(\Omega) =0.
\end{equation}
Suppose that $Z \subseteq L$ has the property $\bigcup_{i=1}^N \phi_i(Z)\subseteq Z$ and is closed and nonempty. Let $z \in Z$ and $\mathtt{i} \in \Sigma_N^*$. It is clear that $\phi_\mathtt{i}^n(z) \in Z$ for every $n \geq 1$. By \eqref{eq:diambound} we have $\lim_{n \to \infty} \diam \phi_\mathtt{i}^n(\Omega)=0$ and clearly $z_\mathtt{i} \in \phi_\mathtt{i}^n(\Omega)$ for every $n\geq 1$, so necessarily $\lim_{n \to \infty} d(\phi_\mathtt{i}^nz,z_\mathtt{i})=0$ and therefore $z_\mathtt{i} = \lim_{n \to \infty} \phi_\mathtt{i}^nz \in Z$. We conclude that $\{z_\mathtt{i} \colon \mathtt{i} \in \Sigma_N^*\}\subseteq Z$ and consequently $L'\subseteq Z$. On the other hand, if $z \in Z$ then since $z \in L$ we may for every $n \geq 1$ find $\mathtt{i}(n) \in \Sigma_N^*$ such that $|\mathtt{i}(n)|=n$ and $z \in \overline{\phi_{\mathtt{i}(n)}(\Omega)}$. Since $z_{\mathtt{i}(n)}\in \phi_{\mathtt{i}(n)}(\Omega)$ we in particular have $d(z,z_{\mathtt{i}(n)}) \leq \diam \phi_{\mathtt{i}(n)}(\Omega)$ for every $n \to \infty$ which by \eqref{eq:diambound} implies $\lim_{n \to \infty} d(z,z_{\mathtt{i}(n)})=0$. We conclude that $z \in \overline{\{z_\mathtt{i} \in \Sigma_N^*\}}$ and therefore $Z \subseteq L'$ as required to prove the claim.
The next piece of groundwork is a lower estimate for the value of $\psi_{\mathtt{i},t}(z)$ when $z \in L$ and $t \in \mathbb{R}^2$. If $t_1 \in \mathbb{R}$ and $z_1 \in \Omega_1 \cap \wedge^k \mathbb{R}^d$ then we have
\[\left(\frac{\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle}{\sign \Re(\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle)}\right)^{t_1}= \left|\langle A_\mathtt{i}^{\wedge k}z_1,w_1\rangle\right|^{t_1} \geq C_1^{- |t_1|}\left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1}\]
using Theorem \ref{th:multicones}(vii), and similarly if $t_2 \in \mathbb{R}$ and $z_2 \in \Omega_2 \cap \wedge^{k+1} \mathbb{R}^d$ then
\[\left(\frac{\langle A_\mathtt{i}^{\wedge (k+1)}z_2,w_2\rangle}{\sign \Re(\langle A_\mathtt{i}^{\wedge (k+1)}z_2,w_2\rangle)}\right)^{t_2}= \left|\langle A_\mathtt{i}^{\wedge (k+1)}z_2,w_2\rangle\right|^{t_2} \geq C_1^{- |t_2|}\left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{t_2}.\]
In particular the values of $\psi_{\mathtt{i},t}^{(1)}(z_1)$ and $\psi_{\mathtt{i},t}^{(2)}(z_2)$ in this case are real and positive.
It follows that if $t \in \mathbb{R}^2$ and $z \in L \subset \Omega \cap ( \wedge^k \mathbb{R}^d\oplus \wedge^{k+1} \mathbb{R}^d)$ then $\psi_{\mathtt{i},t}(z)$ is real and positive and satisfies
\begin{equation}\label{eq:+}
\psi_{\mathtt{i},t}(z) \geq C_1^{-|t_1|-|t_2|}\left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1}\left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{t_2},
\end{equation}an estimate which will be repeatedly found useful later.
The final piece of groundwork is the investigation of the rate of growth of the sums $\sum_{|\mathtt{i}|=n} \|A_\mathtt{i}^{\wedge k}\|^{t_1} \|A_\mathtt{i}^{\wedge (k+1)}\|^{t_2}$ for $t=(t_1,t_2)\in\mathbb{R}^2$. We claim that the limit
\[\mathscr{P}(t):=\lim_{n \to \infty}\frac{1}{n}\log\sum_{|\mathtt{i}|=n}\left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1} \left\|A_\mathtt{i}^{\wedge( k+1)}\right\|^{t_2}\]
exists and that there is $C_4>1$ such that
\begin{equation}\label{eq:pressbound}\sum_{|\mathtt{i}|=n} \left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1} \left\|A_\mathtt{i}^{\wedge( k+1)}\right\|^{t_2} \leq C_4^{\|t\|} e^{n\mathscr{P}(t)}\end{equation}
for every $n \geq 1$ and $t \in \mathbb{R}^2$. By Theorem \ref{th:multicones}(ii) there are constants $\tau_1, \tau_2 \in (0,1]$ such that
\[ \tau_1 \left\|A_\mathtt{i}^{\wedge k}\right\| \cdot \left\|A_\mathtt{j}^{\wedge k}\right\| \leq \left\|A_\mathtt{i}^{\wedge k}A_\mathtt{j}^{\wedge k}\right\| \leq \left\|A_\mathtt{i}^{\wedge k}\right\| \cdot \left\|A_\mathtt{j}^{\wedge k}\right\|,\]
\[ \tau_2 \left\|A_\mathtt{i}^{\wedge (k+1)}\right\| \cdot \left\|A_\mathtt{j}^{\wedge (k+1)}\right\| \leq \left\|A_\mathtt{i}^{\wedge (k+1)}A_\mathtt{j}^{\wedge (k+1)}\right\| \leq \left\|A_\mathtt{i}^{\wedge( k+1)}\right\| \cdot \left\|A_\mathtt{j}^{\wedge (k+1)}\right\|\]
for all $\mathtt{i},\mathtt{j} \in \Sigma_N^*$. In particular we have
\[ \left\|A_{\mathtt{i}\mathtt{j}}^{\wedge k}\right\|^{t_1}\left\|A_{\mathtt{i}\mathtt{j}}^{\wedge (k+1)}\right\|^{t_2}
\geq \tau_1^{|t_1|}\tau_2^{|t_2|}\left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1} \left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{t_2} \left\|A_\mathtt{j}^{\wedge k}\right\|^{t_1}\left\|A_\mathtt{j}^{\wedge (k+1)}\right\|^{t_2} \]
for all $\mathtt{i},\mathtt{j} \in \Sigma_N^*$. It follows that the sequence
\[a_n:= \sum_{|\mathtt{i}|=n}\left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1} \left\|A_\mathtt{i}^{\wedge( k+1)}\right\|^{t_2} \]
satisfies $a_{n+m} \geq \tau_1^{|t_1|}\tau_2^{|t_2|}a_na_m$ for every $n,m \geq 1$. In particular the sequence $\log (\tau_1^{|t_1|}\tau_2^{|t_2|}a_n)$ is superadditive and therefore
\[\lim_{n \to \infty} a_n^{1/n} = \lim_{n \to \infty} \left(\tau_1^{|t_1|}\tau_2^{|t_2|}a_n\right)^{1/n} = \sup_{n \geq 1} \left(\tau_1^{|t_1|}\tau_2^{|t_2|}a_n\right)^{1/n}\]
which yields the existence of the limit
\[\mathscr{P}(t)=\lim_{n \to \infty}\frac{1}{n}\log\left(\sum_{|\mathtt{i}|=n}\left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1} \left\|A_\mathtt{i}^{\wedge( k+1)}\right\|^{t_2}\right)\]
together with the inequality
\[\sum_{|\mathtt{i}|=n} \left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1} \left\|A_\mathtt{i}^{\wedge( k+1)}\right\|^{t_2} \leq \tau_1^{-|t_1|}\tau_2^{-|t_2|} e^{n\mathscr{P}(t)}\]
as desired.
The groundwork being completed, we may now begin to analyse the spectrum of the operator $\mathscr{L}_t$ for $t\in \mathbb{R}^2$. In the first step in this process we show at once that $\rho(\mathscr{L}_t) \leq e^{\mathscr{P}(t)}$ and also that any eigenfunctions of $\mathscr{L}_t$ corresponding to sufficiently large eigenvalues are everywhere approximately bounded by their values on the limit set $L$. Specifically, we claim that there exists $C_5>1$ such that if $\mathscr{L}_tf=\lambda f$, $|\lambda|\geq e^{\mathscr{P}(t)}$ and $f \in \mathcal{A}^2(\Omega)$ then
\begin{equation}\label{eq:soup}\sup_{z \in \Omega} |f(z)| \leq C_5^{\|t\|} \sup_{z \in L}|f(z)|,\end{equation}
and if $f$ is nonzero then additionally $|\lambda|= e^{\mathscr{P}(t)}$. To prove the claim let $z_0 \in \Omega$ be arbitrary: for each $n\geq 1$ we have
\begin{align*}|\lambda^n|\cdot|f(z_0)|=\left|\left(\mathscr{L}_t^nf\right)(z_0)\right|&=\left|\sum_{|\mathtt{i}|=n}\psi_{\mathtt{i},t}(z_0)f\left(\phi_\mathtt{i}(z_0)\right)\right|\\
&\leq C_2^{\|t\|}\sum_{|\mathtt{i}|=n} \left\|A^{\wedge k}_\mathtt{i}\right\|^{t_1} \left\|A^{\wedge (k+1)}_\mathtt{i}\right\|^{t_2}\left|f\left(\phi_\mathtt{i}(z_0)\right)\right|\\
&\leq C_4^{\|t\|}C_2^{\|t\|}e^{n\mathscr{P}(t)} \sup_{z \in \overline{\bigcup_{|\mathtt{i}|=n} \phi_\mathtt{i}(\Omega)}} |f(z)|\end{align*}
using \eqref{eq:psibound} and \eqref{eq:pressbound} so that
\[\sup_{z \in \Omega}|f(z)| \leq C_4^{\|t\|}C_2^{\|t\|}|\lambda|^{-n}e^{n\mathscr{P}(t)} \sup_{z \in \overline{\bigcup_{|\mathtt{i}|=n} \phi_\mathtt{i}(\Omega)}} |f(z)|\]
for all $n\geq 1$. If $|\lambda|=e^{\mathscr{P}(t)}$ then we obtain \eqref{eq:soup} by taking the infimum of the right-hand side with respect to $n \in \mathbb{N}$; if $|\lambda|>e^{\mathscr{P}(t)}$ the same mechanism yields $f=0$. Since $\mathscr{L}_t$ is compact its spectral radius is the maximum of the absolute values of its eigenvalues and we conclude in particular that $\rho(\mathscr{L}_t)\leq e^{\mathscr{P}(t)}$.
Our next task is to construct an eigenfunction corresponding to the largest eigenvalue of $\mathscr{L}_t$, which will be needed in order to show that the leading eigenvalue is real and simple. Following the standard lines of a Ruelle-Perron-Frobenius theorem on a holomorphic function space (see e.g. \cite{Ma91}) the natural approach is to specialise to the real Hilbert space of elements of $\mathcal{A}^2(\Omega)$ which take real values throughout the limit set $L$, use the Kre{\u\i}n-Rutman theorem to construct an eigenfunction which is positive on $L$, and then use \eqref{eq:soup} to show that such a function must be unique. However, in general the limit set $L$ may be so small that an element of $\mathcal{A}^2(\Omega)$ can be identically zero on $L$ without necessarily being the zero function. (For example, if all of the matrices $A_i^{\wedge k}$ have a common leading eigenvector and so do all the matrices $A_i^{\wedge (k+1)}$, then $L$ will be a singleton set.) When this is the case the set $\mathcal{C}$ of functions taking real non-negative values on $L$ does not satisfy condition (ii) of Theorem \ref{th:kr} and the Kre{\u\i}n-Rutman theorem is inapplicable. This obstruction can be circumvented by working on a suitable quotient space, but this results in the construction of a function which is positive on $L$ but satisfies the eigenfunction equation only in its restriction to the set $L$. These considerations are responsible for the length of the arguments which follow. Throughout the remainder of the argument we fix $t \in \mathbb{R}^2$.
Let us define $\mathcal{H}:=\{f \in \mathcal{A}^2(\Omega) \colon f(z)\in \mathbb{R} \text{ for all }z \in L\}$ and note that $\mathcal{H}$ is a closed subset of $\mathcal{A}^2(\Omega)$ as a consequence of Lemma \ref{le:supbound}. Observe also that $\mathcal{H}$ is a real Hilbert space when equipped with the norm $\|\cdot\|_{\mathcal{A}^2(\Omega)}$. In proving the estimate \eqref{eq:+} we showed that $\psi_{i,t}(z)$ is real and positive when $z \in L$ and $i\in\{1,\ldots,N\}$, and it follows from this that $\mathscr{L}_t$ preserves $\mathcal{H}$. Clearly $\mathscr{L}_t$ is a compact operator on $\mathcal{H}$ as a direct consequence of the compactness of $\mathscr{L}_t$ acting on $\mathcal{A}^2(\Omega)$. Define $\mathcal{Z}:=\{f \in \mathcal{H} \colon f(z)=0\text{ for all }z \in L\}$ and note that $\mathcal{Z}$ is a vector subspace of $\mathcal{H}$ and is closed as a consequence of Lemma \ref{le:supbound}. Since $\bigcup_{i=1}^N\phi_i(L)\subseteq L$ it follows that $\mathscr{L}_t\mathcal{Z}\subseteq \mathcal{Z}$ by inspection of the definition of $\mathscr{L}_t$. The quotient space $\mathcal{H}/\mathcal{Z}$ is a Hilbert space when equipped with norm $\|[f]\|_{\mathcal{H}/\mathcal{Z}}:=\inf\{\|f-g\|_{\mathcal{A}^2(\Omega)}\colon g \in \mathcal{Z}\}$, being isometrically isomorphic to the orthogonal complement of $\mathcal{Z}$ in $\mathcal{H}$. Since $\mathscr{L}_t \mathcal{Z}\subseteq \mathcal{Z}$ it is not difficult to see that the operator $\mathscr{L}_t$ induces a compact operator on the real Hilbert space $\mathcal{H}/\mathcal{Z}$ which we also denote by $\mathscr{L}_t$.
We observe that for each $z \in L$ the functional $[f]\mapsto f(z)$ is a well-defined continuous linear functional $\mathcal{H}/\mathcal{Z} \to \mathbb{R}$. Indeed, if $[f] \in \mathcal{H}/\mathcal{Z}$ and $g \in \mathcal{Z}$ then we have $f(z)=(f+g)(z)$ and
\[|f(z)|=|(f+g)(z)| \leq C_L\|f+g\|_{\mathcal{A}^2(\Omega)}\]
where $C_L>0$ is the constant given by Lemma \ref{le:supbound} in respect of the nonempty compact set $L$. In particular $f(z)$ is independent of the choice of representative $f \in [f]$. Since $g \in \mathcal{Z}$ was arbitary we obtain
\[|f(z)|\leq C_L\inf\{\|f+g\|_{\mathcal{A}^2(\Omega)} \colon g \in \mathcal{Z}\}=C_L\|[f]\|_{\mathcal{H}/\mathcal{Z}}\]
and the functional $[f] \mapsto f(z)$ is continuous as required. Now define
\[\mathcal{C}:=\{[f] \in \mathcal{H}/\mathcal{Z} \colon f(z)\geq 0\text{ for all }z \in L\} = \bigcap_{z \in L}\left\{[f]\in \mathcal{H}/\mathcal{Z} \colon f(z) \geq 0\right\}.\]
This set is clearly well-defined, positively homogenous, convex, and closed. The constant function $\mathbf{1}$ belongs to $\mathcal{A}^2(\Omega)$ since $\Omega$ is bounded, and therefore $\mathbf{1} \in \mathcal{H}$. Clearly also $[\mathbf{1}]\in \mathcal{C}$, and if $\|[f]-[\mathbf{1}]\|_{\mathcal{H}/\mathcal{Z}}<C_L^{-1}$ then $f(z)>0$ for all $z \in L$ and therefore $[f] \in \mathcal{C}$. We conclude that $[\mathbf{1}]$ is an interior point of $\mathcal{C}$ and it follows that the span of $\mathcal{C}$ is dense in $\mathcal{H}/\mathcal{Z}$. If $[f] \in \mathcal{C} \cap -\mathcal{C}$ then $f(z)=0$ for all $z \in L$ so that $f \in \mathcal{Z}$ and therefore the only element of $\mathcal{C}\cap-\mathcal{C}$ is $[0]$. In particular the set $\mathcal{C}$ satisfies conditions (i)--(iii) of Theorem \ref{th:kr}. We observe also that $\mathscr{L}_t\mathcal{C}\subseteq \mathcal{C}$ as an easy consequence of \eqref{eq:+}.
Let $z_0 \in L$ be arbitary. For every $g \in \mathcal{Z}$ we have
\begin{align*}\left|\left(\mathscr{L}_t^n\mathbf{1}\right)(z_0)+g(z_0)\right| &=\left(\sum_{|\mathtt{i}|=n} \psi_{\mathtt{i},t}(z_0)\mathbf{1}(\phi_\mathtt{i}(z_0))\right)+g(z_0)\\
&= \sum_{|\mathtt{i}|=n}\psi_{\mathtt{i},t}(z_0)\\
&\geq C_1^{-|t_1|-|t_2|}\sum_{|\mathtt{i}|=n} \left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1}\left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{t_2}\end{align*}
using \eqref{eq:+} so that
\[ C_1^{-|t_1|-|t_2|}\sum_{|\mathtt{i}|=n} \left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1}\left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{t_2}
\leq \left|\left(\mathscr{L}_t^n\mathbf{1}\right)(z_0) + g(z_0)\right| \leq C_L \|\mathscr{L}_t^n\mathbf{1}+g\|_{\mathcal{A}^2(\Omega)}\]
and by taking the infimum over all $g \in \mathcal{Z}$ we obtain
\[ C_1^{-|t_1|-|t_2|}\sum_{|\mathtt{i}|=n} \left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1}\left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{t_2} \leq C_L\|[\mathscr{L}_t^n\mathbf{1}]\|_{\mathcal{H}/\mathcal{Z}} \leq C_L\|\mathscr{L}_t^n\|_{\mathcal{H}/\mathcal{Z}} \|[\mathbf{1}]\|_{\mathcal{H}/\mathcal{Z}}\]
for every $ n\geq 1$. It follows that the spectral radius of $\mathscr{L}_t$ acting on $\mathcal{H}/\mathcal{Z}$ is at least
\[\lim_{n \to\infty} \left(\sum_{|\mathtt{i}|=n} \left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1}\left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{t_2}\right)^{\frac{1}{n}}
=e^{\mathscr{P}(t)}.\]
Let $R \geq e^{\mathscr{P}(t)}>0$ denote this spectral radius.
By Theorem \ref{th:kr} it follows that there exists $\xi_t \in \mathcal{H}$ such that $[\mathscr{L}_t\xi_t]=R[\xi_t]$, $[\xi_t]\in \mathcal{C}$ and $[\xi_t] \neq [0]$. In particular we have $\mathscr{L}_t^n\xi_t-R^n\xi_t \in \mathcal{Z}$ for all $n \geq 1$ and $\xi_t \notin \mathcal{Z}$. Since $[\xi_t]\in \mathcal{C}$ and $\xi_t \notin \mathcal{Z}$ there exists $z_0 \in L$ such that $\xi_t(z_0)> 0$. We have $(\mathscr{L}_t^n\xi_t)(z_0)=R^n\xi_t(z_0)$ for all $n \geq1$ since $\mathscr{L}_t\xi_t-R^n\xi_t \in \mathcal{Z}$, so
\[0<R^n |\xi_t(z_0)|=\left|(\mathscr{L}_t^n\xi_t)(z_0)\right| \leq C_L \|\mathscr{L}_t^n\xi_t\|_{\mathcal{A}^2(\Omega)} \leq C_L \|\mathscr{L}_t^n\|_{\mathcal{A}^2(\Omega)}\|\xi_t\|_{\mathcal{A}^2(\Omega)} \]
for all $n \geq 1$ and therefore the spectral radius of $\mathscr{L}_t$ acting on $\mathcal{A}^2(\Omega)$ must be at least $R \geq e^{\mathscr{P}(t)}$. On the other hand we showed earlier that the spectral radius of $\mathscr{L}_t$ acting on $\mathcal{A}^2(\Omega)$ is not greater than $e^{\mathscr{P}(t)}$, and therefore both that spectral radius and $R$ must equal $e^{\mathscr{P}(t)}$. We conclude that $\mathscr{L}_t\xi_t-e^{\mathscr{P}(t)}\xi_t$ is identically zero on $L$ and the spectral radius of $\mathscr{L}_t$ on $\mathcal{A}^2(\Omega)$ is precisely $e^{\mathscr{P}(t)}$.
We next claim that in fact $\xi_t(z) >0$ for all $z \in L$. Since $[\xi_t]\in\mathcal{C}$ this is equivalent to the statement that $\xi_t(z)\neq 0$ for all $z \in L$. Let $Z:=\{z \in L \colon \xi_t(z)=0\}$. If $Z$ contains a point $z_0$ then since
\[0 \leq e^{\mathscr{P}(t)}\sum_{i=1}^N \psi_{i,t}(z_0)\xi_t(\phi_i(z_0))= \xi_t(z_0)=0\]
and $\psi_{i,t}(z_0)>0$ for all $i=1,\ldots,N$ using \eqref{eq:+}, we have $\phi_1(z_0),\ldots,\phi_N(z_0)\in Z$. It follows that $\bigcup_{i=1}^N \phi_i(Z)\subseteq Z$. We showed earlier that the only closed nonempty subset of $L$ with this property is $L$ itself, so if $Z \neq \emptyset$ then necessarily $Z=L$. It follows that if $\xi_t(z)=0$ for any $z \in L$ then $\xi_t$ is identically zero on $L$, contradicting $\xi_t\notin \mathcal{Z}$. We conclude that $\xi_t(z)$ cannot be zero for any $z \in L$ which proves the claim.
We claim now that if $\mathscr{L}_t\eta = \lambda\eta$ with $|\lambda| = e^{\mathscr{P}(t)}$, $\eta \in \mathcal{A}^2(\Omega)$ and $\eta \neq 0$, then $\eta$ must coincide on $L$ with a scalar multiple of $\xi_t$ and consequently $\lambda=e^{\mathscr{P}(t)}$. Let us fix such a function $\eta$ and number $\lambda$ and begin the proof of the claim. Since $\xi_t(z)>0$ for all $z \in L$ and $L$ is compact, we may multiply $\eta$ by a real scalar in such a way that $\sup_{z \in L} |\eta(z)/\xi_t(z)|=1$. Multiplying $\eta$ in turn by a suitable complex unit we may assume that $\eta(z_0)/\xi_t(z_0)=1$ for some $z_0 \in L$. Now, for all $n \geq 1$ we have
\begin{align*}\eta(z_0)=\lambda^{-n}(\mathscr{L}_t^n\eta)(z_0) &= \sum_{|\mathtt{i}|=n} \lambda^{-n}\psi_{\mathtt{i},t}(z_0)\eta(\phi_\mathtt{i}(z_0))\\
&=\left|\sum_{|\mathtt{i}|=n} \lambda^{-n}\psi_{\mathtt{i},t}(z_0)\eta(\phi_\mathtt{i}(z_0))\right|\\
&=e^{-n\mathscr{P}(t)}\left|\sum_{|\mathtt{i}|=n} \psi_{\mathtt{i},t}(z_0)\eta(\phi_\mathtt{i}(z_0))\right|\\
&\leq e^{-n\mathscr{P}(t)}\sum_{|\mathtt{i}|=n} \left|\psi_{\mathtt{i},t}(z_0)\eta(\phi_\mathtt{i}(z_0))\right|\\
&\leq e^{-n\mathscr{P}(t)}\sum_{|\mathtt{i}|=n} \psi_{\mathtt{i},t}(z_0) \xi_t(\phi_\mathtt{i}(z_0))\\
&=e^{-n\mathscr{P}(t)}\mathscr{L}_t^n\xi_t(z_0) = \xi_t(z_0)=\eta(z_0)\end{align*}
using the positivity of each $\psi_{\mathtt{i},t}$ on $L$. Since the first and last terms in this chain of inequalities are equal, none of the inequalities can be strict. It follows that the argument of $\psi_{\mathtt{i},t}(z_0)\eta(\psi_\mathtt{i}(z_0))$ is independent of $\mathtt{i}$ when the length of $\mathtt{i}$ is fixed, since otherwise the first inequality would be strict due to cancellation. Moreover we must have $|\eta(\phi_\mathtt{i}(z_0))|=\xi_t(\phi_\mathtt{i}(z_0))$ for all $\mathtt{i}$ with length $n$ since $|\eta(\phi_\mathtt{i}(z_0))|\leq\xi_t(\phi_\mathtt{i}(z_0))$ for all such $\mathtt{i}$ and if $|\eta(\phi_\mathtt{i}(z_0))|<\xi_t(\phi_\mathtt{i}(z_0))$ for some $\mathtt{i}$ then the second inequality above would be strict, which it is not. We deduce that for every $n\geq 1$ the quantity $\eta(\phi_\mathtt{i}(z_0))/\xi_t(\phi_\mathtt{i}(z_0))$ takes a constant value on the complex unit circle for all $\mathtt{i}$ such that $|\mathtt{i}|=n$. Since
\[\sum_{|\mathtt{i}|=n} \lambda^{-n}\psi_{\mathtt{i},t}(z_0)\eta(\phi_\mathtt{i}(z_0)) = \sum_{|\mathtt{i}|=n}e^{-n\mathscr{P}(t)} \psi_{\mathtt{i},t}(z_0) \xi_t(\phi_\mathtt{i}(z_0))\]
it must be the case that $\lambda^{-n}\psi_{\mathtt{i},t}(z_0)\eta(\phi_\mathtt{i}(z_0))$ is real and positive for all $\mathtt{i}$ such that $|\mathtt{i}|=n$, so we have $\eta(\phi_\mathtt{i}(z_0))/\xi_t(\phi_\mathtt{i}(z_0))=\lambda^ne^{-n\mathscr{P}(t)}$ for all $\mathtt{i}$ such that $|\mathtt{i}|=n$. It follows that if $z_1 \in L$ denotes the fixed point of $\phi_1$ then
\[\eta(z_1)=\lim_{n\to \infty} \eta(\phi_1^n(z_0)) = \lim_{n \to \infty} \left(\lambda e^{-\mathscr{P}(t)}\right)^n \xi_t(\phi_1^n(z_0))\]
but since $\lim_{n\to \infty} \xi_t(\phi_1^n(z_0))=\xi_t(z_1)\neq 0$ this is only possible if $\lim_{n \to \infty} \left(\lambda e^{-\mathscr{P}(t)}\right)^n$ exists, which implies $\lambda e^{-\mathscr{P}(t)}=1$. We conclude that $\lambda=e^{\mathscr{P}(t)}$ and $\eta(z)=\xi_t(z)$ for all $z$ in the set $Z:=\overline{\{\phi_\mathtt{i}(z_0) \colon \mathtt{i} \in \Sigma_N^*\}}$. Clearly this set $Z$ is a closed nonempty subset of $L$ such that $\bigcup_{i=1}^N\phi_i(Z)\subseteq Z$, and we know that such a set must equal $L$. We conclude that if $\mathscr{L}_t\eta = \lambda\eta$ where $|\lambda| = e^{\mathscr{P}(t)}$ and $\eta \neq 0$ then $\lambda=e^{\mathscr{P}(t)}$ and $\eta$ coincides with a scalar multiple of $\xi_t$ when restricted to $L$.
We have seen that $\mathscr{L}_t$ has no eigenvalues with modulus $e^{\mathscr{P}(t)}$ which are not real and positive, and since $\mathscr{L}_t$ is compact and $\rho(\mathscr{L}_t)=e^{\mathscr{P}(t)}$, by elimination $e^{\mathscr{P}(t)}$ must itself be an eigenvalue. If $\xi_t^{(1)}$, $\xi_t^{(2)}$ are eigenfunctions for this eigenvalue then both coincide on $L$ with a scalar multiple of $\xi_t$, so some nontrivial linear combination of the two eigenfunctions must be identically zero on $L$. This linear combination is also an eigenfunction for the same eigenvalue, so by \eqref{eq:soup} this linear combination must be identically zero on $\Omega$. The two eigenfunctions are therefore proportional to one another. We conclude that there is a one-dimensional eigenspace associated to the eigenvalue $e^{\mathscr{P}(t)}$, that no other eigenvalues of equal or greater modulus exist, and the associated eigenfunction may be chosen positive on $L$.
Let $\hat\xi_t \in \mathcal{A}^2(\Omega)$ be nonzero with $\mathscr{L}_t\hat\xi_t=e^{\mathscr{P}(t)}\hat\xi_t$ and note that $\hat\xi_t$ coincides on $L$ with a nonzero scalar multiple of $\xi_t$. In particular $\hat\xi_t$ is nowhere zero on $L$. To complete the proof of the theorem we wish to show that the algebraic multiplicity of the eigenvalue $e^{\mathscr{P}(t)}$ is $1$. If this is not the case then necessarily $\dim \ker (\mathscr{L}_t-e^{\mathscr{P}(t)}I)^2>1$, and in particular we may choose $\eta \in \ker (\mathscr{L}_t-e^{\mathscr{P}(t)}I)^2 \setminus \ker (\mathscr{L}_t-e^{\mathscr{P}(t)}I)$. The function $\mathscr{L}_t\eta-e^{\mathscr{P}(t)}\eta$ belongs to $\ker (\mathscr{L}_t-e^{\mathscr{P}(t)}I)$ and hence is proportional to $\hat\xi_t$, and is not identically zero since $\eta \notin \ker (\mathscr{L}_t-e^{\mathscr{P}(t)}I)$. Multiplying $\eta$ by a suitable scalar we may assume $\mathscr{L}_t\eta-e^{\mathscr{P}(t)}\eta=e^{\mathscr{P}(t)}\hat\xi_t$. A simple induction shows that $\mathscr{L}_t^n\eta = e^{n\mathscr{P}(t)}(\eta+n\hat\xi_t)$ for all $n \geq 1$. If $z_0 \in L$ is arbitrary we obtain
\begin{align*}\left|\mathscr{L}_t^n\eta(z_0)\right| &=\left|\sum_{|\mathtt{i}|=n}\psi_{\mathtt{i},t}(z_0)\eta(\phi_\mathtt{i}(z_0))\right| \leq \sum_{|\mathtt{i}|=n}\left|\psi_{\mathtt{i},t}(z_0)\eta(\phi_\mathtt{i}(z_0))\right|\\
& \leq C_2^{\|t\|}\left(\sum_{|\mathtt{i}|=n}\left\|A_\mathtt{i}^{\wedge k}\right\|^{t_1}\left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{t_2}\right)\sup_{z \in L}|\eta(z)|\\
& \leq C_4^{\|t\|}C_2^{\|t\|}e^{n\mathscr{P}(t)}\sup_{z \in L}|\eta(z)|\end{align*}
and therefore
\[n|\hat\xi_t(z_0)| =\left|e^{-n\mathscr{P}(t)}\left(\mathscr{L}_t^n\eta\right)(z_0)-\eta(z_0)\right|\leq \left(C_4^{\|t\|}C_2^{\|t\|}+1\right)\sup_{z \in L}|\eta(z)|\]
for all $n \geq 1$, which is impossible since $\hat\xi_t(z_0)\neq 0$. We conclude that $\dim \ker (\mathscr{L}_t-e^{\mathscr{P}(t)}I)^2=1$ and therefore $e^{\mathscr{P}(t)}$ is a simple eigenvalue of $\mathscr{L}_t$ as required.
\end{proof}
\section{Proof of Theorem \ref{th:main} }\label{se:proofs}
Before starting the proof of Theorem \ref{th:main} we require two preliminary lemmas, one concerning the behaviour of the leading eigenvalue of the operator $\mathscr{L}_s$ of Theorems \ref{th:opter} and \ref{th:topaff} and one an abstract result concerning sequences of implicit functions in two complex variables.
\begin{lemma}\label{le:do-not-press}
Let $\mathsf{A}=(A_1,\ldots,A_N)\in M_d(\mathbb{R})^N$ be $k$- and $(k+1)$-multipositive with $N,d \geq 2$ and $0 \leq k<d$, and for each $s \in \mathbb{C}$ let $\mathscr{L}_s \colon \mathscr{H} \to \mathscr{H}$ be as given by Theorem \ref{th:opter}.
Define
\[p(s):=\log\rho(\mathscr{L}_s)=\lim_{n\to \infty} \frac{1}{n}\log \left(\sum_{|\mathtt{i}|=n}\left\|A_\mathtt{i}^{\wedge k}\right\|^{k+1-s} \left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{s-k}\right)\]
for all $s \in \mathbb{R}$. Then $p$ is convex. If additionally there exists a norm $\vertle{\cdot}$ on $\mathbb{R}^d$ with respect to which $\max_{1 \leq i \leq N}\vertle{A_i}<1$, then there exists $c>0$ such that
\[\frac{p(s_2)-p(s_1)}{s_2-s_1}\leq -c<0\]
for all pairs of distinct points $s_1,s_2 \in \mathbb{R}$.
\end{lemma}
\begin{proof}
If $s_1,s_2 \in \mathbb{R}$, $\lambda \in (0,1)$ and $n \geq 1$ then
\begin{eqnarray*}\lefteqn{\sum_{|\mathtt{i}|=n}\left\|A_\mathtt{i}^{\wedge k}\right\|^{k+1-\lambda s_1 -(1-\lambda)s_2} \left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{\lambda s_1 + (1-\lambda )s_2-k}}& &\\
& &=\sum_{|\mathtt{i}|=n}\left(\left\|A_\mathtt{i}^{\wedge k}\right\|^{k+1-s_1}\left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{s_1-k }\right)^\lambda \left(\left\|A_\mathtt{i}^{\wedge k}\right\|^{k+1-s_2}\left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{s_2-k }\right)^{1-\lambda} \\
& & \leq \left(\sum_{|\mathtt{i}|=n}\left\|A_\mathtt{i}^{\wedge k}\right\|^{k+1-s_1}\left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{s_1-k }\right)^\lambda \left(\sum_{|\mathtt{i}|=n}\left\|A_\mathtt{i}^{\wedge k}\right\|^{k+1-s_2}\left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{s_2-k }\right)^{1-\lambda} \end{eqnarray*}
using H\"older's inequality with $p=1/\lambda$ and $q=1/(1-\lambda)$. Taking $n^{\mathrm{th}}$ roots and letting $n \to \infty$ it follows directly that $\rho(\mathscr{L}_{\lambda s_1+(1-\lambda)s_2}) \leq \rho(\mathscr{L}_{s_1})^\lambda \rho(\mathscr{L}_{s_2})^{1-\lambda}$ and the convexity of $p$ follows by taking logarithms.
To complete the proof suppose that there exists a norm $\vertle{\cdot}$ on $\mathbb{R}^d$ with respect to which $\max_{1 \leq i \leq N}\vertle{A_i}<1$, and choose $C>0$ such that $\|B\| \leq C\vertle{B}$ for all $B \in M_d(\mathbb{R})$. Observe that in particular $\sigma_{k+1}(A_\mathtt{i})\leq \sigma_1(A_\mathtt{i})=\|A_\mathtt{i}\|\leq C\vertle{A_\mathtt{i}}$ for all $\mathtt{i} \in \Sigma_N^*$. If $s_1<s_2 \in \mathbb{R}$ and $n \geq 1$ then
\begin{eqnarray*}\lefteqn{\sum_{|\mathtt{i}|=n}\left\|A_\mathtt{i}^{\wedge k}\right\|^{k+1-s_2} \left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{s_2-k}}& &\\
& =&\sum_{|\mathtt{i}|=n} \sigma_1(A_\mathtt{i})\cdots \sigma_k(A_\mathtt{i}) \sigma_{k+1}(A_\mathtt{i})^{s_2-k}\\
& =&\sum_{|\mathtt{i}|=n} \sigma_1(A_\mathtt{i})\cdots \sigma_k(A_\mathtt{i}) \sigma_{k+1}(A_\mathtt{i})^{s_1-k}\sigma_{k+1}(A_\mathtt{i})^{s_2-s_1}\\
& \leq& \left(\max_{|\mathtt{i}|=n}\sigma_{k+1}(A_\mathtt{i})\right)^{s_2-s_1} \sum_{|\mathtt{i}|=n} \sigma_1(A_\mathtt{i})\cdots \sigma_k(A_\mathtt{i}) \sigma_{k+1}(A_\mathtt{i})^{s_1-k}\\
& \leq& \left(\max_{|\mathtt{i}|=n}C\vertle{A_\mathtt{i}}\right)^{s_2-s_1} \sum_{|\mathtt{i}|=n} \sigma_1(A_\mathtt{i})\cdots \sigma_k(A_\mathtt{i}) \sigma_{k+1}(A_\mathtt{i})^{s_1-k}\\
& \leq& C^{s_2-s_1}\left(\max_{1 \leq i \leq N}\vertle{A_i}\right)^{n(s_2-s_1)} \sum_{|\mathtt{i}|=n}\left\|A_\mathtt{i}^{\wedge k}\right\|^{k+1-s_1} \left\|A_\mathtt{i}^{\wedge (k+1)}\right\|^{s_1-k}\end{eqnarray*}
so that by taking the $n^{\mathrm{th}}$ root and letting $n \to \infty$ we obtain
\[\rho(\mathscr{L}_{s_2})\leq \left(\max_{1 \leq i \leq N}\vertle{A_i}\right)^{s_2-s_1}\rho(\mathscr{L}_{s_1}) \]
for all such $s_1$ and $s_2$. Taking logarithms and rearranging yields the claim with $c:=-\log \max_{1\leq i \leq N} \vertle{A_i}>0$.
\end{proof}
As in \S\ref{se:complex-cones} we shall say that $X_1$ is compactly contained in $X_2$ if the closure of $X_1$ is a compact subset of the interior of $X_2$, and express this relation with the notation $X_1 \Subset X_2$.
\begin{lemma}\label{le:disco}
Let $D_1,D_2 \subset \mathbb{C}$ be open discs, let $f_n \colon D_1 \times D_2 \to \mathbb{C}$ be a bounded holomorphic function for each $n\geq 1$, and let $f \colon D_1 \times D_2 \to \mathbb{C}$ be bounded and holomorphic. Suppose that there exists a holomorphic function $g \colon D_1 \to D_2$ such that for all $s \in D_1$, $g(s)$ is a simple zero of the function $z \mapsto f(s,z)$ and is the unique zero of that function in $D_2$. Suppose also that
\[\lim_{n \to \infty} \sup_{s \in D_1} \sup_{z \in D_2} \left|f_n(s,z)-f(s,z)\right| =0.\]
Let $D_1'$ be any open disc which is compactly contained in $D_1$. Then there exist a disc $D_2' \subseteq D_2$, which may be chosen concentric with $D_2$ and with radius arbitrarily close to that of $D_2$, an integer $n_0\geq 1$ and holomorphic functions $g_n \colon D_1' \to D_2'$ defined for all $n\geq n_0$ such that:
\begin{enumerate}[(i)]
\item
For all $n \geq n_0$ and $s \in D_1'$, $g_n(s)$ is a simple zero of $z \mapsto f_n(s,z)$ and is the unique zero of that function in $D_2'$.
\item
For every integer $\ell \geq 0$ there exists $C_\ell>0$ such that
\[\sup_{z \in D_1'} \left|g_n^{(\ell)}(s)-g^{(\ell)}(s)\right| \leq C_\ell\sup_{s \in D_1}\sup_{z \in D_2}|f_n(s,z)-f(s,z)|\]
for all $n \geq n_0$, where $h^{(\ell)}$ denotes the $\ell^{\mathrm{th}}$ derivative of the function $h$.
\end{enumerate}
\end{lemma}
\begin{proof}
Throughout the proof let $D_3$ be an open disc such that $D_1' \Subset D_3 \Subset D_1$. By compactness and continuity we have $g(\overline{D_3})\Subset D_2$. Let $D_2' \Subset D_2$ be any disc which is concentric with $D_2$ and has radius large enough that $g(\overline{D_3})\Subset D_2'$. By compactness and continuity we obtain
\[\inf_{s \in \overline{D_3}} \inf_{z \in \partial D_2'} |f(s,z)|>0\]
and hence by uniform convergence there exists $n_1 \geq 1$ such that for all $n\geq n_1$
\[\sup_{s \in \overline{D_3}} \sup_{z \in \partial D_2'} |f(s,z)-f_n(s,z)|<\inf_{s \in \overline{D_3}} \inf_{z \in \partial D_2'} |f(s,z)|.\]
It follows by Rouch\'e's theorem that for every $s \in \overline{D_3}$ and $n\geq n_1$ there exists a unique zero $g_n(s)$ of the function $z \mapsto f_n(s,z)$ in $D_2'$ and this zero is simple. Since each $f_n$ is holomorphic it follows by the holomorphic implicit function theorem (see e.g. \cite[p.34]{FrGr02}) that each $g_n \colon \overline{D_3} \to D_2'$ is holomorphic on $D_3$.
We claim now that
\[\lim_{n \to \infty}\sup_{s \in \overline{D_3}} |g_n(s)-g(s)|=0.\]
Indeed, let $\varepsilon>0$ be any number which is small enough that for every $s \in \overline{D_3}$ the closed $\varepsilon$-ball centred at $g(s)$ is a subset of $D_2'$. By compactness and the absence of zeros of $z \mapsto f(s,z)$ in $D_2 \setminus \{g(s)\}$ we have
\[\inf_{s \in \overline{D_3}} \inf_{|z-g(s)|=\varepsilon} |f(s,z)|>0\]
so that in the same manner if $n$ is large enough
\[\sup_{s \in \overline{D_3}} \sup_{|z-g(z)|=\varepsilon} |f(s,z)-f_n(s,z)|<\inf_{s \in\overline{D_3}} \inf_{|z-g(s)|=\varepsilon} |f(s,z)|.\]
Applying Rouch\'e's theorem again it follows that if $n$ is sufficiently large then for all $s \in \overline{D_3}$ there is a unique zero of the function $z \mapsto f_n(s,z)$ in the region $0 \leq |z-g(s)|<\varepsilon$. This zero belongs to $D_2'$ and hence is necessarily equal to $g_n(s)$, and we therefore have $\sup_{s \in \overline{D_3}}|g_n(s)-g(s)|\leq \varepsilon$. Since $\varepsilon$ was arbitrary we conclude that
\begin{equation}\label{eq:clams}\lim_{n \to \infty}\sup_{s \in \overline{D_3}} |g_n(s)-g(s)|=0\end{equation}
as claimed.
For each $s \in D_1$ the value $g(s)$ is a simple zero of the function $z \mapsto f(s,z)$, so we have $\frac{\partial f}{\partial z}(s,g(s))\neq0$ for all $s \in D_1$. Define
\[c:=\inf_{s \in \overline{D_3}} \left|\frac{\partial f}{\partial z}(s,g(s))\right|>0.\]
Since $g(\overline{D_3}) \Subset D_2' \Subset D_2$ we may choose $\tau>0$ small enough that for every $z \in \partial D_2'$ the closed ball of radius $2\tau$ centred at $z$ is a subset of $D_2$ which does not intersect $g(\overline{D_3})$. Using \eqref{eq:clams} take $n_2 \geq n_1$ large enough that
\[\sup_{s \in \overline{D_3}} |g_n(s)-g(s)|<\tau\]
for all $n \geq n_2$. Observe that if $s \in \overline{D_3}$ and $n \geq n_2$ then $|g_n(s)-g(s)|<\tau$ and therefore $|g(s)-\omega|>\tau$ and $|g_n(s)-\omega|>\tau$ for all $\omega \in \partial D_2'$. Using Cauchy's integral formula, for any two distinct points $z_1, z_2 \in D_2'$ we have
\begin{eqnarray*}\lefteqn{\frac{f(s,z_1)-f(s,z_2)}{z_1-z_2}-\frac{\partial f}{\partial z}(s,z_2)}& &\\
& =& \frac{1}{2\pi i}\int_{\partial D_2'} \frac{f(s,\omega)}{(z_1-z_2)(\omega-z_1)}- \frac{f(s,\omega)}{(z_1-z_2)(\omega-z_2)} - \frac{f(s,\omega)}{(\omega-z_2)^2}d\omega\\
& =&\frac{1}{2\pi i}\int_{\partial D_2'} \frac{f(s,\omega)((\omega-z_2)^2-(\omega-z_1)(\omega-z_2)-(z_1-z_2)(\omega-z_1))}{(z_1-z_2)(\omega-z_1)(\omega-z_2)^2}d\omega\\
& =&\frac{1}{2\pi i}\int_{\partial D_2'} \frac{f(s,\omega)(z_1^2-2z_1z_2+z_2^2)}{(z_1-z_2)(\omega-z_1)(\omega-z_2)^2}d\omega\\
&=&\frac{1}{2\pi i}\int_{\partial D_2'} \frac{f(s,\omega)(z_1-z_2)}{(\omega-z_1)(\omega-z_2)^2}d\omega.\end{eqnarray*}
Hence if $s \in \overline{D_3}$, $n \geq n_2$ and $g_n(s) \neq g(s)$ then since $g(s),g_n(s)\in D_2'$
\[\left|\frac{f(s,g_n(s))-f(s,g(s))}{g_n(s)-g(s)}-\frac{\partial f}{\partial z}(s,g(s))\right|\leq \frac{R|g_n(s)-g(s)|}{\tau^3} \cdot \sup_{t \in D_1} \sup_{z \in D_2} |f(t,z)|\]
where $R$ denotes the radius of $D_2'$. Now take $n_3 \geq n_2$ large enough that additionally
\[\left(\sup_{s \in \overline{D_3}} |g_n(s)-g(s)| \right)\left(\frac{R}{\tau^3} \sup_{s \in D_1} \sup_{z \in D_2} |f(s,z)|\right)<\frac{c}{2}.\]
If $n \geq n_3$, $s \in \overline{D_3}$ and $g_n(s) \neq g(s)$ then since $f_n(s,g_n(s))=0=f(s,g(s))$ we have
\begin{eqnarray*}
\lefteqn{\left|\frac{f(s,g_n(s))-f_n(s,g_n(s))}{g_n(s)-g(s)}\right|}& &\\
& =&\left|\frac{f(s,g_n(s))-f(s,g(s))}{g_n(s)-g(s)}\right|\\
&\geq& \left|\frac{\partial f}{\partial z}(s,g(s))\right|-\left|\frac{f(s,g_n(s))-f(s,g(s))}{g_n(s)-g(s)} -\frac{\partial f}{\partial z}(s,g(s))\right|>\frac{c}{2}.
\end{eqnarray*}
It follows that when $n \geq n_3$
\[\sup_{s \in \overline{D_3}} |g_n(s)-g(s)| \leq \frac{2}{c} \sup_{s \in \overline{D_3}}\sup_{z \in D_2}|f_n(s,z)-f(s,z)|.\]
To complete the proof of the lemma let $\delta>0$ be small enough that for every $s \in D_1'$ the closed $\delta$-ball centred at $s$ is a subset of $D_3$. By the Cauchy integral formula we have for each integer $\ell\geq 0$ and every $n \geq n_3$
\begin{align*}\sup_{s \in D_1'} \left|g^{(\ell)}_n(s)-g^{(\ell)}(s)\right| &\leq \sup_{s \in D_1'} \left|\frac{\ell!}{2\pi i} \int_{|s-t|=\delta} \frac{g_n(t)-g(t)}{(t-s)^{\ell+1}} dt \right|\\
&\leq \delta^{-\ell}\ell! \sup_{s \in \overline{D_3}} |g_n(s)-g(s)|\\
&\leq \frac{2\ell!}{c\delta^\ell} \sup_{s \in D_1}\sup_{z \in D_2} |f_n(s,z)-f(s,z)|\end{align*}
as required. The proof is complete.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:main}]
Let $\mathsf{A}=(A_1,\ldots,A_N)\in M_d(\mathbb{R})^N$ be $k$- and $(k+1)$-multipositive where $N,d \geq 2$ and $0\leq k<d$. For all $s \in \mathbb{C}$ let $\mathscr{L}_s \colon\mathscr{H}\to \mathscr{H}$ be as given by Theorem \ref{th:opter}.
Define
\[t_n(s):=\sum_{|\mathtt{i}|=n} \frac{\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)^{{d \choose k} -1} \lambda_1\left(A_\mathtt{i}^{\wedge\left(k+1\right)}\right)^{{d \choose k+1} -1} \rho\left(A_\mathtt{i}^{\wedge k}\right)^{k+1-s} \rho\left(A_\mathtt{i}^{\wedge \left(k+1\right)}\right)^{s-k} }{ p'_{A_\mathtt{i}^{\wedge k}}\left(\lambda_1\left(A_\mathtt{i}^{\wedge k}\right)\right)p'_{A_\mathtt{i}^{\wedge \left(k+1\right)}}\left(\lambda_1\left(A_\mathtt{i}^{\wedge \left(k+1\right)}\right)\right) }
\]
and
\[a_n(s):=\frac{(-1)^n}{n!}\det\begin{pmatrix}
t_1(s)& n -1& 0&\cdots &0 &0\\
t_2(s)&t_1(s)&n-2 &\cdots &0 &0\\
t_3(s)&t_2(s)&t_1(s) &\ddots &0 &0\\
\vdots & \vdots & \vdots & \ddots &\ddots& \vdots\\
t_{n-1}(s) &t_{n-2}(s)&t_{n-3}(s)&\cdots &t_1(s) &1\\
t_n(s) &t_{n-1}(s)&t_{n-2}(s)&\cdots &t_2(s) &t_1(s)
\end{pmatrix}\]
for every $s \in \mathbb{C}$ and $n \geq 1$. Define also $a_0(s):=1$. We claim that there exist $\tilde K, \tilde \gamma,\kappa>0$ such that
\begin{equation}\label{eq:astrosheep-6000}|a_n(s)|\leq \tilde{K}^n e^{n \kappa|s|} \exp\left(-\tilde\gamma n^\alpha\right)\end{equation}
for all $n \geq 1$ and $s \in \mathbb{C}$, where $\tilde{K},\tilde{\gamma}$ and $\kappa$ do not depend on $s$ or $n$ and where
\[\alpha:= \frac{{d+1 \choose k+1}-1}{{d+1\choose k+1}-2}.\]
By Theorem \ref{th:opter} there exist constants $C,\gamma,\kappa>0$ such that
\begin{equation}\label{eq:todos-os-garotos-e-todos-as-garotas-querem}\mathfrak{s}_n(\mathscr{L}_s)\leq C\exp\left(\kappa|s| - \gamma n^\beta\right) \end{equation}
for all $n \geq 1$ and $s \in \mathbb{C}$ where $\beta:=({d +1\choose k+1}-2)^{-1}=\alpha-1$, and $\mathscr{L}_s$ is trace-class with $\tr \mathscr{L}_s^n=t_n(s)$ for all $s \in \mathbb{C}$. By Theorem \ref{th:ofundity} we have
\begin{equation}\label{eq:se-voce-procura-a-amy}|a_n(s)| \leq \sum_{i_1<\cdots<i_n}\mathfrak{s}_{i_1}(\mathscr{L}_s) \cdots \mathfrak{s}_{i_n}(\mathscr{L}_s)\end{equation}
for all $n \geq 1$. In order to proceed further we require two elementary inequalities. We first note that for every integer $m\geq 1$
\begin{align}\label{eq:bohnenstadt}\sum_{\ell=m}^\infty e^{-\gamma \ell^\beta} \leq \int_{m-\frac{1}{2}}^\infty e^{-\gamma t^\beta}dt&=\frac{1}{\beta}\int_{\left(m-\frac{1}{2}\right)^\beta}^\infty u^{\frac{1}{\beta}-1}e^{-\gamma u}du\\\nonumber
&\leq \frac{K}{\beta}\int_{\left(m-\frac{1}{2}\right)^\beta}^\infty e^{-\frac{\gamma}{2}u}du\\\nonumber
&\leq \frac{2K}{\beta\gamma} e^{-\frac{\gamma}{2} \left(m-\frac{1}{2}\right)^\beta}\leq \frac{2K}{\beta\gamma}e^{-\frac{\gamma}{2^{1+\beta}}m^\beta} \end{align}
where $K:=\sup\{ x^{\frac{1}{\beta}-1} e^{-\gamma x/2}\colon x\geq \frac{1}{2}\}>0$ depends only on $\beta$ and $\gamma$, and where $e^{-\gamma \ell^\beta}\leq \int_{\ell-\frac{1}{2}}^{\ell+\frac{1}{2}} e^{-\gamma t^\beta}dt$ follows from Jensen's inequality. Secondly we notice that
\begin{equation}\label{eq:haricotville}\sum_{\ell=1}^m \ell^\beta \geq \int_0^m t^\beta dt = \frac{m^{1+\beta}}{1+\beta}\end{equation}
for all integers $m\geq 1$ since the series is an upper Riemann sum for the integral. Combining \eqref{eq:todos-os-garotos-e-todos-as-garotas-querem}, \eqref{eq:se-voce-procura-a-amy}, \eqref{eq:bohnenstadt} and \eqref{eq:haricotville} we may now obtain
\begin{align*}|a_n(s)|&\leq \sum_{i_1<\cdots<i_n} \prod_{\ell=1}^n C\exp(\kappa |s|-\gamma i_\ell^\beta)\\
& = \left(Ce^{\kappa |s|}\right)^n\sum_{i_1<\cdots<i_n}\exp\left(-\gamma \left(i_1^\beta+\cdots + i_n^\beta\right)\right)\\
&\leq\left(Ce^{\kappa |s|}\right)^n\sum_{i_1=1}^\infty \sum_{i_2=2}^\infty \cdots \sum_{i_n=n}^\infty \exp\left(-\gamma \left(i_1^\beta+\cdots + i_n^\beta\right)\right)\\
&=\left(Ce^{\kappa |s|}\right)^n\prod_{m=1}^n \sum_{\ell=m}^\infty \exp\left(-\gamma \ell^\beta\right) \\
&\leq \left(\frac{2KCe^{\kappa |s|}}{\beta\gamma}\right)^n\prod_{m=1}^n \exp\left(-\frac{\gamma}{2^{1+\beta}} m^\beta\right) \\
&\leq \left(\frac{2KCe^{\kappa |s|}}{\beta\gamma}\right)^n \exp\left(-\frac{\gamma}{(1+\beta)2^{1+\beta}} n^{1+\beta}\right)\end{align*}
which establishes the claimed inequality \eqref{eq:astrosheep-6000} with $\tilde\gamma :=\gamma/(2^{1+\beta}(1+\beta))$ and $\tilde{K}:=2KC/\beta\gamma$.
Now define a function $d_n \colon \mathbb{C}^2 \to \mathbb{C}$ for each $n\geq 1$ by $d_n(s,z):=\sum_{m=0}^n a_m(s)z^m$, and define also $d_\infty(s,z):=\sum_{m=0}^\infty a_m(s)z^m$, the convergence of the series being guaranteed by \eqref{eq:astrosheep-6000}. As a consequence of \eqref{eq:astrosheep-6000} it is clear that
\begin{equation}\label{eq:electric-head-part-196883}
\left|d_n(s,z)-d_\infty(s,z)\right| =\left|\sum_{m=n+1}^\infty a_m(s)z^m\right|=O\left(\exp\left(-\frac{\tilde\gamma}{2}n^\alpha\right)\right)
\end{equation}
uniformly on compact subsets of $\mathbb{C}^2$. It is clear by inspection that each $d_n$ is holomorphic, and using the convergence of $d_n$ to $d_\infty$ uniformly on compact sets together with Cauchy's theorem and Morera's theorem it follows easily that $d_\infty \colon \mathbb{C}^2 \to \mathbb{C}$ is holomorphic. By Theorem \ref{th:ofundity} we have $d_\infty(s,z)=\det (I-z\mathscr{L}_s)$ for every $(s,z) \in \mathbb{C}^2$. In particular for every $s \in \mathbb{C}$ the zeros of $z \mapsto d_\infty(s,z)$ are precisely the reciprocals of the nonzero eigenvalues of $\mathscr{L}_s$, with the degree of each zero being equal to the algebraic multiplicity of the corresponding eigenvalue.
For each $s \in \mathbb{R}$ define $r_\infty(s):=\rho(\mathscr{L}_s)^{-1} \in (0,+\infty)$. We observe that $p(s)=-\log r_\infty(s)$ is a continuous function of $s$ by Lemma \ref{le:do-not-press} since it is a convex function of $s \in \mathbb{R}$, so $r_\infty(s) \colon \mathbb{R} \to (0,+\infty)$ is continuous. By the combination of Theorem \ref{th:opter} and Theorem \ref{th:ofundity}, for each $s \in \mathbb{R}$ the function $z\mapsto d_\infty(s,z)$ has a simple zero at $r_\infty(s)$ and has no zeroes with equal or smaller absolute value. We claim that there exist $n_0 \geq 1$, an open set $U \subset \mathbb{C}$ containing $[k,k+1]$, a holomorphic extension of $r_\infty|_{[k,k+1]}$ to $U$ and a sequence of holomorphic functions $r_n \colon U \to \mathbb{C}$ defined for all $n \geq n_0$ such that
\begin{equation}\label{eq:fun-to-funky}\sup_{s \in U} \left|r_n^{(\ell)} (s)-r_\infty^{(\ell)}(s)\right| = O\left(\exp\left(-\frac{\tilde \gamma}{2} n^\alpha\right)\right)\end{equation}
for all integers $\ell \geq 0$ and such that for all $n \geq n_0$ and $s \in [k,k+1]$, $r_n(s)$ is the smallest positive real number $x$ such that $d_n(s,x)=0$.
To prove the claim it is clearly sufficient, by the compactness of $[k,k+1]$, to show that every $s_0 \in [k,k+1]$ admits an open neighbourhood $U(s_0)$ such that $r_\infty$ extends holomorphically from $U(s_0)\cap [k,k+1]$ to all of $U(s_0)$, such that there exists a sequence of functions $r_n \colon U(s_0) \to \mathbb{C}$ defined for all large enough $n$ such that for all $s \in [k,k+1] \cap U(s_0)$, $r_n(s)$ is the smallest positive real number $x$ such that $d_n(s,x)=0$, and such that
\[\sup_{s \in U(s_0)} \left|r_n^{(\ell)} (s)-r_\infty^{(\ell)}(s)\right| = O\left(\exp\left(-\frac{\tilde \gamma}{2} n^\alpha\right)\right)\]
for all integers $\ell \geq 0$. The open set $U$ can then be taken equal to the union of a finite cover of $[k,k+1]$ by different sets $U(s)$, and the characterisation of $r_n(s)$ as the smallest positive root of $d_n(s,x)=0$ ensures that for each $n$ the local functions $r_n \colon U(s) \to \mathbb{C}$ extend consistently to a single well-defined function $r_n \colon U \to \mathbb{C}$.
Let us therefore prove this local version of the preceding claim. Fix $s_0 \in [k,k+1]$. Since $z \mapsto d_\infty(s_0,z)$ has a unique zero in the closed disc with centre $0$ and radius $r_\infty(s_0)$, and all of its zeros are isolated, we may choose an open disc $D_2(s_0)$ with centre $z_0 \in \mathbb{R}$ and radius $R>0$ such that $[0,r_\infty(s_0)] \subset D_2(s_0)$ and such that $\overline{D_2(s_0)}$ contains no other zeros of $z \mapsto d_\infty(s_0,z)$. A simple argument using compactness shows that we may choose a small open disc $D_1(s_0)$ centred at $s_0$ such that
\[\sup_{s \in D_1(s_0)} \sup_{|z-z_0|=R} \left|d_\infty(s,z)-d_\infty(s_0,z)\right|<\inf_{|z-z_0|=R} \left|d_\infty(s_0,z)\right|\]
and by shrinking the neighbourhood $D_1(s_0)$ further if necessary we may assume using continuity that additionally $r_\infty(s) \in D_2(s_0)$ for all $s \in D_1(s_0)\cap [k,k+1]$.
By Rouch\'e's theorem, for all $s \in D_1(s_0)$ the function $z \mapsto d_\infty(s,z)$ has a unique zero in $D_2(s_0)$ and this zero is simple. When $s \in D_1(s_0) \cap[k,k+1]$ this zero must be equal to $r_\infty(s) \in D_2(s_0)$ by uniqueness. Extend $r_\infty \colon D_1(s_0)\cap [k,k+1] \to \mathbb{R}$ to a function $D_1(s_0)\to \mathbb{C}$ by defining $r_\infty(s)$ to be the unique zero of $z \mapsto d_\infty(s,z)$ in $D_2(s_0)$ for each $s \in D_1(s_0)$. By the holomorphic implicit function theorem and the simplicity of the zero $r_\infty \colon D_1(s_0)\to D_2(s_0)$ is holomorphic. Applying Lemma \ref{le:disco} we find, shrinking $D_1(s_0)$ and $D_2(s_0)$ if necessary, that there exist constants $C_\ell>0$, an integer $n_1 \geq 1$ and holomorphic functions $r_n \colon D_1(s_0) \to D_2(s_0)$ defined for all $n \geq n_1$ such that
\begin{align*}\sup_{s \in D_1(s_0)} \left|r_n^{(\ell)}(s)-r_\infty^{(\ell)}(s)\right| &\leq C_\ell \sup_{s \in D_1(s_0)}\sup_{z \in D_2(s_0)} \left| d_n(s,z)-d_\infty(s,z)\right|\\
&= O\left(\exp\left(-\frac{\tilde \gamma}{2} n^\alpha\right)\right)\end{align*}
for every integer $\ell \geq 0$, such that $r_n(s)$ is the unique zero of $z \mapsto d_n(s,z)$ in $D_2(s_0)$ for all $s \in D_1(s_0)$ and $n \geq n_1$ and is a simple zero for all such $s$ and $n$, such that $[0,r_\infty(s_0)] \subseteq D_2(s_0)$, and such that $D_2(s_0)$ is an open disc centred on the real axis. For all $s \in D_1(s_0) \cap [k,k+1]$ and $n \geq n_0$ the numbers $r_n(s)$ and $r_n(s)^*$ both lie in $D_2(s_0)$ and are both zeros of the polynomial $d_n(s,z)=\sum_{m=0}^n a_n(s)z^m$ since the coefficients of that polynomial are real and since $D_2(s_0)$, being a disc centred on the real axis, is symmetric with respect to complex conjugation. By the uniqueness of the zero $r_n(s)$ in $D_2(s_0)$ this is possible only if $r_n(s)=r_n(s)^*$, which is to say if $r_n(s)$ is real. Since $D_2(s_0)$ contains the interval from $0$ to $r_n(s)$, it follows that if $r_n(s)$ is positive then it is the smallest positive real root of $\sum_{m=0}^n a_n(s)x^m$ for all $s \in D_1(s_0)\cap [k,k+1]$. To complete the proof of the claim it therefore suffices to show that if $n$ is sufficiently large then $r_n(s)>0$ for all $s \in D_1(s_0)$. To see this choose $\delta \in (0,r_\infty(s_0))$ small enough that the open $\delta$-ball centred at $r_\infty(s_0)$ is contained in $D_2(s_0)$, and observe that by shrinking $D_1(s_0)$ further if necessary we may obtain
\[\inf_{s \in D_1(s_0)} \inf_{|z-r_\infty(s_0)|=\delta} |d_\infty(s,z)|>0\]
and hence for all large enough $n$
\[\sup_{s \in D_1(s_0)} \sup_{|z-r_\infty(s_0)|=\delta} |d_n(s,z)-d_\infty(s,z)|< \inf_{s \in D_1(s_0)} \inf_{|z-r_\infty(s_0)|=\delta} |d_\infty(s,z)|.\]
By Rouch\'e's theorem this implies that there exists $n_0\geq n_1$ such that for all $n \geq n_0$ and all $s \in D_1(s_0)$ there is a unique zero of $z \mapsto d_n(s,z)$ inside the circle of radius $\delta$ and centre $r_\infty(s_0)$, and since this region is a subset of $D_2(s_0)$ this root must equal $r_n(s)$ by the uniqueness of that root in $D_2(s_0)$. In particular for all $n \geq n_0$ and $s \in D_1(s_0) \cap [k,k+1]$ we have $r_n(s)>r_\infty(s_0)-\delta>0$ and no other root lies in $(0,r_n(s)) \subset D_2(s_0)$. Hence $r_n(s)$ is the smallest positive real root of $\sum_{m=0}^n a_n(s)x^m$ for all $s \in D_1(s_0)\cap [k,k+1]$ as required to prove the local version of the claim with $U(s_0):=D_1(s_0)$. The full statement of the claim follows.
We may now complete the proof of the theorem. Define $P_n(s):=r_n(s)^{-1}>0$ for all $s \in [k,k+1]$ and $n \geq n_0$, and $P(s):=r_\infty(s)^{-1}>0$ for all $s \in \mathbb{R}$. Observe that by Theorem \ref{th:opter} we have $e^{P(A_1,\ldots,A_N;s)}=P(s)$ for all $s \in [k,k+1]$. Since $r_\infty \colon U \to \mathbb{C}$ is holomorphic, $P$ is real-analytic at least on a neighbourhood of $[k,k+1]$. Since $r_\infty(s)$ is positive for all real $s$ and $[k,k+1]$ is compact it follows that
\begin{equation}\label{eq:cheesefish}\inf_{s \in [k,k+1]}r_\infty(s)>0\end{equation}
and by the case $\ell=0$ of \eqref{eq:fun-to-funky} we deduce that
\begin{equation}\label{eq:toastfish}\lim_{n\to \infty} \inf_{s \in [k,k+1]}r_n(s)>0.\end{equation}
Using \eqref{eq:fun-to-funky}, \eqref{eq:cheesefish}, \eqref{eq:toastfish} and the expressions
\[\left|P_n(s)-P(s)\right| = \left|\frac{1}{r_n(s)}-\frac{1}{r_\infty(s)}\right|,\]
\[\left|P'_n(s)-P'(s)\right| = \left|\frac{r_n'(s)}{r_n(s)^2}-\frac{r_\infty'(s)}{r_\infty(s)^2}\right|\]
and
\[\left|P''_n(s)-P''(s)\right| = \left|\frac{r_n''(s)r_n(s)-r_n'(s)^2}{r_n(s)^4}-\frac{r_\infty''(s)r_\infty(s)-r_\infty'(s)^2}{r_\infty(s)^4}\right|\]
it follows by elementary manipulations that
\begin{equation}\label{eq:cnut}\sup_{s \in [k,k+1]}\left|P_n(s)-P(s)\right| = O\left(\exp\left(-\frac{\tilde\gamma}{2}n^\alpha\right)\right),\end{equation}
\begin{equation}\label{eq:wnaker}\sup_{s \in [k,k+1]}\left|P'_n(s)-P'(s)\right| = O\left(\exp\left(-\frac{\tilde\gamma}{2}n^\alpha\right)\right)\end{equation}
and
\begin{equation}\label{eq:avocado-toast-is-the-end-of-civilisation-according-to-the-daily-telegraph}\sup_{s \in [k,k+1]}\left|P''_n(s)-P''(s)\right| = O\left(\exp\left(-\frac{\tilde\gamma}{2}n^\alpha\right)\right).\end{equation}
In the case where we do not assume that $\max_{1 \leq i \leq N}\vertle{A_i}<1$ for some norm on $\mathbb{R}^d$ the estimate \eqref{eq:cnut} already completes the proof of Theorem \ref{th:main}. Otherwise, we claim that $\inf_{s \in [k,k+1]}P''(s)>0$ and $\sup_{s \in [k,k+1]}P'(s)<0$. Let $p(s):=\log P(s)$ for $s \in \mathbb{R}$ so that $P'(s)=p'(s)P(s)$ and $P''(s)=p''(s)P(s)+p'(s)^2P(s)$. Obviously $p$ is real-analytic on $[k,k+1]$ since $P$ is positive and real-analytic there, and $p$ is convex by Lemma \ref{le:do-not-press}, so necessarily $p''(s)\geq 0$ for all $s \in [k,k+1]$. By Lemma \ref{le:do-not-press} we have $p'(s)<0$ for all $s \in [k,k+1]$ and therefore
\begin{equation}\label{eq:hi-ms-coutts}\sup_{s \in [k,k+1]}P'(s)=\sup_{s \in [k,k+1]} p'(s)P(s)<0.\end{equation}
Similarly we observe that $\inf_{s \in [k,k+1]}|p'(s)|>0$, and since $P''(s)=p''(s)P(s)+p'(s)^2P(s)\geq p'(s)^2P(s)$ we likewise deduce that $\inf_{s \in [k,k+1]}P''(s)>0$ as claimed.
Combining the previous claim with \eqref{eq:avocado-toast-is-the-end-of-civilisation-according-to-the-daily-telegraph} we find in particular that $\inf_{s \in [k,k+1]}P''_n(s)>0$ for all large enough $n$, which proves that each such function $P_n \colon [k,k+1] \to \mathbb{R}$ is convex. By the hypothesis $\dimaff(A_1,\ldots,A_N) \in (k,k+1)$ of Theorem \ref{th:main} there exists a solution $s \in (k,k+1)$ to $P(s)=1$, and since $P$ has negative derivative on $[k,k+1]$ this implies that $P(k)>1>P(k+1)$. Combining this observation with \eqref{eq:cnut} we find that $P_n(k)>1>P_n(k+1)$ for all large enough $n$, and by the combination of \eqref{eq:hi-ms-coutts} and \eqref{eq:wnaker} we find that $\sup_{s \in [k,k+1]}P'_n(s)\leq -c<0$ for all large enough $n$ where $c>0$ is some positive constant. It follows that for all large enough $n$ there exists a unique $s_n \in [k,k+1]$ such that $P_n(s_n)=1$. Let $s_\infty:=\dimaff(A_1,\ldots,A_N) \in [k,k+1]$ be the unique solution to $P(s_\infty)=1$. If $s_n \neq s_\infty$ then by the Mean Value Theorem there exists $t$ strictly between $s_n$ and $s_\infty$ such that
\[P'(t)=\frac{P(s_n)-P(s_\infty)}{s_n-s_\infty}\]
and therefore since $P_n(s_n)=1=P(s_\infty)$ we obtain
\[\left|s_n-s_\infty\right|=\frac{|P(s_n)-P(s_\infty)|}{|P'(t)|} =\frac{|P(s_n)-P_n(s_n)|}{|P'(t)|}\\\leq c^{-1}|P(s_n)-P_n(s_n)|.\]
The inequality $|s_n-s_\infty| \leq c^{-1}|P(s_n)-P_n(s_n)|$ obviously also holds when $s_n=s_\infty$, so
\[\left|s_n-s_\infty\right|=O\left(\exp\left(-\frac{\tilde\gamma}{2}n^\alpha\right)\right)\]
as $n \to \infty$ using \eqref{eq:cnut}. The proof of the theorem is complete.
\end{proof}
\section{Examples}\label{se:apps}
\subsection{Methodology}\label{ss:urk}
There are two intuitively natural mechanisms by which to make the approximations given in Theorem \ref{th:main} yield an approximation to the affinity dimension. On the one hand since $e^{P(A_1,\ldots,A_n;s)}$ is decreasing in $s$ and since the affinity dimension is the unique $s \in [k,k+1]$ such that $1$ is the leading eigenvalue of $\mathscr{L}_s$, the affinity dimension corresponds to the smallest $s \in [k,k+1]$ such that $\det(I-\mathscr{L}_s)=0$, which is to say the smallest $s \in [k,k+1]$ such that $\sum_{m=0}^\infty a_m(s)=0$. It is therefore natural to attempt to approximate the affinity dimension by looking for the smallest solution $s$ to the equation $\sum_{m=0}^na_m(s)=0$ for each fixed $n$. In practice however this is problematic since $\mathscr{L}_s$ may in general have infinitely many positive real eigenvalues and the number of solutions to $\sum_{m=0}^na_m(s)=0$ may therefore be extremely large and the function itself highly oscillatory.
\begin{table}
\begin{center
\begin{tabular}{*3c}
\toprule
$n$ & Approximation to affinity dimension&CPU time\\
\midrule
2& 1.14341 79598 76019 95000 60486 91827 85789 60135 &0.043s \\
3&\underline{1.1}1827 23247 08006 28499 89060 66409 13091 47143&0.044s\\
4 &\underline{1.11}538 89736 67461 99644 51849 00512 18003 54788&0.053s\\
5 &\underline{1.115}60 42107 66261 56209 11669 09958 04069 77087&0.075s\\
6 &\underline{1.11560} 31850 39305 08475 98379 83168 80085 68510&0.11s\\
7 &\underline{1.11560} \underline{3}2522 24751 03699 38823 87724 66623 37012&0.16s\\
8 & \underline{1.11560} \underline{325}79 27402 64806 11546 27227 11083 45893&0.30s\\
9 &\underline{1.11560} \underline{3257}7 86505 71154 77556 50836 85812 53178&0.39s\\
10 &\underline{1.11560} \underline{32577} \underline{8}7028 88533 65835 00045 83936 61000& 0.67s\\
11 &\underline{1.11560} \underline{32577} \underline{870}30 91898 36777 33249 49956 17495&1.2s\\
12 &\underline{1.11560} \underline{32577} \underline{87030} 89197 97928 71446 51257 73313&2.0s\\
13 & \underline{1.11560} \underline{32577} \underline{87030} \underline{89}218 88050 96492 48585 23429 &4.3s\\
14 &\underline{1.11560} \underline{32577} \underline{87030} \underline{89218} \underline{8}4942 17623 75680 33697&8.8s\\
15 &\underline{1.11560} \underline{32577} \underline{87030} \underline{89218} \underline{849}37 14660 75123 27001 &20s\\
16 & \underline{1.11560} \underline{32577} \underline{87030} \underline{89218} \underline{84937} \underline{14}840 85419 85122 &44s\\
17& \underline{1.11560} \underline{32577} \underline{87030} \underline{89218} \underline{84937} \underline{14840} 24544 08248&100s\\
18 & \underline{1.11560} \underline{32577} \underline{87030} \underline{89218} \underline{84937} \underline{14840} \underline{245}74 24137&210s\\
19 &\underline{1.11560} \underline{32577} \underline{87030} \underline{89218} \underline{84937} \underline{14840} \underline{24574} \underline{2}5551&440s\\
20 &\underline{1.11560} \underline{32577} \underline{87030} \underline{89218} \underline{84937} \underline{14840} \underline{24574} \underline{25551}&990s\\
\bottomrule
\end{tabular}\bigskip\medskip
\caption{Approximations to the affinity dimension of Example 1 calculated using Theorem \ref{th:main} and the secant method as described in \S\ref{ss:urk}, implemented in Wolfram Mathematica. The CPU time used in each approximation is as reported by Mathematica's {\tt{Timing}} function. For $n=1$ the approximation to the pressure function has no root in $(1,2)$ and this line is therefore omitted from the table. Digits which are empirically observed to have converged to a stable value are underlined.
}\label{ta:blen}
\end{center}
\end{table}
In practice, therefore, we adopt the following alternative approach. For large $n$ the smallest positive real root $x=r_n(s)$ of $\sum_{m=0}^n a_m(s)x^m$ approximates the reciprocal of the leading eigenvalue of $\mathscr{L}_s$. Moreover, for large $n$ the function $s \mapsto r_n(s)^{-1}$ is convex and strictly decreasing with a unique root in $[k,k+1]$ by virtue of Theorem \ref{th:main}. Computing the unique root of a convex decreasing function is a far more tractable enterprise than finding the smallest root of an oscillating function, and for this reason our application of Theorem \ref{th:main} follows the approach of solving $r_n(s)=1$. For this problem we use the \emph{secant method}: for fixed $n$ we evaluate $r_n(s)^{-1}$ firstly at $s_1:=k+1$ and secondly at $s_2:=k$. Given two approximations $s_m$, $s_{m-1}$ we then define $s_{m+1}$ by extrapolating the location of the root from $s_m$, $s_{m-1}$ as if the function $r_n(s)^{-1}$ were affine:
\[s_{m+1}:=\frac{s_{m-1} -s_m - r_n(s_m)^{-1}s_{m-1}+r_n(s_{m-1})^{-1}s_m}{r_n(s_{m-1})^{-1}-r_n(s_{m})^{-1}}.\]
When $r_n^{-1}$ is convex and decreasing the convergence of the sequence $(s_m)$ is guaranteed with super-exponential rate $O(\theta^{m^{(1+\sqrt{5})/2}})$ for some $\theta \in (0,1)$. In practical instances we found that the sequence $(s_m)$ consistently converged empirically to 40 decimal places by around $m\simeq 12$ independently of $n$. The results of this procedure applied to some examples of two- and three-dimensional affine iterated function systems are presented in this section.
For large $n$ one may show that the trace $t_n(s)$ appearing in Theorem \ref{th:main} approximates the value $e^{nP(A_1,\ldots,A_N;s)}$ whereas the coefficients $a_n(s)$ are shown in Theorem \ref{th:main} to decrease to zero with super-exponential speed. The small size of $a_n(s)$ is thus attributable to additive cancellation between potentially very large summands. It is therefore important in implementation to record the traces $t_n(s)$ to much higher accuracy than is desired for the ultimate approximation. In the computations which follow the traces $t_n(s)$ were calculated in arbitrary precision, reducing to finite precision only for the outcome of the calculation of the coefficients $a_n(s)$.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{self-affine-example.png}
\caption{The attractor of the iterated function system $(T_1,T_2,T_3)$ defined in Example 2. It has Hausdorff dimension equal to the affinity dimension of its defining iterated function system, which we compute in Table \ref{ta:aff1} to be approximately 1.4467637738623842971562010827909722280436\ldots}\label{fi:bbing}
\end{figure}
\begin{table}
\begin{center
\begin{tabular}{*3c}
\toprule
n & Approximation to affinity dimension& CPU time\\
\midrule
1& 1.57850 39107 24303 42569 39013 22778 88907 20542 & 0.046s\\
2& \underline{1}.43428 20777 82633 21247 87188 76730 31996 86014& 0.044s\\
3&\underline{1.4}4698 63740 68855 64166 13462 60397 02738 95013&0.064s\\
4 & \underline{1.446}76 23250 25528 19736 40628 61933 67159 40086& 0.11s\\
5 &\underline{1.44676} 37772 54098 43296 70430 41085 33834 29566&0.20s\\
6 &\underline{1.44676} \underline{377}38 59463 32542 61749 94490 38856 75805&0.47s\\
7 &\underline{1.44676} \underline{37738} 62385 23207 08694 66251 21939 16812&1.1s\\
8 &\underline{1.44676} \underline{37738} \underline{6238}4 29704 44057 28444 21373 26314& 3.3s\\
9& \underline{1.44676} \underline{37738} \underline{62384} \underline{297}15 62060 91538 64348 53245&9.7s\\
10&\underline{1.44676} \underline{37738} \underline{62384} \underline{29715} \underline{620}10 82706 74910 92449&36s\\
11 & \underline{1.44676} \underline{37738} \underline{62384} \underline{29715} \underline{62010} \underline{827}90 97276 01619& 114s\\
12 & \underline{1.44676} \underline{37738} \underline{62384} \underline{29715} \underline{62010} \underline{82790} \underline{972}22 80423&360s\\
13 & \underline{1.44676} \underline{37738} \underline{62384} \underline{29715} \underline{62010} \underline{82790} \underline{97222} \underline{804}36&1300s\\
14 & \underline{1.44676} \underline{37738} \underline{62384} \underline{29715} \underline{62010} \underline{82790} \underline{97222} \underline{80436}&4400s\\
\bottomrule
\end{tabular}\bigskip\medskip
\caption{Approximations to the affinity dimension of Example 2 calculated using Theorem \ref{th:main} and the secant method as described in \S\ref{ss:urk}, implemented in Wolfram Mathematica. The CPU time used in each approximation is as reported by Mathematica's {\tt{Timing}} function. Digits which are empirically observed to have converged to a stable value are underlined.}\label{ta:aff1}
\end{center}
\end{table}
\subsection{Example 1: a pair of dominated matrices}\label{ss:slary}
Define
\[A_1:=
\begin{pmatrix}
-\frac{4}{7}&\frac{5}{7}\\
0&\frac{1}{7}
\end{pmatrix},\qquad
A_2:=\begin{pmatrix}
\frac{1}{7}&0\\
-\frac{5}{7}&-\frac{4}{7}
\end{pmatrix}.
\]
We claim that the pair $(A_1,A_2)$ is $1$-dominated. Indeed, define
\[\mathcal{C}_1:=\left\{\begin{pmatrix}x\\y\end{pmatrix} \in \mathbb{R}^2 \colon |x|\geq 2|y|\right\},\]
\[\mathcal{C}_2:=\left\{\begin{pmatrix}x\\y\end{pmatrix} \in \mathbb{R}^2 \colon |y|\geq 2|x|\right\}.\]
If $(x,y)^\top \in \mathcal{C}_1$ then
\[\left|\frac{5}{7}y-\frac{4}{7}x\right|\geq \frac{4}{7}|x|-\frac{5}{7}|y|\geq \frac{3}{7}|y|\geq \left|\frac{2}{7}y\right|\]
and equality of the first and last terms is only possible if $y=0$ and consequently $x=0$. In particular if $(x,y)^\top \in \mathcal{C}_1$ is nonzero we obtain $A_1(x,y)^\top \in \Int \mathcal{C}_1$. Moreover for $(x,y)^\top \in \mathcal{C}_1$ we also have
\[\left|\frac{4}{7}y+\frac{5}{7}x\right|\geq \frac{5}{7}|x|-\frac{4}{7}|y|\geq \frac{3}{7}|x| \geq \left|\frac{2}{7}x\right|\]
which yields $A_2(x,y)^\top \in \Int \mathcal{C}_2$ when $(x,y)^\top$ is nonzero. In a similar manner, if $(x,y)^\top \in \mathcal{C}_2$ then
\[\left|\frac{5}{7}y-\frac{4}{7}x\right|\geq \frac{5}{7}|y|-\frac{4}{7}|x|\geq \frac{3}{7}|y|\geq \left|\frac{2}{7}y\right|\]
and
\[\left|\frac{4}{7}y+\frac{5}{7}x\right|\geq \frac{4}{7}|y|-\frac{5}{7}|x|\geq \frac{3}{7}|x|\geq \left|\frac{2}{7}x\right|\]
which respectively give $A_1(x,y)^\top \in \Int \mathcal{C}_1$ and $A_2(x,y)^\top \in \Int \mathcal{C}_2$ when $(x,y)^\top$ is nonzero.
If we now let $w=(1,1)^\top$ then $\langle u,w\rangle$ is never zero for any nonzero $u \in \mathcal{C}_1 \cup \mathcal{C}_2$, so defining
\[\mathcal{K}_i:=\left\{u \in \mathcal{C}_i\colon \langle u,w\rangle>0\right\}\]
for $i=1,2$ it is not difficult to see that $(\mathcal{K}_1,\mathcal{K}_2)$ is a multicone for $(A_1,A_2)$. In particular Theorem \ref{th:main} may be applied to estimate the affinity dimension of the pair $(A_1,A_2)$. Let $(B_1,B_2):=(A_1,-A_2)$. Since
\begin{align*}e^{P(A_1,A_2;1)}=e^{P(B_1,B_2;1)}&=\lim_{n \to \infty} \left(\sum_{|\mathtt{i}|=n} \left\|B_\mathtt{i}\right\|\right)^{\frac{1}{n}} \\
&\geq \lim_{n \to \infty} \left\|\sum_{|\mathtt{i}|=n} B_\mathtt{i}\right\|^{\frac{1}{n}}\\
&=\lim_{n \to \infty} \left\|(B_1+B_2)^n\right\|^{\frac{1}{n}}=\rho(B_1+B_2)=\frac{\sqrt{50}}{7}>1\end{align*}
and
\[e^{P(A_1,A_2;2)}=|\det A_1|+|\det A_2|=\frac{8}{49}<1\]
we infer that $\dimaff(A_1,A_2) \in (1,2)$. The first 20 approximations to the affinity dimension of $(A_1,A_2)$ are tabulated in Table \ref{ta:blen}.
\subsection{Example 2: a positive planar iterated function system}
Define
\begin{align*}T_1\begin{pmatrix}x\\y\end{pmatrix} &:=\begin{pmatrix}\frac{1}{3} & \frac{1}{9}\\ \frac{1}{2}&\frac{1}{2}\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix},\\
T_2\begin{pmatrix}x\\y\end{pmatrix} &:=\begin{pmatrix}-\frac{1}{2} & -\frac{1}{3}\\ -\frac{1}{3}&-\frac{1}{2}\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix}1\\1\end{pmatrix},\\
T_3\begin{pmatrix}x\\y\end{pmatrix}& :=\begin{pmatrix}\frac{1}{2} & \frac{1}{2}\\ \frac{1}{9}&\frac{1}{3}\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix},\end{align*}
and let $A_1,A_2,A_3 \in M_2(\mathbb{R})$ denote the linear parts of $T_1,T_2,T_3$ respectively. Let $X \subset \mathbb{R}^2$ denote the attractor of $(T_1,T_2,T_3)$. One may show that the strong open set condition is satisfied by $(T_1,T_2,T_3)$ and using \cite{BaHoRa18} one may show that $\dim_H X = \dim_BX= \dimaff (A_1,A_2,A_3)$. It is easily verified that $\rho(A_1-A_2+A_3)>1$ and $|\det A_1|+|\det A_2|+|\det A_3|<1$ and in a similar manner to the previous example we deduce that $\dimaff(A_1,-A_2,A_3)=\dimaff(A_1,A_2,A_3) \in (1,2)$. The Hausdorff dimension $\dim_HX = \dimaff (A_1,A_2,A_3)$ may thus be approximated using Theorem \ref{th:main}; the first 14 approximations are presented in Table \ref{ta:aff1}.
\subsection{Example 3: a three-dimensional iterated function system}
\begin{table}
\begin{center
\begin{tabular}{*3c}
\toprule
n & Approximation to affinity dimension&CPU time\\
\midrule
3&1.74010 38961 34544 64381 66016 57752 82592 79145&0.067s\\
4 & \underline{1}.53612 13489 34570 18769 13237 56458 61628 45041&0.10s\\
5 &\underline{1.5}8779 31446 44939 17928 98900 28708 16065 92496&0.15s\\
6 & \underline{1.58}459 23810 06597 43285 21249 54866 32813 68839&0.22s\\
7 &\underline{1.584}77 97771 44149 34557 48903 92413 22985 52229&0.33s\\
8 & \underline{1.58477} 17757 07488 53767 71488 42424 52891 52003&0.63s\\
9 & \underline{1.58477} 20386 65944 76377 72361 85895 44529 09738&0.80s\\
10 & \underline{1.58477} \underline{203}18 53062 52952 58955 36166 25319 46959&1.4s\\
11 & \underline{1.58477} \underline{2031}9 95110 47059 43620 26740 31575 13317&2.4s\\
12 & \underline{1.58477} \underline{20319} \underline{9}2686 60697 00747 19778 01115 41015&5.4s\\
13 & \underline{1.58477} \underline{20319} \underline{92}720 93370 05697 62846 36869 58071&12s\\
14 & \underline{1.58477} \underline{20319} \underline{92720} 52545 02878 00445 78535 74528&27s\\
15 & \underline{1.58477} \underline{20319} \underline{92720} \underline{52}956 88351 89418 63989 50927&59s\\
16 & \underline{1.58477} \underline{20319} \underline{92720} \underline{5295}3 32862 81715 84179 24019&130s\\
17& \underline{1.58477} \underline{20319} \underline{92720} \underline{52953} \underline{3}5507 79078 84111 41677&270s\\
18 & \underline{1.58477} \underline{20319} \underline{92720} \underline{52953} \underline{35}490 71502 87276 30757&560s\\
19 & \underline{1.58477} \underline{20319} \underline{92720} \underline{52953} \underline{35490} 81124 12318 84553&1200s\\
20 & \underline{1.58477} \underline{20319} \underline{92720} \underline{52953} \underline{35490} \underline{81}076 56294 07542&2800s\\
21 & \underline{1.58477} \underline{20319} \underline{92720} \underline{52953} \underline{35490} \underline{81076} 77018 06325&5900s\\
\bottomrule
\end{tabular}\bigskip\medskip
\caption{Approximations to the affinity dimension of Example 3 calculated using Theorem \ref{th:main} and the secant method as described in \S\ref{ss:urk}, implemented in Wolfram Mathematica. The CPU time used in each approximation is as reported by Mathematica's {\tt{Timing}} function. Digits which are empirically observed to have converged to a stable value are underlined. Convergence is noticeably slower than for two-dimensional examples: in this context our bound for the error in the $n^{\mathrm{th}}$ approximation is $O(\exp(-\gamma n^{5/4}))$ as opposed to $O(\exp(-\gamma n^{2}))$ in the other examples. For $n=1,2$ the approximation to the pressure function has no root in $(1,2)$ and these lines are therefore omitted.
}\label{ta:aff2}
\end{center}
\end{table}
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{3d.png}
\caption{A projection of the attractor of the iterated function system defined by Example 3. Approximations to the affinity dimension computed using Theorem \ref{th:main} are listed in Table \ref{ta:aff2}. It is known from work of Falconer \cite[\S5]{Fa88} that the upper box dimension $\overline{\dim}_BX$ is bounded above by $\dimaff(A_1,A_2)$, but unlike the planar example in Figure \ref{fi:bbing} current techniques are not powerful enough to determine whether or not $\dim_H X= \dimaff (A_1,A_2)$.}\label{fi:sting}
\end{figure}
Consider $(A_1,A_2)$ where
\[A_1:=\frac{1}{12}\begin{pmatrix}5 & 4 & 1\\ 5& 5 & 4\\ 0 &1 &5\end{pmatrix},\qquad A_2:=\frac{1}{12}\begin{pmatrix}5 & 5 & 0\\ 4& 5 & 1\\ 1 &4 &5\end{pmatrix}=A_1^\top \]
and note that $A_1$ and $A_2$ are contractions in the Euclidean norm. It is easily checked that $(A_1A_1,A_1A_2,A_2A_1,A_2A_2)$ is a tuple of positive invertible matrices and is therefore $1$-dominated. By consideration of Theorem \ref{th:bogo} it follows that \eqref{eq:horse} holds for $(A_1A_1,A_1A_2,A_2A_1,A_2A_2)$ and obviously \eqref{eq:horse} therefore also holds for $(A_1,A_2)$. We conclude that $(A_1,A_2)$ is likewise $1$-dominated.
We identify each $A_i$ with the corresponding linear map $\mathbb{R}^3 \to \mathbb{R}^3$ defined by $A_i$ with respect to the standard basis $e_1,e_2,e_3$ of $\mathbb{R}^3$. With respect to the basis $e_1\wedge e_2, e_1\wedge e_3,e_2\wedge e_3$ for $\wedge^2\mathbb{R}^3$ we have
\[A_1^{\wedge 2}=\frac{1}{144}\begin{pmatrix}5&15&11 \\5&25&19\\5&25&21\end{pmatrix},\qquad A_2^{\wedge 2}=\frac{1}{144}\begin{pmatrix}5&5&5\\15&25&25\\11&19&21\end{pmatrix}.\]
Since $(A_1^{\wedge 2},A_2^{\wedge 2})$ is thus representable by a pair of positive matrices we see that $(A_1,A_2)$ is both $1$-and $2$-dominated. Using non-negativity it follows by a theorem of Yu. V. Protasov (\cite{Pr10}) that
\[\lim_{n \to \infty} \left(\sum_{|\mathtt{i}|=n} \|A_\mathtt{i}\| \right)^{\frac{1}{n}}=\rho(A_1+A_2)>1\]
and
\[\lim_{n \to \infty} \left(\sum_{|\mathtt{i}|=n} \left\|A_\mathtt{i}^{\wedge 2}\right\| \right)^{\frac{1}{n}}=\rho\left(A_1^{\wedge 2}+A_2^{\wedge 2}\right)<1.\]
Thus $P(A_1,A_2;1)>0>P(A_1,A_2;2)$ and consequently $\dimaff (A_1,A_2) \in (1,2)$, and we conclude that Theorem \ref{th:main} is applicable to the computation of $\dimaff (A_1,A_2)$. The first 21 approximations to $\dimaff (A_1,A_2)$ are presented in Table \ref{ta:aff2}. An illustration of the attractor of the iterated function system
\[T_1\begin{pmatrix}x\\y\\z\end{pmatrix} := \frac{1}{12}\begin{pmatrix}5 & 4 & 1\\ 5& 5 & 4\\ 0 &1 &5\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}+\begin{pmatrix}1\\0\\0\end{pmatrix}\]
\[T_2\begin{pmatrix}x\\y\\z\end{pmatrix} :=\frac{1}{12}\begin{pmatrix}5 & 5 & 0\\ 4& 5 & 1\\ 1 &4 &5\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}+\begin{pmatrix}0\\0\\1\end{pmatrix}\]
is given in Figure \ref{fi:sting}.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{morris-shmerkin.png}
\caption{This self-affine set was shown in \cite[\S6.6]{MoSh17} to have Hausdorff dimension equal to the affinity dimension of the defining iterated function system. However, the linear parts of the defining affine transformations have non-real eigenvalues and Theorem \ref{th:main} is not applicable. Non-rigorous estimates using the discretisation method described in \S\ref{se:nondom} as tabulated in Table \ref{ta:pdance} suggest that the affinity dimension is equal to approximately 1.522688.}\label{fi:gs}
\end{figure}
\section{Non-dominated matrices}\label{se:nondom}
If $(A_1,\ldots,A_N) \in M_2(\mathbb{R})^N$ is a tuple of invertible matrices which is not $1$-dominated then by a line of reasoning due to A. Avila \cite{Yo04} there exist tuples $(A_1',\ldots,A_N')$ arbitrarily close to $(A_1,\ldots,A_N)$ with the property that some product $A_{i_1}'\cdots A_{i_n}'$ has complex eigenvalues. For such matrices the formula for $t_n(s)$ in Theorem \ref{th:main} has no clear meaning, and also for such matrices no open subset of $\mathbb{RP}^1$ may be found which is mapped strictly inside itself by the action of the matrices $A_i'$, preventing the construction of a trace-class transfer operator in direct mimicry of Theorem \ref{th:main}. For such matrices it is therefore difficult to see how any reasonable adaptation of Theorem \ref{th:main} could be made. In this sense we believe that $1$-domination, or multipositivity, is the weakest open condition on the matrices $A_1,\ldots,A_N$ which permits a version of Theorem \ref{th:main} to be proved.
\begin{table}
\begin{center}
\begin{tabular}{*3c}
\toprule
Mesh size & \makecell{Approximation to \\affinity dimension} & CPU time\\
\midrule
$2$&1.45960943& 0.0048s\\
$2^2$&\underline{1}.\underline{4}3685698 & 0.0061s\\
$2^3$&\underline{1}.\underline{4}3981279 & 0.0096s\\
$2^4$&\underline{1}.\underline{44}686353 & 0.047s\\
$2^5$&\underline{1}.\underline{44}707711 & 0.070s\\
$2^6$&\underline{1}.\underline{44}682259 & 0.11s\\
$2^7$&\underline{1}.\underline{446}69990 & 0.21s\\
$2^8$&\underline{1}.\underline{446}78054 & 0.42s\\
$2^9$&\underline{1}.\underline{4467}7211 & 0.83s\\
$2^{10}$&\underline{1}.\underline{4467}6722 & 1.5s\\
$2^{11}$&\underline{1}.\underline{44676}500 & 3.2s\\
$2^{12}$&\underline{1}.\underline{44676}360 & 6.0s\\
$2^{13}$&\underline{1}.\underline{446763}39 & 12s\\
$2^{14}$&\underline{1}.\underline{446763}78 & 31s\\
$2^{15}$&\underline{1}.\underline{446763}85 & 96s\\
$2^{16}$&\underline{1}.\underline{446763}76 & 460s\\
$2^{17}$&\underline{1}.\underline{4467637}5 & 2400s\\
\bottomrule
\end{tabular}\bigskip\medskip
\caption{Estimates of the affinity dimension of Example 2 calculated using the non-rigorous discretisation method described in \S\ref{se:nondom}. At mesh sizes above around $2^{10}$ the result shows good agreement with Table \ref{ta:aff1} but convergence is slow thereafter. Digits which are empirically observed to have converged to a stable value are underlined.}\label{ta:nkie}
\end{center}
\end{table}
However, for non-dominated matrices it is still possible to obtain non-rigorous estimates of the affinity dimension by other techniques. Given $A_1,\ldots,A_N \in GL_2(\mathbb{R})$ and $s \in [0,1]$ we may define an operator $\mathscr{L}_s \colon C^\alpha(\mathbb{RP}^1) \to C^\alpha(\mathbb{RP}^1)$ by
\[\left(\mathscr{L}_sf\right)(\overline{u}):=\sum_{i=1}^N \left(\frac{\|A_iu\|}{\|u\|}\right)^s f\left(\overline{A_iu}\right),\]
and for $s \in [1,2]$ by
\[\left(\mathscr{L}_sf\right)(\overline{u}):=\sum_{i=1}^N \left(\frac{\|A_iu\|}{\|u\|}\right)^{2-s} |\det A_i|^{s-1} f\left(\overline{A_iu}\right),\]
in such a manner that
\[\rho(\mathscr{L}_s)=\lim_{n\to\infty}\left(\sum_{i_1,\ldots,i_n=1}^N \varphi^s\left(A_{i_1}\cdots A_{i_n}\right)\right)^{\frac{1}{n}}\]
and such that $\rho(\mathscr{L}_s)$ is a simple eigenvalue of $\mathscr{L}_s$, as long as $\alpha \in (0,1)$ is chosen suitably small (in a manner which in general will depend on $s$) and mild algebraic non-degeneracy conditions on $(A_1,\ldots,A_N)$ are met. (These spectral properties are guaranteed by, for example, \cite[Th\'eor\`eme 8.8]{GuLe04}.) We could then hope to estimate the spectral radius of $\mathscr{L}_s$ for different values of $s$ by discretising the phase space $\mathbb{RP}^1$, constructing a large matrix representing a discretised action of $\mathscr{L}_s$, and working on the supposition that the spectral radius of the matrix is a good approximation to $\rho(\mathscr{L}_s)$ and hence to $e^{P(A_1,\ldots,A_N;s)}$. In practical experiments we were able to obtain around five decimal places of accuracy for the affinity dimension by discretising $\mathbb{RP}^1$ into approximately $10^5$ evenly-spaced mesh points: see Tables \ref{ta:nkie} and \ref{ta:pdance}. We observe in particular that the results obtained in Table \ref{ta:nkie} show good agreement with Theorem \ref{th:main} when tested on the multipositive matrix set described in Example 2. However, we have not been able to make this method of estimation rigorous. This approach could also be applied to higher-dimensional affine iterated function systems but we have not investigated the matter of finding suitable discretisations of the more complicated phase spaces required in this context.
\begin{table}
\begin{center}
\begin{tabular}{*3c}
\toprule
Mesh size & \makecell{Approximation to \\affinity dimension} & CPU time\\
\midrule
$2$&1.50000000& 0.0028s\\
$2^2$&\underline{1}.\underline{5}1578683 & 0.0025s\\
$2^3$&\underline{1}.\underline{5}1254065 & 0.0047s\\
$2^4$&\underline{1}.\underline{52}070716 & 0.033s\\
$2^5$&\underline{1}.\underline{52}415711 & 0.059s\\
$2^6$&\underline{1}.\underline{52}305542 & 0.079s\\
$2^7$&\underline{1}.\underline{52}290806 & 0.13s\\
$2^8$&\underline{1}.\underline{522}62668 & 0.26s\\
$2^9$&\underline{1}.\underline{522}69395 & 0.61s\\
$2^{10}$&\underline{1}.\underline{522}70408 & 1.1s\\
$2^{11}$&\underline{1}.\underline{522}69152 & 2.2s\\
$2^{12}$&\underline{1}.\underline{5226}8717 & 4.5s\\
$2^{13}$&\underline{1}.\underline{52268}810 & 7.7s\\
$2^{14}$&\underline{1}.\underline{52268}795 & 18s\\
$2^{15}$&\underline{1}.\underline{522687}80 & 55s\\
$2^{16}$&\underline{1}.\underline{5226878}0 & 220s\\
$2^{17}$&\underline{1}.\underline{5226878}2 & 1400s\\
\bottomrule
\end{tabular}\bigskip\medskip
\caption{Estimates of the affinity dimension of the iterated function system defined in \cite[\S6.6]{MoSh17} and illustrated in Figure \ref{fi:gs}, calculated using the non-rigorous discretisation method described in \S\ref{se:nondom}. Digits which are empirically observed to have converged to a stable value are underlined. No rigorous estimate of the affinity dimension of this IFS is currently available.}\label{ta:pdance}
\end{center}
\end{table}
\section{Acknowledgements}
This research was supported by the Leverhulme Trust (Research Project Grant number RPG-2016-194). The author thanks O. Bandtlow for helpful comments and suggestions.
|
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For a limited time only, you can get OpenCV with Python Blueprints and any other eBook or video for only $10 on the Packt website. That's a staggering 77% discount! The flash sale will be going on for only a few days, so if you've been toying with the idea of getting some of these books, make sure to act now.
In the latest edition of this monthly digest series you can learn how your brain activity changes under the influence of psychedelic drugs, why brain games won't actually make you smarter, and how gene therapy might be able to treat patients blinded from retinitis pigmentosa.
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"redpajama_set_name": "RedPajamaC4"
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\section{Introduction and background}\label{sec:intro}
\subsection{Preamble}
Mathematics is fundamental to the design and analysis of voting systems
(see, for example, the books \cite{BY1982,Birk11, FM, LV, Puk2017}).
Mathematical models for human behaviour frequently involve probability, and they
invariably rely upon assumptions whose validity is ripe for debate. As a general rule,
the greater are the assumptions, the more precise are the conclusions.
A balance needs be struck between tractability and applicability:
excessive assumptions tend to undermine practical relevance, whereas \lq\lq nothing will come of nothing"
[Shakespeare].
Assumption-based conclusions must be exposed to a robustness analysis: to what degree are they
robust when the underlying assumptions are perturbed?
These issues are illustrated here in a study of the so-called Jagiellonian Compromise (JagCom)
of \SZ\ \cite{SZ04,SZ06,SZ07,SZ11}. The JagCom is a proposal for a two-tier voting system such as
that of the Council of the European Union (see Section \ref{ssec13} for further details).
It is based on (i) \emph{square-root weights} coupled with (ii) a certain formula for the \emph{quota}.
While the current work was born out of an interest in learning about the JagCom, it has developed
into (i) a broader study of the notion of power (or influence) for general probability distributions,
combined with (ii) a critical analysis of the arguments leading to the given quota.
The conclusions of this article illuminate the balance
between theory and practical relevance.
The political context of this paper is as follows. The debate rolls on concerning
the allocation of seats in the European Parliament (EP)
between the Member States of the EU (see, for example, \cite{Cart,PukG17}).
It has been argued by members of the Constitutional Affairs Committee (AFCO) of the EP that strategic
reform of the somewhat \emph{ad hoc} method of allocation of parliamentary seats should be considered only in parallel
to a review of the two-tier voting system of the EU Council.
The JagCom is a leading theoretical contender for implementation in the Council.
It has been supported in two open letters to EU governments, \cite{openl2,openl1}, signed by significant
numbers of prominent theoreticians, and it has been the subject of a volume of
positive publicity including \cite{KSZ07,Mach07,comp,SZ10,ZSZ}.
The discussion in Brussels is likely to intensify in the months and years to come,
and this is a propitious moment to re-examine the JagCom in some detail.
There are two principal parts to this paper, as outlined in the following two subsections.
The first concerns the definition of the \lq power' of an individual (as introduced by Penrose)
for general probability distributions. This is connected to the choice of weights in a two-tier voting
system such as the JagCom. The second is a discussion of the choice of quota in the JagCom.
\subsection{Power and influence}\label{sec:1.2}
Lionel Penrose's 1946 paper \cite{Pen} is a fundamental work in the mathematical theory of voting,
and it has received a great deal of attention. Penrose found it convenient to assume that members
of a population choose their votes \emph{independently} at random, and are
\emph{equally likely} to choose either of the two possible outcomes.
These assumptions of independence and unbiasedness lead to a mathematically sophisticated
theory based around the classical study of the sums of so-called independent and identically
distributed (\lq iid'), symmetric random variables (see \cite{Pet1975},
or the less sophisticated account \cite[Sect.\ 5.10]{GS01}).
That said, independence and unbiasedness may, in practice, be far from the truth
in specific cases.
The square-root voting system of Penrose \cite{Pen} is prominent in
discussion of two-tier voting systems in general,
and in specific of that of the Council of the European Union
(see, for example, \cite{Kir16,SZ11}).
The challenge confronted by Penrose is to devise a system for pooling the views of a number of Member States with varying
population sizes. What weight $w_j$ should be assigned
to the opinion of State $j$, having a population of size $N_j$?
The Penrose system amounts to the proposal $w_j \propto \sqrt{N_j}$.
The essence of Penrose's argument is the observation that the number $H$ of heads shown
in $N_j$ independent, unbiased coin tosses satisfies
\begin{equation}\label{1}
\EE\left|H-\tfrac12 N_j\right| \sim \sqrt {2N_j/\pi}, \qq \text{for large } N_j.
\end{equation}
(Here and later, $\EE$ denotes expectation, and $\PP$ denotes probability.)
We shall refer to \eqref{1} as \emph{Penrose's second square-root law}.
The reader is referred back to \cite{Pen} for the deduction of square-root weights from \eqref{1},
although s/he may prefer to read Kirsch's least-squares argument as presented in Section \ref{sec:3.2}.
(Although \eqref{1}, and the subsequent \eqref{4}, are \emph{asymptotic} relations, sharp bounds
may be obtained by elementary methods.)
\begin{remark}\label{rem12}
Penrose defines the \lq edge' as $|N_F-N_A|$, where $N_F$ (\resp, $N_A$) is
the number of votes in favour (\resp, against) the motion. It is immediate
that $|N_F-N_A|=|H-(N_j-H)| = |2H-N_j|$, so that the mean edge equals
$2 \EE|H-\frac12 N_j|$.
\end{remark}
Penrose \cite{Pen} discussed also the concept of the \lq power'
(termed \lq influence' in the current work, after \cite{BOL,Ru81})
of an individual voter within a given election or vote.
He noted that, in a vote within a State containing $N_j$ individuals whose votes are independent
and unbiased, this
power, denoted $\al_j$, has order
\begin{equation}\label{4}
\al_j \sim \sqrt{2/(\pi N_j)}, \qq \text{for large } N_j,
\end{equation}
(see also Banzhaf \cite{Bz}).
We shall refer to \eqref{4} as \emph{Penrose's first square-root law}.
Only one square-root is sometimes attributed to Penrose. In \cite{Pen}, he stated \eqref{1} and
he proved \eqref{4},
and he did not note their inter-relationship. Some later authors
have linked \eqref{1} and \eqref{4} by proposing a weight $w_j'$ for State $j$
such that the product $\al_j w_j'$
does not depend on population-size, that is, $w_j' \propto \sqrt{N_j}$, in agreement with \eqref{1}
(see, for example, the discussion at \cite[p.\ 48 ff.]{SZ10} and \cite[Sect.\ 1]{SZ11}).
This argument appears to assume that: (i) in a population with size $N_j$ and individual
power $\al_j$, the collective power is $\al_j N_j$, and (ii) $1/\al_j$ has, generically, the same order
as $\EE|H-\tfrac12 N_j|$. The first assumption here is open to discussion, and the second
is false for general distributions. Proposition \ref{prop1} and Remark \ref{rem33} explain the
true relationship between \eqref{1} and \eqref{4} in the context of general
probability distributions.
The square-root laws of this article are those of Penrose \cite{Pen}. It is not the current purpose to
discuss in detail the relationship between voting power and voting weight (see, for example,
\cite{AE,FM,Napel-ox}).
In Section \ref{sec:inf}, we introduce the notions of the \emph{absolute} and the \emph{conditional influences}
of an individual in an election. The absolute influence is that considered by Penrose and later authors;
the conditional influence is sometimes considered more appropriate in situations where
individuals' votes are \emph{dependent} random variables. The two quantities are equal
in the independent case, but not generally so.
\subsection{The Jagiellonian Compromise, a two-tier voting system}\label{ssec13}
In a method since dubbed the \lq Jagiellonian Compromise' (JagCom), \SZ\ \cite{SZ04,SZ06,SZ07,SZ11}
have proposed the following two-tier voting system using square-root weights
together with a particular value $q^*$ for the quota $q$.
Writing $N_1,N_2,\dots,N_s$ for the populations of the $s$ States of the Union,
under the JagCom a motion is passed if and only if
\begin{equation}\label{2}
\sum_{j\in J} \sqrt{N_j} - \sum_{j\in \ol J} \sqrt{N_j} > q^*W,
\end{equation}
where $J$ is the set of States voting in favour of the motion, $\ol J$ is the set voting against, and
\begin{equation*}\label{2a}
q^*:= \frac{\sqrt N}W,\q W= \sum_{j=1}^s \sqrt {N_j},
\q N=\sum_{j=1}^s N_j.
\end{equation*}
The value $q=q^*$ given in \eqref{2} is supported by a
heuristic argument based on approximation
by a Gaussian distribution. Although no rigorous justification of this approximation is
yet available (see Appendix \ref{sec:4.2} of the current work), its conclusions gain
some support using exact numerical methods (see Section \ref{sec:43}).
Note that the $q^*$ of \eqref{2a} is not quite the quota of \cite{SZ07}, but rather that
of \cite{Kir07}.
Salient features of two-tier voting systems are summarised in Section \ref{sec:ttv}, with
special attention to the work of Kirsch \cite{Kir07,Kir16} and \SZ\
\cite{SZ04,SZ06,SZ07,SZ11}. This is followed by
a discussion in Section \ref{sec:sinf} and Appendix \ref{sec:4.2} of the
influences of the weighted States within the Council, and of the use
and potential misuse of the Gaussian approximation
in estimating certain related probabilities.
The closing Section \ref{sec:rems} contains some reflections on the JagCom,
and in particular the following conclusions.
\begin{numlist}
\item The JagCom is derived via a set of principles that can be stated
unambiguously and analysed rigorously, and the resulting system is
robust with respect to population changes. Nevertheless,
these principles are arguably fragile and unrealistic, and insufficiently
sensitive to political realities.
\item Despite fragility in the assumptions about voting patterns
made by the JagCom,
we offer no superior alternative here. The
problem of allocating weights is more than just a mathematical puzzle,
but demands a more extensive political vision.
\item The justification for the proposed JagCom quota $q^*$ is empirical and numerical rather
than rigorous. These numerics provide only equivocal guidance which
is satisfied by a range of values of the quota, including the simpler value $q=0$.
Indeed, $q=0$ is the value for which individual powers are maximized.
Given the very modest differences in performance between such values of the quota,
the final choice is best determined through \emph{political} input.
\end{numlist}
\subsection{Resum\'e}
Certain assumptions appear to be necessary for the above analyses, and these are examined
in the current article. There are four areas that receive special attention, namely:
\begin{letlist}
\item the underlying model in which individuals vote according to an unbiased coin toss,
independently of other voters [Section \ref{sec:23}],
\item an alternative interpretation of the concept of \lq voting power' or \lq influence'
[Section \ref{sec:22}],
\item the assumptions of mathematical smoothness under which the Gaussian approximation is
suitable for finite populations [Appendix \ref{sec:4.2}],
\item some implications of exact computations of voting powers in two-tier systems
[Section \ref{sec:43}].
\end{letlist}
Numerous earlier authors have of course considered some of these issues, namely (a),
(c), and (d), and we mention \cite{Kir16, KLMT, KN, LM04, ER, SZ07}.
\begin{remark}[on the literature]
There is an extensive existing literature on the matters considered in this article, and the author
has attempted to include appropriate references.
Apologies are offered to those authors whose work is not listed explicitly.
\end{remark}
\section{Absolute and conditional influence}\label{sec:inf}
\subsection{The history and context of influence}
The concept of \lq influence' is central in the probability theory of disordered systems.
Consider a system that comprises $m$ sub-systems. These could be, for example, individual voters in an election,
nodes in a disordered medium (as in the percolation model), or
particles in a model for the ferromagnet (such as the Ising/Potts models). In studying the behaviour
of the collective system, it is often key to understand the effect of a variation within a given sub-system.
That is, what is the probability that a change in
a given sub-system has a substantial effect on the collective system?
The quantification of influence is long recognised as
being central to the understanding of complex random systems.
For example, influence in voting systems was studied by Ben-Or and Linial \cite{BOL} in
work that played an important role in stimulating a systematic theory of influence and sharp threshold with
many applications in random systems (see \cite{KalS} for a review).
In percolation theory, the influence of a node is the probability that the
node is pivotal for a given global event
(see \eqref{eq:pivotal} for
the definition of pivotality). Estimates for influence are key to most of the principal results
for percolation (see \cite{G99}, for example). In these two areas above, the sub-systems are generally taken
to be \emph{independent} random variables. This is, however, not so for
a number of important processes of statistical mechanics including the Ising and Potts models,
in which the sub-systems are dependent but usually positively correlated. For such systems,
\lq influence' requires a new definition, and this is provided in \cite{GG,GG11} in the context of the
Ising and random-cluster models (see \cite{G-RC}).
The origins of influence are rather older than the above work, and go back at least to the work of
Penrose \cite{Pen} in 1946 and possibly the reliability literature surveyed by Barlow
and Proschan \cite{BP65} in 1965. The two definitions of influence, referred to above, are presented next
in the context of a voting system (we shall use the standard terminology of
probability theory and the theory of interacting systems).
\subsection{Definitions of absolute and conditional influences}\label{sec:22}
There is a population $P$ containing $m$ individuals, and a vote is taken between two possible
outcomes, labelled $+1$ and $-1$. Each individual votes either $+1$ or $-1$.
We write $X(i)$ for the vote of person $i$, and we assume the $X(i)$ are random variables.
The \emph{vote-vector} $X=(X(1),X(2),\dots,X(m))$ takes values in the so-called
\lq configuration space' $\Si=\{-1,1\}^m$.
There exists a predetermined subset $A \subseteq \Si$, and the vote is declared to \emph{pass} if
and only if $X \in A$. It is normal to consider sets $A$ which are \emph{increasing} in that
\begin{equation}\label{eq:inc}
\si\in A,\ \si\le\si'\q \Rightarrow \q \si'\in A.
\end{equation}
The inequality $\si\le\si'$ refers to the natural partial order on $\Si$ given by
$$
\si\le\si' \q\text{if and only if}\q \si(i)\le \si'(i) \text{ for all }i \in P.
$$
For concreteness, \emph{we assume henceforth that $A$ is an increasing subset of $\Si$}
(that is, an \lq increasing event').
For $i\in P$ and a configuration $\si=(\si(1),\si(2),\dots,\si(m))\in\Si$,
we define the vectors $\si^i$, $\si_i$ by
\begin{equation}\label{eq:siud}
\si^i(j) = \begin{cases} 1 &\text{if } j=i,\\ \si(j) &\text{otherwise,}\end{cases}\qq
\si_i(j) = \begin{cases} -1 &\text{if } j=i,\\ \si(j) &\text{otherwise.}\end{cases}\qq
\end{equation}
That is, $\si^i$ (\resp, $\si_i$) agrees with $\si$ except possibly at $i$,
with $i$'s vote set to $1$ (\resp, $-1$).
Individual $i$ is called \emph{pivotal} if the outcome of the vote changes when s/he changes opinion
(the words \emph{decisive} and \emph{critical} are
sometimes used in the voting literature). More formally,
$i$ is called \emph{pivotal} for the configuration $\si$ if
\begin{equation}\label{eq:pivotal}
\si_i\notin A, \qq \si^i \in A.
\end{equation}
This holds since $A$ is assumed increasing.
In all situations considered in this paper, the individual votes $X(i)$ are assumed
to be identically distributed and symmetric in that
\begin{equation}\label{eq:Xsym}
\PP(X(i)=1) = \PP(X(i)=-1)=\tfrac12,
\end{equation}
where $\PP$ denotes the probability measure that governs the vote-vector $X$.
Assumptions of independence will be introduced where appropriate.
\begin{definition}\label{defn2-1}
We say that the vector $X$ is \emph{symmetric} if
\begin{romlist}
\item $X$ and $-X$ have the same distributions, and
\item for all $i \ne j$ there exists a permutation $\pi$ of $\{1,2,\dots,m\}$
with $\pi_i=j$ such that $X$ and $\pi X$ have the same distribution,
where $\pi X$ denotes the
permuted vector $(X_{\pi_1}, X_{\pi_2}, \dots, X_{\pi_m})$.
\end{romlist}
\end{definition}
Condition (ii) above is weaker than requiring
that $X$ be exchangeable, and an example is included in Appendix \ref{ex:1}.
\begin{definition}
Let $A$ be an increasing event.
\begin{letlist}
\item The \emph{absolute influence} of voter $i$ is
\begin{align*}
\al(i) &:= \PP(X^i\in A) - \PP(X_i \in A)\\
&= \PP(X_i\notin A,\, X^i \in A) .
\end{align*}
\item The \emph{conditional influence} of voter $i$ is
$$
\kappa(i):= \PP(A \mid X(i)=1) - \PP(A\mid X(i)=-1).
$$
\end{letlist}
\end{definition}
The so-called \lq power index', or the Banzhaf (or Banzhaf--Penrose) power index in the impartial culture
context, is the term used by many authors for the absolute influence given above.
When $\PP$ is a product measure (that is, the $X(i)$ are independent), it may be seen that
$\al(i)=\kappa(i)$, and the common value $\al$ is termed simply
\emph{influence} by Russo \cite{Ru81} and Ben-Or and Linial \cite{BOL}
(and \emph{power} by Penrose \cite{Pen}).
Equality does not generally hold when $\PP$ is not a product measure.
The above concept of \lq conditional influence' was identified
in \cite{GG}, where it was shown to be the correct adaptation of absolute influence
in proofs of sharp-threshold theorems for certain families
of \emph{dependent} measures arising in stochastic geometry and statistical
physics. It has featured recently in work \cite{KKLN} on so-called prediction values within probabilistic games.
\begin{remark}[Success probability]\label{rem:Puk}
The \emph{success probability} $\eta(i)$
of voter $i$ is the probability that $i$ votes on
the winning side, that is,
$$
\eta(i) := \PP\bigl(A \cap \{X(i)=1\}\bigr) + \PP\bigl(\ol A \cap \{X(i)=-1\}\bigr),
$$
where $\ol A$ is the complement of $A$. See, for example, \cite{PukB14,LV}.
If the $X(i)$ satisfy \eqref{eq:Xsym},
the conditional influence is related to the success probability
by the relation
\begin{equation}\label{succ}
\eta(i)=\tfrac12(1+\kappa(i)),
\end{equation}
see \cite[Thm 3.2.16]{FM} and \cite[Sect.\ 2(a)]{Pen}. This relation is, in fact,
the key step in the proof of the forthcoming Proposition \ref{prop1}.
The success probabilities feature in the recent work of Kirsch \cite{Kir17}.
\end{remark}
\subsection{Examples of influences}\label{sec:23}
There follow three examples of calculations that illustrate the differences between
absolute and conditional influence. For convenience,
we assume $m=2r+1$ is an odd number, and take $A$ to be the \emph{majority event}, that is,
\begin{equation}
\label{majev}
A=\left\{\si: \sum_i \si(i)> 0\right\}.
\end{equation}
It is clear that $A$ is an increasing set.
We shall take the $X(i)$ to be Bernoulli random variables \emph{with a shared parameter $u$
which may itself be random}. Thus, the $X(i)$ are not generally independent.
Let $U$ be a random variable taking values in the interval $(0,1)$. Conditional on the event $U=u$,
the $X(i)$ are defined to be independent random variables with
\begin{equation}\label{eq:collb}
X(i) = \begin{cases} 1 &\text{with probability } u,\\
-1 &\text{with probability } 1-u.
\end{cases}
\end{equation}
If $U$ has a symmetric distribution (in that $U$ and $1-U$ are equally distributed),
then the ensuing vote-vector $X$ is symmetric (and, indeed, exchangeable),
and this is called the \lq collective bias' model by Kirsch \cite{Kir07,Kir16}
(see also \cite{KirL}).
Here are three examples of collective bias in which the absolute and conditional influences vary greatly.
\begin{numlist}
\item \emph{Independent voting.}
Let $\PP(U=\frac12)=1$. The $X(i)$ are independent, unbiased Bernoulli variables,
and
\begin{equation}\label{eq:absinf}
\al(i)=\kappa(i)=\binom {2r}r \left(\frac12\right)^{2r}
\sim \frac1{\sqrt{\pi r}} = \sqrt{\frac2{\pi (m-1)}} \qq \text{as } m \to\oo.
\end{equation}
\item
\emph{Uniform bias.}
Let $U$ be uniform on the interval $(0,1)$.
Then
\begin{equation}\label{eq:unifinf}
\al(i) = \int_0^1 \binom {2r}r u^r(1-u)^r\,du =\frac1m,\qq
\kappa(i)=\frac12+\o(1).
\end{equation}
\item
\emph{Polarised bias.}
Let $\PP(U=\frac13)=\PP(U=\frac23)=\frac12$. There
exists $\gamma>0$ such that
\begin{equation}\label{eq:polbias}
\al(i)=\o(e^{-\gamma m}), \qq \kappa(i)=
\frac13 +\o(1).
\end{equation}
\end{numlist}
We remind the reader that $f(m)=\o(g(m))$ means $f(m)/g(m) \to 0$ as $m\to\oo$.
The success probabilities in Cases 1--3 follow by \eqref{succ} from the above calculations.
Cases 2 and 3 are exemplars of more general situations in which: ($2'$) the distribution
of $U$ on some neighbourhood of $\frac12$ is absolutely continuous
(see, for example, \cite[p.\ 674]{GKB}), and $(3')$
$U$ is almost surely bounded away from $\frac12$ (see, for example,
the formulation of \cite[p.\ 592]{KLMT}).
\begin{remark}\label{rem24}
Only in the case of independence does the absolute
influence have the order of the square root $1/\sqrt m$.
In the two other situations above, the absolute influence
is as small as $1/m$ and $e^{-\g m}$, \resp.
This illustrates the fragility of the square-root laws \eqref{1}, \eqref{4} and
their consequences for voting (see \cite{KMN}).
\end{remark}
\begin{comment}
Correlations are easily computed in the collective bias model of \eqref{eq:collb}.
For example, for $i \ne j$, the covariance $\rho$ between $X(i)$ and $X(j)$ satisfies
\begin{alignat*}{2}
\rho &= \EE\bigl[\EE(X(i)X(j)\mid U)\bigr]\\
&= \EE\bigl[\EE(X(i)\mid U) \EE(X(j)\mid U)\bigr] \q&&\text{by conditional independence}\\
&=\EE[(2U-1)^2] &&\text{by \eqref{eq:collb}}\\
&= 4\, \var(U).
\end{alignat*}
In particular,
$$
\rho=\begin{cases}
\tfrac 13 &\text{for uniform bias},\\
\tfrac 19 &\text{for the above polarised bias}.
\end{cases}
$$
\end{comment}
Further discussion of the relationship between absolute and conditional
influence may be found in \cite[Sect.\ 2]{GG}. A review of influence for product
measures is found at \cite{KalS}, see
also \cite[Sect.\ 4.5]{G-pgs}. Uniform bias
was discussed in \cite{ER}, and polarised bias in \cite{GG}.
Conditional influence is essentially the prediction value of \cite{KKLN}.
\subsection{Penrose's two square-root laws unified.}
We present next an elementary application of conditional influence (the proof is found
at the end of the section). We will see its relevance in
the discussion of the Penrose square-root laws in Remark \ref{rem33}.
\begin{proposition}\label{prop1}
Let $m=2r+1$ be odd.
Assume that $X$ and $-X$ have the same distributions, and let $A$ be the majority event of \eqref{majev}.
Then $S=\sum_{i=1}^m X(i)$ satisfies
$$
\EE|S| = \sum_{i=1}^m \kappa(i).
$$
If $X$ is symmetric, then $\kappa=\kappa(i)$ is constant and $\EE|S|=m\kappa$.
\end{proposition}
We next interpret Proposition \ref{prop1} in the language of Penrose, see Section \ref{sec:1.2}.
Consider a population of size $N_j$, and suppose the corresponding vote-vector $X=(X(1), X(2), \dots, X(N_j))$ is
symmetric. Then $\kappa=\kappa(i)$ does not depend on $i$. The number $H$ of people voting $1$
satisfies $H=\frac12(S+N_j)$, so that, by Proposition \ref{prop1} with $m=N_j$,
\begin{equation}\label{12}
\EE\left|H-\tfrac12 N_j\right| =\tfrac12 N_j\kappa .
\end{equation}
In conclusion, $\EE|H-\tfrac12 N_j|$ grows in the manner of $\sqrt {N_j}$ if and
only if $\kappa$ behaves in the manner of $1/\sqrt{N_j}$.
\begin{remark}\label{rem33}
In the language of Penrose \cite{Pen},
the mean \lq edge' differs from the \emph{conditional influence} by the constant multiple $N_j$.
Thus, in the context of general distributions,
conditional influence takes precedence over absolute influence. Penrose's argument implies that, in a two-tier
voting system, the appropriate weight of state $j$ satisfies $w_j \propto N_j \kappa$,
where $\kappa=\kappa_j$ is the conditional influence.
In this sense, Penrose's \lq\lq two" square-root laws are in reality only one,
so long as one uses \emph{conditional} rather than \emph{absolute} influences:
the asymptotic edge addressed by the
second square-root law is simply a multiple of the asymptotic power addressed by
the first square-root law.
When voting is truly independent, the distinction between absolute and conditional influence is nominal only.
Seen in the light of Remark \ref{rem:Puk},
Proposition \ref{prop1} supports the thesis that, for general probability measures,
the success probability is a more central quantity than the absolute influence
(cf.\ \cite[Sect.\ 3.6]{LV}).
\end{remark}
The question arises of deciding the \lq correct' definition of influence in the general voting context.
There does not seem to be a simple answer to this somewhat philosophical question,
which lies beyond the scope of this mathematical paper. Some minor reflections are offered,
within the context of the symmetric voting model of Definition \ref{defn2-1}.
\begin{letlist}
\item If we are trying to capture the probability that an individual can, as a
theoretical exercise in free will, affect the outcome of a vote, then
we might favour absolute influence. This interpretation
requires stepping outside the mathematical model of Definition \ref{defn2-1},
by postulating the existence of a unique individual
P who votes independently of the rest of the population. Then the absolute influence of P equals the
probability that P's vote is pivotal.
\item When different votes are correlated, a sample of one vote contains information about the other
votes. If we wish to gain such information, then
conditional influence is a way to do so. On the other hand, conditional influence contains very little information
about the power of any given individual. When votes are correlated, individual power tends
to be rather small, and sometimes so small that it ceases to have great value as a discriminator.
\item Only with the help of a probabilistic model can we calculate influence.
An analysis of the above question depends on the interpretation of \lq chance' or \lq randomness' in such a model.
Does it make practical sense to model votes as unbiased random variables?
Possibly in response to this question, some authors have argued that the views of
voters may not themselves be considered random, but it is rather the
\emph{proposals} that are random (see, for example, \cite[p.\ 38]{FM} and \cite[p.\ 360]{Kir07}).
This interesting suggestion poses
some philosophical challenges in its own right, not least
arising from correlation between the responses of a given
voter to different proposals.
\end{letlist}
\begin{proof}[Proof of Proposition \ref{prop1}]
Let $1_A$ denote the indicator function of
an event $A$. Then, since $X$ and $-X$ have the same distribution,
\begin{align*}
\EE|S| &= \EE(S1_{S>0}) - \EE(S1_{S<0})\\
&= 2\EE(S1_{S>0})\\
&= 2\sum_{i=1}^m \EE(X_i1_{S>0}) \\
&= \sum_{i=1}^m\bigl[\PP(S>0\mid X_i=1) - \PP(S>0 \mid X_i=-1)\bigr]\\
&=\sum_{i=1}^m \kappa(i).
\end{align*}
Subject to symmetry, the constantness of $\kappa(i)$ holds by choosing suitable permutations
of $\{1,2,\dots.,m\}$.
\end{proof}
\section{Two-tier voting}\label{sec:ttv}
\subsection{Two-tier voting systems}
We assume there
exist $s$ States with respective populations $N_1, N_2, \dots, N_s$
(which we take for simplicity to be odd numbers).
States are each allowed one representative on the Council of States.
Each State is assumed to conduct
a ballot on a given issue, and the vote of voter $i$ in State $j$ is denoted $X_j(i)\in\{-1,1\}$. The outcome of
the vote in state $j$ is taken to be
\begin{equation}\label{eq:sjdef}
\chi_j:= \begin{cases} 1 &\text{if } S_j := \sum_{i=1}^{N_j} X_j(i) >0,\\
-1 &\text{otherwise}.
\end{cases}
\end{equation}
That is, $\frac12(1+\chi_j)$ is the indicator function of the event that $S_j>0$.
\begin{assumption}[\cite{Kir07}]\label{ass1}
We assume the vectors $X_j=(X_j(i): i=1,2,\dots,N_j)$, $j=1,2,\dots,s$, are independent, which is to say that
the votes of different States are independent. We make no assumption
for the moment about the voters of any given State
beyond that, for given $j$, the vectors $X_j$ are symmetric in that
$X_j$ and $-X_j$ have the same distribution.
\end{assumption}
To the State $j$ is assigned a \emph{weight} $w_j>0$, and we write $W=\sum_j w_j$
for the aggregate weight of the States.
The representative of state $j$ votes $\chi_j$, and the weighted sum
$$
V:=\sum_{j=1}^s w_j \chi_j,
$$
is calculated. The motion is said to \emph{pass} if
\begin{equation}\label{eq:pass}
V > q W,
\end{equation}
and to \emph{fail} otherwise,
where $q$ is a predetermined \emph{quota} (this is not the quota of \cite{SZ07}, but rather that
of \cite{Kir07}, see also \eqref{2}).
This voting system depends on the weights $w=(w_j)$ and the quota $q$,
and we refer to it as the $(w,q)$ system.
Since votes are assumed independent \emph{between} States, the
absolute and conditional influences coincide at the level of the Council.
\begin{question}\label{qn1}
How should the weights $w_j$ and the quota $q$ be chosen?
\end{question}
We summarise two approaches.
\subsection{Penrose/Kirsch and least squares \cite{Kir07,Pen}}\label{sec:3.2}
Penrose has argued that, within any given state,
the strength of a vote is proportional to the mean \lq edge', that is, the quantity
$\EE|N_F-N_A|$
where $N_F$ is the number voting for the successful outcome and $N_A$
is the number voting against. Now, $N_F-N_A = S_j$, where $S_j$ is given in
\eqref{eq:sjdef}.
This motivates the \lq Penrose' proposal that $w_j = \EE|S_j|$.
Kirsch \cite{Kir07} has proposed choosing the $w_j$ in such a way as to
minimise the mean sum of squared errors
$$
Q:= \EE\left(\left[\sum_{j=1}^s (S_j-w_j\chi_j)\right]^2\right).
$$
A quick proof of the following proposition is given at the end of the subsection.
\begin{proposition}[{\cite[Thm 2.1]{Kir07}}]\label{prop:kir}
Subject to Assumption \ref{ass1}, the quantity $Q$ is minimised when
$w_j = \EE|S_j|$ for $ j=1,2,\dots,s$.
\end{proposition}
Thus, Kirsch's least-squares principle leads to the Penrose solution $w_j = \EE|S_j|$,
which we call the \emph{majority rule}.
As explained by Kirsch, this motivates the choice
\begin{equation}\label{eq:kirsch}
w_j = \begin{cases} \sqrt {N_j} &\text{if there is no long-range order},\\
N_j &\text{if there is long-range order},
\end{cases}
\end{equation}
where \lq long-range order' is interpreted
as the non-decay of correlations (see the related Appendix \ref{A-ferro}).\footnote{The term \lq long-range order'
arises in statistical mechanics in situations where random spins $\sigma_v$ are located at the vertices $v$
of a lattice, such as the $d$-dimensional integer lattice $\ZZ^d$, and the correlation between two spins
$\sigma_v$, $\sigma_w$ does not approach $0$ as the distance between $v$ and $w$ diverges.}
For example,
Case 1 of Section \ref{sec:23} has no long-range order, but Cases 2 and 3 possess long-range order.
See also Remark \ref{rem33}.
\begin{proof}[Proof of Proposition \ref{prop:kir}]
By Assumption \ref{ass1},
\begin{alignat*}{3}
Q&=\var\left(\sum_j (S_j-w_j\chi_j)\right)\qq &&\text{since $\EE(S_j)=\EE(\chi_j)=0$}\\
&=\sum_{j=1}^s \var (S_j-w_j\chi_j)\qq&&\text{since the $X_j$ are independent}.
\end{alignat*}
By calculus, the last summand is a minimum when $w_j=\EE(S_j\chi_j)$ as claimed.
\end{proof}
\subsection{S\l omczy\'nski/{\. Z}yczkowski and influence \cite{SZ06, SZ07,SZ11}} \label{sec:SZ}
Let us concentrate on the situation in which the entire vote-set
$(X_j(i) : i =1,2,\dots, N_j,\ j=1,2,\dots,s)$ is a family of independent random variables.
By independence, the absolute and conditional influences (within States) are equal.
The influence $\al_j:=\al_j(i)$ of a member of State $j$ is (asymptotically as $N_j\to\oo$)
proportional to $1/\sqrt{N_j}$, by \eqref{eq:absinf}.
The Penrose/Kirsch proposal of Section \ref{sec:3.2} amounts to $w_j = \EE|S_j| \propto \sqrt {N_j}$.
The product $\al_j w_j$ is (asymptotically, for large $N_j$) constant across the States.
This may be seen as evidence that the voting system with this set of weights is \lq fair' across the union of the States.
How does one calculate the so-called \lq total influence' of a given voter in the $(w,q)$ system?
A given voter is pivotal overall if s/he is pivotal within the relevant State vote,
and furthermore the outcome of that vote is pivotal in the Council's vote.
By Assumption \ref{ass1}, the \emph{total influence} $I_j$ of voter $i$ in State $j$ is the product
\begin{equation}\label{eq:Inf}
I_j = \al_j\be_j,
\end{equation}
where $\be_j=\be_j(w,q)$ is the influence of State $j$ in the Council's vote.
(See \cite[p.\ 67]{FM}.)
\emph{We seek a pair $(w,q)$ such that the total influences are equal (or nearly so)
across the States $j$.}
The total influences $I_j$ of \eqref{eq:Inf} need not be proportional to
the products $\al_j w_j$ of the previous paragraph,
since the ratios $\be_j / w_j$ are in general non-constant across the States.
A number of authors including \SZ\ \cite{SZ11} have developed the following approach.
\begin{numlist}
\item Allocate to State $j$ the weight $w_j=\sqrt{N_j}$.
\item Calculate or estimate the State-influences $\be_j$ as functions of $(w,q)$.
\item Identify a quota $q$ such that $\be_j$ is an approximately linear function of $w_j$.
\item The ensuing products $I_j=\al_j\be_j$ are approximately constant across States.
\end{numlist}
They have proposed choosing the quota $q$
in \eqref{eq:pass} in such a way that, for the given weights $(w_j)$, the sum of squared differences
$$
Q:= \sum_{j=1}^s (\ol w_j-\ol \be_j)^2
$$
is a minimum, where
$$
\ol w_j= \frac{100 w_j}{\sum_k w_k}, \qq \ol\be_j =\frac{100 \be_j}{\sum_k \be_k},
$$
are the \lq normalised' influences and weights, \resp\ (see Table \ref{table:1}).
They present numerical, empirical,
and theoretical evidence that this is often achieved when $q$ is near
\begin{equation}\label{eq:q*}
q^* := \frac {\sqrt N}{\sum_j \sqrt{N_j}}, \qq\text{where } N=\sum_{j=1}^s N_j.
\end{equation}
The theoretical foundation for this proposal lies in: (i) approximating $\be_j$ by a Gaussian integral,
and (ii)
picking $q$ such that the integrand is close to linear in $w_j$. The latter step is achieved by finding the
point at which the $N(\mu,\si^2)$ Gaussian density function
has an inflection, and is thus locally closest to being locally linear. This inflection is easily
found by calculus to be at $q:= \mu\pm\si$, and this leads to the formula \eqref{eq:q*}.
In summary, they argue that, when $w_j=\sqrt{N_j}$ and $q=q^*$,
the $\be_j=\be_j(w,q)$ are close to the $w_j$, and hence
the total influences $I_j = \al_j\be_j$ are close to the products $\al_jw_j$. Finally,
since $\al_j \sim C/ \sqrt {N_j}$ and $w_j = \sqrt{N_j}$, the last product is asymptotically constant
across the States.
The above procedure is termed the Jagiellonian Compromise (or JagCom).
We discuss in Section \ref{sec:sinf} some aspects of the derivation of the quota $q^*$
in \eqref{eq:q*}. Some peripheral support for this choice of quota may be found in
the analyses of \lq toy models' in \cite{SZ07, tomski}.
\begin{remark}
\label{rem34}
The weights $w_j$ are chosen first in the JagCom, and then the quota $q$ is chosen according to a
linearisation argument. It may instead
be preferable to choose the parameters $(w,q)$ in such a way
that the deviation in the total influences $I_j$ is a minimum.
See, for example, \cite{Kurz12}.
\end{remark}
\section{\lq Total influences' in a two-tier system}\label{sec:sinf}
\subsection{Total influences}
A mathematical derivation of the JagCom
quota $q^*$, \eqref{eq:q*}, seems to require certain approximations
which we discuss next. The first issue is to identify the purpose of the analysis. Let $I_j$ be the
total influence of a member of State $j$, as in \eqref{eq:Inf}. One extreme way of
achieving the near-equality of the $I_j$ is to set the quota $q$ on the left side of
\eqref{2} to be either $-\eps+\sum_j \sqrt{N_j}$ or its negation, where $\eps>0$ is small.
If we insist on such unanimity, we achieve
$$
I_j =\al_j\left(\frac12\right)^s \sim \frac{C}{\sqrt{N_j}} \left(\frac12\right)^s.
$$
For large $s$, these influences are nearly equal, indeed nearly equal to $0$. Their ratios
however can be as large as $\sqrt{\Nmax/\Nmin}$, where $\Nmax$ (\resp, $\Nmin$) is
the maximum (\resp, minimum) population size.
An alternative target is that the ratios $I_j/I_k$ be as close to unity as possible,
and a secondary target might be that the total influences are as large as possible.
We consider this next.
Consider a vote of the Council in which each State $k$ has a preassigned weight $w_k>0$.
Let $j \in \{1,2,\dots,s\}$.
By \eqref{eq:pass}, State $j$ is pivotal for the outcome if:
the set $J$ of States (other than $j$) voting for the motion is such that
\begin{equation}\label{eq:3}
w_J+w_j-w_{\ol J} > qW, \qq w_J-w_j-w_{\ol J} \le qW,
\end{equation}
where $J\subseteq \{1,2,\dots,s\}\setminus\{j\}$, $\ol J = \{1,2,\dots,s\}\setminus(J\cup\{j\})$, and
$$
w_K:= \sum_{k\in K} w_k, \qq K \subseteq\{1,2,\dots,s\}.
$$
Inequalities \eqref{eq:3} may be written in the form $qW-w_j < Z_j \le qW+w_j$ where
\begin{equation}\label{eq:Z}
Z_j=w_J-w_{\ol J} = \sum_{k\ne j} w_k \chi_k, \qq j\in\{1,2,\dots,s\},
\end{equation}
and $(\chi_k:k=1,2,\dots,s)$ is a family of independent Bernoulli random variables with
$$
\PP(\chi_k=1)=\PP(\chi_k=-1)=\tfrac12.
$$
Therefore, State $j$ is pivotal in the Council with probability
\begin{align}\label{eq:diff}
\be_j &:= \PP\bigl(qW-w_j < Z_j \le qW+w_j\bigr)\\
&= F_{Z_j}(qW+w_j)-F_{Z_j}(qW-w_j),\nonumber
\end{align}
where $F_{Z_j}$ is the distribution function of $Z_j$.
(Similar formulae appear in \cite[App.]{SZ07}.)
\SZ\ \cite{SZ11} argue that the $F_{Z_j}$ are
\lq nearly' Gaussian, and they consider the appoximation
\begin{equation}\label{eq:app}
\be_j \approx 2w_j \phi_{\mu_j,\si_j}(qW)
\end{equation}
where $\phi_{\mu,\si}$ is the $N(\mu,\si^2)$ Gaussian density function, and
\begin{equation}\label{eq:sigma2}
\mu_j=\EE(Z_j) = 0,\qq \si_j^2=\var(Z_j) =\sum_{k\ne j}w_k^2.
\end{equation}
More precisely, \eqref{eq:app} expresses the hypothesis that the distributions are nearly Gaussian, and that
the approximation is sufficiently uniform to be accurate to the first order, in terms of the density function.
They argue that the approximation is most accurate when $q$ is chosen in such a way that
$qW$ is a point of inflection of $\phi_{\mu_j,\si_j}$, and this leads to the choice
$q=q^*$, with $q^*$ as in \eqref{eq:q*}.
It is explained at the end of Appendix \ref{sec:4.2}, to which the reader is referred for
further details, that the ensuing approximations are good
in the case of the population-sizes of the EU Member States, but no proof is known to sufficient accuracy to
permit the \emph{rigorous} deduction of the quota, $q^*$.
The currently best theoretical tool for Gaussian approximations
is the so-called Berry--Esseen bound, and this is not good enough for
our purpose here. See Table \ref{table:2}.
\subsection{The argument via numerical methods}\label{sec:43}
Once one has accepted the thesis that voters are independent and unbiased,
there is a transparent logic to the choice of weights $w_j = \sqrt{N_j}$.
Attention then turns to the choice of quota $q$. It is shown in Appendix \ref{sec:4.2}
that the mathematical argument of \SZ\ \cite{SZ07}, while neat, is at best incomplete.
In contrast, the \emph{numerical} evidence of \cite{SZ04}, in favour of $q=q^*$, retains some persuasive power.
Similar numerical work has been carried out for the current article using
QMV2017 population data taken from \cite{PukG17}, with the results
reported in Table \ref{table:1}. These results are exact rather than being based on simulation.
\begin{table}
\begin{center}
\definecolor{mygray}{gray}{0.9}
\newcolumntype{s}{>{\columncolor{mygray}} }
\setlength{\arrayrulewidth}{0.5mm}
\begin{tabular}{r |s l || c c | c c sc|c c sc }
\rowcolor{mygray} \multicolumn{2}{c||}{Member State} & \multicolumn{2}{c|}{weights} & \multicolumn{3}{c|}{$q=0$} &
\multicolumn{3}{c}{$q=q^*$} \\
\rowcolor{mygray} $j$ && $w_j$ & $\ol w_j$ & $\be_j$ & $\ol\be_j$ & $\ol\be_j/\ol w_j$ & $\be_j$ & $\ol\be_j$ & $\ol\be_j/\ol w_j$\\
\hline\
1 & Germany & 9.059 & 9.963 &0.357& 10.414 & 1.045 &0.211& 9.937 & 0.997\\
2 & France & 8.165 & 8.979 &0.317& 9.239 & 1.029 &0.191& 8.984 & 1.001\\
3 & Italy & 7.830 & 8.611 &0.302& 8.816 & 1.024 &0.183& 8.619 & 1.001\\
4 & Spain & 6.815 & 7.495 &0.260& 7.575 & 1.011 &0.159& 7.507 & 1.002 \\
5 & Poland & 6.162 & 6.777 &0.233& 6.802 & 1.004 &0.144& 6.787 & 1.001 \\
6 & Romania & 4.445 & 4.888 &0.166& 4.839 & 0.990 &0.104& 4.891 & 1.001 \\
7 & Netherlands & 4.152 & 4.566 &0.155& 4.512 & 0.988 &0.097& 4.568 & 1.000\\
8 & Belgium & 3.360 & 3.695 &0.125& 3.636 & 0.984 &0.078& 3.696 & 1.000 \\
9 & Greece & 3.285 & 3.613 &0.122& 3.554 & 0.984 &0.077& 3.613 & 1.000 \\
\hline
10 & Czech Rep. & 3.232 &3.554 &0.120& 3.495 & 0.983 &0.075& 3.554 & 1.000 \\
11 & Portugal & 3.216 & 3.537 &0.119& 3.478 & 0.983 &0.075& 3.537 & 1.000 \\
12 & Sweden & 3.162 & 3.477 &0.117& 3.418 & 0.983 &0.074& 3.477 & 1.000 \\
13 & Hungary & 3.135 & 3.448 &0.116& 3.389 & 0.983 &0.073& 3.447 & 1.000 \\
14 & Austria & 2.952 & 3.246 &0.109& 3.189 & 0.982 &0.069& 3.246 & 1.000 \\
15 & Bulgaria & 2.675 & 2.942 &0.099& 2.886 & 0.981 &0.062& 2.941 & 1.000 \\
16 & Denmark & 2.388 & 2.626& 0.088& 2.574 & 0.980 &0.056& 2.625 & 1.000 \\
17 & Finland & 2.338 & 2.571 &0.086& 2.520 & 0.980 &0.055& 2.570 & 1.000 \\
18 & Slovakia & 2.326 & 2.558 &0.086& 2.507 & 0.980 &0.054& 2.557 & 1.000 \\
\hline
19 & Ireland & 2.160 & 2.375 &0.080& 2.327 & 0.980 &0.050& 2.374 & 1.000 \\
20 & Croatia & 2.047 & 2.251 &0.076& 2.204 & 0.979 &0.048& 2.250 & 1.000 \\
21 & Lithuania & 1.700 & 1.870 &0.063& 1.829 & 0.978 &0.040& 1.868 & 0.999 \\
22 & Slovenia & 1.437 & 1.580 &0.053& 1.545 & 0.978 &0.034& 1.579 & 0.999 \\
23 & Latvia & 1.403 & 1.543 &0.052& 1.508 & 0.977 &0.033& 1.542 & 0.999\\
24 & Estonia & 1.147 & 1.261 &0.042& 1.233 & 0.978 &0.027& 1.260 & 0.999 \\
25 & Cyprus & 0.921 & 1.013 &0.034& 0.990 & 0.977 &0.021& 1.012 & 0.999 \\
26 & Luxembourg & 0.759 & 0.835 &0.028& 0.815 & 0.976 &0.018& 0.834 & 0.999 \\
27 & Malta & 0.659 & 0.725 &0.024& 0.708 & 0.977 &0.015& 0.724 & 0.999 \\
\hline
& Totals & 90.930 & 100 &3.429 & 100 &&2.123& 100&\\
\end{tabular}
\end{center}
\vskip12pt
\caption{Member State $j$ has weight $w_j=\sqrt{N_j}$ and normalised weight
$\ol w_j=100w_j/W$, where $W=\sum_j w_j$.
Two values of the quota $q$ are considered, namely, $q=0$ and $q=q^*$
(see \eqref{eq:q*}). For each, the influences $\be_j$ have been computed, and the normalised influences
$\ol\be_j=100\be_j/B$ are given above, where $B=\sum_j \be_j$.
The ratios $\ol\be_j/\ol w_j$ are presented alongside the $\ol\be_j$.
The ratios lie in the interval $[0.976,1.045]$ when $q=0$, and in the interval
$[0.997,1.002]$ when $q=q^*$.}
\label{table:1}
\end{table}
Table \ref{table:1} lends some support to the choice $q=q^*$.
\begin{letlist}
\item The ratios of normalised influences $\ol\be_j$
to normalised weights $\ol w_j$ are very close to $1$ when $q=q^*$.
\item Further calculations show that the sum of squared differences $Q=\sum_j(\ol w_j-\ol\be_j)^2$, considered
as a function of $q=0, \frac12 q^*, q^*, \frac32 q^*$, is a minimum when $q=q^*$.
(More refined calculations are possible.)
\end{letlist}
We note, however, the following.
\begin{romlist}
\item The choice $q=q^*$ lacks transparency.
In contrast, the choice $q=0$ is simple and easy to explain.
\item The ratios $\ol\be_j/\ol w_j$ are also close to $1$ when $q=0$.
The agreement is not quite so perfect as when $q=q^*$,
but the differences are minor.
\item The sum $Q$ is similarly close to $0$ when $q=0$, albeit not so close as when $q=q^*$.
\item The influences $\be_j$ are largest when $q=0$. (See also \cite[App.]{SZ07}.)
\end{romlist}
In summary, the numerics are best when $q=q^*$, but the improvements relative to
the more transparent choice of $q=0$ are minor. The numerical differences between
these two cases (and indeed other reasonable values of $q$) are so
small that they are unlikely to be separated by any technical analysis of the type of
Appendix \ref{sec:4.2}. (See also \cite[Fig.\ 7]{widgren}.) We conclude the following.
\begin{numlist}
\item On the basis of the theoretical and numerical
evidence concerning the ratios $\ol\be_j/\ol w_j$, there is no convincing evidence that any one value of the quota
is materially preferable to any other.\footnote{Large positive or negative values are
evidently poor, but
we consider here only values $q$ such that $qW/\sqrt N$ has order $1$.
Other choices for $q$ have been considered in, for example, \cite{Birk11,CCM}.}
\item The total influences $I_j$ are largest when $q=0$.
\end{numlist}
\section{Some remarks on the Jagiellonian Compromise}\label{sec:rems}
Theoreticians propose, politicians dispose (and certain Presidents of the United States have historically
played on both teams). Members of each group have interests and potential conflicts.
The theoretician earns respect through honest assessment of the virtues
(or not) of, and principles underlying, a particular proposal. They
hope that politicians will accord fair weight and balance to principled proposals irrespective of
personal advantage. While theoreticians are usually free of conflicts arising out of employment
within a politically aligned organization, politicians are usually heavily conflicted
(see, for example, \cite{PukG17}).
Communication between the two groups can be challenging.
The use of language such as \lq local limit theorem'
and \lq Berry--Esseen bound' tends to create barriers.
Such methodology is however key to proper
study of the two-tier voting system of Sections
\ref{sec:ttv}--\ref{sec:sinf},
and practitioners have worked diligently to communicate its relevance.
The JagCom proposes the use of square-root weights $w_j = \sqrt{N_j}$
with a specific choice of the quota $q$.
Each of these two proposals will rightly continue to attract critical discussion.
The square-root weights of equation \eqref{1} and Proposition \ref{prop:kir}
may be justified if: (i) there is no bias, and (ii) there is no \lq long-range order'
(in the language of statistical mechanics).
Each of these assumptions seems over-perfectionist.
Issues before the Council may be systematically more popular in some States than in others,
and the consequent biases risk undermining either or both of the above two assumptions.
The \lq collective bias' model of Kirsch and others (see Section \ref{sec:inf})
is both more flexible and more empirical, at serious cost to the square-root laws for
influence and majority (see \cite{KirL}). Other authors have considered the effect on
weights of introducing a concave utility function (see Koriyama et al.\ \cite{KLMT}).
Such approaches give rise to weight distributions which, in turn,
benefit from calibration against the politics and practical workings of the Council.
The right choice of weights is not a simple matter of finding some neat mathematics.
That said, no concrete proposal
to displace square-root weights in the JagCom is made in the current work.
Having chosen the weights, the identification of the quota is subject to similar ambiguities.
The JagCom \lq exact'
quota $q^*$ of \eqref{eq:q*} hinges on the assumptions of square-root weights and equality of
absolute influence, in combination
with numerical data and the Gaussian approximation of Appendix \ref{sec:4.2}.
The last is unproven in the current context of the
QMV2017 population data
of the States of the EU.
In their favour, the proposed square-root weights and the exact
quota $q^*$ of the JagCom have been derived via a set of principles that can be stated unambiguously and analysed
fairly rigorously, and which are robust with respect to changes in population data.
Proponents argue that they are thus less susceptible than many
alternatives to \lq political meddling'.
We turn now to the numerics of the JagCom quota $q^*$.
If the ratios $\ol \be_j/\ol w_j$ in Table \ref{table:1} are close to $1$, then
the total influences $I_j = \al_j\be_j$ of \eqref{eq:Inf} are almost constant across Member States.
As indicated in the shaded columns of the table, this holds for both $q=0$ and $q=q^*$
(they are nearly perfect when $q=q^*$, and very close for other values of $q$).
Indeed this holds for a range of values of $q$ including the values $0$ and $q^*$
(see also \cite{widgren}).
\emph{One may deduce that, from a practical point of view, there
is little to choose between different values of $q$. This may be a situation in
which political considerations may have the final word.}
It seems generally considered to be the case that there is no enthusiasm amongst politicians
for the choice $q=0$, on the grounds that a body such as the EU Council should
seek a compromise between simple majority and unanimity.
Overall, the details of the JagCom rely on a number of assumptions
that are arguably fragile and/or unrealistic. This potential
weakness needs to be acknowledged when making the case for the JagCom.
The JagCom is a legitimate proposal for the two-tier voting system of the Council of the EU,
whose finer details may profit from input by politicians in choosing a system
judged to serve well the needs of the nearly 450 million residents of the 27 Member States
of the European Union without the United Kingdom. Our closing quote (Machover \cite[Abs.]{Mach07})
accords a balanced responsibility to both theoreticians and politicians:
\lq\lq This is essentially a political matter; but a political decision
ought to be made in a theoretically enlightened way."
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La Jeune Peinture Belge
La Jeune Peinture Belge was founded in 1945 by Robert Delevoy and René Lust in support of a new generation of artists. The movement of La Jeune Peinture Belge played an important role in the post-war art history and in the revaluation of abstraction.
James Ensor as honorary chairman, some progressive key figures and art collectors created a stimulating framework for thirteen young artists. They made a series of international exhibitions and managed to give attention to give to artists with very diverse visual languages. Soon the movement became a springboard for many of them. It also laid the first seeds of a new abstraction. Later the second generation geometric abstracts emerged under the impulse of the groups Art Abstrait and Formes.
In 1948, the movement had to end because lender René Lust dies. In that short time, the movement made its mark on the postwar awakening of Belgian art.
Luc Peire, Pierre Alechinsky, Jan Cockx, Willy Anthoons, Jo Delahaut, Jean Rets, Marc Mendelson, Antoine Mortier, Louis Van Lint, Anne Bonnet, Gaston Bertrand, Rik Slabbinck and Jules Lismonde
La Jeune Peinture Belge - Artists
Luc Peire
|
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|
\section{Conclusion}~\label{sec:conclusion}
In this paper, we introduced STFNet, a principled way of designing neural networks from the time-frequency perspective. STFNet endows time-frequency analysis with additional flexibility and capability. In addition to just parameterizing the frequency manipulations with deep neural networks, we bring two key insights into the design of STFNet. On one hand, STFNet leverages and preserves the frequency domain semantics that encode time and frequency information. On the other hand, STFNet circumvents the uncertainty principle through multi-resolution transform and processing. Evaluations show that STFNet consistently outperforms the state-of-the-art deep learning models with a clear margin under diverse sensing modalities, and our two designing insights significantly contribute to the improvement. The designs and evaluations of STFNet unveil the benefits of incorporating domain-specific modeling and transformation techniques into neural network design.
\section{Discussion}~\label{sec:discussion}
This paper provides a principled way of designing neural networks for sensing signals inspired by the fundamental nature of the underlying physical processes. STFNet, operates directly in the frequency domain, in which the measured physical phenomena are best exposed. We propose three types of learnable frequency manipulations that are able to operate on multi-resolution representations, while preserving the underlying time-frequency information. Although extensive experiments have illustrated the superior performance of STFNet,
further research is needed to better understand design choices for neural networks from the time-frequency perspective.
One challenge is to explore the possibility of integrating neural networks with other time-frequency transformations. In this paper, STFNet focuses on the short-time Fourier Transform. However, STFT is the most basic one. There are plenty of other transformation basis functions in traditional time-frequency analysis. How to naturally integrate them with neural network while keeping the underlying physical meaning within transformed representations? How to choose or design the most suitable transformation basis functions that meet the corresponding mathematical requirements? Answers to these questions can greatly impact the way researchers design neural networks for sensing signal processing.
\begin{figure}[!htb]
\vspace{-0.5cm}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.675\linewidth]{ablation_pool_acc-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{Accuracy with $95\%$ confidence interval. }
\label{fig:ablation_pool_acc}
\end{subfigure}%
\vspace{-0.1cm}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.675\linewidth]{ablation_pool_f1-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{F1 score with $95\%$ confidence interval.}
\label{fig:ablation_pool_f1}
\end{subfigure}
\vspace{-0.4cm}
\caption{STFNet Pooling v.s. Mean/Max Pooling}
\label{fig:ablation_pool}
\vspace{-0.3cm}
\end{figure}
Another challenge is to empower the frequency manipulations to have heterogeneous behaviours over the time. In STFNet, all designed operations are learnable frequency manipulations, which perform identically over time. In order to fully exploit the potential of time-frequency analysis, further research is needed on designing time-varying time-frequency manipulations, that adapt to current temporal patterns.
Furthermore, a better experimental and theoretical understanding is needed of the basic settings of neural networks to support computation in time-frequency domain. For traditional real-value neural networks, researchers have good intuitions about the basic configurations of initialization, activation functions, dropout and normalization techniques, and optimization methods. However, for neural network in the time-frequency domain, our understanding is limited. Although the reseach community started to study the basic settings of neural networks with complex values~\cite{trabelsi2017deep}, the current understanding remains preliminary. Time-frequency analysis can have operations in both the real and complex domains. At the same time, the underlying time-frequency information within the internal representations can make the related studies even more complicated. We believe that this understanding will greatly facilitate future design of deep learning systems for IoT.
In addition, outside the IoT context, there exists a large number of transformations and dimension reduction techniques, such as SVD and PCA, that have made great impact in revealing useful features of complex phenomena. Our study of deep learning with STFT suggests that integrating deep neural networks with other common transformations may facilitate learning in domains where such transformations reveal essential features of the input signal domain. Future work is needed to explore this conjecture.
\section{Evaluation}~\label{sec:evaluation}
In this section, we evaluate the STFNet with diverse sensing modularities. We focus on the device-based and device-free human activity recognitions with motion sensors (accelerometer and gyroscope), WiFi, ultrasound, and visible light. We first introduce the experimental setting, including data collection and baseline algorithms. Next, we show the performance metrics of leave-one-user-out evaluation of human activity recognition with different modularities. Finally, we analyze the effectiveness of STFNet through several ablation studies.
\subsection{Experimental Settings}~\label{sec:eval_setting}
In this subsection, we first introduce detailed information of the dataset we used or collected for each evaluation task. Then we specify the way to test the performance of evaluation task.
\textbf{\textit{Motion Sensor:}} In this experiment, we recognize human activity with motion sensors on smart devices.
We use the dataset collected by Allan et al.~\cite{stisen2015smart}. This dataset contains readings from two motion sensors (accelerometer and gyroscope). Readings were recorded when users executed activities scripted in no specific order, while carrying smartwatches and smartphones. The dataset contains 9 volunteers, 6 activities (biking, sitting, standing, walking, climbStair-up, and climbStair-down).
We align two sensor readings, linear interpolate two readings by 100Hz, and segment them into non-overlapping data samples with time interval 5.12s.
Therefore, each data sample is a $512\times 6$ matrix, where both accelerometer and gyroscope have readings on $x$, $y$, and $z$ axis.
\textbf{\textit{WiFi:}} In this experiment, we make use of Channel State Information (CSI) to analyze human activities. CSI refers to the known channel properties of a communication link, which can be affected by the presence of humans and their activities. We employ 11 volunteers (including both men and women) as the subjects and collect CSI data from 6 different rooms in two different buildings.
In particular, we build a WiFi infrastructure, which includes a transmitter (a wireless router) and two receivers.
We use the tool to report CSI values of 30 OFDM subcarriers~\cite{halperin2011tool}.
The experiment contains 6 activities (wiping the whiteboard, walking, moving a suitcase, rotating the chair. sitting, as well as standing up and sitting down).
We linearly interpolate the CSI data with a uniform sampling period, and down-sample the measurements into 100Hz. Then we segment the down-sampled CSI data into non-overlapping data samples with time interval 5.12s. Therefore, each data sample is a $512\times 30$ matrix, where each CSI measurement has readings from 30 subcarriers.
\textbf{\textit{Ultrasound:}} In this experiment, we conduct human activity recognition based on ultrasound. We employ 12 volunteers as the subjects to conduct the 6 different activities. The activity data are collected from 6 different rooms in two different buildings. The transmitter is an iPad on which an ultrasound generator app is installed, and it can emit an ultrasound signal of approximately 19 KHz. The receiver is a smartphone and we use the installed recorder app to collect the sound waves. We demodulate the received signal with carrier frequency 19KHz, and down-sample the measurement into 100Hz. Then we segment the down-sampled ultrasound data into non-overlapping data samples with time interval 5.12s. Therefore, each sample is a $512\times1$ matrix.
\textbf{\textit{Visible light:}} In this experiment, we capture the human activity in the visible light system. We build an optical system using photoresistors to capture the in-air body gesture, which can detect the illuminance change (lux) caused by the body interaction.
In the experiment, there are three light conditions (natural mode, warm mode, and cool mode) and 4 hand gestures (drawing an anticlockwise circle, drawing a clockwise circle, drawing a cross, and shaking hand side to side).
We employ 6 volunteers as the subjects and each of them performs 20 trials of every gesture under a given lighting condition. We linearly interpolate and down-sample the measurements into 25Hz. Then we segment the data into non-overlapping data samples with time interval 5.12s. Therefore, each sample is a $128\times6$ matrix, where each measurement contains readings from 6 CdS cells.
\begin{table}[!htp]
\vspace{-0.3cm}
\begin{center}
\caption {Illustration of models with two sensor inputs.}
\vspace{-0.3cm}
\label{tab:exp_model}
\scriptsize
\begin{tabular}{| c | c | c | c |}
\hline
\multicolumn{2}{ |c| }{STFNet-Filter/Conv} & \multicolumn{2}{ c| }{DeepSense/ComplexNet}\\
\hline
\hline
\multirow{ 2}{*}{Sensor Data 1} & \multirow{ 2}{*}{Sensor Data 2} & Chunked & Chunked \\
& & Sensor Data 1 & Sensor Data 2 \\
\hline
STFNet1-1 & STFNet1-2 & Conv Layer1-1 & Conv Layer1-2\\
\hline
STFNet2-1 & STFNet2-2 & Conv Layer2-1 & Conv Layer2-2\\
\hline
STFNet3-1 & STFNet3-2 & Conv Layer3-1 & Conv Layer3-2\\
\hline
\multicolumn{2}{ |c| }{STFNet-pooling} & \multicolumn{2}{ c| }{Max pooling}\\
\hline
\multicolumn{2}{ |c| }{STFNet4} & \multicolumn{2}{ c| }{Conv Layer4}\\
\hline
\multicolumn{2}{ |c| }{STFNet5} & \multicolumn{2}{ c| }{Conv Layer5}\\
\hline
\multicolumn{2}{ |c| }{STFNet6} & \multicolumn{2}{ c| }{Conv Layer6}\\
\hline
\multicolumn{2}{ |c| }{Averaging} & \multicolumn{2}{ c| }{GRU}\\
\hline
\multicolumn{2}{ |c| }{Softmax} & \multicolumn{2}{ c| }{Softmax}\\
\hline
\end{tabular}
\end{center}
\end{table}
\setlength{\textfloatsep}{0pt}
\textbf{\textit{Testing:}} In the whole evaluation, to illustrate the generalization ability of STFNet and other baseline models, we perform leave-one-user-out cross validation for every task. For each time, we choose the data from one user as testing data with the left as training data. We then compare the performance of models according to their accuracy and F1 score with $95\%$ confidence interval.
\subsection{Models in Comparison}~\label{sec:baseline}
In order to evaluate, when compared to conventional deep learning components (\emph{i.e.}, convolutional and recurrent layers), whether our proposed STFNet component is better at decoding information and extracting features from sensing inputs, we substitute components in the state-of-the-art neural network structure for IoT applications with STFNet. In the whole evaluation, we choose DeepSense as the state-of-the-art structure, which has shown signifiant improvements on various sensing tasks~\cite{yao2017deepsense}. The illustration of structures of five comparing models with two sensor inputs are shown in Table~\ref{tab:exp_model}. Detailed information of our comparing models are listed as follows,
\begin{enumerate}
\item \textbf{\textit{STFNet-Filter:}} This model integrates the proposed STFNet component and the DeepSense structure. Within the STFNet component, we use the STFNet-filtering operation designed in Section~\ref{sec:STFNet_filtering}. The intuition of DeepSense structure is to first perform local processing within each sensor and then perform global sensor fusion over multiple sensors. In this model, we replace all convolutional layers used in local/global sensor data processing with our time-frequency analyzing component, STFNet. Since our model has already incorporated time-domain analysis within the STFNet component through multi-resolution processing, we replace the Gated Recurrent Units (GRU) with simple feature averaging time at last.
\item \textbf{\textit{STFNet-Conv:}} This model is almost the same as the STFNet-Filter, except that we use the STFNet-convolution operation designed in Section~\ref{sec:STFNet_convolution}.
\item \textbf{\textit{DeepSense-Freq:}} This model is the original DeepSense~\cite{yao2017deepsense}. It divides the input sensing data into chunks, and processes each chunk with DFT. It treats the real and imagery parts of discrete Fourier transformed time chunks as the additional feature dimensions. This is the state-of-the-art deep learning model for sensing data modelling and IoT applications.
\begin{figure}[!htb]
\vspace{-0.5cm}
\centering
\includegraphics[width=0.675\linewidth]{HHAR-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{The accuracy and F1 score with $95\%$ confidence interval for motion sensors.}
\label{fig:HHAR_acc_f1}
\vspace{-0.4cm}
\end{figure}
\vspace{-0.1cm}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.675\linewidth]{WIFI-eps-converted-to.pdf}
\vspace{-0.4cm}
\caption{The accuracy and F1 score with $95\%$ confidence interval for WiFi.}
\label{fig:WIFI_acc_f1}
\end{figure}
\item \textbf{\textit{DeepSense-Time:}} This model is almost the same as the DeepSense-Freq, except that it directly takes the chunked raw sensing data without DFT as input.
\item \textbf{\textit{ComplexNet:}} This model is a complex-value neural network~\cite{trabelsi2017deep} that can operate on complex-value inputs.
Instead of using simple CNN and RNN structure as originally proposed~\cite{trabelsi2017deep}, we cheat in their favor by using the DeepSense structure, which improves the performance in all tasks.
The network inputs are chunked sensing data with DFT.
\end{enumerate}
\subsection{Effectiveness}
In this section, we discuss about the effectiveness of our proposed STFNet based on extensive experiments and diverse sensing modalities, compared with other state-of-the-art deep learning models.
As we mentioned in Section~\ref{sec:eval_setting}, all models are evaluated through leave-one-user-out cross validation with accuracy and F1 score accompanied by the $95\%$ confidence interval. STFNet-based models (STFNet-Filter and STFNet-Conv) take a sliding window set for multi-resolution short-time Fourier transform.
We choose the set to be $\{16, 32, 64, 128\}$ for activity recognition with motion sensors, WiFi, and ultrasound; and choose set to be $\{8, 16, 32, 64\}$ for activity recognition with visible light. DeepSense-based models (DeepSense-Freq and DeepSense-Time) need a sliding window for chunking input signals. In the evaluation, we cheat in their favor by choosing the best-performing window size from $\{8, 16, 32, 64, 128\}$ according to the accuracy metric.
In addition, we consistently configure STFNet-filtering operation with linear interpolation, and STFNet-convolution operation with spectral padding. We will show further evaluations on multi-resolution operations and the effects of diverse operation settings in Section~\ref{sec:ablation}.
\subsubsection{Motion Sensors}
For device-based activity recognition with motion sensors, there are 9 users. The accuracy and F1 score with the $95\%$ confidence interval for leave-one-user-out cross validation are illustrated in Figure~\ref{fig:HHAR_acc_f1}.
\begin{figure}[!htb]
\vspace{-0.5cm}
\centering
\includegraphics[width=0.675\linewidth]{acoustic-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{The accuracy and F1 score with $95\%$ confidence interval for Ultrasound.}
\label{fig:acoustic_acc_f1}
\end{figure}
\begin{figure}[!htb]
\vspace{-0.4cm}
\centering
\includegraphics[width=0.675\linewidth]{light-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{The accuracy and F1 score with $95\%$ confidence interval for Visible light.}
\label{fig:light_acc_f1}
\end{figure}
STFNet based models, \emph{i.e.}, STFNet-Filter and STFNet-Conv, outperform all other baseline models with a large margin. The confidence interval lower bound of STNet-Filter and STFNet-Conv is even better than the confidence interval upper bound of DeepSense-Freq and DeepSense-Time.
STFNet-Filter performs better than STFNet-Conv in this experiment, indicating that different activities have distinct global profiling patterns with motion sensor readings in the frequency domain, even among different users. STFNet-Filter is able to learn the accurate global frequency profiling, which makes it the top-performance model in this task. In addition, compared to ComplexNet, STFNet based models show clear improvements. Therefore, using just complex-value neural network for sensing signal is far from enough. The multi-resolution processing and operations that are spectral-compatible are all crucial designs.
\subsubsection{WiFi}
For device-free activity recognition with WiFi signal, there are 11 users. The accuracy and F1 score with the $95\%$ confidence interval for leave-one-user-out cross validation are illustrated in Figure~\ref{fig:WIFI_acc_f1}.
STFNet based models still outperform all others with a clear margin, illustrating the effectiveness of principled design of STFNet from time-frequency perspective. DeepSense-Freq outperforms DeepSense-Time in this experiment, which means that even having time-frequency transformation as pre-processing can help. The complex-value network, ComplexNet, performs worse than its real-value counterpart, DeepSense-Freq. This indicates that blindly processing time-frequency representations without preserving their physical meanings can even hurt the final performance. STNet-Conv performs better than STNet-Filter in the WiFi experiment, indicating that local shiftings in the frequency domain are more representative for diverse activities profiled with WiFi CSI.
\begin{figure}[!htb]
\vspace{-0.5cm}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.675\linewidth]{ablation_multi_single_acc-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{Accuracy with $95\%$ confidence interval. }
\label{fig:ablation_multi_single_acc}
\end{subfigure}%
\vspace{-0.1cm}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.675\linewidth]{ablation_multi_single_f1-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{F1 score with $95\%$ confidence interval.}
\label{fig:ablation_multi_single_f1}
\end{subfigure}
\vspace{-0.3cm}
\caption{Multi-Resolution v.s. Single-Resolution}
\label{fig:ablation_multi_single}
\end{figure}
\subsubsection{Ultrasound}
There are 12 users in device-free activity recognition with ultrasound experiment. The accuracy and F1 score with the $95\%$ confidence interval for leave-one-user-out cross validation are illustrated in Figure~\ref{fig:acoustic_acc_f1}.
STFNet based models still significantly outperforms all other baselines. An interesting observation is that ComplexNet performs even worse than both DeepSense-Freq and DeepSene-Time, which again validates the importance of designing neural networks for sensing signal with multi-resolution processing as well as preserving the time and frequency information.
\subsubsection{Visible Light}
There are 6 users in the experiment of device-free activity recognition with visible light. The accuracy and F1 score with the $95\%$ confidence interval are illustrated in Figure~\ref{fig:light_acc_f1}. Except for the DeepSense-Time, all other models can can achieve an accuracy of approximately 90\% or higher. STFNet based models still do the best. There is no significant difference between STFNet-Filter and STFNet-Conv, which indicates that measured visible light readings have quite clean representations in the frequency domain.
\subsection{Ablation Studies}~\label{sec:ablation}
In the previous section, we illustrate the performance of STFNet compared to other state-of-the-art baselines. In this section, we focus mainly on the STFNet design. We conduct several ablation studies by deleting one designing feature from STFNet at a time.
\subsubsection{Multi-Resolution v.s. Single-Resolution}
First, we validate the effectiveness of our design of multi-resolution processing in STFNet block. As shown in Figure~\ref{fig:STFNet_overview}, this includes multi-resolution STFT, hologram interleaving, and weights sharing techniques in STFNet-Filtering and STFNet-Convolution operations. In this experiment, we add two more baseline models, STFNet-Single-Filter and STFNet-Single-Conv, generated by deleting the multi-resolution processing in STFNet-Filter and STFNet-Conv respectively. These two models pick the best-performing window size from $\{8, 16, 32, 64, 128\}$ according to the accuracy metric. The results for all four tasks are illustrated in Figure~\ref{fig:ablation_multi_single}, where DeepSense-Freq severs as a decent performance low-bound. The design of multi-resolution processing significantly impacts the performance of STFNet. STFNet-Single-Filter and STFNet-Single-Conv show clear performance degradation compared to their multi-resolution counterparts. In addition, STFNet-Single-Filter and STFNet-Single-Conv still consistently outperform DeepSense-Freq with a clear margin. This is because our other designed operations, including STFNet-Filtering, STFNet-Convolution, STFNet-Pooling still facilitate the learning in time-frequency domain.
\begin{figure}[!htb]
\vspace{-0.5cm}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.675\linewidth]{ablation_padding_acc-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{Accuracy with $95\%$ confidence interval. }
\label{fig:ablation_padding_acc}
\end{subfigure}%
\vspace{-0.1cm}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.675\linewidth]{ablation_padding_f1-eps-converted-to.pdf}
\vspace{-0.35cm}
\caption{F1 score with $95\%$ confidence interval.}
\label{fig:ablation_padding_f1}
\end{subfigure}
\vspace{-0.4cm}
\caption{Spectral Padding v.s. Zero Padding}
\label{fig:ablation_padding}
\end{figure}
\subsubsection{Spectral Padding v.s. Zero Padding} Next, we validate our design of spectral padding in the STFNet-Convolution operation as shown in Figure~\ref{fig:STFNet_convolution}. In this experiment, we add a new baseline algorithm, STFNet-Conv-zPad, by replacing spectral padding with traditional zero padding in the STFNet-Conv. The accuracy and F1 score of all four tasks are shown in Figure~\ref{fig:ablation_padding}. Here, DeepSense-Freq is still treated as a performance low-bound. By comparing STFNet-Conv-zPad and STFNet-Conv, we can see that spectral padding consistently helps improving the model performance. In most cases, the improvement is limited. However, in the case of visible light, spectral padding significantly improves both accuracy and F1 score. Therefore, designing neural network by preserving the time-frequency semantics of sensing signal is an important rule to follow.
\vspace{-0.2cm}
\subsubsection{Linear Interpolation v.s. Spectral Interpolation} Then, we compare our two designs of weight interpolation method in the STFNet-Filtering operation, linear interpolation and spectral interpolation, as shown in Figure~\ref{fig:STFNet_filtering}. The STFNet-Filter defined in Section~\ref{sec:baseline} uses linear interpolation, so we rename it as STFNet-Filter-LinearInpt in this experiment. We add a new baseline model called STFNet-Filter-SpectralInpt by using spectral interpolation instead of linear interpolation in STFNet-Filter. The results of all four tasks are illustrated in Figure~\ref{fig:ablation_interp}. In general, the performance of two design choices are almost the same. At most of time, linear interpolation performs slightly better. In addition, we recommend using linear interpolation, since its implementation is easier,
\begin{figure}[!htb]
\vspace{-0.5cm}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.675\linewidth]{ablation_interp_acc-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{Accuracy with $95\%$ confidence interval. }
\label{fig:ablation_interp_acc}
\end{subfigure}%
\vspace{-0.1cm}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.675\linewidth]{ablation_interp_f1-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{F1 score with $95\%$ confidence interval.}
\label{fig:ablation_interp_f1}
\end{subfigure}
\vspace{-0.4cm}
\caption{Linear Interpolation v.s. Spectral Interpolation}
\label{fig:ablation_interp}
\end{figure}
\vspace{-0.2cm}
\subsubsection{STFNet Pooling v.s. Mean/Max Pooling} Finally, we validate our design of STFNet-Pooling (low-pass deisgn) as shown in Figure~\ref{fig:STFNet_pooling}. In this experiment, we add two new baseline algorithms, STFNet-Filter-mPad and STFNet-Conv-mPad, by replacing STFNet-Pooling in STFNet-Filter and STFNet-Conv with traditional max/mean pooling in the time domain (through choosing the one has better accuracy). The results are illustrated in Figure~\ref{fig:ablation_pool}. In all settings, STFNet-Pooling shows better performance. In some cases, the improvement are significant. We believe that STFNet-Pooling can achieve even better performance if given the detailed signal-to-noise ratio over the frequency domain for each specific sensor. Then we can employ other pooling strategies instead of the low-pass design.
\section{Introduction}
Motivated by the needs of IoT applications, this paper presents a principled way of designing deep neural networks that learn (from IoT sensing signals) features inspired by the fundamental properties of the underlying domain of measurements; namely, properties of physical signals. We refer by IoT applications to those where sensors measure some physical quantities, generating (possibly complex and multi-dimentional) time-series data, typically reflecting some underlying physical process. The human brain (whose wiring inspires the structure of conventional neural networks) extracts features well-suited for {\em external perceptual\/} tasks, which explains the great success of such networks at those tasks. In contrast, the internal {\em physical processes\/} underlying sensor measurements in IoT systems have properties (such as physical intertia, characteristics of wireless signal propagation, and signal resonance) that depend more on signal {\em frequency\/}, motivating feature extraction in the {\em frequency domain\/}. It is no coincidence that much of classical signal processing literature works by transforming time-series data to the frequency domain first. To help capture signatures of internal physical processes the way a brain captures their externally perceived properties, this paper develops a new neural network block designed specifically for learning in the frequency domain.
The design of neural network structures greatly influences efficiency of signal modelling and ease of extraction of hidden patterns. Convolutional neural networks (CNNs) for image recognition, for example, align perfectly with biological studies of the visual cortex~\cite{hubel1968receptive} and with domain knowledge in digital image processing~\cite{gonzalez2002digital}. We thus ask a fundamental question: what structures are well-suited for the domain of physical sensor measurements, which we henceforth call the domain of IoT?
Previous research on customizing deep learning models to the needs of IoT applications~\cite{yao2018deep,lane2015deepear,yao2017deepsense} mainly focused on designing neural network structures that integrate conventional deep learning components, such as convolutional and recurrent layers, to extract spatial and temporal properties of inputs. On the other hand, since the physics of measured phenomena are best expressed in the frequency domain, decades of research on signal processing developed powerful techniques for time-frequency analysis of signals, including motion sensor signals~\cite{stisen2015smart, hemminki2013accelerometer}, radio frequency signals~\cite{wang2015understanding,pu2013whole}, acoustic signals~\cite{gupta2012soundwave,chen2014airlink}, and visible light signals~\cite{li2016practical}. A popular transform that maps time-series measurements to the frequency domain is the
Short-Time Fourier Transform (STFT). We, therefore, propose a new neural network model, namely, Short-Time Fourier Neural Networks (STFNets) that operate directly in the frequency domain.
One potential approach for learning in the frequency domain might simply be to convert sensing signals into the frequency domain first, and then apply conventional neural network components, possibly extending them to support operations on complex-numbers so they can represent frequency-domain quantities~\cite{trabelsi2017deep}. These approaches miss two key opportunities for improvement, described below, that we take advantage of in this work. As a result, our work leads to more accurate results, as shown in the evaluation section. The two reasons that account for our improvements are as follows.
First, different from traditional neural networks, where the internal representations constitute features with no physical meaning, the internal representations in STFNet leverage frequency domain semantics that encode time and frequency information.
All operations and learnable parameters we propose are explicitly made compatible with the basic properties of spectral data, and align corresponding frequency and time components. In our design, we categorize spectral manipulations into three main types: filtering, convolution, and pooling. Filtering refers to the general spectral filtering and global template matching operation; convolution refers to the local motif detection including shift detection and local template detection; and pooling refers to dimension reduction over the frequency domain. We then design the spectral-compatible parameters and operating rules for these three manipulation categories respectively, which have shown superior performance in evaluations compared to the application of {\em conventional\/} neural networks in the domain of complex-numbers.
Second, transforming signals to the frequency domain is governed by the {\em uncertainty principle\/}~\cite{smith2007mathematics}. The transformed representation cannot achieve both a high frequency resolution and a high time resolution at the same time. In STFT, the time-frequency resolution is controlled by the length of the sliding window (the length of the part of the time-series being converted at a time). With a longer window, we can obtain a finer-grained frequency representation. However, we then cannot achieve a time resolution smaller than the window size. The uncertainty principle causes a dilemma in traditional time-frequency analysis. One often needs to guess the best time-frequency resolution using trial and error. In STFNet, we circumvent this dilemma by simultaneously computing multiple STFTs with different time-frequency resolutions. The representations with different time-frequency resolutions are then mutually enhanced in a data-driven manner. The network then automatically learns the best resolution or resolutions, where the most useful features are present.
STFNet defines a formal way to extract features from multiple time-frequency transformations with the same set of spectral-compatible operations and parameters, which greatly reduces model complexity while improving accuracy.
We demonstrate the effectiveness of STFNet through extensive experiments with various sensing modalities. During the evaluation, we focus on device-based and device-free human activity recognition with a broad range of sensing modalities, including motion sensors (accelerometer and gyroscopes), WiFi, ultrasound, and visible light. The experimental results validate the design settings of STFNets and illustrate their superior accuracy compared to the state-of-the-art deep learning frameworks for IoT applications.
Broadly speaking, the main contributions of this paper to the general research landscape of deep learning and IoT are twofold:
\begin{enumerate}
\item STFNet presents a principled way of designing neural networks that reveal the key properties of physical processes underlying the sensing signals from the time-frequency perspective.
\item STFNet unveils the benefit of incorporating domain-specific analytic modelling and transformation techniques into the neural network design.
\end{enumerate}
The rest of paper is organized as follows. Section~\ref{sec:related} introduces related work on deep learning in the context of mobile sensing as well as deep learning for spectral representations. We introduce the detailed technical design of STFNet in Section~\ref{sec:model}. The evaluation is presented in Section~\ref{sec:evaluation}. Finally, we discuss the results in Section~\ref{sec:discussion} and conclude in Section~\ref{sec:conclusion}.
\section{Short-Time Fourier Neural Networks}~\label{sec:model}
We introduce the technical details of STFNets in this section. We separate the technical descriptions into six parts. In the first two subsections, we provide some background followed by a high-level overview of STFNet components, including (i) hologram interleaving, (ii) STFNet-filtering, (iii) STFNet-convolution, and (iv) STFNet-pooling. In the remaining four subsections, we describe the technical details of each of these components, respectively.
\subsection{Background and STFNet Overview}
IoT devices sample the physical environment generating time-series data. Discrete Fourier Transform (DFT) is a mathematical tool that converts $n$ samples over time (with a sampling rate of $f_s$) into a $n$ components in frequency (with a frequency step of $f_s/n$). The more samples are selected, the finer the component resolution is in frequency.
\begin{figure}[htb]
\vspace{-0.3cm}
\centering
\includegraphics[width=0.75\linewidth]{dataflow-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{Data Flow within a block of STFNet.}
\label{fig:dataflow}
\end{figure}
We can always transform the whole sequence of data with DFT, achieving a high frequency resolution. However, we then lose information on signal evolution over time, or the time resolution. In order to solve this problem, Short-Time Fourier Transform (STFT) divides a longer time signal into shorter segments of equal length and computes DTF separately on each shorter segment. By losing a certain degree of frequency resolution, STFT helps us regain the time resolution to some extent. In choosing $n$, there arises a fundamental trade-off between the attainable time and frequency resolution, which is called the {\em uncertainty principle\/}~\cite{smith2007mathematics}.
For the purposes of learning to pedict a given output, the optimal trade-off point depends on the time and frequency granularity of the features that best determine the outputs we want to reproduce. The goal of STFNets is thus to learn frequency domain features that predict the output, while at the same time learn the best resolution trade-off point in which the relevant features exist.
The building component of an STFNet is an {\em STFNet block\/}, shown in Figure~\ref{fig:STFNet_overview}. An STFNet block is the layer-equivalent in our neural network. The larger network would normally be composed by stacking such layers. Within each block, STFNet circumvents the uncertainty principle by computing multiple STFT representations with different time-frequency resolutions. Collectively, these representations constitute what we call the {\em time-frequency hologram\/}. And we call an individual time-frequency signal representation, a hologram
representation. They are then used to mutually enhance each other by filling-in missing frequency components in each.
Candidate frequency-domain features are then extracted from these enhanced representations via general spectral manipulations that come in two flavors; filtering and convolution. They represent global and local feature extraction operations, respectively. The filtering and convolution kernels are learnable, making each STFNet layer a building block for spectral manipulation and learnable frequency domain feature extraction. In addition, we also design a new mechanism, called pooling, for frequency domain dimensionality reduction in STFNets. Combinations of features extracted using the above manipulations then pass through activation functions and an inverse STFT transform to produce (filtered) outputs in the time domain. Stacking STFNet blocks has the effect of producing progressively sharper (i.e., higher order) filters to shape the frequency domain signal representation into more relevant and more fine-tuned features.
Figure~\ref{fig:dataflow} gives an example of an SFTNet block that accepts as input a two-dimensional time-series signal (e.g., 2D accelerometers data). Each dimension is then transformed to the frequency domain at four different resolutions using STFT, generating four different internal nodes, each representing the signal in the frequency domain at a different time-frequency resolution. Collectively, the four representations constitute the hologram. In the next step, mutual enhancements are done improving all representations. Each representation then undergoes a variety of alternative spectral manipulations (called ``filters" in the figure). Two filters are shown in the figure for each dimension. The parameters of these filters are the weights multiplied by the frequency components of the filter input; a different weight per component. These parameters are what the network learns. Note that, a filter does not change the time-frequency resolution of the corresponding input. Filter outputs of the same time-frequency resolution are then combined additively across all dimensions and passed through a non-linear activation function (as in a conventional convolutional neural network). An inverse STFT brings each such combined output back to the time domain, where it becomes an input to the next STFNet block. (Alternatively, the inverse STFT can be applied after dimension combination and before the activation function.) Hence, each output time-series is produced by applying spectral manipulation and fusion to one particular time-frequency resolution of all input time-series. Once converted to the time domain, however, the output time-series can be resampled in the next block at different time-frequency resolutions again. The goal of STFNet is to learn the weighting of different frequency components within each filter in each block such that features are produced that best predict final network outputs.
\subsection{STFNet Block Fundamentals}~\label{sec:STFNet_overview}
In this subsection, we introduce the formulation of our design elements wihin each STFNet block. In the rest of this paper, all vectors are denoted by bold lower-case letters (e.g., $\mathbf{x}$ and $\mathbf{y}$), while matrices and tensors are represented by bold upper-case letters (e.g., $\mathbf{X}$ and $\mathbf{Y}$). For a vector $\mathbf{x}$, the $j^{th}$ element is denoted by $\mathbf{x}_{[j]}$. For a tensor $\mathbf{X}$, the $t^{th}$ matrix along the third axis is denoted by $\mathbf{X}_{[\cdot, \cdot, t]}$, and other slicing denotations are defined similarly. We use calligraphic letters to denote sets (e.g., $\mathcal{X}$ and $\mathcal{Y}$). For set $\mathcal{X}$, $|\mathcal{X}|$ denotes the cardinality.
We denote the input to the STFNet block as $\mathbf{X}\in \mathbb{R}^{T \times D}$,
where we divide the input $D$-dimension time-series into windows of size $T$ samples. We call $T$ the signal length and $D$ the signal dimension.
Since we concentrate on sensing signals, we assume that all the raw and internal-manipulated sensing signals are real-valued in time domain.
As shown in Figure~\ref{fig:STFNet_overview}, the input signal $\mathbf{X}$ first goes through a multi-resolution short-time Fourier transform ($\text{Multi\_STFT}$), which is a compound traditional short-time Fourier transform ($\text{STFT}$), to provide a time-frequency hologram of the signal. $\text{STFT}$ breaks the original signal up into chunks with a sliding window, where sliding window $\mathbf{W}(t)$ with width $\tau$ only has non-zero values for $1 \le t\le \tau$. Then each chunk is Discrete-Fourier transformed,
\begin{equation}
\small
\mathbf{STFT}^{(\tau, s)}(\mathbf{X})_{[m, k, d]} = \sum_{t=1}^T \mathbf{X}_{[t, d]} \cdot \mathbf{W}(t - s\cdot m) \cdot e^{-j\frac{2\pi k}{\tau} (t - s\cdot m)},
\label{eqn:STFT}
\end{equation}
where $\mathbf{STFT}^{(\tau, s)}(\mathbf{X}) \in \mathbb{C}^{M\times K\times D}$ denotes the short-time Fourier transform with width $\tau$ and sliding step $s$. $M$ denotes the number of time chunks. $K$ denotes the number of frequency components. Since input signal $\mathbf{X}$ is real-valued, its discrete Fourier transform is conjugate symmetric. Therefore, we only need the $\lfloor\tau/2\rfloor + 1$ frequency components to represent the signal, \emph{i.e.}, $K = \lfloor\tau/2\rfloor + 1$.
In this paper, we focus on sliding chunks with rectangular window and no overlaps to simplify the formulation, \emph{i.e.}, $s = \tau$ and $M = T/\tau$. We therefore denote of short-time Fourier transform as $\mathbf{STFT}^{(\tau)}(\mathbf{X})$.
\begin{figure}[!htb]
\vspace{-0.3cm}
\centering
\includegraphics[width=0.7\linewidth]{STFNet_hologram_interleaving2-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{The design of hologram interleaving.}
\label{fig:hologram_interleaving}
\vspace{-0.1cm}
\end{figure}
The $\text{Multi\_STFT}$ operation is composed of multiple short-time Fourier transform with different window widths $\mathcal{T} = \{\tau_i\}$. The window width, $\tau_i$, determines the time-frequency resolution of $\text{STFT}$. Larger $\tau_i$ provides better frequency resolution, while smaller $\tau_i$ provides better time resolution. In this paper, we set the window widths to be powers of $2$, \emph{i.e.}, $\tau_i = 2^{p_i}$ $\forall p_i \in \mathbb{Z}_0^+$, to simplify the design later. We can thus formulate $\text{Multi\_STFT}$ as:
\begin{equation}
\small
\begin{split}
\mathbf{Multi\_STFT}^{(\mathcal{T})}\{\mathbf{X}\} & = \big\{\mathbf{STFT}^{(\tau_i)}(\mathbf{X})\big\} \text{ for }2^{p_i} \in \mathcal{T}.
\end{split}
\label{eqn:Multi_STFT}
\end{equation}
Next, according to Figure~\ref{fig:STFNet_overview}, the multi-resolution representations go into the hologram interleaving component, which
enables the representations to compensate and balance their time-frequency resolutions with each other. The technical details of the hologram interleaving component are introduced in Section~\ref{sec:STFNet_hologram}.
The STFNet block then manipulates multiple hologram representations with the same set of spectral-compatible operation(s), including STFNet-filtering, STFNet-convolution, and STFNet-pooling. We will formulate these operations in Section~\ref{sec:STFNet_filtering},~\ref{sec:STFNet_convolution}, and~\ref{sec:STFNet_pooling}, respectively.
Finally, the STFNet block converts the manipulated frequency representations back into the time domain with the inverse short-time Fourier transform. The resulting representations from different views of the hologram are weighted and merged as the input ``signal" for the next block. Since we merge the output representations from different views of the hologram, we reduce the output feature dimension of STFNet-filtering and convolution operations by the factor of $1/|\mathcal{T}|$ to prevent the dimension explosion.
\subsection{STFNet Hologram Interleaving}~\label{sec:STFNet_hologram}
In this subsection, we introduce the formulation of hologram interleaving.
Due to the Fourier uncertainty principle, the representations in time-frequency hologram either have high time resolution or high frequency resolution.
The hologram interleaving tries to use representations with high time resolution to instruct the representations with low time resolution to highlight the important components over time.
This is done by two steps:
\begin{enumerate}
\item Revealing the mathematical relationship of aligned time-frequency components among different representations in the time-frequency hologram.
\item Updating the original relationship in a data-driven manner through neural-network attention components.
\end{enumerate}
We start from the definition of time-frequency hologram, generated by $\text{Multi\_STFT}$ defined in~\eqref{eqn:Multi_STFT}. Note that, the window width set $\mathcal{T}$ is defined as $ \{2^{p_i}\}$, $ \forall p_i \in \mathbf{Z}_0^{+}$.
Without loss of generality, an illustration of multi-resolution short-time Fourier transformed representations with input signal having length $16$ and signal dimension $3$ as well as $\mathcal{T} = \{4, 8, 16\}$ are illustrated in Figure~\ref{fig:hologram_interleaving}.
In order to find out the relationship of aligned time-frequency components, we start with the frequency-component dimension. Since different representations only change the window width $\tau_i$ of STFT but not the sampling frequency $f_s$ of input signal, these frequency components represent frequencies from $0$ to $f_s/2$ (Nyquist frequency) with step $f_s/\tau_i$. Then we can first obtain the relationship of frequency ranging steps among different representations,
\vspace{-0.15cm}
\begin{equation}
\small
\forall p_i > p_j \textrm{ , } \frac{f_s/\tau_j}{f_s/\tau_i} = 2^{p_i - p_j} \in \mathbf{Z}_0^{+}.
\vspace{-0.15cm}
\label{eqn:freq_align}
\end{equation}
Therefore, a low frequency-resolution representation (with window width $2^{p_j}$) can find their frequency-equivalent counterparts for every $2^{p_i - p_j}$ frequency components in a high frequency-resolution representation (with window width $2^{p_i}$). The upper part of Figure~\ref{fig:hologram_interleaving} provides a simple illustration of such relationship. In the following analysis, we will use the original index $k$ and corresponding frequency $k\cdot f_s/\tau_i$ interchangeably to recall the frequency component from the time-frequency hologram $\mathbf{STFT}^{(\tau)}(\mathbf{X})_{[m, k, d]} $.
Next, we analyze the relationship over the time-chunk dimension, when two representations have frequency-equivalent components.
Note that time chunks in $\mathbf{STFT}^{(\tau)}(\mathbf{X})$ are generated by sliding rectangular window without overlap. Based on~\eqref{eqn:STFT}, for representations having window widths $\tau_i=2^{p_i}$ and $\tau_j=2^{p_j}$ ($p_i > p_j$),
\vspace{-0.3cm}
\begin{equation}
\small
\begin{split}
\mathbf{STFT}^{(\tau_i)}(\mathbf{X})_{[m, 2^{p_i - p_j}k, d]} = \sum_{t=2^{p_i} m + 1}^{2^{p_i}(m+1)} \mathbf{X}_{[t, d]} \cdot e^{-j\frac{2\pi 2^{p_i - p_j} k}{2^{p_i}} (t - m\cdot 2^{p_i})}, \\
= \sum_{m_j = 2^{p_i-p_j}m}^{2^{p_i-p_j}(m+1) -1} \sum_{t=m_j +1}^{ 2^{p_j}(m_j+1)} \mathbf{X}_{[t, d]} \cdot e^{-j\frac{2\pi k}{2^{p_j}} (t - m\cdot 2^{p_j})},\\
= \sum_{m_j = 2^{p_i-p_j}m}^{2^{p_i-p_j}(m+1) -1} \mathbf{STFT}^{(\tau_j)}(\mathbf{X})_{[m_j, k, d]}. \phantom{aaaaaaaaaaa}
\end{split}
\label{eqn:time_align}
\end{equation}
\vspace{-0.3cm}
Therefore, given the equivalent frequency component, a time component in low time-resolution representation (with window width $2^{p_i}$) is the sum of $2^{p_i - p_j}$ aligned time components of the high time-resolution representation (with window width $2^{p_j}$). As a toy example in Figure~\ref{fig:hologram_interleaving}, the first row of the middle tensor is equal to the sum of first two rows of the left tensor for frequencies $0$, $f_s/4$, and $f_s/2$. The row of the right tensor is equal to the sum of four rows of the left tensor for frequencies $0$, $f_s/4$, and $f_s/2$. The row of the right tensor is equal to the sum of two rows of the middle tensor for frequencies $f_s/8$ and $3f_s/8$, etc.
According to the analysis above, the high frequency-resolution representations lose their fine-grained time resolutions at certain frequencies by summing the corresponding frequency components up over a range of time. However, the high time-resolution representations preserve these information.
The idea of hologram interleaving is to replace the sum operation in high frequency-resolution representation with a weighted merge operation to highlight the important information over time. For a certain frequency component, the weight of merging is learnt through the most fine-grained information preserved
in the time-frequency hologram. In this paper, we implement the weighted merge operation as a simple attention module. For a merging input $\mathbf{z} \in\mathbb{C}^{S \times 1}$, where $S$ is the number of elements to be merged, the merge operation is formulated as:
\vspace{-0.4cm}
\begin{equation}
\small
\begin{split}
\mathbf{a} &= \mathrm{softmax}( |\mathbf{W}_m \mathbf{z}| ), \\
y &= S\times \mathbf{a}^\intercal \mathbf{z},
\end{split}
\vspace{-0.2cm}
\label{eqn:time_align}
\end{equation}
\begin{figure}[!htb]
\vspace{-0.3cm}
\centering
\includegraphics[width=0.7\linewidth]{STFNet_filtering-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{The STFNet-filtering operation.}
\label{fig:STFNet_filtering}
\end{figure}
where $|\cdot|$ is the piece-wise magnitude operation for complex-number vector; and $\mathbf{W}_m \in \mathbb{C}^{S\times S}$ is the learnable weight matrix. Notice that the final merged result is rescaled by the factor $S$ to imitate the ``sum" property of Fourier transform.
\subsection{STFNet-Filtering Operation}~\label{sec:STFNet_filtering}
Starting from this subsection, we will introduce our three spectral-compatible operations in STFNet. In each subsection, the introduction includes two main parts: 1) the basic formulation of proposed spectral-compatible operation, and 2) extending a single operation to multi-resolution data.
Spectral filtering is a widely-used operation in time-frequency analysis. The STFNet-filtering operation replaces the traditional manually designed spectral filter with a learnable weight that can update during the training process.
Although the spectral filtering is equivalent to the time-domain convolution according to convolution theorem~\footnote{\url{https://en.wikipedia.org/wiki/Convolution_theorem}}, the filtering operation helps to handle the multi-resolution time-frequency analysis, and facilitates the parameterization and modelling. We denote the input tensor as $\mathbf{X} \in \mathbf{C}^{M\times K\times D}$, where $M$ is the number of time chunk, $K$ frequency component number, and $D$ input feature dimension. The STFNet-filtering operation is formulated as:
\begin{equation}
\small
\begin{split}
\mathbf{Y}_{[m, k, \cdot]} = \mathbf{X}_{[m, k, \cdot]} \mathbf{W}_{f[k, \cdot, \cdot]},
\end{split}
\label{eqn:STFNet_filtering}
\end{equation}
where $\mathbf{W}_f \in \mathbb{C}^{K\times D\times O}$ is the learnable weight matrix, $O$ the output feature dimension, and $\mathbf{Y}\in\mathbb{C}^{M\times K\times O}$ the output representation.
The function of STFNet-filtering operation is providing a set of learnable global frequency template matchings over the time. However, it is not straightforward to extend the matching operation to the representations with different time-frequency resolutions. Although we can create multiple $\mathbf{W}_f$ with different frequency resolutions $K$, it can introduce unnecessary complexity and redundancy.
STFNet-filtering solves this problem by interpolating the frequency components in weight matrix. As we mentioned in Section~\ref{sec:STFNet_hologram}, data in hologram with different frequency resolutions have the same frequency range (from $0$ to $f_s/2$) but different frequency steps ($f_s/\tau$). Therefore, STFNet-filtering operation only has one weight matrix $\mathbf{W}_f$ with $K = \lfloor\tau/2\rfloor + 1$ frequency components. When the operation input has $K' = \lfloor\tau'/2\rfloor + 1$ frequency components with $K' < K$, we can subsample the frequency components in $\mathbf{W}_f$.
When $K' > K$, we interpolate the frequency components of $\mathbf{W}_f$. STFNet provides two kind of interpolation methods: 1) linear interpolation and 2) spectral interpolation.
\begin{figure}[!htb]
\vspace{-0.2cm}
\centering
\includegraphics[width=0.7\linewidth]{STFNet_convolution-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{The STFNet-convolution operation with dilated configuration.}
\label{fig:STFNet_convolution}
\end{figure}
The linear interpolation generates the missing frequency components in extended weight matrix $\mathbf{W}'_{f} \in\mathbb{C}^{K'\times D\times O}$ from the two neighbouring frequency components in $\mathbf{W}_f$:
\begin{equation}
\small
\begin{split}
&k_l = \Big\lfloor k' \frac{\tau}{\tau'} \Big\rfloor \quad k_r = k_l +1, \\
&\mathbf{W}'_{f[k',\cdot, \cdot]} = \mathbf{W}_{f[k_l,\cdot, \cdot]} \Big(k_r - k'\frac{\tau}{\tau'}\Big) + \mathbf{W}_{f[k_r,\cdot, \cdot]} \Big( k'\frac{\tau}{\tau'} - k_l \Big).
\end{split}
\label{eqn:linear_interpolate}
\end{equation}
The spectral interpolation utilizes the relationship between discrete-time Fourier transform (DTFT) and discrete Fourier transform (DFT). For a time-limited signal (with length $\tau$), DTFT regards it as a infinite-length data with zeros outside the time-limited range, while DFT regards it as a $\tau$-periodic data. As a result, DTFT generates a continuous function over the frequency domain, while DFT generates a discrete function.
Therefore, DFT can be regarded as a sampling of DTFT with step $f_s/\tau$. In order to increase the frequency resolution of $\mathbf{W}_f$, we can increase the sampling step from $f_s/\tau$ to $f_s/\tau'$, which is called spectral interpolation. Spectral interpolation can be done through zero padding in the time domain~\cite{smith2007mathematics},
\begin{equation}
\small
\begin{split}
\mathbf{W}'_{f[\cdot, d, o]} = \mathbf{DFT}\Big(\mathbf{ZeroPad}_{\tau' - \tau} \mathbf{IDFT}\big( \mathbf{W}_{f[\cdot, d, o]} \big) \Big),
\end{split}
\label{eqn:spectral_interpolate}
\end{equation}
where $\mathbf{ZeroPad}_t$ denotes padding $t$ zeros at the end of sequence, and $\mathbf{IDFT}(\cdot)$ denotes the inverse discrete Fourier transform. Please note that, if we pad infinite zeros to the IDFT result, then DFT turns into DTFT. An simple illustration of STFNet-filtering operation is shown in Figure~\ref{fig:STFNet_filtering}.
\subsection{STFNet-Convolution Operation}~\label{sec:STFNet_convolution}
In this subsection, we introduce our design of STFNet-convolution operation. Other than filtering operation that handles global pattern matching, we still need the convolution operation to deal with local motifs in the frequency domain.
We denote the input tensor as $\mathbf{X} \in \mathbf{C}^{M\times K\times D}$, where $M$ is the number of time chunk, $K$ number of frequency component, and $D$ input feature dimension.
The convolution operation involves two steps: 1) padding the input data, and 2) convolving with kernel weight matrix $\mathbf{W}_c \in \mathbb{C}^{1\times S \times D \times O}$, where $S$ is the kernel size along the frequency axis and $O$ is still the output feature dimension.
Without the padding step, the output of convolution operation will shrink the number of frequency components, which may break the underlying structure and information in the frequency domain. Therefore, we need to pad extra ``frequency component" to keep the shape of output tensor unchanged compared to that of the input data. In the deep learning research, padding zeros is a common practice. Zero padding is reasonable for inputs such as images and signal in the time domain, meaning no additional information in the padding range.
However, padding zero-valued frequency component introduces additional information in the frequency domain.
Therefore, STFNet-convolution operation proposes the spectral padding for time-frequency analysis.
According to the definition of DFT, transformed data is periodic within the frequency domain. In addition, if the original signal is real-valued, then the transformed data is conjugate symmetric within each period.
Previously, we cut the number of frequency components of a $\tau$-length signal to $K = \lfloor\tau/2\rfloor +1$ for reducing the redundancy. In the spectral padding, we add these frequency components back according to the rule
\begin{figure}[!htb]
\vspace{-0.2cm}
\centering
\includegraphics[width=0.7\linewidth]{STFNet_pooling-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{The low-pass STFNet-pooling operation.}
\label{fig:STFNet_pooling}
\end{figure}
\begin{equation}
\mathbf{X}_{[\cdot, \tau-k, \cdot]} = \mathbf{X}_{[\cdot,-k, \cdot]} = \mathbf{X}_{[\cdot,k, \cdot]}^*,
\label{eqn:spectral_padding}
\end{equation}
where $\mathbf{X}^*$ denotes complex conjugation. In addition, the number of padding before and after the input tensor is same as the previous padding techniques.
Then we can define the basic convolution operation in STFNet
\begin{equation}
\mathbf{Y} = \mathbf{SpectralPad}( \mathbf{X} ) \circledast \mathbf{W}_c,
\label{eqn:STFNet_conv}
\end{equation}
where $\mathbf{SpectralPad}(\cdot)$ denotes our spectral padding operation, and $\circledast$ denotes the convolution operation.
Next, we discuss the way to share the kernel weight matrix $\mathbf{W}_c$ with multi-resolution data. Other than interpolating the kernel weight matrix as shown in~\eqref{eqn:linear_interpolate} and~\eqref{eqn:spectral_interpolate}, we propose another solution for the STFNet-convolution operation. The convolution operation concerns more about the pattern of relative positions on the frequency domain.
Therefore, instead of providing additional kernel details on fine-grained frequency resolution, we can just ensure that the convolution kernel is applied with the same frequency spacing on representations with different frequency resolutions.
Such idea can be implemented with the dilated convolution~\cite{yu2015multi}.
If $\mathbf{W}_c$ is applied to a input tensor with $K=\lfloor \tau/2\rfloor +1$ frequency components, for a input tensor with $K'=\lfloor \tau'/2\rfloor +1$ frequency components ($\tau' > \tau$), the dilated rate $r$ is set to $\tau'/\tau - 1$.
A simple illustration of STFNet-convolution with dilated configuration is shown in Figure~\ref{fig:STFNet_convolution}.
\subsection{STFNet-Pooling Operation}~\label{sec:STFNet_pooling}
In order to provide a dimension reduction method for sensing series within STNet, we introduce the STFNet-pooling operation. STFNet-pooling truncates the spectral information over time with a pre-defined frequency pattern. As a widely-used processing technique, filtering zeroes unwanted frequency components in the signal. Various filtering techniques have been designed, including low-pass filtering, high-pass filtering, and band-pass filtering, which serve as templates for our STFNet-pooling. Instead of zeroing unwanted frequency components, STFNet-pooling removes unwanted components and then concatenates the left pieces.
For applications with domain knowledge about signal-to-noise ratio over the frequency domain, specific pooling strategy can be designed. In this paper, we focus on low-pass STFNet-pooling as an illustrative example.
To extend the STFNet-pooling operation to multiple resolutions and preserving spectral information, we make sure that all representations have the same cut-off frequency according to their own frequency resolutions. A simple example of low-pass STFNet-pooling operation is shown in Figure~\ref{fig:STFNet_pooling}. We can see that our three tensors are truncated according to the same cut-off frequency, $f_s/4$.
\section{Related Work}~\label{sec:related}
The impressive achievements in image classification using deep neural networks at the turn of the decade~\cite{krizhevsky2012imagenet} precipitated a re-emergence of interest in deep learning. Deep neural networks have achieved significant accuracy improvements in a broad spectrum of areas, including computer vision~\cite{simonyan2014very,he2016deep}, natural language processing~\cite{collobert2011natural,bahdanau2014neural}, and network analysis~\cite{perozzi2014deepwalk,kipf2016semi}.
Recent efforts applied deep learning in the context of IoT.
In order to improve the predictive accuracy of IoT applications, researchers employed deep learning to model complicated sensing tasks~\cite{lane2015deepear,yao2017deepsense}. In order to improve system efficiency at executing neural networks on low-end IoT devices, efforts have been made to compress model parameters and/or structures in a manner that does not entail (almost any) accuracy loss~\cite{yao2017deepiot,yao2018fastdeepiot,bhattacharya2016sparsification,han2015deep}.
Recent work in the context of IoT also addressed mathematical foundations for quantifying confidence in deep learning predictions to support mission-critical applications. The work produced deep neural networks that offer well-calibrated uncertainty estimates in results~\cite{yao2018rdeepsense,yao2018apdeepsense,gal2015dropout,gal2016theoretically}.
Finally, the challenge of insufficient labeling of IoT data was addressed by introducing semi-supervised approaches for deep learning that allow neural network training using mostly {\em unlabeled\/} data~\cite{yao2018sensegan}.
However, none of the aforementioned IoT-inspired efforts addressed the customization of learning machinery to a different signal space inspired by the physics of measured processes; namely, the frequency domain.
\begin{figure}[htb]
\vspace{-0.3cm}
\centering
\includegraphics[width=0.7\linewidth]{STFNet_overview-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{The overview design of STFNet block.}
\label{fig:STFNet_overview}
\end{figure}
\setlength{\textfloatsep}{0pt}
To fill the above gap, recent work in machine learning focused on extending deep neural networks to complex numbers and spectral representations.
Trabelsi et al. propose deep complex networks, investigating the complex-value neural network structure~\cite{trabelsi2017deep}. However, they mainly concentrate on the problems of initialization, normalization, and activation functions when extending real-valued operations directly into the complex-value domain. Their designs focus more on complex-value representations than spectral representations, and do not take the properties of spectral data into consideration.
Rippel et al. study spectral representations for convolutional neural networks~\cite{rippel2015spectral}. However, their study focuses on spectral parametrizing of standard CNNs, instead of designing operations customized for spectral data. In addition, their work treats input data fully from the frequency perspective instead of the time-frequency perspective.
Yao et al. propose a neural network that takes short-time Fourier transformed data as inputs~\cite{yao2017deepsense}. Yet their design uses traditional CNNs and RNNs, combining the real and imagery parts of complex-value inputs as additional features.
To the best of our knowledge, STFNet is the first work that integrates neural networks with traditional time-frequency analysis, and designs fundamental spectral-compatible operations for Fourier-transformed representations. Our study shows that the approach leads to improved accuracy compared to the state of the art. It implies that integrating neural networks with domain-inspired transformation techniques (in our case, the Fourier Transform of physical time-series signals) projects input signals into a space that significantly facilitates the learning process.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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**CONTENTS**
_TITLE PAGE_
_DEDICATION_
_EPIGRAPH_
_MAP_
_HISTORICAL NOTE_
BOOK ONE: XERXES
CHAPTER ONE
CHAPTER TWO
CHAPTER THREE
CHAPTER FOUR
CHAPTER FIVE
CHAPTER SIX
CHAPTER SEVEN
BOOK TWO: ALEXANDROS
CHAPTER EIGHT
CHAPTER NINE
CHAPTER TEN
CHAPTER ELEVEN
BOOK THREE: ROOSTER
CHAPTER TWELVE
CHAPTER THIRTEEN
CHAPTER FOURTEEN
BOOK FOUR: ARETE
CHAPTER FIFTEEN
CHAPTER SIXTEEN
CHAPTER SEVENTEEN
BOOK FIVE: POLYNIKES
CHAPTER EIGHTEEN
CHAPTER NINETEEN
CHAPTER TWENTY
CHAPTER TWENTY-ONE
CHAPTER TWENTY-TWO
CHAPTER TWENTY-THREE
BOOK SIX: DIENEKES
CHAPTER TWENTY-FOUR
CHAPTER TWENTY-FIVE
CHAPTER TWENTY-SIX
CHAPTER TWENTY-SEVEN
CHAPTER TWENTY-EIGHT
CHAPTER TWENTY-NINE
BOOK SEVEN: LEONIDAS
CHAPTER THIRTY
CHAPTER THIRTY-ONE
CHAPTER THIRTY-TWO
CHAPTER THIRTY-THREE
CHAPTER THIRTY-FOUR
BOOK EIGHT: THERMOPYLAE
CHAPTER THIRTY-FIVE
CHAPTER THIRTY-SIX
CHAPTER THIRTY-SEVEN
CHAPTER THIRTY-EIGHT
_ACKNOWLEDGMENTS_
_ALSO BY STEVEN PRESSFIELD_
_Excerpt from The Profession_
_COPYRIGHT_
FOR MY MOTHER AND FATHER
_Although extraordinary valor was displayed by the entire corps of Spartans and Thespaians, yet bravest of all was declared the Spartan Dienekes. It is said that on the eve of battle, he was told by a native of Trachis that the Persian archers were so numerous that, when they fired their volleys, the mass of arrows blocked out the sun. Dienekes, however, quite undaunted by this prospect, remarked with a laugh, "Good. Then we'll have our battle in the shade."_
—HERODOTUS, _THE HISTORIES_
The fox knows many tricks;
the hedgehog one good one.
—ARCHILOCHUS
**HISTORICAL NOTE**
_In 480 B.C. the forces of the Persian Empire under King Xerxes, numbering according to Herodotus two million men, bridged the Hellespont and marched in their myriads to invade and enslave Greece._
_In a desperate delaying action, a picked force of three hundred Spartans was dispatched to the pass of Thermopylae, where the confines between mountains and sea were so narrow that the Persian multitudes and their cavalry would be at least partially neutralized. Here, it was hoped, an elite force willing to sacrifice their lives could keep back, at least for a few days, the invading millions._
_Three hundred Spartans and their allies held off the invaders for seven days, until, their weapons smashed and broken from the slaughter, they fought "with bare hands and teeth" (as recorded by Herodotus) before being at last overwhelmed._
_The Spartans and their Thespaian allies died to the last man, but the standard of valor they set by their sacrifice inspired the Greeks to rally and, in that fall and spring, defeat the Persians at Salamis and Plataea and preserve the beginnings of Western democracy and freedom from perishing in the cradle._
_Two memorials remain today at Thermopylae. Upon the modern one, called the Leonidas monument in honor of the Spartan king who fell there, is engraved his response to Xerxes' demand that the Spartans lay down their arms. Leonidas' reply was two words,_ Molon labe. _"Come and get them."_
_The second monument, the ancient one, is an unadorned stone engraved with the words of the poet Simonides. Its verses comprise perhaps the most famous of all warrior epitaphs:_
_Go tell the Spartans, stranger passing by,
that here obedient to their laws we lie._
_BY ORDER OF HIS MAJESTY, Xerxes son of Darius, Great King of Persia and Media, King of Kings, King of the Lands; Master of Libya, Egypt, Arabia, Ethiopia, Babylonia, Chaldea, Phoenicia, Elam, Syria, Assyria and the nations of Palestine; Ruler of Ionia, Lydia, Phrygia, Armenia, Cilicia, Cappadocia, Thrace, Macedonia and the trans-Caucasus, Cyprus, Rhodes, Samos, Chios, Lesbos and the islands of the Aegean; Sovereign Lord of Parthia, Bactria, Caspia, Sousiana, Paphlagonia and India; Lord of all men from the rising to the setting sun, His Most Holy, Reverend and Exalted, Invincible, Incorruptible, Blessed of God Ahura Mazda and Omnipotent among Mortals. Thus decreeth His Magnificence, as recorded by Gobartes the son of Artabazos, His historian:_
_That, following the glorious victory of His Majesty's forces over the arrayed Peloponnesian foe, Spartans and allies, at the pass of Thermopylae, having extinguished the enemy to the last man and erected trophies to this valorous conquest, yet was His Majesty in His God-inspired wisdom desirous of further intelligence, both of certain infantry tactics employed by the enemy which proved of some effect against His Majesty's troops, and of the type of foemen these were who, though unbound by liege law or servitude, facing insuperable odds and certain death, yet chose to remain at their stations, and perished therein to the final man._
_His Majesty's regret having been expressed at the dearth of knowledge and insight upon these subjects, then did intercede God Ahura Mazda on His Majesty's behalf. A survivor of the Hellenes (as the Greeks call themselves) was discovered, grievously wounded and in a state of extremis, beneath the wheels of a battle waggon, being unseen theretofore due to the presence of numerous corpses of men, horses and beasts of transport being heaped upon the site. His Majesty's surgeons being summoned and charged under pain of death to spare no measure to preserve the captive's life, God yet granted His Majesty's desire. The Greek survived the night and the_ _morning following. Within ten days the man had recovered speech and mental faculty and, though yet confined to a litter and under direct care of the Royal Surgeon, was able not only at last to speak but to express his fervent desire to do so._
_Several unorthodox aspects of the captive's armor and raiment were noted by the detaining officers. Beneath the man's battle helmet was found not the traditional felt cawl of the Spartan hoplite, but the dogskin cap associated with the race of helots, the Lakedaemonian slave class, serfs of the land. In contrast inexplicable to His Majesty's officers, the prisoner's shield and armor were of the finest bronze, etched with rare Hibernian cobalt, while his helmet bore the transverse crest of a full Spartiate, an officer._
_In preliminary interviews, the man's manner of speech proved to be a compound of the loftiest philosophical and literary language, indicative of a deep familiarity with the epics of the Hellenes, intermingled with the coarsest and most crude gutter argot, much of which was uninterpretable even to His Majesty's most knowledgeable translators. The Greek, however, willingly agreed to translate these himself, which he did, utilizing scraps of profane Aramaic and Persian which he claimed to have acquired during certain sea travels beyond Hellas. I, His Majesty's historian, seeking to preserve His Majesty's ears from the foul and often execrable language employed by the captive, sought to excise the offensive material before His Majesty was forced to endure hearing it. Yet did His Majesty in His God-inspired wisdom instruct His servant so to translate the man's speech as to render it in whatever tongue and idiom necessary to duplicate the precise effect in Greek. This have I attempted to do. I pray that His Majesty recall the charge He imparted and hold His servant blameless for those portions of the following transcription which will and must offend any civilized hearer._
_Inscribed and submitted this sixteenth day of Ululu, Fifth Year of His Majesty's Accession._
**ONE**
**T** _hird day of Tashritu, Fifth Year of His Majesty's Accession, south of the Lokrian border, the Army of the Empire having continued its advance unopposed into central Greece, establishing an encampment opposite the eastern fall of Mount Parnassus, the sum of whose watercourses, as numerous others before upon the march from Asia, failed and was drunk dry by the troops and horses._
_The following initial interview took place in His Majesty's campaign tent, three hours after sunset, the evening meal having been concluded and all court business transacted. Field marshals, advisors, household guards, the Magi and secretaries being present, the detaining officers were instructed to produce the Greek. The captive was brought in upon a litter, eyes cloth-bound so as to dissanction sight of His Majesty. The Magus performed the incantation and purification, permitting the man to speak within the hearing of His Majesty. The prisoner was instructed not to speak directly toward the Royal Presence but to address himself to the officers of the household guard, the Immortals, stationed upon His Majesty's left._
_The Greek was directed by Orontes, captain of the Immortals, to identify himself. He responded that his name was Xeones the son of Skamandridas of Astakos, a city in Akarnania. The man Xeones stated that he wished first to thank His Majesty for preserving his life and to express his gratitude for and admiration of the skill of the Royal Surgeon's staff. Speaking from his litter, and yet struggling with weakness of breath from several as-yet-unhealed wounds of the lungs and thoracic organs, he offered the following disclaimer to His Majesty, stating that he was unfamiliar with the Persian style of discourse and further stood unfortunately lacking in the gifts of poesy and story-spinning. He declared that the tale he could tell would not be of generals or kings, for the political machinations of the great, he said, he was and had been in no position to observe. He could only relate the story as he himself had lived it and witnessed it, from the_ _vantage of a youth and squire of the heavy infantry, a servant of the battle train. Perhaps, the captive declared, His Majesty would discover little of interest in this narrative of the ordinary warriors, the "men in the line," as the prisoner expressed it._
_His Majesty, responding through Orontes, Captain of the Immortals, asserted to the contrary that this was precisely the tale he wished most to hear. His Majesty was, He declared, already possessed of abundant intelligence of the intriguings of the great; what He desired most to hear was this, "the infantryman's tale."_
_What kind of men were these Spartans, who in three days had slain before His Majesty's eyes no fewer than twenty thousand of His most valiant warriors? Who were these foemen, who had taken with them to the house of the dead ten, or as some reports said, as many as twenty for every one of their own fallen? What were they like as men? Whom did they love? What made them laugh? His Majesty knew they feared death, as all men. By what philosophy did their minds embrace it? Most to the point, His Majesty said, He wished to acquire a sense of the individuals themselves, the real flesh-and-blood men whom He had observed from above the battlefield, but only indistinctly, from a distance, as indistinguishable identities concealed within the blood- and gore-begrimed carapaces of their helmets and armor._
_Beneath his cloth-bound eyes, the prisoner bowed and offered a prayer of thanksgiving to some one of his gods. The story His Majesty wished to hear, he asserted, was the one he could truly tell, and the one he most wished to._
_It must of necessity be his own story, as well as that of the warriors he had known. Would His Majesty be patient with this? Nor could the telling confine itself exclusively to the battle, but must proceed from events antecedent in time, for only in this light and from this perspective would the lives and actions of the warriors His Majesty observed at Thermopylae be given their true meaning and significance._
_His Majesty, field marshals, generals and advisors being satisfied, the Greek was given a bowl of wine and honey for his thirst and asked to commence where he pleased, to tell the story in whatever manner he deemed appropriate. The man, Xeones, bowed once upon his litter and began:_
**I** had always wondered what it felt like to die.
There was an exercise we of the battle train practiced when we served as punching bags for the Spartan heavy infantry. It was called the Oak because we took our positions along a line of oaks at the edge of the plain of Otona, where the Spartiates and the Gentleman-Rankers ran their field exercises in fall and winter. We would line up ten deep with body-length wicker shields braced upon the earth and they would hit us, the shock troops, coming across the flat in line of battle, eight deep, at a walk, then a pace, then a trot and finally a dead run. The shock of their interleaved shields was meant to knock the breath out of you, and it did. It was like being hit by a mountain. Your knees, no matter how braced you held them, buckled like saplings before an earthslide; in an instant all courage fled our hearts; we were rooted up like dried stalks before the ploughman's blade.
That was how it felt to die. The weapon which slew me at Thermopylae was an Egyptian hoplite spear, driven in beneath the plexus of the ribcage. But the sensation was not what one would have anticipated, not being pierced but rather slammed, like we sparring fodder felt beneath the oaks.
I had imagined that the dead would be detached. That they would look upon life with the eyes of objective wisdom. But the experience proved the opposite. Emotion ruled. It seemed nothing remained but emotion. My heart ached and broke as never it could on earth. Loss encompassed me with a searing, all-mastering pain. I saw my wife and children, my dear cousin Diomache, she whom I loved. I saw Skamandridas, my father, and Eunike, my mother, Bruxieus, Dekton and "Suicide," names which mean nothing to His Majesty to hear, but which to me were dearer than life and now, dying, dearer still.
Away they flew. Away I flew from them.
I was keenly conscious of the comrades-in-arms who had fallen with me. A bond surpassing by a hundredfold that which I had known in life bound me to them. I felt a sense of inexpressible relief and realized that I had feared, more than death, separation from them. I apprehended that excruciating war survivor's torment, the sense of isolation and self-betrayal experienced by those who had elected to cling yet to breath when their comrades had let loose their grip.
That state which we call life was over.
I was dead.
And yet, titanic as was that sense of loss, there existed a keener one which I now experienced and felt my brothers-in-arms feeling with me. It was this.
That our story would perish with us.
That no one would ever know.
I cared not for myself, for my own selfish or vainglorious purposes, but for them. For Leonidas, for Alexandros and Polynikes, for Arete bereft by her hearth and, most of all, for Dienekes. That his valor, his wit, his private thoughts that I alone was privileged to share, that these and all that he and his companions had achieved and suffered would simply vanish, drift away like smoke from a woodland fire, this was unbearable.
We had reached the river now. We could hear with ears that were no longer ears and see with eyes that were no longer eyes the stream of Lethe and the hosts of the long-suffering dead whose round beneath the earth was at last drawing to a period. They were returning to life, drinking of those waters which would efface all memory of their existence here as shades.
But we from Thermopylae, we were aeons away from drinking of Lethe's stream. We remembered.
A cry which was not a cry but only the multiplied pain of the warriors' hearts, all feeling what I, too, felt, rent the baleful scene with unspeakable pathos.
Then from behind me, if there can be such a thing as "behind" in that world where all directions are as one, came a glow of such sublimity that I knew, we all knew at once, it could be nothing but a god.
Phoebus Far Darter, Apollo himself in war armor, moved there among the Spartiates and Thespaians. No words were exchanged; none were needed. The Archer could feel the men's agony and they knew without speech that he, warrior and physician, was there to succor it. So quickly that surprise was impossible I felt his eye turn toward me, me the last and least who could expect it, and then Dienekes himself was beside me, my master in life.
I would be the one. The one to go back and speak. A pain beyond all previous now seized me. Sweet life itself, even the desperately sought chance to tell the tale, suddenly seemed unendurable alongside the pain of having to take leave of these whom I had come so to love.
But again, before the god's majesty, no entreaty was possible.
I saw another light, a sicklier, cruder, more coarse illumination, and knew that it was the sun. I was soaring back. Voices came to me through physical ears. Soldiers' speech, in Egyptian and Persian, and leather-gauntleted fists pulling me from beneath a sheaf of corpses.
The Egyptian marines told me later that I had uttered the word _lokas,_ which in their tongue meant "fuck," and they had laughed even as they dragged my shattered body out into the light of day.
They were wrong. The word was _Loxias_ —the Greek title of respect for Apollo the Cunning, or Apollo Crabwise, whose oracles arise ever elusive and oblique—and I was half crying to him, half cursing him for laying this terrible responsibility on me who had no gift to perform it.
As poets call upon the Muse to speak through them, I croaked my inarticulate grunt to the Striker From Afar.
If indeed you have elected me, Archer, then let your fine-fletched arrows spring from my bow. Lend me your voice, Far Darter. Help me to tell the tale.
**TWO**
**T** hermopylae is a spa. The word in Greek means "hot gates," from the thermal springs and, as His Majesty knows, the narrow and precipitous defiles which form the only passages by which the site may be approached—in Greek, _pylae_ or _pylai,_ the East and West Gates.
The Phokian Wall around which so much of the most desperate fighting took place was not constructed by the Spartans and their allies in the event, but stood in existence prior to the battle, erected in ancient times by the inhabitants of Phokis and Lokris as defense against the incursions of their northern neighbors, the Thessalians and Macedonians. The wall, when the Spartans arrived to take possession of the pass, stood in ruins. They rebuilt it.
The springs and pass themselves are not considered by the Hellenes to belong to the natives of the area, but are open to all in Greece. The baths are thought to possess curative powers; in summer the site teems with visitors. His Majesty beheld the charm of the shaded groves and pool houses, the oak copse sacred to Amphiktyon and that pleasantly meandering path bounded by the Lion's Wall, whose stones are said to have been set in place by Herakles himself. Along this in peacetime are customarily arrayed the gaily colored tents and booths used by the vendors from Trachis, Anthela and Alpenoi to serve whatever adventurous pilgrims have made the trek to the mineral baths.
There is a double spring sacred to Persephone, called the Skyllian fountain, at the foot of the bluff beside the Middle Gate. Upon this site the Spartans established their camp, between the Phokian Wall and the hillock where the final tooth-and-nail struggle took place. His Majesty knows how little drinking water is to hand from other sources in the surrounding mountains. The earth between the Gates is normally so parched and dust-blown that servants are employed by the spa to oil the walkways for the convenience of the bathers. The ground itself is hard as stone.
His Majesty saw how swiftly that marble-hard clay was churned into muck by the contending masses of the warriors. I have never seen such mud and of such depth, whose moisture came only from the blood and terror-piss of the men who fought upon it.
When the advance troops, the Spartan rangers, arrived at Thermopylae prior to the battle, a few hours before the main body which was advancing by forced march, they discovered, incredibly, two parties of spa-goers, one from Tiryns, the other from Halkyon, thirty in all, men and women, each in their separate precincts, in various states of undress. These pilgrims were startled, to say the least, by the sudden appearance in their midst of the scarlet-clad armored Skiritai, all picked men under thirty, chosen for speed of foot as well as prowess in mountain fighting. The rangers cleared the bathers and their attendant perfume vendors, masseurs, fig-cake and bread sellers, bath and oil girls, strigil boys and so forth (who had ample intelligence of the Persian advance but had thought that the recent down-valley storm had rendered the northern approaches temporarily impassable). The rangers confiscated all food, soaps, linens and medical accoutrements and in particular the spa tents, which later appeared so grimly incongruous, billowing festively above the carnage. The rangers reerected these shelters at the rear, in the Spartan camp beside the Middle Gate, intending them for use by Leonidas and his royal guard.
The Spartan king, when he arrived, refused to avail himself of this shelter, deeming it unseemly. The Spartiate heavy infantry likewise rejected these amenities. The tents fell, in one of the ironies to which those familiar with war are accustomed, to the use of the Spartan helots, Thespaian, Phokian and Opountian Lokrian slaves and other attendants of the battle train who suffered wounds in the arrow and missile barrages. These individuals, too, after the second day refused to accept shelter. The brightly colored spa tents of Egyptian linen, now in tatters, came as His Majesty saw to protect only the beasts of transport, the mules and asses supporting the commissariat, who became terrorized by the sights and smells of the battle and could not be held by their teamsters. In the end the tents were torn to rags to bind the wounds of the Spartiates and their allies.
When I say Spartiates, I mean the formal term in Greek, _Spartiatai,_ which refers to Lakedaemonians of the superior class, full Spartans—the _homoioi_ —Peers or Equals. None of the class called Gentleman-Rankers or of the _perioikoi,_ the secondary Spartans of less than full citizenship, or those enlisted from the surrounding Lakedaemonian towns, fought at the Hot Gates, though toward the end when the surviving Spartiates became so few that they could no longer form a fighting front, a certain "leavening element," as Dienekes expressed it, of freed slaves, armor bearers and battle squires, was permitted to fill the vacated spaces.
His Majesty may nonetheless take pride in knowing that his forces defeated the flower of Hellas, the cream of her finest and most valiant fighting men.
As for my own position within the battle train, the explanation may require a certain digression, with which I hope His Majesty will be patient.
I was captured at age twelve (or, more accurately, surrendered) as a _heliokekaumenos,_ a Spartan term of derision which means literally "scorched by the sun." It referred to a type of nearly feral youth, burned black as Ethiopians by their exposure to the elements, with which the mountains abounded in those days preceding and following the first Persian War. I was cast originally among the Spartan helots, the serf class that the Lakedaemonians had created from the inhabitants of Messenia and Helos after they in centuries past had conquered and enslaved them. These husbandmen, however, rejected me because of certain physical impairments which rendered me useless for field labor. Also the helots hated and mistrusted any foreigner among them who might prove an informer. I lived a dog's life for most of a year before fate, luck or a god's hand delivered me into the service of Alexandros, a Spartan youth and protege of Dienekes. This saved my life. I was recognized at least ironically as a freeborn and, evincing such qualities of a wild beast as the Lakedaemonians found admirable, was elevated to the status of _parastates pais,_ a sort of sparring partner for the youths enrolled in the _agoge,_ the notorious and pitiless thirteen-year training regimen which turned boys into Spartan warriors.
Every heavy infantryman of the Spartiate class travels to war attended by at least one helot. _Enomotarchai,_ the platoon leaders, take two. This latter was Dienekes' station. It is not uncommon for an officer of his rank to select as his primary attendant, his battle squire, a freeborn foreigner or even a young _mothax,_ a noncitizen or bastard Spartan still in _agoge_ training. It was my fortune, for good or ill, to be chosen by my master for this post. I supervised the care and transport of his armor, maintained his kit, prepared his food and sleeping site, bound his wounds and in general performed every task necessary to leave him free to train and fight.
My childhood home, before fate set me upon the road which found its end at the Hot Gates, was originally in Astakos in Akarnania, north of the Peloponnese, where the mountains look west over the sea toward Kephallinia and, beyond the horizon, to Sikelia and Italia.
The island of Ithaka, home of Odysseus of lore, lay within sight across the straits, though I myself was never privileged to touch the hero's sacred soil, as a boy or later. I was due to make the crossing, a treat from my aunt and uncle, on the occasion of my tenth birthday. But our city fell first, the males of my clan were slaughtered and females sold into slavery, our ancestral land taken, and I cast out, alone save my cousin Diomache, without family or home, three days before the start of my tenth year to heaven, as the poet says.
**THREE**
**W** e had a slave on my father's farm when I was a boy, a man named Bruxieus, though I hesitate to use the word "slave," because my father was more in Bruxieus' power than the other way round. We all were, particularly my mother. As lady of the house she refused to make the most trifling domestic decision—and many whose scope far exceeded that—without first securing Bruxieus' advice and approval. My father deferred to him on virtually all matters, save politics within the city. I myself was completely under his spell.
Bruxieus was an Elean. He had been captured by the Argives in battle when he was nineteen. They blinded him with fiery pitch, though his knowledge of medicinal salves later restored at least a poor portion of his sight. He bore on his brow the ox-horn slave brand of the Argives. My father acquired him when he was past forty, as compensation for a shipment of hyacinth oil lost at sea.
As nearly as I could tell, Bruxieus knew everything. He could pull a bad tooth without clove or oleander. He could carry fire in his bare hands. And, most vital of all to my boy's regard, he knew every spell and incantation necessary to ward off bad luck and the evil eye.
Bruxieus' only weakness as I said was his vision. Beyond ten feet the man was blind as a stump. This was a source of secret, if guilty, pleasure to me because it meant he needed a boy with him at all times to see. I spent weeks never leaving his side, not even to sleep, since he insisted on watching over me, slumbering always on a sheepskin at the foot of my little bed.
In those days it seemed there was a war every summer. I remember the city's drills each spring when the planting was done. My father's armor would be brought down from the hearth and Bruxieus would oil each rim and joint, rewarp and reshaft the "two spears and two spares" and replace the cord and leather gripware within the _hoplon_ 's oak and bronze sphere. The drills took place on a broad plain west of the potters' quarter, just below the city walls. We boys and girls brought sunshades and fig cakes, scrapped over the best viewing positions on the wall and watched our fathers drill below us to the trumpeters' calls and the beat of the battle drummers.
This year of which I speak, the dispute of note was over a proposal made by that session's _prytaniarch,_ an estate owner named Onaximandros. He wanted each man to efface the clan or individual crest on his shield and replace it with a uniform _alpha,_ for our city Astakos. He argued that Spartan shields all bore a proud _lambda,_ for their country, Lakedaemon. Fine, came the derisive response, but we're no Lakedaemonians. Someone told the story of the Spartiate whose shield bore no crest at all, but only a common housefly painted life-size. When his rankmates made sport of him for this, the Spartan declared that in line of battle he would get so close to his enemy that the housefly would look as big as a lion.
Every year the military drills followed the same pattern. For two days enthusiasm reigned. Every man was so relieved to be free of farm or shop chores, and so delighted to be reunited with his comrades (and away from the children and women around the house), that the event took on the flavor of a festival. There were sacrifices morning and evening. The rich smells of spitted meat floated over everything; there were wheaten buns and honey candies, fresh-rolled fig cakes, and bowls of rice and barley grilled in sweet new-pressed sesame oil.
By the third day the militiamen's blisters started. Forearms and shoulders were rubbed raw by the heavy _hoplon_ shields. The warriors, though most were farmers or grovers and supposedly of stout seasoned limb, had in fact passed the bulk of their agricultural labor in the cool of the counting room and not out behind a plough. They were getting tired of sweating. It was hot under those helmets. By the fourth day the sunshine warriors were presenting excuses in earnest. The farm needed this, the shop needed that, the slaves were robbing them blind, the hands were screwing each other silly. "Look at how straight the line advances now, on the practice field," Bruxieus would chuckle, squinting past me and the other boys. "They won't step so smartly when heaven starts to rain arrows and javelins. Each man will be edging to the right to get into his rankmate's shadow." Meaning the shelter of the shield of the man on his right. "By the time they hit the enemy line, the right wing will be overlapped half a _stade_ and have to be chased back into place by its own cavalry!"
Nonetheless our citizen army (we could put four hundred heavy-armored hoplites into the field on a full call-up), despite the potbellies and wobbly shins, had acquitted itself more than honorably, at least in my short lifetime. That same _prytaniarch,_ Onaximandros, had two fine span of oxen, got from the Kerionians, whose countryside our forces allied with the Argives and Eleuthrians had plundered ruthlessly three years running, burning a hundred farms and killing over seventy men. My uncle Tenagros had a stout mule and a full set of armor got in those seasons. Nearly every man had something.
But back to our militia's maneuvers. By the fifth day, the city fathers were thoroughly exhausted, bored and disgusted. Sacrifices to the gods redoubled, in the hope that the immortals' favor would make up for any lack of _polemike techne,_ skill at arms, or _empeiria,_ experience, on the part of our forces. By now there were huge gaps in the field and we boys had descended upon the site with our own play shields and spears. That was the signal to call it a day. With much grumbling from the zealots and great relief from the main body, the call was issued for the final parade. Whatever allies the city possessed that year (the Argives had sent their _strategos autokrater,_ that great city's supreme military commander) were marshaled gaily into the reviewing stands, and our reinvigorated citizen-soldiers, knowing their ordeal was nearly over, loaded themselves up with every ounce of armor they possessed and passed in glorious review.
This final event was the greatest excitement of all, with the best food and music, not to mention the raw spring wine, and ended with many a farm cart bearing home in the middle watch of the night sixty-five pounds of bronze armor and a hundred and seventy pounds of loudly snoring warrior.
This morning, which initiated my destiny, came about because of ptarmigan eggs.
Among Bruxieus' many talents, foremost was his skill with birds. He was a master of the snare. He constructed his traps of the very branches his prey favored to roost upon. With a pop! so delicate you could hardly hear it, his clever snares would fire, imprisoning their mark by the "boot" as Bruxieus called it, and always gently.
One evening Bruxieus summoned me in secrecy behind the cote. With great drama he lifted his cloak, revealing his latest prize, a wild ptarmigan cock, full of fight and fire. I was beside myself with excitement. We had six tame hens in the coop. A cock meant one thing—eggs! And eggs were a supreme delicacy, worth a boy's fortune at the city market.
Sure enough, within a week our little banty had become the strutting lord of the walk, and not long thereafter I cradled in my palms a clutch of precious ptarmigan eggs.
We were going to town! To market. I woke my cousin Diomache before the middle watch was over, so eager was I to get to our farm's stall and put my clutch up for sale. There was a _diaulos_ flute I wanted, a double-piper that Bruxieus had promised to teach me coot and grouse calls on. The proceeds from the eggs would be my bankroll. That double-piper would be my prize.
We set out two hours before dawn, Diomache and I, with two heavy sacks of spring onions and three cheese wheels in cloth loaded on a half-lame female ass named Stumblefoot. Stumblefoot's foal we had left home tied in the barn; that way we could release mama in town when we unloaded, and she would make a beeline on her own, straight home to her baby.
This was the first time I had ever been to market without a grown-up and the first with a prize of my own to sell. I was excited, too, by being with Diomache. I was not yet ten; she was thirteen. She seemed a full-grown woman to me, and the prettiest and smartest in all that countryside. I hoped my friends would fall in with us on the road, just to see me on my own beside her.
We had just reached the Akarnanian road when we saw the sun. It was bright flaring yellow, still below the horizon against the purple sky. There was only one problem: it was rising in the north.
"That's not the sun," Diomache said, stopping abruptly and jerking hard on Stumblefoot's halter. "That's fire."
It was my father's friend Pierion's farm.
The farm was burning.
"We've got to help them," Diomache announced in a voice that brooked no protest, and, clutching my cloth of eggs in one hand, I started after her at a fast trot, hauling the bawling gimpy-foot ass. How can this happen before fall, Diomache was calling as we ran, the fields aren't tinder-dry yet, look at the flames, they shouldn't be that big.
We saw a second fire. East of Pierion's. Another farm. We pulled up, Diomache and I, in the middle of the road, and then we heard the horses.
The ground beneath our bare feet began to rumble as if from an earthquake. We saw the flare of torches. Cavalry. A full platoon. Thirty-six horses were thundering toward us. We saw armor and crested helmets. I started running toward them, waving in relief. What luck! They would help us! With thirty-six men, we'd have the fires out in—
Diomache yanked me back hard. "Those aren't our men."
They came past at a near gallop, looking huge and dark and ferocious. Their shields had been blackened, soot smeared on the blazes and stockings of their horses, their bronze greaves caked with dark mud. In the torchlight I saw the white beneath the soot on their shields. Argives. Our allies. Three riders reined in before us; Stumblefoot bawled in terror and stamped to break; Diomache held the halter fast.
"What you got there, girlie?" the burliest of the horsemen demanded, wheeling his lathered, mud-matted mount before the onion sacks and the cheeses. He was a wall of a man, like Ajax, with an open-faced Boeotian helmet and white grease under his eyes for vision in the dark. Night raiders. He leaned from his saddle and made a lunging swipe for Stumblefoot. Diomache kicked the man's mount, hard in the belly; the beast bawled and spooked.
"You're burning our farms, you traitorous bastards!"
Diomache slung Stumblefoot's halter free and slapped the fear-stricken ass with all her strength. The beast ran like hell and so did we.
I have sprinted in battle, racing under arrow and javelin fire with sixty pounds of armor on my back, and countless times in training have I been driven up steep broken faces at a dead run. Yet never have my heart and lungs labored with such desperate necessity as they did that terror-filled morning. We left the road at once, fearing more cavalry, and bolted straight across country, streaking for home. We could see other farms burning now. "We've got to run faster!" Diomache barked back at me. We had come beyond two miles, nearly three, on our trek toward town, and now had to retrace that distance and more across stony, overgrown hillsides. Brambles tore at us, rocks slashed our bare feet, our hearts seemed like they must burst within our breasts. Dashing across a field, I saw a sight that chilled my blood. Pigs. Three sows and their litters were scurrying in single file across the field toward the woods. They didn't run, it wasn't a panic, just an extremely brisk, well-disciplined fast march. I thought: those porkers will survive this day, while Diomache and I will not.
We saw more cavalry. Another platoon and another, Aetolians of Pleuron and Kalydon. This was worse; it meant the city had been betrayed not just by one ally but by a coalition. I called to Diomache to stop; my heart was about to explode from exertion. "I'll leave you, you little shit!" She hauled me forward. Suddenly from the woods burst a man. My uncle Tenagros, Diomache's father. He was in a nightshirt only, clutching a single eight-foot spear. When he saw Diomache, he dropped the weapon and ran to embrace her. They clung to each other, gasping. But this only struck more terror into me. "Where's Mother?" I could hear Diomache demanding. Tenagros' eyes were wild with grief. "Where's _my_ mother?" I shouted. "Is my father with you?"
"Dead. All dead."
"How do you know? Did you see them?"
"I saw them and you don't want to."
Tenagros retrieved his spear from the dirt. He was breathless, weeping; he had soiled himself; there was liquid shit on the inside of his thighs. He had always been my favorite uncle; now I hated him with a murderous passion. "You ran!" I accused him with a boy's heartlessness. "You showed your heels, you coward!"
Tenagros turned on me with fury. "Get to the city! Get behind the walls!"
"What about Bruxieus? Is he alive?"
Tenagros slapped me so hard he bowled me right off my feet. "Stupid boy. You care more about a blind slave than your own mother and father."
Diomache hauled me up. I saw in her eyes the same rage and despair. Tenagros saw it too.
"What's that in your hands?" he barked at me.
I looked down. There were my ptarmigan eggs, still cradled in the rag in my palms.
Tenagros' callused fist smashed down on mine, shattering the fragile shells into goo at my feet.
"Get into town, you insolent brats! Get behind the walls!"
**FOUR**
**H** is Majesty has presided over the sack of numberless cities and has no need to hear recounted the details of the week that followed. I will append the observation only, from the horror-benumbed apprehension of a boy shorn at one blow of mother and father, family, clan, tribe and city, that this was the first time my eyes had beheld those sights which experience teaches are common to all battles and all slaughters.
This I learned then: there is always fire.
An acrid haze hangs in the air night and day, and sulphurous smoke chokes the nostrils. The sun is the color of ash, and black stones litter the road, smoking. Everywhere one looks, some object is afire. Timber, flesh, the earth itself. Even water burns. The pitilessness of flame reinforces the sensation of the gods' anger, of fate, retribution, deeds done and hell to pay.
All is the obverse of what it had been.
Things are fallen which had stood upright. Things are free which should be bound, and bound which should be free. Things which had been hoarded in secret now blow and tumble in the open, and those who had hoarded them watch with dull eyes and let them go.
Boys have become men and men boys. Slaves now stand free and freemen slaves. Childhood has fled. The knowledge of my mother and father's slaughter struck me less with grief for them or fear for myself than with the imperative to assume at once their station. Where had I been on the morn of their murder? I had failed them, trotting off on my boyish errand. Why had I not foreseen their peril? Why was I not standing at my father's shoulder, armed and possessed of a man's strength, to defend our hearth or die honorably before it, as he and my mother had?
Bodies lay in the road. Mostly men, but women and children too, with the same dark blot of fluid sinking into the pitiless dirt. The living trod past them, grief-riven. Everyone was filthy. Many had no shoes. All were fleeing the slave columns and the roundup which would be starting soon. Women carried infants, some of them already dead, while other dazed figures glided past like shades, bearing away some pitifully useless possession, a lamp or a volume of verse. In peacetime the wives of the city walked abroad with necklaces, anklets, rings; now one saw none, or it was secreted somewhere to pay a ferryman's toll or purchase a heel of stale bread. We encountered people we knew and didn't recognize them. They didn't recognize us. Numb reunions were held along roadsides or in copses, and news was traded of the dead and the soon to be dead.
Most piteous of all were the animals. I saw a dog on fire that first morning and ran to snuff his smoking fur with my cloak. He fled, of course; I couldn't catch him, and Diomache snatched me back with a curse for my foolishness. That dog was the first of many. Horses hamstrung by sword blades, lying on their flanks with their eyes pools of numb horror. Mules with entrails spilling; oxen with javelins in their sides, lowing pitifully yet too terrified to let anyone near to help. These were the most heartbreaking: the poor dumb beasts whose torment was made more pitiful by their lack of faculty to understand it.
Feast day had come for crows and ravens. They went for the eyes first. They peck a man's asshole out, though God only knows why. People chased them off at first, rushing indignantly at the blandly feeding scavengers, who would retreat as far only as necessity dictated, then hop back to the banquet when the coast was clear. Piety demanded that we bury our fallen countrymen, but fear of enemy cavalry pushed us on. Sometimes bodies would be dragged into a ditch and a few pitiful handfuls of dirt cast over them, accompanied by a miserable prayer. The crows got so fat they could barely fly a foot off the ground.
We did not go into the city, Diomache and I.
We had been betrayed from within, she instructed me, speaking slowly as one would to a simpleton, to make sure I understood. Sold out by our own citizens, some faction seeking power, then they themselves had been double-crossed by the Argives. Astakos was a port, a poor one, but a western harbor nonetheless, which Argos had long coveted. Now she had it.
We found Bruxieus on the morn of the second day. His slave brand had saved him. That, and his blindness, which the conquerors mocked even as he cursed and swung at them with his staff. "You're free, old man!" Free to starve or beg from his belly's necessity for the victor's yoke.
The rain came that evening. This, too, seems a constant coda to slaughter. What had been ash was now gray mud, and the stripped bodies which had not been reclaimed by sons and mothers now glistened a ghastly white, cleansed by the gods in their remorseless way.
Our city no longer existed. Not alone the physical site, the citizens, the walls and farms. But the very spirit of our nation, the _polis_ itself, that ideal of mind called Astakos that, yes, had been smaller than a _deme_ of Athens or Corinth or Thebes, that, yes, had been poorer than Megara or Epidauros or Olympia, but that existed as a city nonetheless. Our city, my city. Now it was effaced utterly. We who called ourselves Astakiots were effaced with it. Without a city, who were we? What were we?
A dislocation of the faculties seemed to unman all. No one could think. A numb shock possessed our hearts. Life had become like a play, a tragedy one had seen enacted on the stage—the fall of Ilium, the sack of Thebes. Only now it was real, performed by actors of flesh and blood, and those actors were ourselves.
East of the Field of Ares, where the fallen in battle were buried, we came upon a man digging a grave for an infant. The baby, wrapped in the man's cloak, lay like a grocer's bundle at the edge of the pit. He asked me to hand it down to him. He was afraid the wolves would get it, he said, that's why he had dug the hole so deep. He didn't know the child's name. A woman had handed it to him during the flight from the city. He had carried the babe for two days; on the third morning it died. Bruxieus wouldn't let me hand the little body down; it was bad luck, he said, for a living young spirit to handle a dead one. He did it himself. We recognized the man now. He was a _mathematikos,_ a tutor of arithmetic and geometry, from the city. His wife and daughter emerged from the woods; we realized they had been hiding till they knew we brought no harm. They had all lost their minds. Bruxieus had instructed Diomache and me in the signs. Madness was contagious, we must not linger.
"We needed Spartans," the teacher declared, speaking softly behind his sad watery eyes. "Just fifty would have saved the city."
Bruxieus was nudging us to go.
"See how numb we are?" the man continued. "We glide about in a daze, disconnected from our reason. You'll never see Spartans in such a state. This"—he gestured to the blackened landscape—"is their element. They move through these horrors with clear eyes and unshaken limbs. And they hate the Argives. They are their bitterest enemies."
Bruxieus pulled us away.
"Fifty of them!" the man still shouted, while his wife struggled to tug him back to the safety of the trees. "Five! One would have saved us!"
We recovered Diomache's mother's body, and my mother's and father's, on the eve of the third day. A squad of Argive infantry had set up camp around the gutted ruins of our farmhouse. Already surveyors and claims markers had arrived from the conquering cities. We watched, hidden, from the woods as the officials marked off the parcels with their measuring rods and scrawled upon the white wall of my mother's kitchen garden and sign of the clan of Argos whose lands ours would now become.
An Argive taking a piss spotted us. We took to flight but he called after us. Something in his voice convinced us that he and the others intended no harm. They had had enough of blood for now. They waved us in, gave us the bodies. I sponged the mud and blood off my mother's corpse, using the singlet she had made for me, for my promised passage to Ithaka. Her flesh was like cold wax. I did not weep, neither shrouding her form in the burial robe she had woven with her own hands and which in its cupboard chest remained miraculously unstolen, nor interring her bones and my father's beneath the stone that bore our ancestors' blazon and signia.
It was my place to know the rites, but I had not been taught them, awaiting my initiation to the tribe when I turned twelve. Diomache lit the flame, and the Argives sang the _paean,_ the only sacred song any of them knew.
_Zeus Savior, spare us_
_Who march into your fire_
_Grant us courage to stand_
_Shield-to-shield with our brothers_
_Beneath your mighty aegis_
_We advance_
_Lord of the Thunder_
_Our Hope and our Protector_
When the hymn was over, the men raped her.
I didn't understand at first what they intended. I thought she had violated some portion of the rite and they were going to beat her for it. A soldier snatched me by the scalp, one hairy forearm around my neck to snap it. Bruxieus found a spear at his throat and the point of a sword pricking the flesh of his back. No one said a word. There were six of them, armorless, in sweat-dark corselets with their rank dirty beards and the rain-sodden hair on their chests and calves coarse and matted and filthy. They had been watching Diomache, her smooth girl's legs and the start of breasts beneath her tunic.
"Don't harm them," Diomache said simply, meaning Bruxieus and me.
Two men took her away behind the garden wall. They finished, then two more followed, and the last pair after that. When it was over, the sword was lowered from Bruxieus' back, and he crossed to carry Diomache away in his arms. She wouldn't let him. She stood to her feet on her own, though she had to brace herself against the wall to do it, both her thighs dark with blood. The Argives gave us a quarter-skin of wine and we took it.
It was clear now that Diomache could not walk. Bruxieus took her up in his arms. Another of the Argives pressed a hard bread into my hands. "Two more regiments will be coming from the south tomorrow. Get into the mountains and go north, don't come down till you're out of Akarnania." He spoke kindly, as if to his own son. "If you find a town, don't bring the girl in or this will happen again."
I turned and spat on his dark stinking tunic, a gesture of powerlessness and despair. He caught my arm as I turned away. "And get rid of that old man. He's worthless. He'll only wind up getting you and the girl killed."
**FIVE**
**T** hey say that ghosts sometimes, those that cannot let go their bond to the living, linger and haunt the scenes of their days under the sun, hovering like substanceless birds of carrion, refusing Hades' command to retire beneath the earth. That is how we lived, Bruxieus, Diomache and I, in the weeks following the sack of our city. For a month and more, for most of that summer we could not quit our vacated _polis._ We roamed the wild country above the _agrotera,_ the marginal wastes surrounding the cropland, sleeping in the day when it was warm, moving at night like the shades we were. From the ridgelines we watched the Argives move in below, repopulating our groves and farmsteads with the excess of their citizenry.
Diomache was not the same. She would wander away by herself, into the dark glades, and do unspeakable things to her womanly parts. She was trying to dispatch the child that might be growing inside her. "She thinks she has given offense to the god Hymen," Bruxieus explained to me when I broke in upon her one day and she chased me with curses and a hail of stones. "She fears that she may never be a man's wife now but only a slave or a whore. I have tried to tell her this is foolishness, but she will not hear it, coming from a man."
There were many others like us in the hills then. We would run into them at the springs and try to resume the fellow-feeling we had shared as Astakiots. But the extinction of our _polis_ had severed those happy bonds forever. It was every man for himself now; every clan, every kin group.
Some boys I knew had formed a gang. There were eleven of them, none more than two years older than I, and they were holy terrors. They carried arms and boasted that they had killed grown men. They beat me up one day when I refused to join them. I wanted to, but couldn't leave Diomache. They would have taken her in too, but I knew she would never go near them.
"This is our country," their boy-lord warned me, a beast of twelve who called himself Sphaireus, "Ball Player," because he had stuffed in hide the skull of an Argive he had slain, and now kicked about with him the way a monarch bears a _skeptron._ He meant his gang's country, the high ground above the city, beyond the reach of Argive armor. "If we catch you trespassing here again, you or your cousin or that slave, we'll cut out your liver and feed it to the dogs."
At last in fall we put our city behind us. In September when Boreas, the North Wind, begins to blow. Without Bruxieus and his knowledge of roots and snares, we would have starved.
Before, on my father's farm, we had caught wild birds for our cote, or to make breeding pairs, or just to hold for an hour before returning them to freedom. Now we ate them. Bruxieus made us devour everything but the feathers. We crunched the little hollow bones; we ate the eyes, and the legs right down to the boot, discarding only the beak and the unchewable feet. We gulped eggs raw. We choked down worms and slugs. We wolfed grubs and beetles and fought over the last lizards and snakes before the cold drove them underground for good. We gnawed so much fennel that to this day I gag at a whiff of that anisey smell, even a pinch flavoring a stew. Diomache grew thin as a reed.
"Why won't you talk to me anymore?" I asked her one night as we tramped across some stony hillside. "Can't I put my head in your lap like we used to?"
She began to cry and would not answer me. I had made myself an infantryman's spear, stout ash and fire-hardened, no longer a boy's toy but a weapon meant to kill. Visions of revenge fed my heart. I would live among the Spartans. I would slay Argives one day. I practiced the way I had seen our warriors do, advancing as if on line, an imaginary shield before me at high port, my spear gripped strong above the right shoulder, poised for the overhand strike. I looked up one dusk and there stood my cousin, observing me coldly. "You will be like them," she said, "when you grow."
She meant the soldiers who had shamed her.
"I will not!"
"You will be a man. You won't be able to help yourself."
One night when we had tramped for hours, Bruxieus inquired of Diomache why she had held herself so silent. He was concerned for the dark thoughts that might be poisoning her mind. She refused to speak at first. Then, at last relenting, told us in a sweet sad voice of her wedding. She had been planning it in her head all night. What dress she would wear, what style of garland, which goddess she would dedicate her sacrifice to. She had been thinking for hours, she told us, of her slippers. She had all the strapping and beadwork worked out in her mind. They would be so beautiful, her bridal slippers! Then her eyes clouded and she looked away. "This shows what a fool I have become. No one will marry me."
"I will," I proffered at once.
She laughed. "You? A fair chance of that!"
Foolish as it sounds to recount, to my boy's heart these careless words stung like no others in my life. I vowed that I would marry Diomache one day. I would be man enough and warrior enough to protect her.
For a time in autumn we tried surviving on the seacoast, sleeping in caves and combing the sloughs and marshes. You could eat there at least. There were shellfish and crabs, mussels and spinebacks to be prised from rocks; we learned how to take gulls on the wing with stakes and nets. But the exposure was brutal as winter came on. Bruxieus began to suffer. He would never let his weakness show to Diomache and me when he thought we were looking, but I would watch his face sometimes when he slept. He looked seventy. The elements were hard on him in his years; all the old wounds ached, but more than that he was donating his substance to preserve ours, Diomache's and mine. Sometimes I would catch him looking at me, studying a tilt to my face or the tone of something I had said. He was making sure I hadn't gone crazy or feral.
As the cold came on, it became more difficult to find food. We must beg. Bruxieus would pick out an isolated farmstead and approach the gate alone; the hounds would converge in a clamorous pack and the men of the farm would emerge, on guard, from the fields or from some rude falling-down outbuilding; brothers and a father, their callused hands resting on the tools which would become weapons if the need arose. The hills were infested with outlaws then; the farmers never knew who would walk up to their gate and with what duplicitous intent. Bruxieus would doff his cap and wait for the woman of the house, making sure she took note of his milky eyes and beaten posture. He would indicate Diomache and me, shivering miserably in the road, and ask the mistress not for food, which would have made us beggars in the landsmen's eyes and prompted them loosing the dogs on us, but for any broken item of use that she could spare—a rake, a thrashing staff, a worn-out cloak, something we could repair and sell in the next town. He made sure to ask directions and appear eager to be moving on. That way they knew any kindness would not make us linger. Almost always the farm wives volunteered a meal, sometimes inviting us in to hear what news we bore from foreign places and to tell us their own.
It was during one of those forlorn feedings that I first heard the word "Sepeia." This is a place of Argos, a wooded area near Tiryns, where a battle had just been fought between the Argives and the Spartans. The boy who bore this tale was a farmer's visiting nephew, a mute, who communicated through signs and whom even his own family could barely comprehend. The Spartans under King Kleomenes, the boy gave us to understand, had achieved a spectacular victory. Two thousand Argives dead was one figure he had heard, though others had it at four thousand and even six. My heart exploded with joy. How I wished I could have been there! To have been a man grown, advancing in that battle line, mowing down in fair fight the men of Argos, as they had cut down by perfidy my own mother and father.
The Spartans became for me the equivalent of avenging gods. I couldn't learn enough about these warriors who had so devastatingly defeated the murderers of my mother and father, the violators of my innocent cousin. No stranger we met escaped my boyish grilling. Tell me about Sparta. Her double kings. The three hundred Knights who protected them. The _agoge_ which trained the city's youth. The _syssitia,_ the warriors' messes. We heard a tale of Kleomenes. Someone had asked the king why he did not raze Argos once and for all when his army had stood at the gates and the city lay prostate before him. "We need the Argives," Kleomenes responded. "Who else will our young men train on?"
In the winter hills we were starving. Bruxieus was getting weaker. I took to stealing. Diomache and I would raid a shepherd's fold at night, fighting off the dogs with sticks and snatching a kid if we could. Most of the shepherds carried bows; arrows would whiz past us in the dark. We stopped to grab them and soon had quite a cache. Bruxieus hated to see us turning into thieves. We got a bow one time, snagging it right out from under a sleeping goatherd's nose. It was a man's weapon, a Thessalian cavalry bow, so stout that neither Diomache nor I could draw it. Then came the event which changed my life and set it on the course that reached its terminus at the Hot Gates.
I got caught stealing a goose. She was a fat prize, her wings pegged for market, and I got careless going over a wall. The dogs got me. The men of the farm dragged me into the mud of the livestock pen and nailed me to a hide board the size of a door, driving tanning spikes through both my palms. I was on my back, screaming in agony, while the farm men lashed my kicking, flailing legs to the board, vowing that after lunch they would castrate me like a sheep and hang my testicles upon the gate as a warning to other thieves. Diomache and Bruxieus crouched, hidden, up the hillside; they could hear everything...
**H** _ere the captive drew up in his narration. Fatigue and the ordeal of his wounds had taken severe toll of the man, or perhaps, his listeners imagined, it was memory of the instance he was recounting. His Majesty, through the captain Orontes, inquired of the prisoner if he required attention. The man declined. The hesitation in the tale, he declared, arose not from any incapacity of its narrator, but at the prompting of the god by whose inner direction the order of events was being dictated, and who now commanded a momentary alteration of tack. The man Xeones resettled himself and, granted permission to wet his throat with wine, resumed._
**T** wo summers subsequent to this incident, in Lakedaemon, I witnessed a different kind of ordeal: a Spartan boy beaten to death by his drill instructors.
The lad's name was Teriander, he was fourteen; they called him Tripod because no one of his age-class could take him down in wrestling. Over the succeeding years I looked on in attendance as two dozen other boys succumbed beneath this same trial, each like Tripod disdaining so much as a whimper in pain, but he, this lad, was the first.
The whippings are a ritual of the boys' training in Lakedaemon, not in punishment for stealing food (at which exploit the boys are encouraged to excel, to develop resourcefulness in war), but for the crime of getting caught. The beatings take place alongside the Temple of Artemis Orthia in a narrow alley called the Runway. The site is beneath plane trees, a shaded and quite pleasant space in less grisly circumstances.
Tripod was the eleventh boy whipped that day. The two _eirenes,_ drill instructors, who administered the beatings, had already been replaced by a fresh pair, twenty-year-olds just out of the _agoge_ and as powerfully built as any youths in the city. It worked like this: the boy whose turn it was grasped a horizontal iron bar secured to the bases of two trees (the bar had been worn smooth by decades, some said centuries, of the ritual) and was flogged with birch rods, as big around as a man's thumb, by the _eirenes_ taking turns. A priestess of Artemis stood at the boy's shoulder, presenting an ancient wooden image which must, tradition dictated, receive the spray of human blood.
Two of the boy's mates from his training platoon kneel at each shoulder to catch the lad when he falls. At any time the boy may terminate the ordeal by releasing the bar and pitching forward to the dirt. Theoretically a boy would only do this when thrashed to unconsciousness, but many pitched simply when they could no longer bear the pain. Between a hundred and two hundred looked on this day: boys of other platoons, fathers, brothers and mentors and even some of the boys' mothers, keeping discreetly to the rear.
Tripod kept taking it and taking it. The flesh of his back had been torn through in a dozen places; you could see tissue and fascia, ribcage and muscle and even the spine. He would not go down. "Pitch!" his two comrades kept urging between blows, meaning let go of the bar and fall. Tripod refused. Even the drill instructors began hissing this between their teeth. One look in the boy's face and you could see he had passed beyond reason. He had made up his mind to die rather than raise the hand for quarter. The _eirenes_ did as they were instructed in such cases: they prepared to wallop Tripod so hard in four rapid successive blows that the impact would knock him unconscious and thus preserve his life. I will never forget the sound those four blows made upon the boy's back. Tripod dropped; the drill instructors immediately declared the ordeal terminated and summoned the next boy.
Tripod managed to lift himself upon all fours. Blood was sheeting from his mouth, nose and ears. He could not see or speak. He managed somehow to turn about and almost stand, then he sank slowly to his seat, held there a moment and then dropped, hard, into the dirt. It was clear at once that he would never rise.
Later that evening when it was over (the ritual was not suspended on account of Tripod's death but continued for another three hours), Dienekes, who had been present, walked apart with his protege, the boy Alexandros whom I mentioned earlier. I served Alexandros at this time. He was twelve but looked no older than ten; already he was a wonderful runner, but extremely slight and of a sensitive disposition. Moreover he had shared a bond of affection with Tripod; the older boy had been a sort of guardian or protector; Alexandros was devastated by his death.
Dienekes walked with Alexandros, alone except for his own squire and myself, to a spot beneath the temple of Athena Protectress of the City, immediately below the slope from the statue of Phobos, the god of fear. At that time Dienekes' age was, I would estimate, thirty-five years. He had already won two prizes of valor, at Erythrae against the Thebans and at Achillieon against the Corinthians and their Arkadian allies. As nearly as I can recall, this is how the older man instructed his protege:
First, in a gentle and loving tone, he recalled his own first sight, when he was a lad in years younger even than Alexandros, of a boy comrade whipped to death. He recounted several of his own ordeals in the Runway, beneath the rod.
Then he began the sequence of query and response which comprises the Lakedaemonian syllabus of instruction.
"Answer this, Alexandros. When our countrymen triumph in battle, what is it that defeats the foe?"
The boy responded in the terse Spartan style, "Our steel and our skill."
"These, yes," Dienekes corrected him gently, "but something more. It is that." His gesture led up the slope to the image of Phobos.
Fear.
Their own fear defeats our enemies.
"Now answer. What is the source of fear?"
When Alexandros' reply faltered, Dienekes reached with his hand and touched his own chest and shoulder.
"Fear arises from this: the flesh. This," he declared, "is the factory of fear."
Alexandros listened with the grim concentration of a boy who knows his whole life will be war; that the laws of Lykurgus forbid him and every other Spartan to know or pursue a trade other than war; that his term of obligation extends from age twenty to age sixty, and that no force under heaven will excuse him from soon, very soon, assuming his place in line of battle and clashing shield-to-shield, helmet-to-helmet with the enemy.
"Now answer again, Alexandros. Did you observe today in the manner of the _eirenes_ delivering the beating any sign or indication of malice?"
The boy answered no.
"Would you characterize their demeanor as barbarous? Did they take pleasure in dealing agony to Tripod?"
No.
"Was their intention to crush his will or break his spirit?"
No.
"What was their intention?"
"To harden his mind against pain."
Throughout this conversation the older man maintained a voice tender and solicitous with love. Nothing Alexandros could do would ever make this voice love him less or abandon him. Such is the peculiar genius of the Spartan system of pairing each boy in training with a mentor other than his own father. A mentor may say things that a father cannot; a boy can confess to his mentor that which would bring shame to reveal to his father.
"It was bad today, wasn't it, my young friend?"
Dienekes then asked the boy how he imagined battle, real battle, compared with what he had witnessed today.
No answer was required or expected.
"Never forget, Alexandros, that this flesh, this body, does not belong to us. Thank God it doesn't. If I thought this stuff was mine, I could not advance a pace into the face of the enemy. But it is not ours, my friend. It belongs to the gods and to our children, our fathers and mothers and those of Lakedaemon a hundred, a thousand years yet unborn. It belongs to the city which gives us all we have and demands no less in requital."
Man and boy moved on, down the slope to the river. They followed the path to that grove of double-boled myrtle called the Twins, sacred to the sons of Tyndareus and to the family to which Alexandros belonged. It would be to this spot, on the night of his final ordeal and initiation, that he would repair, alone save his mother and sisters, to receive the salve and sanction of the gods of his line.
Dienekes sat upon the earth beneath the Twins. He gestured to Alexandros to take the place beside him.
"Personally I think your friend Tripod was foolish. What he displayed today contained more of recklessness than true courage, _andreia._ He cost the city his life, which could have been spent more fruitfully in battle."
Nonetheless it was clear Dienekes respected him.
"But to his credit he showed us something of nobility today. He showed you and every boy watching what it is to pass beyond identification with the body, beyond pain, beyond fear of death. You were horrified to behold his _agonisma,_ but it was awe that struck you truly, wasn't it? Awe of that boy or whatever _daimon_ animated him. Your friend Tripod showed us contempt for this." Again Dienekes indicated the flesh. "A contempt which approached the stature of the sublime."
From my spot, above on the bank, I could see the boy's shoulders shudder as the grief and terror of the day at last purged themselves from his heart. Dienekes embraced and comforted him. When at last the boy had recomposed himself, his mentor gently released him.
"Have your instructors taught you why the Spartans excuse without penalty the warrior who loses his helmet or breastplate in battle, but punish with loss of all citizenship rights the man who discards his shield?"
They had, Alexandros replied.
"Because a warrior carries helmet and breastplate for his own protection, but his shield for the safety of the whole line."
Dienekes smiled and placed a hand upon his protege's shoulder.
"Remember this, my young friend. There is a force beyond fear. More powerful than self-preservation. You glimpsed it today, in a crude and unself-aware form, yes. But it was there and it was genuine. Let us remember your friend Tripod and honor him for this."
**I** was screaming upon the hide board. I could hear my cries bounce off the walls of the livestock enclosure and shriek off, multiplied, up the hillsides. I knew it was disgraceful but I could not stop.
I begged the farm men to release me, to end my agony. I would do anything, and I described it all at the top of my lungs. I cried out to the gods in a shameful little boy's voice piping up the mountainside. I knew Bruxieus could hear me. Would his love for me impel him to dash in and be nailed alongside me? I didn't care. I wanted the pain to end. I begged the men to kill me. I could feel the bones in both hands shattered by the spikes. I would never hold a spear or even a gardening spade. I would be a cripple, a clubfist. My life was over and in the meanest, most dishonorable way.
A fist shattered my cheek. "Shut your pipehole, you sniveling little shitworm!" The men set the tanning board upright, angled against a wall, and there I squirmed, impaled, for the sun's endless crawl across the sky. Urchins from the up-valley farms clustered to watch me scream. The girls tore my rags and poked at my privates; the boys pissed on me. Dogs snuffed my bare soles, emboldening themselves to make a meal of me. I only stopped wailing when my throat could cry no longer. I was trying to tear my palms free right through the spikes, but the men lashed my wrists tighter so I couldn't move. "How does that feel, you fucking thief? Let's see you pick off another prize, you night-creeping little rat."
When at last their own growling bellies drove my tormentors indoors for supper, Diomache slipped down from the hill and cut me free. The spikes would not come out of my palms; she had to blade the wood off the frame with her dagger. My hands came away with the tanning nails still through them. Bruxieus carried me off, as he had borne Diomache earlier, after her violation.
"Oh God," my cousin said when she saw my hands.
**SIX**
**T** hat winter, Bruxieus said, was the coldest he could remember. Sheep froze in the high pastures. Twenty-foot drifts sealed the passes. Deer were driven so desperate with hunger that they straggled down, skeleton-thin and blind from starvation, all the way to the shepherds' winter folds, where they presented themselves for slaughter, point-blank before the herdsmen's bows.
We stayed in the mountains, so high up that martens' and foxes' fur grew white as the snow. We slept in dugouts that shepherds had abandoned or in ice caves we chopped out with stone axes, lining their floors with pine boughs and huddling together beneath our triple cloaks in a pile like puppies. I begged Bruxieus and Diomache to abandon me, let me die in peace in the cold. They insisted that I allow them to carry me down to a town, to a physician. I refused absolutely. Never again would I place myself before a stranger, any stranger, without a weapon in my hand. Did Bruxieus imagine that doctors possessed a more exalted sense of honor than other men? What payment would some hill-town quack demand? What profitable turn would he discover in a slave and a crippled boy? What use would he make of a starving thirteen-year-old girl?
I had another reason for refusing to go to a town. I hated myself for the shameless way I had cried out, and could not make myself stop, during the hours I was put to the trial. I had seen my own heart and it was the heart of a coward. I despised myself with a blistering, pitiless scorn. The tales I had cherished of the Spartans only made me loathe myself more. None of them would beg for his life as I had, absent every scrap of dignity. The dishonor of my parents' murder continued to torment me. Where was I in their hour of desperation? I was not there when they needed me. In my mind I imagined their slaughter again and again, and always myself absent. I wanted to die. The only thought that lent me solace was the certainty that I would die, soon, and thereby exit this hell of my own dishonored existence.
Bruxieus intuited these thoughts and tried in his gentle way to disarm them. I was only a child, he told me. What prodigies of valor could be expected from a lad of ten? "Boys are men at ten in Sparta," I declared.
This was the first and only time I saw Bruxieus truly, physically angry. He seized me by both shoulders and shook me violently, commanding me to face him. "Listen to me, boy. Only gods and heroes can be brave in isolation. A man may call upon courage only one way, in the ranks with his brothers-in-arms, the line of his tribe and his city. Most piteous of all states under heaven is that of a man alone, bereft of the gods of his home and his _polis._ A man without a city is not a man. He is a shadow, a shell, a joke and a mockery. That is what you have become now, my poor Xeo. No one may expect valor from one cast out alone, cut off from the gods of his home."
He drew up then; his eyes broke away in sorrow. I saw the slave brand upon his brow. I understood. Such was the state he had endured, all these years, in the house of my father. "But you have acted the man, little old uncle," I said, employing the fondest Astakiot term of affection. "How have you done it?"
He looked at me with sad, gentle eyes. "The love I might have given my own children, I gave to you, little nephew. That was my answer to the unknowable ways of God. But it seems the Argives are dearer to Him than I. He has let them rob me of my life not once, but twice."
These words, intended to bring comfort, only reinforced further my resolve to die. My hands had swollen now to twice their normal size. Pus and poison oozed from them, then froze in a hideous icy mass that I had to chip away each morning to reveal the mangled flesh beneath. Bruxieus did everything he could with salves and poultices, but it was no use. Both central metacarpals had been shattered in my right hand. I could not close the fingers nor form a fist. I would never hold a spear nor grip a sword. Diomache sought to comfort me by equating my ruin to hers. I scorned her bitterly. "You can still be a woman. What can I do? How can I ever take my place in the line of battle?"
At night, bouts of fever alternated with fits of teeth-rattling ague. I curled contorted in Diomache's arms, with Bruxieus' bulk enwrapping us both for warmth. I called out again and again to the gods but received no whisper in reply. They had abandoned us, it was clear, now that we no longer possessed ourselves or were possessed by our _polis._
One fever-racked night, perhaps ten days after the incident at the farmstead, Diomache and Bruxieus wrapped me in skins and set off foraging. It had begun to snow and they hoped to use the silence, perhaps with luck to take unawares a hare or a gone-to-ground covey of grouse.
This was my chance. I resolved to take it. I waited till Bruxieus and Diomache had moved off beyond sight and sound. Leaving cloak and furs and foot wraps behind for them, I set out barefoot into the storm.
I climbed for what seemed like hours but was probably no more than five minutes. The fever had me in its grip. I was blind like the deer, yet guided by an infallible sense of direction. I found a place amid a stand of pines and knew this was my spot. A profound sense of decorum possessed me. I wanted to do this properly and, above all, to be no trouble to Bruxieus and Diomache.
I picked out a tree and settled my back against it so that its spirit, which touched both earth and sky, would conduct mine safely out of this world. Yes, this was the tree. I could feel Sleep, brother of Death, advancing up from the toes. Feeling ebbed from my loins and midsection. When the numbness reaches the heart, I imagined, I will pass over. Then a terrifying thought struck me.
What if this is the wrong tree? Perhaps I should be leaning against that one. Or that other, over there. A panic of indecision seized me. I was in the wrong spot! I had to get up but could no longer command my limbs to move. I groaned. I was failing even in my own death. Just as my panic and despair reached their apex, I was startled to discover a man standing directly above me in the grove!
My first thought was that he could help me move. He could advise me. Help me decide. Together we would pick out the correct tree and he would place my back against it. From some part of my mind the numb thought arose: what is a man doing up here at this hour, in this storm?
I blinked and tried with all my failing power to focus. No, this was not a dream. Whoever this was, he was really here. The thought came foggily that he must be a god. It occurred to me that I was acting impiously toward him. I was giving offense. Surely propriety demanded that I respond with terror or awe, or prostrate myself before him. Yet something in his posture, which was not grave but oddly whimsical, seemed to say, Don't give yourself the bother. I accepted this. It seemed to please him. I knew he was going to speak, and that whatever words came forth would be of paramount importance for me, in this my earthly life or the life I was about to pass into. I must listen with all my faculties and forget nothing.
His eyes met mine with a gentle, amused kindness.
"I have always found the spear to be," he spoke with a quiet majesty that could be nothing other than the voice of a god, "a rather inelegant weapon."
What a queer thing to say, I thought.
And why "inelegant"? I had the sense that the word was absolutely deliberate, the one precise term the god sought. It seemed to carry significance for him in level upon level, though I myself had no idea what this meaning could be. Then I saw the silver bow slung over his shoulder.
The Archer.
Apollo Far Striker.
In a flash that was neither thunderbolt nor revelation but the plainest, least adorned apprehension in the world, I understood all that his words and presence implied. I knew what he meant, and what I must do.
My right hand. Its severed sinews would never produce the warrior's grasp upon the shank of a spear. But its forefingers could catch and draw the twined gut of a bowstring. My left, though ever denied power to close upon the gripcord of a _hoplon_ shield, could yet hold stable the handpiece of a bow and extend it to full stretch.
The bow.
The bow would preserve me.
The Archer's eyes probed mine, gently, for one final instant. Had I understood? His glance seemed to inquire not so much "Will you now serve me?" as to confirm the fact, unknown to me heretofore, that I had been in his service all my life.
I felt warmth returning to my midsection and the blood surging like a tide into my legs and feet. I heard my name being called from below and knew it was my cousin, she and Bruxieus in alarm, scouring the hillside for me.
Diomache reached me, scrabbling over the snowy crest and lurching into the grove of pines. "What are you doing up here all alone?" I could feel her slapping my cheeks, hard, as if to bring me around from a vision or transport; she was crying, clutching and hugging me, tearing off her cloak to wrap about me. She called back to Bruxieus, who in his blindness was clambering as fast as he could up the slope below.
"I'm all right," I heard my voice assuring her. She slapped me again and then, weeping, cursed me for being such a fool and scaring them so to death. "It's all right, Dio," I heard my voice repeating. "I'm all right."
**SEVEN**
**I** beg His Majesty's patience with this recounting of the events following the sack of a city of which he has never heard, an obscure _polis_ without fame, spawner of no hero of legend, without link to the greater events of the present war and of the battle which His Majesty's forces fought with the Spartans and their allies at the pass of Thermopylae.
My intent is simply to convey, through the experiences of two children and a slave, some poor measure of the soul terror and devastation which a vanquished population, any population, is forced to endure in the hour of its nation's extinction. For though His Majesty has commanded the sack of empires, yet, if one may speak plainly, he has witnessed the sufferings of their peoples only at a remove, from atop a purple throne or mounted on a caparisoned stallion, protected by the gold-pommeled spears of his royal guard.
Over the following decade more than six score battles, campaigns and wars were fought between and among the cities of Greece. At least forty _poleis,_ including such inpregnably founded citadels as Knidos, Arethusa, Kolonaia, Amphissa and Metropolis, were sacked in whole or in part. Numberless farms were torched, temples burned, warships sunk, men-at-arms slaughtered, wives and daughters carried off into slavery. No Hellene, however mighty his city, could state with certainty that even one season hence he would still find himself above the earth, with his head still upon his shoulders and his wife and children slumbering in safety by his side. This state of affairs was unexceptional, neither better nor worse than any era in a thousand years, back to Achilles and Hektor, Theseus and Herakles, to the birth of the gods themselves. Business as usual, as the _emporoi,_ the merchants, say.
Each man of Greece knew what defeat in war meant and knew that sooner or later that bitter broth would complete its circuit of the table and settle at last before his own place.
Suddenly, with the rise of His Majesty in Asia, it seemed that hour would be sooner.
Terror of the sack spread throughout all Greece as word began coming, from the lips of too many to be disbelieved, of the scale of His Majesty's mobilization in the East and his intent to put all Hellas to the torch.
So all-pervasive was this dread that it had even been given a name.
_Phobos._
The Fear.
Fear of you, Your Majesty. Terror of the wrath of Xerxes son of Darius, Great King of the Eastern Empire, Lord of all men from the rising to the setting sun, and the myriads all Greece knew were on the march beneath his banner to enslave us.
Ten years had passed since the sack of my own city, yet the terror of that season lived on, indelible, within me. I was nineteen now. Events which will in their course be related had parted me from my cousin and from Bruxieus and carried me, as was my wish, to Lakedaemon and there, after a time, into the service of my master, Dienekes of Sparta. In this capacity I was dispatched (myself and a trio of other squires) in attendance upon him and three other Spartiate envoys—Olympieus, Polynikes and Aristodemos—to the island of Rhodes, a possession of His Majesty's empire. It was there that these warriors, and I myself, glimpsed for the first time a fraction of the armored might of Persia.
The ships came first. I had been given the afternoon free and, making use of the time to learn what I could of the island, had attached myself to a company of Rhodian slingers in their practice. I watched as these ebullient fellows hurled with astonishing velocity their lead sling bullets thrice the size of a man's thumb. They could drill these murderous projectiles through half-inch pine planks at a hundred paces and strike a target the size of a man's chest three times out of four. One among them, a youth my own age, was showing me how the slingers carved with their dagger points into the soft lead of their bullets whimsical greetings—"Eat this" or "Love and kisses"—when another of the platoon looked up and pointed out to the horizon, toward Egypt. We saw sails, perhaps a squadron, at least an hour out. The slingers forgot them and continued their drill. What seemed like moments later, the same fellow sang out again, this time with startlement and awe. All drew up and stared. Here came the squadron, triple-bankers with their sails brailed up for speed, already turning the cape and bearing fast upon the breakwater. None had ever seen vessels of such size moving so fast. They must be skimmers, someone said. Racing shells. No full-size ship, and certainly no man-of-war, could slice the water at speeds like that.
But they were warships. Tyrian triremes so tight to the surface that the swells seemed to crest no more than a handbreadth beneath their thalamites' benches. They were racing each other for sport beneath His Majesty's banner. Training for Greece. For war. For the day their bronze-sheathed rams would send the navies of Hellas to the bottom.
That evening Dienekes and the other envoys made their way on foot to the harbor at Lindos. The warships were drawn up upon the strand, within a perimeter manned by Egyptian marines. These recognized the Spartans by their scarlet cloaks and long hair. A wry scene ensued. The captain of the marines motioned the Spartiates forward, calling them forth with a smile from the throng who had assembled to gawk at the vessels and taking them through a full inspecting admiral's tour. The men speculated, through an interpreter, about how soon they would be at war with each other, and whether fate would bring them again face-to-face across the line of slaughter.
The Egyptian marines were the tallest men I had ever seen and burned nearly black by the sun of their desert land. They were under arms, in doeskin boots, with bronze fish-scale cuirasses and ostrich-plume helmets detailed with gold. Their weapons were the pike and scimitar. They were in high spirits, these marines, comparing the muscles of their buttocks and thighs with those of the Spartans, while each laughed in his tongue unintelligible to the other.
"Pleased to meet you, you hyena-jawed bastards." Dienekes grinned at the captain, speaking in Doric and clapping the fellow warmly upon the shoulder. "I'm looking forward to carving your balls and sending them home in a basket." The Egyptian laughed uncomprehending and replied, beaming, with some foreign-tongued insult no doubt equally menacing and obscene.
Dienekes asked the captain's name, which the man replied was Ptammitechus. The Spartan tongue was defeated by this and settled upon "Tommie," which seemed to please the officer just as well. He was asked how many more warships like these the Great King numbered in his navy. "Sixty" came the translated response.
"Sixty ships?" asked Aristodemos.
The Egyptian loosed a brilliant smile. "Sixty squadrons."
The marines conducted the Spartans upon a more detailed examination of the warships, which, hauled up on the sand, had been canted onto careening beams, exposing the undersides of their hulls for cleaning and sealing, which chores the Tyrian seamen were now enthusiastically performing. I smelled wax. The sailors were greasing the boats' bellies for speed. The vessels' planks were butted end-to-end with mortise-and-tenon joinery of such precision that it seemed the work not of shipwrights, but of master cabinetmakers. The conjoining plates between the ram and the hull were glazed with speed-enhancing ceramic and waxed with some kind of naphtha-based oil which the mariners applied molten, with paddles. Alongside these speedsters, the Spartan state galley _Orthia_ looked like a garbage scow. But the items which commanded the most animated attention bore no bearing to concerns of the sea.
These were the mail loincloths worn by the marines to protect their private parts.
"What are these, diapers?" Dienekes inquired, laughing and tugging at the hem of the captain's corselet.
"Be careful, my friend," the marine responded with a mock-theatric gesture, "I have heard about you Greeks!"
The Egyptian inquired of the Spartans why they wore their hair so long. Olympieus replied, quoting the lawmaker Lykurgus, "Because no other adornment makes a handsome man more comely or an ugly one more terrifying. And it's free."
The marine next began teasing the Spartans about their notoriously short _xiphos_ swords. He refused to believe that these were the actual weapons the Lakedaemonians carried into battle. They must be toys. How could such diminutive apple-corers possibly work harm to an enemy?
"The trick is"—Dienekes demonstrated, pressing himself chest-to-chest to the Egyptian Tommie—"to get nice and cozy."
When they parted, the Spartans presented the marines with two skins of Phalerian wine, the finest they had, a gift intended for the Rhodian consulate. The marines gave each Spartan a gold daric (a month's pay for a Greek oarsman) and a sack apiece of fresh Nile pomegranates.
The mission returned to Sparta unsuccessful. The Rhodians, as His Majesty knows, are Dorian Hellenes; they speak a dialect similar to the Lakedaemonians and call their gods by the same Doric-derived names. But their island had been since before the first Persian War a protectorate of the Empire. What option other than submission did the Rhodians possess, their nation lying as it does within the very shadow of the masts of the imperial fleet? The Spartan embassy had sought, against all expectation, to detach through ancient bonds of kinship some portion of the Rhodian navy from service to His Majesty. It found no takers.
Nor had there been, our embassy learned upon its return to the mainland, from simultaneous missions dispatched to Crete, Cos, Chios, Lesbos, Samos, Naxos, Imbros, Samothrace, Thasos, Skyros, Mykonos, Paros, Tenos and Lemnos. Even Delos, birthplace of Apollo himself, had offered tokens of submission to the Persian.
_Phobos._
This terror could be inhaled in the air of Andros, where we touched upon the voyage home. One felt it like a sweat on the skin at Keos and Hermione, where no harbor inn or beaching ground lacked for ship's masters and oarsmen with terror-inspiring tales of the scale of mobilization in the East and eyewitness reports of the uncountable myriads of the enemy.
_Phobos._
This stranger accompanied the embassy as it landed at Thyrea and began the dusty, two-day hump across Parnon to Lakedaemon. Trekking up the eastern massif, the envoys could see landsmen and city folk evacuating their possessions to the mountains. Boys drove asses laden with sacks of corn and barley, protected by the men of the family under arms. Soon the old ones and the children would follow. In the high country, clan groups were burying jars of wine and oil, building sheepfolds and carving crude shelters out of the cliffsides.
_Phobos._
At the frontier fort of Karyai, our party fell in with an embassy from the Greek city of Plataea, a dozen men including a mounted escort, headed for Sparta. Their ambassador was the hero Arimnestos of Marathon. It was said that this gentleman, though well past fifty, had in that famous victory ten years past waded in full armor into the surf, slashing with his sword at the oars of the Persian triremes as they backed water, fleeing for their lives. The Spartans loved this sort of thing. They insisted on Arimnestos' party joining ours for supper and accompanying us on the remaining march to the city itself.
The Plataean shared his intelligence of the enemy. The Persian army, he reported, comprised of two million men drawn from every nation of the Empire, had assembled at the Great King's capital, Susa, in the previous summer. The force had advanced to Sardis and wintered there. From this site, as the greenest lieutenant could not fail to project, the myriads would proceed north along the coast highways of Asia Minor, through Aeolis and the Troad, crossing the Hellespont by either bridge of boats or massive ferrying operation, then proceed west, traversing Thrace and the Chersonese, southwest across Macedonia and then south into Thessaly.
Greece proper.
The Spartans recounted what they had learned at Rhodes; that the Persian army was already on the march from Sardis; the main body stood even now at Abydos, readying to cross the Hellespont.
They would be in Europe within a month.
At Selassia a messenger from the ephors in Sparta awaited my master with an ambassadorial pouch. Dienekes was to detach himself from the party and proceed at once to Olympia. He took his leave at the Pellana road and, accompanied by myself alone, set out at a fast march, intending to cover the fifty miles in two days.
It is not uncommon upon these treks to have fall in with one as he tramps various high-spirited hounds and even half-wild urchins of the vicinity. Sometimes these carefree comrades remain on the troop all day, trotting in merry converse at the trekker's heels. Dienekes loved these ranging strays and never failed to welcome them and take cheer in their serendipitous companionship. This day, however, he sternly dismissed all we encountered, canine as well as human, striding resolutely onward, glancing neither left nor right.
I had never seen him so troubled or so grave.
An incident had occurred at Rhodes which I felt certain lay at the source of my master's disquiet. This event transpired at the harbor, immediately after the Spartans and Egyptian marines had completed their exchange of gifts and were making ready to take leave of one another. There arose then that interval when strangers often discard that formality of intercourse with which they have heretofore conversed and speak instead man-to-man, from the heart. The captain Ptammitechus had clearly taken to my master and the _polemarch_ Olympieus, Alexandros' father. He summoned these now aside, declaring that he had something be wished to show them. He led them into the naval commander's campaign tent, erected there upon the strand, and with this officer's permission produced a marvel the like of which the Spartans, and of course I myself, had never beheld.
This was a map.
A geographer's representation not merely of Hellas and the islands of the Aegean but of the entire world.
The chart spread in breadth nearly two meters, of consummate detail and craftsmanship and inscribed upon Nile papyrus, a medium so extraordinary that though held to the light one could see straight through it, yet even the strongest man's hands could not rend it, save by first opening a tear with the edge of a blade.
The marine rolled the map out upon the squadron commander's table. He showed the Spartans their own homeland, in the heart of the Peloponnese, with Athens 140 miles to the north and east, Thebes and Thessaly due north of there, and Mounts Ossa and Olympus at the northernmost extremity of Greece. West of this the mapmaker's stylus had depicted Sikelia, Italia and all the leagues of sea and land clear to the Pillars of Herakles. Yet the bulk of the chart had barely begun to be unfurled.
"I wish only to impress upon you, for your own preservation, gentlemen," Ptammitechus addressed the Spartans through his interpreter, "the scale of His Majesty's Empire and the resources he commands to bring against you, that you may make your decision to resist or not, based upon fact and not fancy."
He unrolled the papyrus eastward. Beneath the lamplight arose the islands of the Aegean, Macedonia, Illyria, Thrace and Scythia, the Hellespont, Lydia, Karia, Cilicia, Phoenicia and the Ionic cities of Asia Minor. "All these nations the Great King controls. All these he has compelled into his service. All these are coming against you. But is this Persia? Have we reached yet the seat of Empire..."
Out rolled more leagues of landmass. The Egyptian's hand swept over the outlines of Ethiopia, Libya, Arabia, Egypt, Assyria, Babylonia, Sumeria, Cappadocia, Armenia and the trans-Caucasus. The fame of each of these kingdoms he recited, quoting the numbers of their warriors and the arms and armaments they carried.
"A man traveling fast may traverse all the Peloponnese in four days. Look here, my friends. Merely to get from Tyre to Susa, the Great King's capital, is three months' march. And all that land, all its men and wealth, belong to Xerxes. Nor do his nations contend one against the other as you Hellenes love so to do, nor disunite into squabbling alliances. When the King says assemble, his armies assemble. When he says march, they march. And still," he said, "we have not reached Persepolis and the heart of Persia."
He rolled the map out farther.
Into sight arose yet more lands covering yet more leagues and called by yet more curious names. The Egyptian reeled off more numbers. Two hundred thousand from this satrapy, 300,000 from that. Greece, in the West, was looking punier and punier. She seemed to be shriveling into a microcosm in contrast to the endless mass of the Persian Empire. The Egyptian spoke now of outlandish beasts and chimera. Camels and elephants, wild asses the size of draught horses. He sketched the lands of Persia herself, then Media, Bactria, Parthia, Caspia, Aria, Sogdiana and India, nations of whose names and existence his listeners had never even heard.
"From these vast lands His Majesty draws more myriads of warriors, men raised under the blistering sun of the East, inured to hardships beyond your imagining, armed with weapons you have no experience in combating and financed by gold and treasure beyond counting. Every article of produce, every fruit, grain, pig, sheep, cow, horse, the yield of every mine, farm, forest and vineyard belongs to His Majesty. And all of it he has poured into the mounting of this army which marches now to enslave you.
"Listen to me, brothers. The race of Egyptians is an ancient one, numbering the generations of its fathers by the hundreds into antiquity. We have seen empires come and go. We have ruled and been ruled. Even now we are technically a conquered people, we serve the Persians. Yet regard my station, friends. Do I look poor? Is my demeanor dishonored? Peer here within my purse. With all respect, brothers, I could buy and sell you and all you own with only that which I bear upon my person."
At that point Olympieus called the Egyptian short and demanded that he speak to his point.
"My point is this, friends: His Majesty will honor you Spartans no less than us Egyptians, or any other great warrior people, should you see wisdom and enlist yourselves voluntarily beneath his banner. In the East we have learned that which you Greeks have not. The wheel turns, and man must turn with it. To resist is not mere folly, but madness."
I watched my master's eyes then. Clearly he perceived the Egyptian's intent as genuine and his words proffered out of friendship and regard. Yet he could not stop anger from flushing his countenance.
"You have never tasted freedom, friend," Dienekes spoke, "or you would know it is purchased not with gold, but steel." He contained his anger swiftly, reaching to rap the Egyptian's shoulder like a friend and to meet his eyes with a smile.
"And as for the wheel you speak of," my master finished, "like every other, it turns both ways."
We arrived at Olympia on the afternoon of the second day from Pellana. The Olympic Games, sacred to Zeus, are the holiest of all Hellenic festivals; during the weeks of their celebration no Greek may take up arms against another, or even against an alien invader. The Games would be held this very year, within weeks; in fact the Olympic grounds and dormitories were already teeming with athletes and trainers from all the Greek cities, preparing on-site as prescribed by heaven's law. These competitors, in their youthful prime and peerless in speed and prowess, surrounded my master on the instant of his arrival, clamorous for intelligence of the Persian advance and torn by the Olympic proscription from bearing arms. It was not my place to inquire of my master's mission; one could only surmise, however, that it entailed a request for dispensation from the priests.
I waited outside the precinct while Dienekes conducted his business within. Several hours of daylight remained when he finished; our two-man party, unescorted as it was, should have turned about and pushed on for Sparta at once. But my master's troubled mood continued; he seemed to be working something out in his mind. "Come on," he said, leading toward the Avenue of the Champions, west of the Olympic stadium, "I'll show you something for your education."
We detoured to the steles of honor, where the names and nations of champions of the Games were recorded. There my own eye located the name of Polynikes, one of my master's fellow envoys to Rhodes, graven twice for successive Olympiads, victor in the armored _stadion_ race. Dienekes pointed out the names of other Lakedaemonian champions, men now in their thirties and forties whom I knew by sight from the city, and others who had fallen in battle decades and even centuries past. Then he indicated a final name, four Olympiads previous, in the victors' lists for the pentathlon.
_Iatrokles_
_Son of Nikodiades_
_Lakedaemonian_
"This was my brother," Dienekes said.
That night my master took shelter at the Spartan dormitory, a cot being vacated for him within and space set aside for me beneath the porticoes. But his mood of disquiet had not abated. Before I had even settled on the cool stones, he appeared from within fully dressed and motioned me to follow. We traversed the deserted avenues to the Olympic stadium, entering via the competitors' tunnel and emerging into the vast and silent expanse of the agonists' arena, purple and brooding now in the starlight. Dienekes mounted the slope above the judges' station, those seats upon the grass reserved during the Games for the Spartans. He selected a sheltered site beneath the pines at the crest of the slope overlooking the stadium, and there he settled.
I have heard it said that for the lover the seasons are marked in memory by those mistresses whose beauty has enflamed his heart. He recalls this year as the one when, moonstruck, he pursued a certain beloved about the city, and that year, when another favorite yielded at last to his charms.
For the mother and father, on the other hand, the seasons are numbered by the births of their children—this one's first step, that one's initial word. By these homely ticks is the calendar of the loving parent's life demarcated and set within the book of remembrance.
But for the warrior, the seasons are marked not by these sweet measures nor by the calendared years themselves, but by battles. Campaigns fought and comrades lost; trials of death survived. Clashes and conflicts from which time effaces all superficial recall, leaving only the fields themselves and their names, which achieve in the warrior's memory a stature ennobled beyond all other modes of commemoration, purchased with the holy coin of blood and paid for with the lives of beloved brothers-in-arms. As the priest with his _graphis_ and tablet of wax, the infantryman, too, has his scription. His history is carved upon his person with the stylus of steel, his alphabet engraved with spear and sword indelibly upon the flesh.
Dienekes settled upon the shadowed earth above the stadium. I began now, as was my duty as his squire, to prepare and apply the warm oil, laced with clove and comfrey, which were required by my master, and virtually every other Peer past thirty years, simply to settle himself upon the earth in sleep. Dienekes was far from an old man, barely two years past forty, yet his limbs and joints creaked like an ancient's. His former squire, a Scythian called "Suicide," had instructed me in the proper manner of kneading the knots and loaves of scar tissue about my master's numerous wounds, and the little tricks in arming him so that his impairments would not show. His left shoulder could not move forward past his ear, nor could that arm rise at the elbow above his collarbone; the corselet had to be wrapped first about his torso, which he would support by pinning it with his elbows while I set the shoulder leathers and thumb-bolted them into place. His spine would not bend to lift his shield, even from its position of rest against his knee; the bronze sleeve had to be held aloft by me and jockeyed into place over the forearm, in the standing position. Nor could Dienekes flex his right foot unless the tendon was massaged until the flow of the nerves had been restored along their axis of command.
My master's most gruesome wound, however, was a lurid scar, the width of a man's thumb, that ran in jagged course across the entire crown of his brow, just below the hairline. This was not visible normally, covered as it was by the fall of his long hair across his forehead, but when he bound his hair to accept the helmet, or tied it back for sleep, this livid gash re-presented itself. I could see it now in the starlight. Apparently the curiosity in my expression struck my master as comical, for he chuckled and lifted his hand to trace the line of the scar.
"This was a gift from the Corinthians, Xeo. An ancient one, picked up around the time you were born. Its history, aptly enough, tells a tale of my brother."
My master glanced away, down the slope that led toward the Avenue of the Champions. Perhaps he felt the proximity of his brother's shade, or the fleeting shards of memory, from boyhood or battle or the _agon_ of the Games. He indicated that I might pour for him a bowl of wine, and that I may take one for myself.
"I wasn't an officer then," he volunteered, still preoccupied. "I wore a banty hat instead of a curry brush." Meaning the front-to-back-crested helmet of the infantry ranker, instead of the transverse-crested helm of a platoon leader. "Would you like to hear the tale, Xeo? As a bedtime story."
I replied that I would, very much. My master considered. Clearly he was debating in his mind if such a retelling constituted vanity or excessive self-revelation. If it did, he would break it off at once. Apparently, however, the incident contained an element of instruction, for, with a barely perceptible nod, my master gave himself permission to proceed. He settled more comfortably against the slope.
"This was at Achilleion, against the Corinthians and their Arkadian allies. I don't even remember what the war was about, but whatever it was, those sons of whores had found their courage. They were putting the steel to us. The line had broken down, the first four ranks were scrambled, it was man against man across the entire field. My brother was a platoon leader and I was a third." Meaning he, Dienekes, commanded the third squad, sixteen positions back in order of march. "So that when we deployed into line by fours, I came up to my third's position beside my brother at the head of my squad. We fought as a _dyas,_ Iatrokles and I; we had trained in the pairs since we were children. Only there was none of that sport now, it was pure blood madness.
"I found myself across from a monster of the enemy, six and a half feet tall, a match for two men and a horse. He was dismasted, his spear had been shivered, and he was so raging with possession he didn't have the presence of mind to go for his sword. I said to myself, man, you better get some iron into this bastard fast, before he remembers he's got that daisy-chopper on his hip.
"I went for him. He met me with his shield as a weapon, swinging it, edge-on like an axe. His first blow splintered my own shield. I had my eight-footer by the haft, trying to uppercut him, but he splintered the shaft clean through with a second blow. I was now bronze-naked in front of this demon. He swung that shield like a relish plate. Took me right here, square above the eye sockets.
"I could feel the crown of the helmet tear up and off, shearing half my skull with it. The bottom lip of the eyehole had opened the muscles beneath the brow, so that my left eye was sheeted with blood.
"I had that helpless feeling you get when you're wounded, when you know it's bad but you don't know how bad, you think you may be dead already but you're not sure. Everything is happening slowly, as in a dream. I was down on my face. I knew this giant was over me, aiming some blow to send me to hell.
"Suddenly he was there beside me. My brother. I saw him take a step and sling his _xiphos_ like a throwing blade. It hit this Corinthian Gorgon right below the nose; the iron smashed the fellow's teeth, blew right through the bone of the jaw and into his throat, lodging there with the grip sticking out before his face."
Dienekes shook his head and released a dark chuckle, the kind one summons recalling a tale at a distance, knowing how close he had come to annihilation and in awe before the gods that he had somehow survived. "It didn't even slow this dick-stroker down. He came right back at Iatrokles, with bare hands and that pig-poker buried square in his jaw. I took him low and my brother took him high. We dropped him like a wrestler. I drove the blade end of my eight-footer that was now a one-footer into his guts, then grabbed the butt-spike end of someone's discarded eight from the dirt and laid all my weight on it, right through his groin all the way into the ground, nailing him there. My brother had grabbed the bastard's sword and hacked half the top of his head off, right through the bronze of his helmet. He still got up. I had never seen my brother truly terrified but this time it was serious. 'Zeus Almighty!' he cried, and it was not a curse but a prayer, a piss-down-your-leg prayer."
The night had grown cool; my master draped his cloak around his shoulders. He took another draught of wine.
"He had a squire, my brother did, from Antaurus in Scythia, of whom you may have heard. This man was called by the Spartans 'Suicide.'"
My expression must have betrayed startlement, for Dienekes chuckled in response. This fellow, the Scythian, had been Dienekes' squire before me; he became my own mentor and instructor. It was all new to me, however, that the man had served my master's brother before him.
"This reprobate had come to Sparta like you, Xeo, on his own, the crazy bastard. Fleeing bloodguilt, a murder; he had killed his father or father-in-law, I forget which, in some hill-tribe dispute over a girl. When he arrived in Lakedaemon, he asked the first man he met to dispatch him, and scores more for days. No one would do it, they feared ritual pollution; finally my brother took him with him to battle, promising he'd get him polished off there.
"The man turned out a holy terror. He wouldn't keep to the rear like the other squires, but waded right in, unarmored, seeking death, crying out for it. His weapon, as you know, was the javelin; he crafted his own, sawed-off specimens no longer than a man's arm, which he called 'darning needles.' He carried twelve of them, in a quiver like arrows, and threw them by the clutch of three, one after the other, at the same man, saving the third for the close work."
This indeed described the man. Even now, what must be twenty years later, he remained fearless to the point of madness and utterly reckless of his life.
"Anyway here he came now, this Scythian lunatic. Hoom, hoom, hoom, he put two darning needles through that Corinthian monster's liver and out his back, and added one for good measure right where the man's fruit hung. That did it. The titan looked straight at me, bellowed once, then dropped like a sack off a waggon. I realized later that half my skull was showing through to the sun, my face a mass of blood, and the whole right side of my beard and chin had been hacked off."
"How did you get out of the battle?" I asked.
"Get out? We had to fight across another thousand yards before the enemy finally turned the creases and it was over. I couldn't tell the state I was in. My brother wouldn't let me touch my face. 'You've got a few scratches,' he said. I could feel the breeze on my skull; I knew it was bad. I remember only this ghoulish surgeon, our friend Suicide, stitching me up with sailor's twine while my brother held my head and cracked jokes. 'You're not going to be too pretty after this one. I won't have to worry any more about you stealing my bride.'"
Here Dienekes drew up, his expression going suddenly sober and solemn. He declared that the story at this point proceeded into the province of the personal. He must put a period to it.
I begged him to continue. He could see the disappointment on my face. Please, sir. You must not carry the tale this far, only to discard it by the wayside.
"You know," he offered in wry admonishment, "what happens to squires who spread tales out of school." He took a draught of wine and, after a thoughtful moment, resumed.
"You are aware that I am not my wife's first husband. Arete was married to my brother first."
I had known this, but never from my master's lips.
"It created a grievous rift in my family, because I habitually declined to share a meal at his home, I always found some excuse. My brother was deeply wounded by this, thinking I disrespected his wife or had found some fault in her which I would not divulge. He had taken her from her family very young, when she was just seventeen, and this overhaste I know troubled him. He wanted her so much he couldn't wait, he was afraid another would claim her. So when I avoided his house, he thought I found fault with him for this.
"He went to our father and even to the ephors over it, seeking to force me to accept his invitations. One day we wrestled in the _palaistra_ and he nearly strangled me (I was never half a match for him) and ordered me that evening to present myself at his home, in my best dress and manners. He swore he would break my back if I gave offense once more.
"It was just getting to be evening when I spotted him approaching me again, beside the Big Ring, as I was finishing training. You know the lady Arete and her tongue. She had had a talk with him. 'You are blind, Iatrokles,' she had said. 'Can't you see that your brother has feelings for me? That is why he declines all invitations to visit with us. He feels shame to experience these passions for his brother's wife.'
"My brother asked me straight out if this was true. I lied like a dog, but he saw through me as he always did. You could see he was profoundly troubled. He stood absolutely still, in a way he had since he was a boy, considering the matter. 'She will be yours when I am slain in battle,' he declared. That seemed to settle the matter for him.
"But not for me. Within a week I found excuse to get myself out of the city, assisting on an embassy overseas. I managed to keep away for the whole winter, returning only when the Herakles regiment was called up for Pellene. My brother was killed there. I didn't even know it in the advance, not until the battle was won and we remustered. I was twenty-four years old. He was thirty-one."
Dienekes' countenance grew even more solemn. All effect of the wine had fled. He hesitated for long moments, as if considering whether to continue or break off the tale at this point. He scrutinized my expression until at last, seeming to satisfy himself that I was listening with the proper attention and respect, he dumped the dregs of his bowl and continued.
"I felt it was my doing, my brother's death, as if I had willed it in secret and the gods had somehow responded to this shameful prayer. It was the most painful thing that had ever happened to me. I felt I couldn't go on living, but I didn't know how honorably to end my life. I had to come home, for my father and mother's sake and for the funeral games. I never went near Arete. I intended to leave Lakedaemon again as soon as the games were over, but her father came to me. 'Aren't you going to say one word to my daughter?' He had no clue of my feelings for her, he simply meant the courtesy of a brother-in-law and my obligation as _kyrios_ to see that Arete was given to a proper husband. He said that husband should be myself. I was Iatrokles' only brother, the families were already profoundly intertwined and since Arete had as yet borne no children, mine with hers would be as if they were my brother's as well.
"I declined.
"This gentleman could make no guess of the real reason, that I couldn't embrace the shame of satisfying my deepest self-interest over the bones of my own brother. Arete's father could not understand; he was deeply hurt and insulted. It was an impossible situation, spawning suffering and sorrow in every quarter. I had no idea how to set it right. I was at wrestling one afternoon, just going through the motions, plagued by internal torment, when there came a commotion at the _Gymnasion_ gate. A woman had entered the precinct. No female, as all know, may intrude upon those grounds. Murmurs of outrage were building. I myself arose from the pit— _gymnos_ as all were, naked—to join the others in throwing the interloper out.
"Then I saw. It was Arete.
"The men parted before her like grain before the reapers. She stopped right beside the lanes, where the boxers were standing naked waiting to enter the ring.
"'Which of you will have me as his wife?' she demanded of the entire assembly, who were by now gaping slack-jawed, dumbstruck as calves. Arete is a lovely woman still, even after four daughters, but then, yet childless and barely nineteen, she was as dazzling as a goddess. Not a man didn't desire her, but they were all too paralyzed to utter a peep. 'Will no man come forward to claim me?'
"She turned and marched then, right up in front of me. 'Then you must make me your wife, Dienekes, or my father will not be able to bear the shame.'
"My heart was wrenched by this, half numb at the sheer brass and temerity of this woman, this girl, to attempt such a stunt, the other half moved profoundly by her courage and wit."
"What happened?" I asked.
"What choice did I have? I became her husband."
Dienekes related several other tales of his brother's prowess in the Games and his valor in battle. In every field, in speed and wit and beauty, in virtue and forbearance, even in the chorus, his brother eclipsed him. It was clear Dienekes revered him, not merely as a younger brother will his elder, but as a man, in sober assessment and admiration. "What a pair Iatrokles and Arete made. The whole city anticipated their sons. What warriors and heroes their combined lines would produce."
But Iatrokles and Arete had had no children, and the lady's with Dienekes had all been girls.
Dienekes gave it no voice, but one could readily perceive the sorrow and regret upon his face. Why had the gods granted him and Arete only daughters? What could it be but their curse, that divinely apportioned requital for the crime of selfish love in my master's heart? Dienekes rose from this preoccupation, or what I felt certain was this preoccupation, and gestured down the slope toward the Avenue of the Champions.
"Thus you see, Xeo, how courage before the enemy may perhaps come more easily to me than to others. I hold the example of my brother before me. I know that no matter what feat of valor the gods permit me to perform, I will never be his equal. This is my secret. What keeps me humble."
He smiled. An odd, sad sort of smile.
"So now, Xeo, you know the secrets of my heart. And how I came to be the handsome fellow you see before you." I laughed, as my master had wanted. All merriment, however, had fled his features.
"And now I am tired," he said, shifting upon the earth. "If you will excuse me, it's time to deflower the straw maiden, as they say."
And with that he curled upon his reed groundbed and settled at once into sleep.
**EIGHT**
**T** _he preceding interviews were transcribed over the course of several evenings as His Majesty's forces continued their still-unopposed advance into Hellas. The defenders at Thermopylae having been vanquished, the Hellenic fleet suffering further severe losses of ships and men at the naval battle fought simultaneously opposite Artemisium, all Greek and allied units, army and navy, now fled the field. The Hellenic land forces retreated south toward the Isthmus of Corinth, across which they and the armies now massing from the other Greek cities, including the forces of Sparta under a full call-up, were constructing a wall to defend the Peloponnese. The sea elements withdrew around Euboea and Cape Sounion to unite with the main body of the Hellenic fleet at Athens and Salamis in the Gulf of Saronika._
_His Majesty's army put all Phokis to the torch. Imperial troops burned to the ground the cities of Drymus, Charada, Erochus, Tethronium, Amphikaea, Neon, Pedies, Trites, Elateia, Hylampolis and Parapotamii. All temples and sanctuaries of the Hellenic gods, including that of Apollo at Abae, were razed and their treasuries looted._
_As for His Majesty Himself, the Royal Person's time now became consumed, nearly twenty hours a day, with urgent matters military and diplomatic. These demands notwithstanding, yet did His Majesty's desire remain undiminished to hear the continuation of the captive Xeones' tale. He ordered the interviews to proceed in His absence, their verbatim record to be transcribed for His Majesty's perusal at such hours as He found free._
_The Greek responded vigorously to this order. The sight of his native Hellas being reduced by the overmastering numbers of the imperial forces caused the man severe distress and seemed to fire his will to commit to record as much of his tale as he could, as expeditiously as possible. Dispatches relating the overrunning of the Temple of the Oracle of Apollo at Delphi seemed only to increase the prisoner's grief. Privately he stated his_ _concern that His Majesty was growing impatient with the tale of his own and other individuals' personal histories and becoming anxious to move on to the more apposite topics of Spartan tactics, training and military philosophy. The Greek begged His Majesty's patience, stating that the tale seemed to be "telling itself" at the god's direction and that he, its narrator, could only follow where it led._
_We began again, His Majesty absent, on the evening of the ninth day of Tashritu, in the tent of Orontes, captain of the Immortals._
**H** is Majesty has requested that I recount some of the training practices of the Spartans, particularly those relating to the youth and their rearing under the Lykurgan warrior code. A specific incident may be illustrative, not only to impart certain details but to convey also the flavor of the thing. This event was in nowise atypical. I report it both for its informative value and because it involved several of the men whose heroism His Majesty witnessed with his own eyes during the struggle at the Hot Gates.
This incident took place some six years prior to the battle at Thermopylae. I was fourteen at the time and not yet employed by my master as his battle squire; in fact I had at that time barely dwelt in Lakedaemon two years. I was serving as a _parastates pais,_ a sparring partner, to a Spartiate youth of my own age named Alexandros. This individual I have mentioned once or twice in other contexts. He was the son of the _polemarch,_ or war leader, Olympieus, and at that time, aged fourteen, the protege of Dienekes.
Alexandros was a scion of one of the noblest families of Sparta; his line descended on the Eurypontid side directly from Herakles. He was, however, not constitutionally suited to the role of warrior. In a gentler world Alexandros might have been a poet or musician. He was easily the most accomplished flute player of his age-class, though he barely touched the instrument to practice. His gifts as a singer were even more exceptional, both as a boy alto and later as a man when his voice stabilized into a pure tenor.
It chanced, unless the hand of a god was at work in it, that he and I when we were thirteen were flogged simultaneously, for separate offenses, on different sides of the same training field. His transgression related to some breach within his _agoge boua,_ his training platoon; mine was for improperly shaving the throat of a sacrificial goat.
In our separate whippings, Alexandros fell before I did. I mention this not as cause for pride; it was simply that I had taken more beatings. I was more accustomed to it. The contrast in our deportment, unfortunately for Alexandros, was perceived as a disgrace of the most egregious order. As a means of rubbing his nose in it, his drill instructors assigned me permanently to him, with instructions that he fight me over and over until he could beat the hell out of me. For my part, I was informed that if I was even suspected of going easy on him, out of fear of the consequences of harming my better, I would be lashed until the bones of my back showed through to the sun.
The Lakedaemonians are extremely shrewd in these matters; they know that no arrangement could be more cunningly contrived to bind two youths together. I was keenly aware that, if I played my part satisfactorily, I would continue in Alexandros' service and become his squire when he reached twenty and took his station as a warrior in line of battle. Nothing could have suited me more. This was why I had come to Sparta in the first place—to witness the training close-up and to endure as much of it as the Lakedaemonians would permit.
The army was at the Oaks, in the Otona valley, a blistering late summer afternoon, on an eight-nighter, what they call in Lakedaemon, the only city which practices it, an _oktonyktia._ These are regimental exercises normally, though in this case it involved a division. An entire _mora,_ more than twelve hundred men with full armor and battle train including an equal number of squires and helots, had marched out into the high valleys and drilled in darkness for four nights, sleeping in the day in open bivouac, by watches, at full readiness with no cover, then drilling day and night for the following three days. Conditions were deliberately contrived to make the exercise as close as possible to the rigor of actual campaign, simulating everything except casualties. There were mock night assaults up twenty-degree slopes, each man bearing full kit and _panoplia,_ sixty-five to eighty pounds of shield and armor. Then assaults down the hill. Then more across. The terrain was chosen for its boulder-strewn aspect and the numerous gnarled and low-branched oaks which dotted the slopes. The skill was to flow around everything, like water over rocks, without breaking the line.
No amenities whatever were brought. Wine was at half-rations the first four days, none the second two, then no liquid at all, including water, for the final two. Rations were hard linseed loaves, which Dienekes declared fit only for barn insulation, and figs alone, nothing hot. This type of exercise is only partially in anticipation of night action; its primary purpose is training for surefootedness, for orientation by feel within the phalanx and for action without sight, particularly over uneven ground. It is axiomatic among the Lakedaemonians that an army must be able to dress and maneuver the line as skillfully blind as sighted, for, as His Majesty knows, in the dust and terror of the _othismos,_ the initial battlefield collision and the horrific scrum that ensues, no man can see more than five feet in any direction, nor hear even his own cries above the din.
It is a common misconception among the other Hellenes, and one deliberately cultivated by the Spartans, that the character of Lakedaemonian military training is brutal and humorless in the extreme. Nothing could be further from the fact. I have never experienced under other circumstances anything like the relentless hilarity that proceeds during these otherwise grueling field exercises. The men bitch and crack jokes from the moment the _sarpinx_ 's blare sounds reveille till the final bone-fatigued hour when the warriors curl up in their cloaks for sleep, and even then you can hear cracks being muttered and punchy laughter breaking out in odd corners of the field for minutes until sleep, which comes on like a hammerblow, overtakes them.
It is that peculiar soldiers' humor which springs from the experience of shared misery and often translates poorly to those not on the spot and enduring the same hardship. "What's the difference between a Spartan king and a mid-ranker?" One man will lob this query to his mate as they prepare to bed down in the open in a cold driving rain. His friend considers mock-theatrically for a moment. "The king sleeps in that shithole over there," he replies. "We sleep in this shithole over here."
The more miserable the conditions, the more convulsing the jokes become, or at least that's how it seems. I have witnessed venerable Peers of fifty years and more, with thick gray in their beards and countenances as distinguished as Zeus', dropping helpless with mirth onto hands and knees, toppling onto their backs and practically pissing down their legs they were laughing so hard. Once on an errand I saw Leonidas himself, unable to get to his feet for a minute or more, so doubled over was he from some otherwise untranslatable wisecrack. Each time he tried to rise, one of his tent companions, grizzled captains in their late fifties but to him just boyhood chums he still addressed by their _agoge_ nicknames, would torment him with another variation on the joke, which would reconvulse him and drop him back upon his knees.
This, and other like incidents, endeared Leonidas universally to the men, not just the Spartiate Peers but the Gentleman-Rankers and _perioikoi_ as well. They could see their king, at nearly sixty, enduring every bit of misery they did. And they knew that when battle came, he would take his place not safely in the rear, but in the front rank, at the hottest and most perilous spot on the field.
The purpose of an eight-nighter is to drive the individuals of the division, and the unit itself, beyond the point of humor. It is when the jokes stop, they say, that the real lessons are learned and each man, and the _mora_ as a whole, make those incremental advances which pay off in the ultimate crucible. The hardship of the exercises is intended less to strengthen the back than to toughen the mind. The Spartans say that any army may win while it still has its legs under it; the real test comes when all strength is fled and the men must produce victory on will alone.
The seventh day had come and gone now, and the army had reached that stage of exhaustion and short-temperedness that the eight-nighter was contrived to produce. It was late afternoon; the men were just rousing themselves from some pitifully inadequate catnap, parched and filthy and stink-begrimed, in anticipation of the final night's drill. Everyone was hungry and tired and drained utterly of fluid. A hundred variations were spun out on the same joke, each man's wish for a real war so he could finally get more than a half hour's snooze and a bellyful of hot chow. The men were dressing their long sweat-matted hair, griping and bitching, while their squires and helots, as miserable and dehydrated as they, handed them the last dry fig cake, without wine or water, and readied them for the sunset sacrifice, while their stacked arms and _panoplia_ waited in perfect order for the night's work to begin.
Alexandros' training platoon was already awake and in formation, with eight others of the fourth age-class, boys thirteen and fourteen under their twenty-year-old drill instructors, on the lower slopes below the army's camp. These _agoge_ platoons were regularly exposed to the sight of their elders and the rigors they endured, as a means of rousing their emulative instincts to even greater levels of exertion. I had been dispatched to the upper camp with a message stick when the commotion came from back down across the plain.
I turned and saw Alexandros singled out at the edge of his platoon, with Polynikes, the Knight and Olympic champion, standing before him, raging. Alexandros was fourteen, Polynikes twenty-three; even at a range of a hundred yards you could see the boy was terrified.
This warrior Polynikes was no man to be trifled with. He was a nephew of Leonidas, with a prize of valor already to his name, and utterly pitiless. Apparently he had come down from the upper camp on some errand, had passed the boys of the _agoge_ in their lineup and spotted some breach of discipline.
Now the Peers on the slope above could see what it was.
Alexandros had neglected his shield, or to use the Doric term, _etimasen,_ "defamed" it. Somehow he had allowed it to lie outside his grasp, facedown, untended on the ground with its big concave bowl pointing at the sky.
Polynikes stood in front of him. "What is this I see in the dirt before me?" he roared. The Spartiates uphill could hear every syllable. "It must be a chamber pot, with its bowl peeking up so daintily."
Is it a chamber pot? he demanded of Alexandros. The boy answered no.
Then what is it?
It is a shield, lord.
Polynikes declared this impossible.
"It can't be a shield, I'm certain of that." His voice carried powerfully up the amphitheater of the valley. "Because not even the dumbest bum-fucked shitworm of a _paidarion_ would leave a shield lying facedown where he couldn't snatch it up in an instant when the enemy came upon him." He towered above the mortified boy.
"It is a chamber pot," Polynikes declared. "Fill it."
The torture began.
Alexandros was ordered to piss into his shield. It was a training shield, yes. But Dienekes knew as he looked down with the other Peers from the slope above that this particular _aspis,_ patched and repatched over decades, had belonged to Alexandros' father and grandfather before him.
Alexandros was so scared and so dehydrated, he couldn't raise a drop.
Now a second factor entered the equation. This was the tendency among the youths in training, those who were not for the moment the object of their superiors' rage, to convulse with perverse glee at the misery of whatever luckless mate now found himself spitted above the coals. Up and down the line of boys, teeth sank into tongues seeking to suppress this fear-inspired hilarity. One lad named Ariston, who was extremely handsome and the fastest sprinter of the fourth class, something of a younger version of Polynikes himself, could not contain himself. A snort escaped his clamped jaws.
Polynikes turned upon him in fury. Ariston had three sisters, all what the Lakedaemonians call "two-lookers," meaning they were so pretty that one look was not enough, you had to look twice to appreciate them.
Polynikes asked Ariston if he thought this was funny.
"No, lord," the boy replied.
"If you think this is funny, wait till you get into combat. You'll think that's hysterical."
"No, lord."
"Oh yes you will. You'll be giggling like your goddam sisters." He advanced a pace nearer. "Is that what you think war is, you fucking come-spot?"
"No, lord."
Polynikes pressed his face inches from the boy's, glowering into his eyes with a look of blistering malice. "Tell me. Which do you think will be the bigger laugh: when you take an enemy spear eighteen inches up the dogblossom, or when your psalm-singing mate Alexandros takes one?"
"Neither, lord." Ariston's face was stone.
"You're afraid of me, aren't you? That's the real reason you're laughing. You're so fucking happy it wasn't you I singled out."
"No, lord."
"What? You're not afraid of me?"
Polynikes demanded to know which it was. Because if Ariston was afraid of him, then he was a coward. And if he wasn't, he was reckless and ignorant, which was even worse.
"Which is it, you miserable mound of shit? 'Cause you'd better fucking well be afraid of me. I'll put my dick in your right ear, pull it out your left and fill that chamber pot myself."
Polynikes ordered the other boys to take up Alexandros' slack. While their pathetic dribbles of urine splotched onto the wood and leather-padded frame, over the good-luck talismans that Alexandros' mother and sisters had made and that hung from the inner frame, Polynikes returned his attention to Alexandros, querying him on the protocol of the shield, which the boy knew and had known since he was three.
The shield must stand upright at all times, Alexandros declaimed at the top of his voice, with its forearm sleeve and handgrip at the ready. If a warrior stand at the rest, his shield must lean against his knees. If he sit or lie, it must be supported upright by the _tripous basis,_ a light three-legged stand which all bore inside the bowl of the concave _hoplon,_ in a carrying nest made for that purpose.
The other youths under Polynikes' orders had now finished urinating as best they could into the hollow of Alexandros' shield. I glanced at Dienekes. His features betrayed no emotion, though I knew he loved Alexandros and wished for nothing more than to dash down the slope and murder Polynikes.
But Polynikes was right. Alexandros was wrong. The boy must be taught a lesson.
Polynikes now had Alexandros' _tripous basis_ in his hand. The little tripod was comprised of three dowels joined at one end by a leather thong. The dowels were the thickness of a man's finger and about eighteen inches long. "Line of battle!" Polynikes bellowed. The platoon of boys formed up. He had them all lay their shields, defamed, facedown in the dirt, exactly as Alexandros had done.
By now twelve hundred Spartiates up the hill were observing the spectacle, along with an equal number of squires and helot attendants.
"Shields, port!"
The boys lunged for their heavy, grounded _hopla._ As they did, Polynikes lashed at Alexandros' face with the tripod. Blood sprung. He swatted the next boy and the next until the fifth at last wrestled his twenty-pound, unwieldy shield off the ground and up into place to defend himself.
He made them do it again and again and again.
Starting at one end of the line, then the other, then the middle. Polynikes, as I have said, was an Agiad, one of the Three Hundred Knights and an Olympic victor besides. He could do anything he liked. The drill instructor, who was just an _eirene,_ had been brushed aside, and could do nothing but look on in mortification.
"This is hilarious, isn't it?" Polynikes demanded of the boys. "I'm beside myself, aren't you? I can hardly wait to see combat, which will be even more fun."
The youths knew what was coming next.
Tree fucking.
When Polynikes tired of torturing them here, he would have their drill instructor march them over to the edge of the plain, to some particularly stout oak, and order them, in formation, to push the tree down with their shields, just the way they would assault an enemy in battle.
The boys would take station in ranks, eight deep, the shield of each pressed into the hollow of the boy's back before him, with the leading boy's shield mashed by their combined weight and pressure against the oak. Then they would do _othismos_ drill.
They would push.
They would strain.
They would fuck that tree for all they were worth.
The soles of their bare feet would churn the dirt, heaving and straining until a rut had been excavated ankle-deep, while they crushed each other's guts humping and hurling, grinding into that unmoveable trunk. When the front-rank boy could stand no more, he would assume the position of the rearmost and the second boy would move up.
Two hours later Polynikes would casually return, perhaps with several other young warriors, who had themselves been through this hell more than once during their own _agoge_ years. These would observe with shock and disbelief that the tree was still standing. "By God, these dog-strokers have been at it half the watch and that pitiful little sapling is still right where it was!"
Now effeminacy would be added to the list of the lads' crimes. It was unthinkable that they be allowed to return to the city while this tree yet defied them; such failure would disgrace their fathers and mothers, brothers, sisters, aunts, uncles and cousins, all the gods and heroes of their line, not to mention their hounds, cats, sheep and goats and even the rats in their helots' barns, who would hang their heads and have to slink off to Athens or some other rump-split _polis_ where men were men and knew how to put out a respectable fucking.
That tree is the enemy!
Fuck the enemy!
On it would go, into all-night shield drill which by mid second watch would have reduced the boys to involuntary regurgitation and defecation; they would be puking and shitting themselves, their bodies shattered utterly from exhaustion, and then, when the dawn sacrifices at last brought clemency and reprieve, the boys would fall in for another full day of training without a minute's sleep.
This torment, the boys knew now as they stood under Polynikes face-lashing, was yet to come. This was what they had to look forward to.
By this point every nose in the formation had been broken. Each boy's face was a sheet of blood. Polynikes was just taking a breath (he had tired his arm with all that swatting) when Alexandros thoughtlessly reached with a hand to the side of his blood-begrimed face.
"What do you think you're doing, buttfuck?" Polynikes turned instantly upon him.
"Wiping the blood, lord."
"What are you doing that for?"
"So I can see, lord."
"Who the fuck told you you had a right to see?"
Polynikes continued his blistering mockery. Why did Alexandros think the division was out here, training at night? Was it not to learn to fight when they couldn't see? Did Alexandros think that in combat he would be allowed to pause to wipe his face? That must be it. Alexandros would call out to the enemy and they would halt politely for a moment, so the boy could pluck a nosenugget from his nostril or wipe a turdberry from his crease. "I ask you again, is this a chamber pot?"
"No, lord. It is my shield."
Again Polynikes' dowels blasted the boy across the face. "'My'?" he demanded furiously. "'My'?"
Dienekes looked on, mortified, from where he stood at the edge of the upper camp. Alexandros was excruciatingly aware that his mentor was watching; he seemed to summon his composure, rally all his senses. The boy stepped forward, shield at high port. He straightened to attention before Polynikes and enunciated in his loudest, clearest voice:
_"This is my shield._
_I bear it before me into battle,_
_but it is not mine alone._
_It protects my brother on my left._
_It protects my city._
_I will never let my brother_
_out of its shadow_
_nor my city out of its shelter._
_I will die with my shield before me_
_facing the enemy."_
The boy finished. The last of his words, shouted at the top of his voice, echoed for a long moment around the valley walls. Twenty-five hundred men stood listening and watching.
They could see Polynikes nod, satisfied. He barked an order. The boys resumed formation, each now with his shield in proper place, upright against its owner's knees.
"Shields, port!"
The boys lunged for their _hopla._
Polynikes swung the tripod.
With a crack that could be heard across the valley, the slashing sticks struck the bronze of Alexandros' shield.
Polynikes swung again, at the next boy and next. All shields were in place. The line protected.
He did it again from the right and from the left. Now all shields leapt into the boys' grips, all swiftly into place before them.
There.
With a nod to the platoon's _eirene,_ Polynikes stepped back. The boys held fast at attention, shields at high port, with the blood beginning to cake dry on their empurpled cheekbones and shattered noses.
Polynikes repeated his order to the drill instructor, that these sheep-stroking sons of whores would do tree-fucking till the end of the second watch, then shield drill till dawn.
He walked once down the line, meeting each boy's eye. Before Alexandros, he halted.
"Your nose was too pretty, son of Olympieus. It was a girl's nose." He tossed the boy's tripod into the dirt at his feet. "I like it better now."
**NINE**
**O** ne of the boys died that night. His name was Hermion; they called him "Mountain." At fourteen he was as strong as any in his age-class or the class above, but dehydration in combination with exhaustion overcame him. He collapsed near the end of the second watch and fell into that state of convulsive torpor the Spartans call _nekrophaneia,_ the Little Death, from which a man may recover if left alone but will die if he tries to rise or exert himself. Mountain understood his extremity but refused to stay down while his mates kept their feet and continued their drill.
I tried to make the platoon take water, I and my helot mate Dekton, whom they later called "Rooster." We snuck a skin to them around the middle of the first watch, but the boys refused to accept it. At dawn they carried Mountain in on their shoulders, the way the fallen in battle are borne.
Alexandros' nose never did heal properly. His father had it broken again, twice, and reset by the finest battle surgeons, but the seam where the cartilage meets the bone never mended quite right. The airway would constrict involuntarily, triggering those spasms of the lungs called by the Greeks _asthma,_ which were excruciating simply to watch and must have been unbearable to endure. Alexandros blamed himself for the death of the boy called Mountain. These fits, he was certain, were the retribution of heaven for his lapse of concentration and unwarrior-like conduct.
The spasms enfeebled Alexandros' endurance and made him less and less a match for his age-mates within the _agoge._ Worse still was the unpredictability of the attacks. When they hit, he was good for nothing for minutes at a stretch. If he could not find a way to reverse this condition, he could not when he reached manhood be made a warrior; he would lose his citizenship and be left to choose between living on in some lesser state of disgrace or embracing honor and taking his own life.
His father, gravely concerned, offered sacrifice again and again and even sent to Delphi for counsel from the Pythia. Nothing helped.
Aggravating the situation further was the fact that, despite what Polynikes had said about the boy's broken nose, Alexandros remained "pretty." Nor did his breathing difficulties, for some reason, affect his singing. It seemed somehow that fear, rather than physical incapacity, was the trigger for these attacks.
The Spartans have a discipline they call _phobologia,_ the science of fear. As his mentor, Dienekes worked with Alexandros privately on this, after evening mess and before dawn, while the units were forming up for sacrifice.
_Phobologic_ discipline is comprised of twenty-eight exercises, each focusing upon a separate nexus of the nervous system. The five primaries are the knees and hams, lungs and heart, loins and bowels, the lower back, and the girdle of the shoulders, particularly the trapezius muscles, which yoke the shoulder to the neck.
A secondary nexus, for which the Lakedaemonians have twelve more exercises, is the face, specifically the muscles of the jaw, the neck and the four ocular constrictors around the eye sockets. These nexuses are termed by the Spartans _phobosynakteres,_ fear accumulators.
Fear spawns in the body, _phobologic_ science teaches, and must be combated there. For once the flesh is seized, a _phobokyklos,_ or loop of fear, may commence, feeding upon itself, mounting into a "runaway" of terror. Put the body into a state of _aphobia,_ fearlessness, the Spartans believe, and the mind will follow.
Under the oaks, in the still half-light before dawn, Dienekes practiced alone with Alexandros. He would tap the boy with an olive bough, very lightly, on the side of the face. Involuntarily the muscles of the trapezius would contract. "Feel the fear? There. Feel it?" The older man's voice crooned soothingly, like a trainer gentling a colt. "Now. Drop your shoulder." He popped the boy's cheek again. "Let the fear bleed out. Feel it?"
Man and boy worked for hours on the "owl muscles," the _ophthalmomyes_ surrounding the eyes. These, Dienekes instructed Alexandros, were in many ways the most powerful of all, for God in His wisdom made mortals' keenest defensive reflex that which protects the vision. "Watch my face when the muscles constrict," Dienekes demonstrated. "What expression is this?"
_"Phobos._ Fear."
Dienekes, schooled in the discipline, commanded his facial muscles to relent.
"Now. What does this expression indicate?"
_"Aphobia._ Fearlessness."
It seemed effortless when Dienekes did it, and the other boys in their training were practicing and mastering this too. But for Alexandros, nothing of the discipline came easy. The only time his heart beat truly without fear was when he mounted the choral stand and stood, solitary, to sing at the _Gymnopaedia_ and the other boy's festivals.
Perhaps his true guardians were the Muses. Dienekes had Alexandros sacrifice to them and to Zeus and Mnemosyne. Agathe, one of the "two-looker" sisters of Ariston, made a charm of amber to Polyhymnia, and Alexandros carried it with him, pended from the crosshatch within his shield.
Dienekes encouraged Alexandros in his singing. The gods endow each man with a gift by which he may conquer fear; Alexandros', Dienekes felt certain, was his voice. Skill in singing in Sparta is counted second only to martial valor and in fact is closely related, through the heart and lungs, within the discipline of the _phobologia._ This is why the Lakedaemonians sing as they advance into battle. They are schooled to open the throat and gulp the air, work the lungs till the accumulators relent and break the constriction of fear.
There are two running courses within the city: the Little Ring, which begins at the _Gymnasion_ and follows the Konooura road beneath Athena of the Brazen House, and the Big Ring, which laps all five villages, past Amyklai, along the Hyakinthian Way and across the slopes of Taygetos. Alexandros ran the big one, six miles barefoot, before sacrifice and after dinner mess. Extra rations were slipped him by the helot cooks. By unspoken compact the boys of his _boua_ protected him in training. They covered for him when his lungs betrayed him, when it seemed he might be singled out for punishment. Alexandros responded with a secret shame which propelled him to even greater exertions.
He began to train in the "all-in," that type of no-holds-barred boys' brawling unique to Lakedaemon, in which the competitor may kick, bite, gouge the eyes, do anything but raise the hand for quarter. Alexandros hurled himself barefoot up the Therai watercourse and bare-handed against the _pankratist_ 's bag; he ran weighted sprints, he pounded his fists into the trainer's boxes of sand. His slender hands became scarred and knuckle-busted. His nose broke again and again. He fought boys from his own platoon and others, and he fought me.
I was growing fast. My hands were getting stronger. Every athletic action Alexandros performed, I could do better. In the fighting square it was all I could do not to break up his face even more. He should have hated me, but it was not in him. He shared his surplus rations and worried that I would be whipped for going easy on him.
We talked for hours in secret on the pursuit of _esoterike harmonia,_ that state of self-composure which the exercises of the _phobologia_ are designed to produce. As a string of the _kithera_ vibrates purely, emitting only that note of the musical scale which is its alone, so must the individual warrior shed all which is superfluous in his spirit, until he himself vibrates at that sole pitch which his individual _daimon_ dictates. The achievement of this ideal, in Lakedaemon, carries beyond courage on the battlefield; it is considered the supreme embodiment of virtue, _andreia,_ of a citizen and a man.
Beyond _esoterike harmonia_ lies _exoterike harmonia,_ that state of union with one's fellows which parallels the musical harmony of the multistringed instrument or of the chorus of voices itself. In battle _exoterike harmonia_ guides the phalanx to move and strike as one man, of a single mind and will. In passion it unites husband to wife, lover to lover, in wordless perfect union. In politics _exoterike harmonia_ produces a city of concord and unity, in which each individual, securing his own noblest expression of character, donates this to each other, as obedient to the laws of the commonwealth as the strings of the _kithera_ to the immutable mathematics of music. In piety _exoterike_ _harmonia_ produces that silent symphony which most delights the ears of the gods.
At the height of that summer there was a war with the Antirhionians. Four of the army's twelve _lochoi_ were mobilized (reinforced by elements of the Skiritai, the mountain rangers who comprised their own main-force regiment) to a call-up of the first ten age-classes, twenty-eight hundred in all. This was no force to be taken lightly, all-Lakedaemonian, commanded by the king himself; the battle train alone would be half a mile long. It would be the first full-scale campaign since the death of Kleomenes and the third in which Leonidas would assume command as king.
Polynikes would go as a Knight of the king's bodyguard, Olympieus with the Huntress battalion in the Wild Olive _lochos_ and Dienekes as a platoon commander, an _enomotarch,_ in the Herakles. Even Dekton, my half-breed friend, would be mobilized as herd boy for the sacrificial beasts.
The entire Deukalion mess in which Alexandros "stood-to," meaning acted as occasional cupbearer and server so he could observe his elders and learn, was called up except the five eldest men, between forty and sixty. For Alexandros, though he was six years too young to go, the mobilization seemed to plunge him even more deeply under his cloud. The uncalled-up Peers twitched about with their own brand of frustration. The air was touchy and ripe for explosiveness.
Somehow an all-in match got started one evening between Alexandros and me, outdoors behind the mess. The Peers gathered eagerly; the action was just what they needed. I could hear Dienekes' voice, cheering the brawl on. Alexandros seemed full of fire; we were bare-handed and his smallish fists flew fast as darts. He kicked me hard, to the temple, and followed with a solid elbow to the gut; I dropped. It was a true fall, I was really hurt, but the Peers had seen Alexandros' friends cover for him so frequently that they now thought I was tanking it. Alexandros did too.
"Get up, you outlander piece of shit!" He straddled me in the dirt and hit me again when I rose. For the first time I heard real killer instinct in his voice. The Peers heard it too and raised a shout of delight. Meanwhile the hounds, of whom there were never fewer than twenty after chow time, howled and bounded from every quarter in the turf-skimming fever that their masters' excited voices now drove them to.
I got up and hit Alexandros. I knew I could beat him easily, despite his crowd-impelled fury; I tried to pull my punch, just slightly so that no one would notice. They did. A howl of outrage rose from the Peers of the mess and others from adjacent _syssitia,_ who had now clustered, forming a ring from which neither Alexandros nor I could escape.
Men's fists cuffed me hard about the ears. "Fight him, you little fucker!" The pack instinct had seized the hounds; they were at the verge of losing themselves to their animal nature. Suddenly two burst into the ring. One got in a nip at Alexandros before the men's sticks sent him scampering. That was it.
A spasm of the lungs seized Alexandros; his throat constricted, he began to choke. My punch hesitated. A three-foot switch burned my back. "Hit him!" I obeyed; Alexandros dropped to one knee. His lungs had frozen, he was helpless. "Pound him, you whore's son!" a voice shouted from behind me. "Finish him!"
It was Dienekes.
His switch lashed me so hard it drove me to my knees. The delirium of voices overwhelmed the senses, all calling for me to polish Alexandros off. It was not anger at him. Nor were they rooting for me. The Peers could not have cared less about me. It was for him, to teach him, to make him eat the thousandth bitter lesson of the ten thousand more he would endure before they hardened him into the rock the city demanded and allowed him to take his place as an Equal and a warrior. Alexandros knew it and rose with the fury of desperation, choking for breath; he charged like a boar. I felt the lash. I swung with everything I had. Alexandros spun and dropped, face-first into the dirt, blood and spittle slinging from the side of his mouth.
He lay there, motionless as a dead man.
The Peers' shouting ceased instantly. Only the ungodly racket of the hounds continued at its maddening shrill pitch. Dienekes stepped across to the fallen form of his protege and knelt to feel his heart. In unconsciousness Alexandros' breath returned. Dienekes' hand scraped the sputum from the boy's lips.
"What are you gaping at!" he barked at the circling Peers. "It's over! Let him be!"
The army marched out next morning for Antirhion. Leonidas strode at the fore, in full _panoplia_ including slung shield, with his brow wreathed and his plumeless, unadorned helmet riding the rolled battle pack atop his scarlet cloak, his long steel-colored hair immaculately dressed and falling to his shoulders. About him marched the companion guard of the Knights, a half call-up, a hundred and fifty, with Polynikes in the forerank of honor beside six other Olympic victors. They marched not rigidly nor in grim silent lockstep, but at ease, talking and joking with one another and their families and friends along the roadside. Leonidas himself, were it not for his years and station of honor, could easily have been mistaken for a common infantryman, so unprepossessing was his armament, so nonchalant his demeanor. Yet all the city knew that this march-out, as the two previous beneath his command, was driven by his will and his will alone. It was aimed at the Persian invasion the king knew would come, perhaps not this year, perhaps not five years from now, but surely and inevitably.
The twin ports of Rhion and Antirhion commanded the western approach to the Gulf of Corinth. This avenue threatened the Peloponnese and all of central Greece. Rhion, the near-side port, stood already within the Spartan hegemony; she was an ally. But Antirhion across the strait remained haughtily aloof, thinking herself beyond the reach of Lakedaemonian power. Leonidas meant to show her the error of her ways. He would bring her to heel and bottle up the gulf, protecting central Hellas from Persian sea assault, at least from the northwest.
Alexandros' father, Olympieus, marched past at the head of the Wild Olive regiment, with Meriones, the fifty-year-old battle captive and former Potidaean captain, beside him as his squire. This gentle fellow possessed a grand beard, white as snow; he used to secrete little treasures within its bushy nest and pluck them forth, as surprise gifts, for Alexandros and his sisters when they were children. He did this now, straying to the roadside, to place in Alexandros' hand a tiny iron charm in the shape of a shield. Meriones clasped the boy's hand with a wink and moved on.
I stood in the crowd before the Hellenion with Alexandros and the other boys of the training platoons, the women and children, the whole city drawn up beneath the acacias and cypresses, singing the hymn to Castor, as the regiments trooped out along the Going-Away Street with their shields slung and spears at the slope, helmets lashed athwart the shoulders of their crimson cloaks, bobbing atop their _polemothylakioi,_ the battle packs which the Peers bore now for show but which, like their armor, would be transferred, with all kit save spears and swords, to the shoulders of their squires when the army assumed column of march and stripped for the long, dusty hump north.
Alexandros' beautiful broken face remained a mask as Dienekes strode into view, flanked by his squire, Suicide, at the head of his platoon of the Herakles _lochos._ The main body of troops passed on. Leading and accompanying each regiment trudged the pack animals laden with the supplies of the commissariat and thwacked merrily on the rumps by the switches of their helot herd boys. The train of armament waggons passed next, already obscured within a churning storm of road dust; then followed the tall victualry waggons with their cargo of oil pots and wine jars, sacks of figs, olives, leeks, onions, pomegranates and the cooking pots and ladles swinging on hooks beneath them, banging into each other musically in the dust of the mules' tread, contributing a ringing metronomic air to the cacophony of cracking whips and squalling wheel rims, teamsters' bawls and groaning axles.
Behind the provisions bearers came the portable forges and armorers' kits with their spare _xiphos_ blades and butt-spikes, "lizard-stickers" and long iron spear blades, then the spare eight-footers, uncured ash and cornel shafts lashed lengthwise along the waggon rails. Helot armorers strode in the cloud alongside, clad in their dogskin caps and aprons, forearms crisscrossed with the burn scars of the smithy.
Last of all trooped the sacrificial goats and sheep, with their horns wrapped and leashes held by the helot herd urchins, led by Dekton in his already road-begrimed altar-boy white, trailering a haltered ass laden with feed grain and two victory roosters in cages, one on either side of the cargo frame. He grinned when he passed, a little flash of contempt escaping his otherwise impeccably pious demeanor.
I was deep into slumber that night, on the stone of the portico behind the ephorate, when I felt a hand shake me awake. It was Agathe, the Spartan girl who had made Alexandros' charm to Polyhymnia. "Get up, you!" she hissed, so as not to alert the score of other youths of the _agoge_ asleep and on watch around these public buildings. I blinked around. Alexandros, who had been asleep beside me, was gone. "Hurry!"
The girl melted at once into shadow. I followed her swiftly through the dark streets to that copse of the double-boled myrtle they called Dioscuri, the Twins, just west of the start of the Little Ring.
Alexandros was there. He had snuck away from his platoon without me (which would have put both of us, if caught, in line for a merciless whipping). He stood now, wearing his black _pais_ ' cloak and battle pack, confronted by his mother, the lady Paraleia, one of their male house helots and his two younger sisters. Hard words flew. Alexandros intended to follow the army to battle. "I'm going," he declared. "Nothing will stop me."
I was ordered by Alexandros' mother to knock him down.
I saw something flash in his fist. His _xyele,_ the sicklelike weapon all the boys carried. The women saw it too, and the deadly-grim look in the lad's eye. For a long moment, every form froze. The preposterousness of the situation was becoming more and more apparent, as was the adamantine resolution of the boy.
His mother straightened before him.
"Go, then," the lady Paraleia addressed her son at last. She didn't need to add that I would go with him. "And may God preserve you in the lashing you receive when you return."
**TEN**
**I** t was not hard to follow the army. The track along the Oenous was churned to dust, ankle-deep. At Selassia the _perioikic_ Stephanos regiment had joined the expedition. Alexandros and I, arriving in the dark, could still make out the trodden-bare marshaling ground and the freshly dried blood upon the altar where the sacrifices had been performed and the omens taken. The army itself was half a day ahead; we could not stop for sleep, but pushed on all night.
At dawn we came upon men we recognized. A helot armorer named Eukrates had broken his leg in a fall and was being helped home by two of his fellows. He informed us that at the frontier fort of Oion fresh intelligence had reached Leonidas. The Antirhionians, far from rolling over and playing dead as the king had hoped, had sent envoys in secret, appealing for aid to the _tyrannos_ Gelon in Sikelia. Gelon could appreciate as well as Leonidas and the Persians the strategic indispensability of the port of Antirhion; he wanted it too. Forty Syrakusan ships bearing two thousand citizen and mercenary heavy infantry were on their way to reinforce the Antirhionian defenders. It would be a real battle after all.
The Spartan force pressed on through Tegea. The Tegeates, member allies of the Peloponnesian League and obligated to "follow the Spartans whithersoever they should lead," reinforced the army with six hundred of their own heavy infantry, swelling its fighting total to beyond four thousand. Leonidas had not been seeking _parataxis,_ a pitched battle, with the Antirhionians. Rather he had hoped to overawe them with a show of such force that they would perceive the folly of defiance and enroll themselves of their own free will in the alliance against the Persians. Among Dekton's herd was a wrapped bull, brought in anticipation of celebration, of festive sacrifice in honor of this new addition to the League. But the Antirhionians, perhaps bought by Gelon's gold, inflamed by the rhetoric of some glory-hungry demagogue or betrayed by a lying oracle, had chosen to make a fight of it.
When Alexandros spoke to the helots on the road, he had queried them for intelligence on the specific makeup of the Syrakusan forces: which units, under which commanders, reinforced by which auxiliaries. The helots didn't know. In any army other than the Spartan, such ignorance would have provoked a fierce tongue-lashing or worse. Yet Alexandros let it go without a thought. Among the Lakedaemonians, it is considered a matter of indifference of whom and in what the enemy consists.
The Spartans are schooled to regard the foe, any foe, as nameless and faceless. In their minds it is the mark of an ill-prepared and amateur army to rely in the moments before battle on what they call _pseudoandreia,_ false courage, meaning the artificially inflated martial frenzy produced by a general's eleventh-hour harangue or some peak of bronze-banging bravado built to by shouting, shield-pounding and the like. In Alexandros' mind, which already at age fourteen mirrored that of the generals of his city, one Syrakusan was as good as the next, one enemy _strategos_ no different from another. Let the foe be Mantinean, Olynthian, Epidaurian; let him come in elite units or hordes of shrieking rabble, crack citizen regiments or foreign mercenaries hired for gold. It made no difference. None was a match for the warriors of Lakedaemon, and all knew it.
Among the Spartans the work of war is demystified and depersonalized through its vocabulary, which is studded with references both agrarian and obscene. Their word which I translated earlier as "fuck," as in the youths' tree-fucking, bears the connotation not so much of penetration as of grinding, like a miller's stone. The front three ranks "fuck" or "mill" the enemy. The verb "to kill," in Doric _theros,_ is the same as "to harvest." The warriors in the fourth through sixth ranks are sometimes called "harvesters," both for the work they do on the trampled enemy with the butt-spike "lizard-stickers" of their eight-footers and for that pitiless threshing stroke they make with the short _xiphos_ sword, which itself is often called a "reaper." To decapitate a man is to "top him off" or "give him a haircut." Chopping off a hand or arm is called "limbing."
Alexandros and I arrived at Rhion, at the bluff overlooking the army's embarkation port, a little after midnight of the third day. The port lights of Antirhion shone, clearly visible across the narrow strait. The embarkation beaches were already packed with men and boys, women and children, a thronging festive mob gathered to watch the spectacle of the fleet of galleys and coasters, conscripted merchantmen, ferries and even fishing boats assembled in advance by the allied Rhionians to transport the army in darkness west along the coast, out of sight of Antirhion, then across the gulf where it stood widest, some five miles down. Leonidas, respecting the sea-fighting reputation of the Antirhionians, had elected to make this passage at night.
Among the blufftop farewell-bawlers Alexandros and I located a boy our age whose father, he claimed, owned a fast smack and would not be averse to pocketing the wad of Attic drachmas clutched in Alexandros' fist in exchange for a swift silent crossing, no questions asked. The boy led us down through the crush of spectators and merrymakers to an obscure launching beach called the Ovens, behind an unlighted breakwater. Not twenty minutes after the last Spartan transport had cast off, we were on the water too, trailing the fleet out of sight to the west.
I fear the sea anytime, but never more than on a moonless night and in the hands of strangers. Our captain had insisted on bringing along his two brothers, though a man and a boy could easily handle the light swift craft. I have known these coasters and man jacks and mistrust them; the brothers, if indeed that's what they were, were hulking louts barely capable of speech, with beards so dense they began just below the eyeline and extended thick as fur to the matted pelts of their chests.
An hour passed. The smack was making far too much speed; across the dark water the plash of the transports' oarblades and even the creaking of looms against tholepins carried easily. Alexandros ordered the pirate twice to retard his progress, but the man tossed it off with a laugh. We were downwind, he said, no one could hear us, and even if they did they would take us for part of the convoy, or one of the spectator boats, trailing to catch the action.
Sure enough, as soon as the belly of the coastline had swallowed the lights of Rhion behind us, a Spartan cutter emerged out of the black and made way to intercept us. Doric voices hailed the smack and ordered her to heave-to. Suddenly our skipper demanded his money. When we land, Alexandros insisted, as agreed. The beards clamped oars in their fists like weapons. Cutter's getting closer, boys. How will it go with you if you're caught?
"Give him nothing, Alexandros," I hissed.
But the boy perceived the precariousness of our predicament. "Of course, Captain. It will be my pleasure."
The pirate accepted his fare, grinning like Charon on the ferry to hell. "Now, lads. Over the side with you."
We were smack in the middle of the widest part of the gulf.
Our boatman indicated the Spartan cutter bearing swiftly down. "Catch a line and keep under the stern while I feed these lubbers a yard of shit." The beards loomed. "Soon as we talk these fools off, we'll haul you back aboard none the worse for wear."
Over we went. Up came the cutter. We heard the scrape of a knife blade through rope.
The line came off in our hands.
"Happy landings, lads!"
In a flash the smack's steering oar bit deep into the swell, the two worthless brutes suddenly showed themselves anything but. Three swift heaves on the driving oars and the smack shot off like a sling bullet.
We were cast adrift in the middle of the channel.
The cutter came up, calling after the smack as she sped from sight. The Spartans still hadn't seen us. Alexandros clamped my arm. We must not sing out, that would mean dishonor.
"I agree. Drowning's a lot more honorable."
"Shut up."
We held silent, treading water while the cutter quartered the area, scanning for other craft that might be spies. Finally she showed her stern and rowed off. We were alone beneath the stars.
As vast as the sea can look from the deck of a ship, it looks even bigger from a single handbreadth above the surface.
"Which shore do we make for?"
Alexandros gave me a look as if I had lost my senses. Of course we would go forward.
We paddled for what seemed like hours. The shore had not crawled one spear-length closer. "What if the current's against us? For all we know we're stuck here in place, or even drifting backward."
"We're closer," Alexandros insisted.
"Your eyes must be better than mine."
There was nothing to do but paddle and pray. What monsters of the sea prowled at this moment beneath our feet, ready to snare our legs in their horrible coils, or shear us off at the kneecaps? I could hear Alexandros gulp water, fighting an asthmatic fit. We pulled closer together. Our eyes were gumming up from the salt; our arms felt like lead.
"Tell me a story," Alexandros said.
For a moment I feared he was going mad.
"To encourage each other. Keep our spirits up. Tell me a story."
I recited some verses from the _Iliad_ which Bruxieus had made Diomache and me commit to memory, our second summer in the hills. I was getting the hexameters out of order but Alexandros didn't care; the words seemed to fortify him greatly.
"Dienekes says the mind is like a house with many rooms," he said. "There are rooms one must not go into. To anticipate one's death is one of those rooms. We must not allow ourselves even to think it."
He instructed me to continue, selecting only verses of valor. He declared that we must under no circumstances give thought to failure. "I think the gods may have dropped us here on purpose. To teach us about those rooms."
We paddled on. Orion the Hunter had stood overhead when we began; now his arc descended, halfway down the sky. The shore stood as far off as ever.
"Do you know Agathe, Ariston's sister?" Alexandros asked out of nowhere. "I'm going to marry her. I've never told anyone that."
"Congratulations."
"You think I'm joking. But my thoughts have kept coming back to her for hours, or however long we've been out here." He was serious. "Do you think she'll have me?"
It made as much sense to debate this in the middle of the ocean as anything else. "Your family outranks hers. If your father asks, hers will have to say yes."
"I don't want her that way. You've watched her. Tell me the truth. Will she have me?"
I considered it. "She made you that amber charm. Her eyes never leave you when you sing. She comes out to the Big Ring with her sisters when we run. She pretends to be training, but she's really sneaking looks at you."
This seemed to cheer Alexandros mightily. "Let's make a push. Twenty minutes as strong as we can, and see how far we can get."
When we hit twenty, we decided to try for another.
"You have a girl you love too, don't you?" Alexandros asked as we paddled. "From your city. The girl you lived in the hills with, your cousin who went to Athens."
I said it was impossible that he could know all that.
He laughed. "I know everything. I hear it from the girls and the goat boys and from your helot friend Dekton." He said he wanted to know more about "this girl of yours."
I told him I wouldn't tell him.
"I can help you to see her. My great-uncle is _proxenos_ for Athens. He can have her found, and brought to the city if you wish."
The swells were getting bigger; a cold wind had gotten up. We were going nowhere. I supported Alexandros again as another choking fit attacked him. He stuck his thumb between his teeth and bit through the flesh till it bled. The pain seemed to steady him. "Dienekes says that warriors advancing into battle must speak steadily and calmly to each other, each man encouraging his mate. We have to keep talking, Xeo."
The mind plays tricks in conditions of such extremity. I cannot tell how much I spoke aloud to Alexandros over the succeeding hours and how much simply swam before memory's eye as we labored endlessly toward the shore that refused to come closer.
I know I told him of Bruxieus. If my knowledge of Homer were worthy, all credit lay with this fortune-cursed man, sightless as the poet himself, and his fierce will that I and my cousin not grow to adulthood wild and unlettered in the hills.
"This man was mentor to you," Alexandros pronounced gravely, "as Dienekes is to me." He wished to hear more. What was it like to lose mother and father, to watch your city burn? How long did you and your cousin remain in the hills? How did you get food, and how protect yourself from the elements and wild beasts?
In gulps and snatches, I told him.
**B** y our second summer in the mountains, Diomache and I had become such accomplished hunters that not only did we no longer need to descend to town or farm for food, we no longer wished to. We were happy in the hills. Our bodies were growing. We had meat, not once or twice a month or on festival occasions only, as in our fathers' houses, but every day, with every meal. Here was our secret. We had found dogs.
Two puppies to be exact, runts of a disowned litter. Arkadian shepherd's hounds we had discovered shivering and suckling-blind, abandoned by their mother, who had untimely given birth in midwinter. We named one Happy and the other Lucky, and they were. By spring both had legs to run, and by summer their instincts had made them hunters. With those dogs our hungry days were over. We could track and kill anything that breathed. We could sleep with both eyes closed and know that nothing could take us unawares. We became such a proficient hunting team, Dio and I and the hounds, that we actually passed up opportunities, came upon game and let it go with the benevolence of gods. We feasted like lords and viewed the sweating valley farmers and plodding highland goatherds with contempt.
Bruxieus began to fear for us. We were growing wild. Cityless. In evenings past, Bruxieus had recited Homer and made it a game how many verses we could repeat without a slip. Now this exercise took on a deadly earnestness for him. He was failing, we all knew it. He would not be with us much longer. Everything he knew, he must pass on.
Homer was our school, the _Iliad_ and _Odyssey_ the texts of our curriculum. Over and over Bruxieus had us recite the verses upon Odysseus' return, when, clad in rags and unrecognizable as the rightly lord of Ithaka, the hero of Troy seeks shelter at the hut of Eumaeus, the swineherd. Though Eumaeus has no idea that the traveler at his gate is his true king, and thinks him only another cityless beggar, yet out of respect to Zeus, who protects the wayfarer, he invites the wanderer kindly in and shares with him his humble fare.
This was humility, hospitality, graciousness toward the stranger; we must imbibe it, sink it deep within our bones. Bruxieus tutored us relentlessly in compassion, that virtue which he saw diminishing each day within our mountain-hardened hearts. We were made to recite the tent scene at the close of the _Iliad,_ when Priam of Troy kneels before Achilles to kiss in supplication the hand of the man who has slain his sons, including the mightiest and dearest to him, Hektor, hero and protector of Ilium. Then Bruxieus grilled us upon it. What would we have done were we Achilles? Were we Priam? Was each man's action proper and pious in the eyes of the gods?
We must have a city, Bruxieus declared.
Without a city we were no better than the wild brutes we hunted and killed.
Athens.
There, Bruxieus insisted, was where Dio and I must go. The city of Athena was the only truly open city in Hellas, her freest and most civilized. The love of wisdom, _philosophia,_ was esteemed in Athens beyond all other pursuits; the life of the mind was cultivated and honored, invigorated by a high culture of theater, music, poetry, architecture and the arts. Nor were the Athenians inferior to any city in Hellas in the practice of war.
The Athenians welcomed immigrants. A bright strong boy like me could take a trade, indenture himself in a shop. And Athens had a fleet. Even with my crippled hands I could pull an oar. With my skill with the bow I could become a _toxotes_ , a marine archer, distinguish myself in war and exploit that service to advance my position.
Athens, too, was where Diomache must go. As a well-spoken freeborn, and with her blooming beauty, she could find service in a respected house and attract no shortage of admirers. She was at just the right age for a bride; it was far from a stretch to imagine her securing betrothal to a citizen. As the wife even of a _metik,_ a resident alien, she could protect me, aid me in securing employment. And we would have each other.
As Bruxieus' strength diminished with the passing weeks, his conviction intensified that we follow his will in these matters. He made us swear that when his time came, we would go down from the hills and make for Attika, to the city of Athena.
In October of that second year Dio and I hunted one long cold-coming day and killed nothing. We tramped back into camp, grumbling at each other, anticipating a mean porridge of mixed pulse and mountain peas and, worse, the sight of Bruxieus, whose slackening constitution was each day becoming more painful to behold, maintaining that all was well with him; he did not need meat. We saw his smoke and watched the dogs bound up the hill as they loved to, sprinting to their friend to receive his hugs and homecoming roughhouse.
From the trail's turn below the camp we heard their barking. Not the usual squeals of play, but something keener, more insistent. Happy scrambled into view a hundred feet above us. Diomache looked at me and we both knew.
It took an hour to build Bruxieus' pyre. When his gaunt slave-branded body lay at last within the purifying flame, I lit a pitched arrow from the hollow above his heart and loosed it, flaming, with all my strength, arcing like a comet down the dark valley.
_...then aged Nestor, peerless in_
_wisdom among the flowing-haired Achaeans,_
_laid himself down in the fullness of years_
_and closed his eyes as if in sleep,_
_slain by Artemis' gentle darts._
Ten dawns later Diomache and I stood at the Three-Cornered Way, on the frontier of Attika and Megara, where the Athens road breaks off to the east, the Sacred Road to Delphi and the west and the Corinthian southwest, to the Isthmus and the Peloponnese. No doubt we looked like the most savage pair of ragamuffins, barefoot, faces scorched by the sun, our long hair tied in horsetails behind us. Both of us carried daggers and bows, and the dogs loped beside us, as burr-coated and filthy as we were.
Traffic lumbered through the Three Corners, the predawn vehicles, freighters and produce waggons, firewood haulers, farm urchins on their way to market with their cheeses and eggs and sacks of onions, just as Dio and I had started out for Astakos that morning that seemed so long ago and yet was only two winters by the calendar. We halted at the crossroads and asked directions. Yes, a teamster pointed, Athens was that way, two hours, no more.
My cousin and I had barely spoken on the weeklong tramp down from the mountains. We were thinking of cities and what our new life would be like. I watched the other travelers when they passed on the highway, how they eyed her. The need was on her to be a woman. "I want babies," she said out of the blue, the last day as we marched. "I want a husband to care for and to care for me. I want a home. I don't care how humble, just someplace I can have a little garden, put flowers on the sill and make it pretty for my husband and our children." This was her way of being kind to me, of drawing a distance beforehand, so I would have time to absorb it. "Can you understand, Xeo?"
I understood. "Which dog do you want?"
"Don't be cross with me. I'm only trying to tell you how things are, and how they must be."
We decided she would take Lucky, and I would keep Happy.
"We can stay together in the city," she thought out loud as we walked. "We'll tell the people we're brother and sister. But you must understand, Xeo, if I find a decent man, someone who will treat me with respect..."
"I understand. You can stop talking now."
Two days before, a gentlewoman of Athens had passed us on the highway, traveling by coach with her husband and a merry party of friends and servants. The lady had been taken by the sight of this wild girl, Diomache, and insisted upon having her serving women bathe and oil her and dress her hair. She wanted to do mine too, but I wouldn't let them near me. Their whole party stopped by a shaded stream and entertained themselves with cakes and wine while the maids took Dio away and groomed her. When my cousin emerged, I didn't recognize her. The Athenian lady was beside herself with delight; she couldn't stop praising Dio's charms, nor anticipating the stir her blossoming beauty would create among the young bloods of the city. The lady insisted that Dio and I proceed straight to her husband's home the moment we arrived in Athens; she would look to our employment and the continuation of our schooling. Her manservant would await us at the Thriasian Gates. Just ask anyone.
We tramped on, that last long day. On the freighters that passed now we could read the words "Phaleron" and "Athens" scrawled on the destination bands of serried wine jars and crated merchandise. Accents were becoming Attic. We stopped to watch a troop of Athenian cavalry, out on a lark. Four seamen marched past, heading for the city, each balancing his oar upon his shoulder and carrying his strap and cushion. That would be me before long.
Always in the hills Dio and I had slept in each other's arms, not as lovers, but for warmth. These final nights on the road, she wrapped herself in her own cloak and took her sleep apart. At last we arrived at dawn before the Three Corners. I had stopped and was watching a freight waggon pass. I could feel my cousin's eyes upon me.
"You're not coming, are you?"
I said nothing.
She knew which fork I would be taking.
"Bruxieus will be angry with you," she said.
Dio and I had learned, from the dogs and on the hunt, how to communicate with just a look. I told her good-bye with my eyes and begged her to understand. She would be well cared for in this city. Her life as a woman was just beginning.
"The Spartans will be cruel to you," Diomache said. The dogs paced impatiently at our feet. They did not yet know that they were parting too. Dio took my hands in both of hers. "And will we never sleep in each other's arms again, cousin?"
It must have seemed a queer spectacle to the teamsters and farm boys passing, the sight of these two wild children embracing upon the roadside, with their slung bows and daggers and their cloaks bound into traveler's rolls upon their backs.
Diomache took her road and I took mine. She was fifteen. I was twelve.
**H** ow much of this I imparted to Alexandros in those hours in the water, I cannot say. Dawn had still not shown her face when I finished. We were clinging to a miserable floating spar, barely big enough to support one, and too exhausted to swim another stroke. The water was getting colder. _Hypothermia_ gripped our limbs; I heard Alexandros cough and sputter, struggling for the strength to speak.
"We have to quit this spar. If we don't, we'll die."
My eyes strained toward the north. Peaks could be made out, but the shore itself remained invisible. Alexandros' cold hand clasped mine.
"Whatever happens," he swore, "I will not abandon you."
He let go of the spar. I followed.
An hour later we collapsed like Odysseus on a rock beach beneath a bawling rookery. We gulped fresh water from a cliff-wall spring, washed the salt from our hair and eyes and knelt in thanksgiving for our deliverance. For half the morning we slept like the dead. I climbed for eggs, which we wolfed raw from the shell, standing on the sand in the rags of our garments.
"Thank you, my friend," Alexandros said very quietly.
He extended his hand; I took it.
"Thank you too."
The sun stood near its zenith; our salt-stiff cloaks had dried upon our backs.
"Let's get moving," Alexandros said. "We've lost half a day."
**ELEVEN**
**T** he battle took place on a dusty plain to the west of the city of Antirhion, within bowshot of the beach and immediately beneath the citadel walls. A desultory stream, the Akanathus, meandered across the plain, bisecting it at the midpoint. Perpendicular to this watercourse, along the seaward flank, the Antirhionians had thrown up a crude battle wall. Rugged hills sealed the enemy's left. A portion of the plain adjacent the wall was occupied by a maritime junkyard; rotting craft lay littered at all angles, extending halfway across the field, amid tumbledown work shacks and stinking mounds of debris squalled over by wheeling flocks of gulls. In addition the enemy had strewn boulders and driftwood to break up the flat over which Leonidas and his men must advance. Their own side, the foe's, had been cleared smooth as a schoolmaster's desk.
When Alexandros and I scurried breathless and tardy upon the site, the Spartan Skiritai rangers had just finished setting the enemy refuse yards ablaze. The armies yet stood in formation, two-fifths of a mile apart, with the burning hulks between them. All native merchantmen and fishing craft had been withdrawn by the enemy, either hauled to safety within the fortified portion of the anchorage or standing offshore beyond the invaders' reach. This did not deter the Skiritai from torching the wharves and warehouses of the harbor. The timbers of the ship sheds the rangers had saturated with naphtha; already they blazed in ruins to the waterline. The defenders of Antirhion, as Leonidas and the Spartans well knew, were militiamen, farmers and potters and fishermen, summertime soldiers like my father. The devastation of their harbor was meant to unnerve them, to dislocate their faculties unaccustomed to such sights and sear into their unseasoned senses the stink and scourge of coming slaughter. It was morning, about market time, and the shore breeze had gotten up. Black smoke from the careened wrecks began to obscure the field; the pitch and encaustic of their timbers blazed with fury, abetted by the wind, which turned the debris-pile smudge burns into howling bonfires.
Alexandros and I had secured a vantage along the landward bluff, no more than a furlong above the site where the massed formations must clash. The smoke was already gagging us. We made our way across the slope. Others had claimed the site before us, boys and older men of Antirhion, armed with bows, slings and missile weapons they meant to hurl down upon the Spartans as they advanced, but these light-armed forces had been cleared early by the Skiritai, whose comrades below would advance as always from their position of honor on the Lakedaemonian left. The rangers took possession of half the face, driving the enemy skirmishers back where their slings and shafts were outranged and could work no harm to the army.
Directly beneath us, an eighth of a mile away, the Spartans and their allies were marshaling into their ranks. Squires armed the warriors from the feet up, starting with the heavy oxhide soles which could tread over fire; then the bronze greaves, which the squires bent into place around the shins of their masters, securing them at the rear of the calf by the flex of the metal alone. We could see Alexandros' father, Olympieus, and the white beard of his squire, Meriones.
The troops bound their private parts next, accompanied by obscene humor as each warrior mock-solemnly saluted his manhood and offered a prayer that he and it would still be acquainted when the day was over.
This process of arming for battle, which the citizen-soldiers of other _poleis_ had practiced no more than a dozen times a year in the spring and summer training, the Spartans had rehearsed and re-rehearsed, two hundred, four hundred, six hundred times each campaigning season. Men in their fifties had done this ten thousand times. It was as second-nature to them as oiling or dusting their limbs before wrestling or dressing their long hair, which they, fitted now with the linen _spolas_ corselet and bronze breastplate, proceeded to do with elaborate care and ceremony, assisting one another like a regiment of dandies preparing for a dress ball, all the while radiating an eerie presence of calm and nonchalance.
Finally the men scribed their names or signs upon _skytalides,_ the improvised twig bracelets they called "tickets," which would distinguish their bodies should they, falling, be maimed too hideously to be identified. They used wood because it was valueless as plunder by the enemy.
Behind the massing men, the omens were being taken. Shields, helmets and foot-long spearpoints had been burnished to a mirror's gleam; they flashed brilliantly in the sun, investing the massed formation with the appearance of some colossal milling machine, made not so much of men as of bronze and iron.
Now the Spartans and Tegeates advanced to their positions in the line. First the Skiritai, on the left, forty-eight shields across and eight deep; next the Selassian Stephanos, the Laurel regiment, eleven hundred _perioikic_ hoplites. To the right of these massed the six hundred heavy infantry of Tegea; then the _agema_ of the Knights in the line's center, Polynikes prominent among them, thirty shields across and five deep, to fight around and protect the person of the king. Right of these, dressing their line, moved into place the Wild Olive regiment, a hundred and forty-four across, with the Panther battalion adjacent the Knights, then the Huntress with Olympieus in the forerank, and the Menelaion. On their right, already to their marks, massed the battalions of the Herakles, another hundred and forty-four across, with Dienekes clearly visible at the head of his thirty-six-man _enomotia,_ dividing now into four nine-man files, or _stichoi,_ anchoring the right. The total, excluding armed squires ranging as auxiliaries, exceeded forty-five hundred and extended wing to wing across the plain for nearly six hundred meters.
From our vantage, Alexandros and I could see Dekton, as tall and muscular as any of the warriors, unarmored in his altar-boy white, leading two she-goats swiftly out to Leonidas, who stood garlanded with the battle priests before the formation in readiness for the sacrifice. Two goats were needed in case the first bled inpropitiously. The commanders' postures, like those of the massed warriors, projected an air of absolute insouciance.
Across from these the Antirhionians and their Syrakusan allies had massed in their numbers, the same width as the Spartans but six or more shields deeper. The scrapyard hulks had now burned down to ashy skeletons, spewing a blanket of smoke across the field. Beyond these, the stones of the harbor sizzled black in the water, while the spikes of burned-black wharf timbers protruded from the flotsam-choked surface like burial stones; a clotted ash-colored haze obscured what was left of the waterfront.
The wind bore the smoke upon the enemy, upon the massed individuals, the sinews of whose knees and shoulders shivered and quaked beneath the weight of their unaccustomed armor, while their hearts hammered in their breasts and the blood sang in their ears. It took no diviner's gift to discern their state of agitation. "Watch their spearpoints," Alexandros said, pointing to the massed foe as they jostled and jockeyed into their ranks. "See them tremble. Even the plumes on their helmets are quaking." I looked. In the Spartan line the iron-bladed forest of eight-footers rose solid as a spike fence, each shaft upright and aligned, dressed straight as a geometer's line and none moving. Across among the enemy, shafts wove and wobbled; all save the Syrakusans in the center were misaligned in rank and file. Some shafts actually clattered against their neighbors', chattering like teeth.
Alexandros was tallying the battalions in the Syrakusans' ranks. He made their total at twenty-four hundred shields, with twelve to fifteen hundred mercenaries and an additional three thousand citizen militiamen from the city of Antirhion herself. The enemy's numbers totaled half again that of the Spartans'. It was not enough and the foe knew it.
Now the clamor began.
Among the enemy's ranks, the bravest (or perhaps the most fear-stricken) began banging the ash of their spear shafts upon the bronze bowls of their shields, creating a tumult of _pseudoandreia_ which reverberated across and around the mountain-enclosed plain. Others reinforced this racket with the warlike thrusting of their spearpoints to heaven and the loosing of cries to the gods and shouts of threat and anger. The roar multiplied threefold, then five, and ten, as the enemy rear ranks and flankers picked the clamor up and contributed their own bluster and bronze-banging. Soon the entire fifty-four hundred were bellowing the war cry. Their commander thrust his spear forward and the mass surged behind him into the advance.
The Spartans had neither moved nor made a sound.
They waited patiently in their scarlet-cloaked ranks, neither grim nor rigid, but speaking quietly to each other words of encouragement and cheer, securing the final preparation for actions they had rehearsed hundreds of times in training and performed dozens and scores more in battle.
Here came the foe, picking up the pace of his advance. A fast walk. A swinging stride. The line was extending and fanning open to the right, "winging out" as men in fear edged into the shadow of the shield of the comrade on their right; already one could see the enemy ranks stagger and fall from alignment as the bravest surged forward and the hesitant shrank back.
Leonidas and the priests still stood exposed out front.
The shallow stream yet waited before the enemy. The foe's generals, expecting the Spartans to advance first, had formed their lines so that this watercourse stood midway between the armies. In the enemy's plan, no doubt, the sinuous defile of the river would disorder the Lakedaemonian ranks and render them vulnerable at the moment of attack. The Spartans, however, had outwaited them. As soon as the bronze-banging began, the enemy commanders knew they could not restrain their ranks longer; they must advance while their men's blood was up, or all fervor would dissipate and terror flood inevitably into the vacuum.
Now the river worked against the enemy. His foreranks descended into the defile, yet a quarter mile from the Spartans. Up they came, their already disordered dress and interval disintegrating further. They were again on the flat now, but with the river to their rear, the most perilous place it could be in the event of a rout.
Leonidas stood patiently watching, flanked by the battle priests and Dekton with his goats. The enemy was now a fifth of a mile off and accelerating the pace of his advance. The Spartans still hadn't moved. Dekton handed over the first she-goat's leash. We could see him glancing apprehensively as the plain began to thunder from the pounding of the enemy's feet and the air commenced to ring with their fear- and rage-inspired cries.
Leonidas performed the _sphagia,_ crying aloud to Artemis Huntress and the Muses, then piercing with his own sword the throat of the sacrificial goat whose haunches he pinned from behind with his knees, his left hand hauling the beast's jaw exposed as the blade thrust through its throat. No eye in the formation failed to see the blood gush and spill into Gaia, maternal earth, splattering as it fell Leonidas' bronze greaves and painting crimson his feet in their oxhide battle soles.
The king turned, with the life-fled victim yet clamped between his knees, to face the Skiritai, Spartiates, _perioikoi_ and Tegeates, who still held, patient and silent, in their massed ranks. He extended his sword, dark and dripping the blood of holy sacrifice, first heavenward toward the gods whose aid he now summoned, then around, toward the fast-advancing enemy.
"Zeus Savior and Eros!" his voice thundered, eclipsed but not unheard in that cacophonous din. "Lakedaemon!"
The _sarpinx_ sounded "Advance!," trumpeters sustaining the eardrum-numbing note ten paces after the men had stepped off, and now the pipers' wail cut through, shrill notes of their _auloi_ piercing the melee like the cry of a thousand Furies. Dekton heaved the butchered goat and the live one over his shoulders and scampered like hell for the safety of the ranks.
To the beat the Spartans and their allies advanced, eight-footers at the upright, their honed and polished spearpoints flashing in the sun. Now the foe broke into an all-out charge. Leonidas, displaying neither haste nor urgency, fell into step in his place in the front rank as it advanced to envelop him, with the Knights flowing impeccably into position upon his right and left.
Now from the Lakedaemonian ranks rose the _paean,_ the hymn to Castor ascending from four thousand throats. On the climactic beat of the second stanza,
_Heaven-shining brother_
_Skyborne hero_
the spears of the first three ranks snapped from the vertical into the attack.
Words cannot convey the impact of awe and terror produced upon the foe, any foe, by this seemingly uncomplex maneuver, called in Lakedaemon "spiking it" or "palming the pine," so simple to perform on the parade ground and so formidable under conditions of life and death. To behold it executed with such precision and fearlessness, no man surging forward out of control nor hanging back in dread, none edging right into the shadow of his rankmate's shield, but all holding solid and unbreakable, tight as the scales on a serpent's flank, the heart stopped in awe, the hair stood straight up upon the neck and shivers coursed powerfully the length of the spine.
As when some colossal beast, brought to bay by the hounds, wheels in his fury, bristling with rage and baring his fangs, and plants himself in the power and fearlessness of his strength, so did the bronze and crimson phalanx of the Lakedaemonians now snap as one into its mode of murder.
The left wing of the enemy, eighty across, collapsed even before the shields of their _promachoi,_ the front-rankers, had come within thirty paces of the Spartans. A cry of dread rose from the throats of the foe, so primal it froze the blood, and then was swallowed in the tumult.
The enemy left broke from within.
This wing, whose advancing breadth had stood an instant earlier at forty-eight shields, abruptly became thirty, then twenty, then ten as panic flared like a gale-driven fire from terror-stricken pockets within the massed formation. Those in the first three ranks who turned in flight now collided with their comrades advancing from the rear. Shield rim caught upon shield rim, spear shaft upon spear shaft; a massive tangle of flesh and bronze ensued as men bearing seventy pounds of shield and armor stumbled and fell, becoming obstacles and impediments to their own advancing comrades. You could see the brave men stride on in the advance, crying out in rage to their countrymen as these abandoned them. Those who still clung to courage pushed past those who had forsaken it, calling out in outrage and fury, trampling the forerankers, or else, as valor deserted them too, jerked free and fled to save their own skins.
At the height of the foe's confusion the Spartan right fell upon them. Now even the bravest of the enemy broke. Why should a man, however valorous, stand and die while right and left, fore and rear, his fellows deserted him? Shields were flung, spears cast wildly to the turf. Half a thousand men wheeled on their heels and stampeded in terror. At that instant the center and right of the enemy's line crashed shields-on into the central corps of the Spartans.
That sound which all warriors know but which to Alexandros' and my youthful ears had been heretofore unknown and unheard now ascended from the clash and collision of the _othismos._
Once, at home when I was a child, Bruxieus and I had helped our neighbor Pierion relocate three of his stacked wooden beehives. As we jockeyed the stack into place upon its new stand, someone's foot slipped. The stacked hives dropped. From within those stoppered confines yet clutched in our hands arose such an alarum, neither shriek nor cry, growl nor roar, but a thrum from the netherworld, a vibration of rage and murder that ascended not from brain or heart, but from the cells, the atoms of the massed _poleis_ within the hives.
This selfsame sound, multiplied a hundred-thousandfold, now rose from the massed compacted crush of men and armor roiling beneath us on the plain. Now I understood the poet's phrase the "mill of Ares" and apprehended in my flesh why the Spartans speak of war as work. I felt Alexandros' fingernails dig into the flesh of my arm.
"Can you see my father? Do you see Dienekes?"
Dienekes waded into the rout below us; we could see his cross-crested "curry brush" at the right of the Herakles, in the fore of the third platoon. As disordered as were the ranks of the enemy, so held the Spartans' intact and cohesive. Their forerank did not charge wildly upon the foe, flailing like savages, nor did they advance with the stolid precision of the parade ground. Rather they surged, in unison, like a line of warships on the ram. I had never appreciated how far beyond the interleaved bronze of the _promachoi_ 's shields the murderous iron of their eight-footers could extend. These punched and struck, overhand, driven by the full force of the right arm and shoulder, across the upper rim of the shield; not just the spears of the front-rankers but those of the second and even the third, extending over their mates' shoulders to form a thrashing engine that advanced like a wall of murder. As wolves in a pack take down the fleeing deer, so did the Spartan right fall upon the defenders of Antirhion, not in frenzied shrieking rage, lip-curled and fang-bared, but predator-like, cold-blooded, applying the steel with the wordless cohesion of the killing pack and the homicidal efficiency of the hunt.
Dienekes was turning them. Wheeling his platoon to take the enemy in flank. They were in the smoke now. It became impossible to see. Dust rose in such quantities beneath the churning feet of the men, commingling with the screen of smoke from the tindered hulks, that the entire plain seemed afire, and from the choking cloud arose that sound, that terrible indescribable sound. We could sense rather than see the Herakles _lochos,_ directly beneath us where the dust and smoke were thinner. They had routed the enemy left; their front ranks now surged into the business of cutting down those luckless bastards who had fallen or been trampled or whose panic-unstrung knees could not find strength to bear them swiftly enough from their own slaughter.
On the center and right, along the whole line the Spartans and Syrakusans clashed now shield-to-shield, helmet-to-helmet. Amid the maelstrom we could catch only glimpses, and those primarily of the rear-rankers, eight deep on the Lakedaemonian side, twelve and sixteen deep on the Syrakusan, as they thrust the three-foot-wide bowls of their _hoplon_ shields flush against the backs of the men in file before them and heaved and ground and shoved with all their strength, the soles of their footgear churning up trenches in the plain and slinging yet more dust into the already choking air.
No longer was it possible to distinguish individual men, or even units. We could see only the tidal surge and back-surge of the massed formations and hear without ceasing that terrible, blood-stilling sound.
As when a flood descends from the mountains and the wall of water crashes down the dry courses, smashing into the stone-founded stakes and woven brush of the husbandman's dam, so did the Spartan line surge against the massed weight of the Syrakusans. The dam's bulk, founded as firmly against the flood as fear and forethought may devise, seems itself to dig in and hold, to plant its force fiercely into the earth, and for long moments displays no sign of buckling. But then, as the anxious planter watches, before his eyes a surge begins to capsize one deep-sunken stake, another rush undermines a stacked stone revetment. Into each fraction of a breach, the force and weight of the downrushing wave thrusts itself irresistibly, hammering deeper, tearing and gouging, widening the gap and exploiting it with each successive ripping surge.
Now the dam wall which had cracked only a handbreadth splits to a foot and then a yard. The mass of the plunging flood builds upon itself, as ton upon ton plummets in from the courses above, adding its weight to the irresistible ever-mounting tide. Along the banked margins of the watercourse, sheets of earth calve into the churning, boiling torrent. So now did the Syrakusan center, pounded and hammered by the Tegeate heavy infantry, the king and the Knights and the massed battalions of the Wild Olive, begin to peel and founder.
The Skiritai had routed the enemy right. From the left the battalions of the Herakles rolled up the enemy flank. Each Syrakusan wingman forced to wheel to defend his unshielded side meant another drawn off from the forward push against the frontally advancing Spartans. The sound of the keening struggle seemed to rise for a moment, then went dead silent as desperate men summoned every reserve of valor from their shrieking, exhausted limbs. An eternity passed in the time it takes to draw a dozen breaths, and then, with the same sickening sound made by the mountain dam as it gives way unable to withstand the onrushing torrent, the Syrakusan line cracked and broke.
Now in the dust and fire of the plain the slaughter began.
A shout, half of joy and half of awe, sprung from the throats of the crimson-tunicked Spartans. Back the Syrakusan line fell, not in rout and riot as their allies the Antirhionians had done, but in still-disciplined squads and bunches, held yet by their officers, or whatever brave men had taken it upon themselves to act as officers, maintaining their shields to the fore and closing ranks as they retreated. It was no use. The Spartan front-rankers, men of the first five age-classes, were the cream of the city in foot speed and strength, none save the officers over twenty-five years old. Many, like Polynikes in the van among the Knights, were sprinters of Olympic and near-Olympic stature with garland after garland won in games before the gods.
These now, loosed by Leonidas and driven on by their own lust for glory, pressed home the sentence of steel upon the fleeing Syrakusans.
When the trumpeters had blown the _sarpinx_ and its mind-numbing wail sounded the call to still the slaughter, even the rawest untrained eye could read the field like a book.
There, on the Spartan right where the Herakles regiment had routed the Antirhionians, one saw the turf unchurned and the field beyond littered with enemy shields and helmets, spears and even breastplates, flung aside by the stampeding foe in his flight. Bodies lay scattered at intervals, facedown, with the shameful gashes of death delivered upon their fleeing backs.
On the right where the stronger troops of the enemy had held longer against the Skiritai, the carnage spread thicker and more dense, the turf chewed more fiercely; along the battle wall which the foe had erected to anchor its flank, clumps of corpses could be seen, slain as they, trapped by their own wall, had struggled in vain to scale it.
Then the eye found the center, where the slaughter had achieved its most savage concentration. Here the earth was rent and torn as if a thousand span of oxen had assaulted it all day with the might of their hooves and the steel of their ploughs' deep-churning blades. The chewed-up dirt, dark with piss and blood, extended in a line three hundred meters across and a hundred deep where the feet of the contending formations had heaved and strained for purchase upon the earth. Bodies sprawled like a carpet upon the earth, mounded in places two and three deep. To the rear, across the plain where the Syrakusans had fled, and along the riven walls of the watercourse, more corpses could be seen in scattered perimeters manned by two and three, five and seven, where these in their flight had closed ranks and made their stand, doomed as castles of sand against the tide. They fell with wounds of honor, facing their Spartan foe, cut down from the front.
A wail arose from the hillsides where the watching Antirhionian skirmishers now looked down upon their comrades' vanquishment, while from the walls of the citadel itself wives and daughters keened in grief as must have Hekube and Andromache upon the battlements of Ilium.
The Spartans were hauling bodies off the stacks of the dead, seeking friend or brother, wounded and clinging yet to life. As each groaning foeman was flung down, a _xiphos_ blade held him captive at the throat. "Hold!" Leonidas cried, motioning urgently to the trumpeters to resound the call to break off. "Attend them! Attend the enemy too!" he shouted, and the officers relayed the order up and down the line.
Alexandros and I, pounding pell-mell down the slope, had reached the plain now. We were on the field. I sprinted two strides behind as the boy ranged in mortal urgency among the blood- and gore-splattered warriors, whose flesh seemed yet to burn with the furnace heat of fury and whose breath appeared to our eyes to steam upon the air.
"Father!" Alexandros cried in the exigency of dread, and then, ahead, he glimpsed the cross-crested officer's helmet and then Olympieus himself, upright and unwounded. The expression of shock upon the _polemarch_ 's face was almost comical when he beheld his son sprinting toward him out of the carnage. Man and boy embraced with wide-flung arms. Alexandros' fingers searched his father's corselet and breastplate, probing to confirm that all four limbs stood intact and no unseen punctures yet leaked dark blood.
Dienekes emerged from the still-seething throng; Alexandros flew into his arms. "Are you all right? Did they wound you?" I raced up. Suicide stood there beside Dienekes, "darning needle" javelins in hand, his own face sprayed with the sling of enemy blood. A knot of staring men had clustered; I saw at their feet the torn and motionless form of Meriones, Olympieus' squire.
"What are you doing here?" Olympieus demanded of his son, his tone turning to anger as he realized the peril the boy had put himself in. "How did you get here?"
Around us other faces reacted with equal wrath. Olympieus swatted his son, hard, across the skull. Then the boy saw Meriones. With a cry of anguish he dropped to his knees in the dirt beside the fallen squire.
"We swam," I announced. A heavy fist cuffed me, then another and another.
"What is this to you, a lark? You come to sightsee?"
The men were furious, as well they should have been. Alexandros, unhearing in his concern for Meriones, knelt over the man, who lay upon his back with a warrior crouched at each side, his helmetless head pillowed upon a _hoplon_ shield and his bushy white beard clotted with blood, snot and sputum. Meriones, as a squire, had no cuirass to shield his breast; he had taken a Syrakusan eight-footer right through the bone of the chest. A seeping wound pooled blood into the bowl of his sternum; his tunic bunched up sodden with the dark, already clotting fluid; we could hear the hissing of air as his sucking lungs fought for breath and inhaled blood instead.
"What was he doing in the line?" Alexandros' voice, cracking with grief, demanded of the gathered warriors. "He's not supposed to be there!"
The boy barked for water. "Bearer!" he shouted, and shouted again. He tore his own tunic and, doubling the linen, pressed it as a dressing against his fallen friend's air-sucking chest. "Why don't you bind him?" his youth's voice cried to the encircled, gravely watching men. "He's dying! Can't you see he's dying?" He bellowed again for water, but none came. The men knew why, and now, watching, it became clear to Alexandros too, as it was already to Meriones.
"I've got one foot in the ferry, little old nephew," the ancient fighter's leaking air pipes managed to croak.
Life was ebbing fast from the warrior's eyes. He was, as I said, not a Spartan but a Potidaean, an officer in his own country, taken captive long years past and never permitted to see his home again. With an effort that was pitiful to behold, Meriones summoned strength to lift one hand, black with blood, and placed it gently upon the boy's. Their parts reversed, the dying man comforted the living youth.
"No happier death than this," his leaking lungs wheezed.
"You will go home," Alexandros vowed. "By all the gods, I will carry your bones myself."
Olympieus knelt now too, taking his squire's hand in his own. "Name your wish, old friend. The Spartans will bear you there."
The old man tried to speak but the pipes of his throat would not obey him. He struggled weakly to elevate his head; Alexandros restrained him, then gently cradled the veteran's neck and lifted it. Meriones' eyes glanced to the front and the sides where, amid the churned and liquid turf, the scarlet cloaks of other fallen warriors could be seen, each surrounded by a knot of comrades and brothers-in-arms. Then, with an effort which seemed to consume all his remaining substance, he spoke:
"Where these lie, plant me there. Here is my home. I ask none better."
Olympieus swore it. Alexandros, kissing Meriones' forehead, seconded the vow.
A dark peace seemed to settle upon the man's eyes. A moment passed. Then Alexandros lifted his own clear pure tenor in the Hero's Farewell:
_"That_ daimon _which God_
_breathed into me at birth_
_I with glad heart_
_return now to Him."_
In victory Dekton brought to Leonidas the rooster which would be sacrificed as thank-offering to Zeus and Nike. The boy himself was flushed with the triumph; his hands shook violently, wishing they had been permitted to hold a shield and spear and stand in the line of battle.
For my own part I could not stop staring about at the faces of the warriors I had known and watched in drill and training but until now had never looked upon in the blood and horror of battle. Their stature in my mind, already elevated beyond the men of any other city I had known, now rose close to that of heroes and demigods. I had witnessed the mere sight of them utterly rout the not-unvaliant Antirhionians, fighting before their own walls in defense of their homes and families, and overcome within minutes the crack troops of the Syrakusans and their mercenaries, trained and equipped by the tyrant Gelon's limitless gold.
Nowhere in all the field had these Spartans faltered. Now even in the hot blood aftermath their discipline maintained them chaste and noble, above all vaunting and boasting. They did not strip the bodies of the slain, as the soldiers of any other city would eagerly and gloatingly do, nor did they erect trophies of vainglory and conceit from the arms of the vanquished. Their austere thank-offering was a single cock, worth less than an obol, not because they disrespected the gods, but because they held them in awe and deemed it dishonorable to overexpress their mortal joy in this triumph that heaven had granted them.
I watched Dienekes, re-forming the ranks of his platoon, listing their losses and summoning aid for the wounded, the _traumatiai._ The Spartans have a term for that state of mind which must at all costs be shunned in battle. They all it _katalepsis,_ possession, meaning that derangement of the senses that comes when terror or anger usurps dominion of the mind.
This, I realized now watching Dienekes rally and tend to his men, was the role of the officer: to prevent those under his command, at all stages of battle—before, during and after—from becoming "possessed." To fire their valor when it flagged and rein in their fury when it threatened to take them out of hand. That was Dienekes' job. That was why he wore the transverse-crested helmet of an officer.
His was not, I could see now, the heroism of an Achilles. He was not a superman who waded invulnerably into the slaughter, single-handedly slaying the foe by myriads. He was just a man doing a job. A job whose primary attribute was self-restraint and self-composure, not for his own sake, but for those whom he led by his example. A job whose objective could be boiled down to the single understatement, as he did at the Hot Gates on the morning he died, of "performing the commonplace under uncommonplace conditions."
The men were collecting their "tickets" now. These, to which I alluded earlier, are the wooden-twig bracelets tied with twine which each man makes for himself before battle, to identify his corpse if necessary in the aftermath. A man writes or scratches his name twice, once on each end of the twig, then breaks it down the middle. The "blood half" he ties with string around his left wrist and wears with him into battle; the "wine half" stays behind in a basket maintained with the train in the rear. The halves are broken off jaggedly on purpose, so that even if the blood name were effaced or defiled in some other way, its twin would still fit in an unequivocally recognizable manner. When the battle is over, each man retrieves his ticket. Those remaining unclaimed in the basket number and identify the slain.
When the men heard their names called and came forward to take their tickets, they could not stop their limbs from quaking.
All up and down the line, one beheld warriors clustering in groups of twos and threes as the terror they had managed to hold at bay throughout the battle now slipped its bonds and surged upon them, overwhelming their hearts. Clasping their comrades by the hand, they knelt, not from reverence alone, though that element was abundant, but because the strength had suddenly fled from their knees, which could no longer support them. Many wept, others shuddered violently. This was not regarded as effeminate, but termed in the Doric idiom _hesma phobou,_ purging or "fear-shedding."
Leonidas strode among the men, letting all see that their king lived and moved unwounded. The men gulped greedily their ration of strong, heavy wine and made no shame to drink water as well and plenty of it. The wine went down fast and produced no effect whatever. Some of the men tried to dress their hair, as if thereby to induce a return to normalcy. But their hands trembled so badly they could not do it. Others would chuckle knowingly at the sight, the veteran warriors who knew better than to try; it was impossible to make the limbs behave, and the frustrated groomers would chuckle back, a dark laughter from hell.
When the tickets had all found their mates and been reclaimed by their owners, those pieces bereft within the basket identified the men who had been killed or were too badly wounded to come forward. These latter were claimed by brothers and friends, fathers and sons and lovers. Sometimes a man would take his own ticket, then another, and sometimes a third besides, weeping as he accepted them. Many returned to the basket, just to look in. In this way they could perceive the numbers of the lost.
This day it was twenty-eight.
His Majesty may set this number in comparison alongside the thousands slain in greater battles and perhaps judge it insignificant. But it seemed like decimation now.
There was a stir, and Leonidas emerged into view along the front of the assembled warriors. "Have you knelt?" He moved down the line, not declaiming like some proud monarch seeking satisfaction from the sound of his own voice, but speaking softly like a comrade, touching each man's elbow, embracing some, placing an arm around others, speaking to each warrior man-to-man, Peer-to-Peer, with no kingly condescension. Assemble, the word spread by murmur without needing to be spoken.
"Does every man have the halves of his ticket? Have your hands stopped shaking enough to fit them together?" He laughed and the men laughed with him. They loved him.
The victors formed up in no particular order, wounded and unwounded, plus squires and helots. They cleared a space for the king, those in front kneeling to allow their comrades behind to see and hear, while Leonidas himself strode informally up and down the line, presenting himself so that his voice would carry and his face be seen by all.
The battle priest, Olympieus in this case, held the basket up before the king. Leonidas took out each unclaimed ticket and read the name. He offered no eulogy. No word was spoken but the name. Among the Spartans, this alone is considered the purest form of consecration.
Alkamenes.
Damon.
Antalkides.
Lysandros.
On down the list.
The bodies, already retrieved by their squires from the field, would be cleansed and oiled; prayers would be offered and sacrifices made. Each of the fallen would be shrouded in his own cloak or that of a friend and interred here upon the site, beside his mates, beneath a mound of honor. Shield, sword, spear and armor alone would be borne home by his comrades, unless the omens declared it more honorable for his corpse to be restored and interred in Lakedaemon.
Leonidas now held up his own bracelet and slid the twin halves together into place. "Brothers and allies, I salute you. Gather, friends, and hear the words of my heart."
He paused for a moment, sober and solemn.
Then, when all stood silent, he spoke:
"When a man seats before his eyes the bronze face of his helmet and steps off from the line of departure, he divides himself, as he divides his 'ticket,' in two parts. One part he leaves behind. That part which takes delight in his children, which lifts his voice in the chorus, which clasps his wife to him in the sweet darkness of their bed.
"That half of him, the best part, a man sets aside and leaves behind. He banishes from his heart all feelings of tenderness and mercy, all compassion and kindness, all thought or concept of the enemy as a man, a human being like himself. He marches into battle bearing only the second portion of himself, the baser measure, that half which knows slaughter and butchery and turns the blind eye to quarter. He could not fight at all if he did not do this."
The men listened, silent and solemn. Leonidas at that time was fifty-five years old. He had fought in more than two score battles, since he was twenty; wounds as ancient as thirty years stood forth, lurid upon his shoulders and calves, on his neck and across his steel-colored beard.
"Then this man returns, alive, out of the slaughter. He hears his name called and comes forward to take his ticket. He reclaims that part of himself which he had earlier set aside.
"This is a holy moment. A sacramental moment. A moment in which a man feels the gods as close as his own breath.
"What unknowable mercy has spared us this day? What clemency of the divine has turned the enemy's spear one handbreadth from our throat and driven it fatally into the breast of the beloved comrade at our side? Why are we still here above the earth, we who are no better, no braver, who reverenced heaven no more than these our brothers whom the gods have dispatched to hell?
"When a man joins the two pieces of his ticket and sees them weld in union together, he feels that part of him, the part that knows love and mercy and compassion, come flooding back over him. This is what unstrings his knees.
"What else can a man feel at that moment than the most grave and profound thanksgiving to the gods who, for reasons unknowable, have spared his life this day? Tomorrow their whim may alter. Next week, next year. But this day the sun still shines upon him, he feels its warmth upon his shoulders, he beholds about him the faces of his comrades whom he loves and he rejoices in their deliverance and his own."
Leonidas paused now, in the center of the space left open for him by the troops.
"I have ordered pursuit of the foe ceased. I have commanded an end to the slaughter of these whom today we called our enemies. Let them return to their homes. Let them embrace their wives and children. Let them, like us, weep tears of salvation and burn thank-offerings to the gods.
"Let no one of us forget or misapprehend the reason we fought other Greeks here today. Not to conquer or enslave them, our brothers, but to make them allies against a greater enemy. By persuasion, we hoped. By coercion, in the event. But no matter, they are our allies now and we will treat them as such from this moment.
"The Persian!"
Suddenly Leonidas' voice rose, booming with such explosive emotion that those closest to him started from its sudden power. "The Persian is why we fought here today. His presence loomed, invisible, over the battlefield. He is why these tickets lie bereft in this basket. Why twenty-eight of the noblest men of the city will never again behold the beauty of her hills or dance again to her sweet music. I know many of you think I am half-cracked, I and Kleomenes the king before me." Laughter from the men. "I hear the whispers, and sometimes they're not such whispers." More laughter. "Leonidas hears voices the rest of us don't. He takes chances with his life in an unkingly manner and prepares for war against an enemy he has never seen and who many say will never come. All this is true..."
The men laughed again. "But hear this and never forget it: the Persian will come. He will come in numbers dwarfing those he sent four years ago when the Athenians and Plataeans defeated him so gloriously on the plain of Marathon. He will come tenfold, a hundredfold, mightier. And he will come soon."
Leonidas paused again, the heat in his breast making his face flush and his eyes burn with fever and conviction.
"Listen to me, brothers. The Persian is not a king as Kleomenes was to us or as I am to you now. He does not take his place with shield and spear amid the manslaughter, but looks on, safe, from a distance, atop a hill, upon a golden throne." Murmured jeers rose from the men's throats as Leonidas spoke this. "His comrades are not Peers and Equals, free to speak their minds before him without fear, but slaves and chattel. Each man, even the noblest, is deemed not an equal before God, but the King's property, counted no more than a goat or a pig, and driven into battle not by love of nation or liberty, but by the lash of other slaves' whips.
"This King has tasted defeat at the Hellenes' hands, and it is bitter to his vanity. He comes now to revenge himself, but he comes not as a man worthy of respect, but as a spoiled and petulant child, in its tantrum when a toy is snatched from it by a playmate. I spit on this King's crown. I wipe my ass on his throne, which is the seat of a slave and which seeks nothing more noble than to make all other men slaves.
"Everything I have done as king and everything Kleomenes has performed before me, every enemy courted, every confederation forged, every weak-kneed ally brought to heel, has been for this single event: the day when Darius, or one of his sons, returns to Hellas to pay us out."
Leonidas lifted now the basket which held the tickets of the fallen.
"That is why these, better men than ourselves, gave their lives here today, why they consecrated this earth with their heroes' blood. This is the meaning of their sacrifice. They have dumped their guts not in this piss-puddle war we fought today, but in the first of many battles in the greater war which God in heaven and all of you in your hearts know is coming. These brothers are heroes of that war, which will be the gravest and most calamitous in history.
"On that day," and Leonidas gestured out over the gulf, to Antirhion below and Rhion across the channel, "on that day when the Persian brings his multitudes against us via this strait, he will find not clear passage and paid-for friends, but enemies united and implacable, Hellene allies who will sally to meet him from both shores. And if he chooses some other route, if his spies report what awaits him here and he elects another passage, some other site of battle where land and sea play to our greater advantage, it will be because of what we did today, because of the sacrifice of these our brothers whose bodies we inter now within a hero's grave.
"Therefore I have not waited for the Syrakusans and the Antirhionians, our enemies this day, to send their heralds to us as is customary to entreat our permission to retrieve the bodies of their slain. I have dispatched our runners to them first, offering them truce without rancor, with generosity. Let our new allies reclaim unprofaned the armor of their fallen, let them recover undefiled the bodies of their husbands and sons.
"Let those we spared this day stand beside us in line of battle on that day when we teach the Persian once and for all what valor free men can bring to bear against slaves, no matter how vast their numbers or how fiercely they are driven on by their child-king's whip."
**TWELVE**
**A** _t this point in the recounting of the tale, an unfortunate incident occurred regarding the Greek Xeones. A subordinate of the Royal Surgeon, during the ongoing attendance upon the captive's wounds, unwittingly informed the fellow of the fate of Leonidas, the Spartan king and commander at Thermopylae, after the battle at the Hot Gates, and what sacrilege, to the Greek's eyes, His Majesty's troops had performed upon the corpse after it was recovered from the heaps of the dead following the slaughter. The prisoner had hitherto been in ignorance of this._
_The man's outrage was immediate and extreme. He forthwith refused to speak any further on the subjects to hand and in fact demanded of his immediate captors, Orontes and the officers of the Immortals, that they put him also to death, and at once. The man Xeones stood clearly in a state of extreme consternation over the beheading and crucifixion of the body of his king. All arguments, threats and blandishments failed to dislodge him from this posture of grief._
_It was clear to the captain Orontes that, should His Majesty be informed of the prisoner's defiance, however much He Himself desired to hear the continuance of the man's tale, the captive Xeones must, for his insolence to the Royal Person, be put to death. The captain, truth to tell, feared as well for his own head and those of his officers, should His Majesty be frustrated by the Greek's intransigence in His desire to learn all He could about the Spartan enemy._
_Orontes had become, through various informal exchanges with the fellow Xeones during the course of the interrogation, something of a confidant and even, if the word's meaning may be stretched to this point, a friend. He sought upon his own initiative to soften the captive's stance. To that end he attempted to make clear to the Greek the following:_
_That the physical desecration performed upon the corpse of Leonidas was regretted keenly by His Majesty almost as soon as He had ordered it._ _The actual command had been issued amid the grief of the battle's aftermath, when His Majesty's blood was raging over the loss before his own eyes of thousands, by some counts as many as twenty thousand, of the Empire's finest warriors slain by the troops of Leonidas, whose defiance of God Ahura Mazda's will could only be perceived through Persian eyes as an outrage against heaven. In addition two of His Majesty's own brothers, Habrocomes and Hyperanthes, and more than thirty royal kinsmen had been sent down to the house of death by the Spartan foe and their allies._
_Moreover, the captain appended, the mutilation of Leonidas' corpse was, when viewed in the apposite light, a testament to the respect and awe in which the Spartan king was held by His Majesty, for against no other commander of the enemy had He ever ordered such extreme and, to Hellenic eyes, barbarous retribution._
_The man Xeones remained unmoved by these arguments and repeated his desire to be dispatched at once. He refused all food and water. It seemed that the telling of his tale would be broken off here and not resumed._
_It was at this point, fearing that the situation could not be kept from His Majesty much longer, that Orontes sought out Demaratos, the deposed king of Sparta residing within the court as a guest exile and advisor, and urged his intercession. Demaratos, responding, betook himself in person to the Royal Surgeon's tent and there spoke alone with the captive Xeones for more than an hour. When he emerged, he informed the captain Orontes that the man had experienced a change of heart and was now willing to continue the interrogation._
_The crisis had passed. "Tell me," the captain Orontes inquired, much relieved, "what argument and persuasion did you employ to effect this turnabout?"_
_Demaratos replied that of all the Hellenes the Spartans were acknowledged the most pious and held the gods most in awe. He declared it his own observation that in this regard among the Lakedaemonians, the lesser rankers and those in service, particularly the outlanders of the captive Xeones' station, were almost without exception, in Demaratos' phrase, "more Spartan than the Spartans."_
_Demaratos had, he said, appealed to the fellow's respect for the gods,_ _specifically Phoebus Apollo, for whom the man clearly evinced the most profound reverence. He suggested that the prisoner pray and sacrifice to determine, as best he could, the god's will. For, he told the fellow, surely the Far Striker has assisted your tale thus far. Why would he now order its discontinuance? Did the man Xeones, Demaratos asked, place himself above the immortal gods, presuming to know their unknowable will and stopper their words at his own whim?_
_Whatever answer the captive received from his gods, it apparently coincided with the counsel proposed to him by Demaratos._
_We picked up the tale again on the fourteenth day of the month of Tashritu._
**P** olynikes was awarded the prize of valor for Antirhion.
This was his second, achieved at the unheard-of age of twenty-four years. No other Peer save Dienekes had been decorated twice, and that not until he was nearly forty. For his heroism Polynikes was appointed Captain of the Knights; it would be his honor to preside over the nomination of the Three Hundred king's companions for the following year. This supremely coveted distinction, coupled with his sprinter's crown from Olympia, established Polynikes as a beacon of fame whose brilliance shone forth far beyond the borders of Lakedaemon. He was perceived as a hero of all Hellas, a second Achilles, who stood now upon the threshold of unbounded and undying glory.
To Polynikes' credit, he refused to become puffed up over this. If any swelling of the head could be discerned, it manifested itself only in a more fiercely applied self-discipline, though this zeal for virtue, as events were to tell, could spill over into excess when applied to others less spectacularly gifted than himself.
As for Dienekes, he had only been honored with inclusion in the company of Knights once, when he was twenty-six, and had declined respectfully all subsequent nominations. He liked the obscurity of a platoon commander, he said. He felt more himself among the ranks. It was his conviction that he could contribute best by leading men directly, and that only to a certain number. He refused all attempts to promote him beyond the platoon level. "I can't count past thirty-six" was his standard disclaimer. "Beyond that, I get dizzy."
I will add, from my own observation, that Dienekes' gift and vocation, more so even than warrior and officer, was that of teacher.
As all born teachers, he was primarily a student.
He studied fear, and its opposite.
But to pursue such an excursus at this time would lead us astray from the narrative. To resume at Antirhion:
On the return passage to Lakedaemon, as punishment for accompanying Alexandros in following the army, I was removed from that youth's company and forced to march in the dust at the rear of the train, with the sacrificial herd and my half-helot friend Dekton. This Dekton had acquired at Antirhion a new nickname—Rooster—from the event that, immediately following the battle, he had delivered the thank-offering cock to Leonidas half-strangled in his own fists, so frenzied was he with excitement from the battle and his own frustrated desire to have participated in it. The name stuck. Dekton _was_ a rooster, bursting with barnyard belligerence and ready to scrap with anything, his own size or three times bigger. This new tag was picked up by the whole army, who began to regard the boy as something of a good-luck talisman, a mascot of victory.
This of course galled Dekton's pride beyond even its accustomed bellicose state. In his eyes the name embodied condescension, yet another reason to hate his masters and to despise his own position in their service. He declared me a blockhead for following the army.
"You should've flown," he hissed sidelong as we trudged in the choking flyblown wake of the train. "You deserve every lash you get, not for what they blame you for, but for not drowning that hymn-singer Alexandros when you had the chance—and churning your shanks straight to the temple of Poseidon." He meant that sanctuary in Tainaron to which runaways could flee and be granted asylum.
My loyalty to the Spartans was rebuked with scorn and ridicule by Dekton. I had been placed in this boy's power shortly after fate had brought me to Lakedaemon, two years earlier, when both he and I were twelve. His family worked the estate of Olympieus, Alexandros' father, who was related to Dienekes via his wife, Arete. Dekton himself was a half-breed helot, illegitimately sired, so rumor had it, by a Peer whose gravestone,
_Idotychides_
_in war at Mantinea_
lay along the Amyklaian Way, opposite the line of _syssitia,_ the common messes.
This half-Spartiate lineage did nothing to advance Dekton's status. He was a helot and that was it. If anything, the youths his age, and the Peers even more so, regarded him with extra suspicion, reinforced by the fact of Dekton's exceptional strength and athletic skill. At fourteen he was built like a grown man and nearly as strong.
He would have to be dealt with someday, and he knew it.
I myself had been in Lakedaemon half a year then, a wild boy just down from the hills and consigned, since it was safer than risking ritual pollution by killing me, to the meanest of farm labor. I proved such an infuriating failure at this that my helot masters took their complaints directly to their lord, Olympieus. This gentleman took pity on me, perhaps for my free birth, perhaps because I had come into the city's possession not as a captive, but of my own uncoerced will.
I was reassigned to the goat and kid detail.
I would be a herd boy for the sacrificial animals, minding the train of beasts that serviced the morning and evening ceremonies and followed the army into the field for training exercises.
The head boy was Dekton. He hated me from the first. He saved his most blistering scorn for my tale, imprudently confessed, of receiving counsel directly from Apollo Far Striker. Dekton thought this hilarious. Did I think, did I dream, did I imagine, that an Olympian god, scion of Zeus Thunderer, protector of Sparta and Amyklai, guardian of Delphi and Delos and who knows how many other _poleis,_ would piss away his valuable time swooping down to chat in the snow with a cityless _heliokekaumenos_ like me? In Dekton's eyes I was the dumbest mountain-mad yokel he had ever seen.
He appointed me the herd's Chief Ass Wiper. "You think I'm going to get my back striped for handing the king a shit-caked goat? Get in there, make that puckerhole spotless!"
Dekton never missed an occasion to humiliate me. "I'm educating you, Bung Boy. These assholes are your academy. Today's lesson is the same as yesterday's: In what does the life of a slave consist? It is in being debased and degraded and having no option but to endure it. Tell me, my freeborn friend. How do you like it?"
I would make no response, but simply obey. He scorned me the more for that.
"You hate me, don't you? You'd like nothing better than to chop me down. What's stopping you? Give it a try!" He stood before me one afternoon when we and the other boys were grazing the animals in the king's pasture. "You've lain awake planning it," Dekton taunted me. "You know just how you'd do it. With that Thessalian bow of yours, if your masters would let you near it. Or with that dagger you keep hidden between the boards in the barn. But you won't kill me. No matter how much disgrace I heap on your head, no matter how miserably I degrade you."
He picked up a rock and threw it at me, point-blank, striking me so hard in the chest it almost knocked me over. The other helot boys clustered to watch. "If it was fear that stopped you, I could respect that. It would at least show sense." Dekton slung another stone that struck me in the neck, drawing blood. "But your reason is more senseless than that. You won't harm me for the same reason you won't hurt one of these miserable, stinking beasts." With that, he kicked a goat furiously in the gut, bowling it over and sending it bawling. "Because it will offend _them._ " He gestured with bitter contempt across the plain to the gymnastic fields, where three platoons of Spartiates were going through spear drill in the sun. "You won't touch me because I'm their property, just like these shit-eating goats. I'm right, aren't I?"
My expression answered for me.
He glared at me with contempt. "What are they to you, moron? Your city was sacked, they say. You hate the Argives and think these sons of Herakles"—he indicated the drilling Peers, spitting the final phrase with sarcastic loathing—"are their enemies. Wake up! What do you think _they_ would have done had they sacked your city? The same and worse! As they did to my country, to Messenia and to me. Look at my face. Look at your own. You've fled slavery only to become lower than a slave yourself."
Dekton was the first person I had ever met, man or boy, who had absolutely no fear of the gods. He didn't hate them as some do, or mock their antics as I had heard the impious freethinkers did in Athens and Corinth. Dekton didn't grant their existence at all. There were no gods, it was as simple as that. This struck me with a kind of awe. I kept watch, waiting for him to be felled by some hideous blow of heaven.
Now, on the road home from Antirhion, Dekton (I should say Rooster) continued the harangue I had heard from him so many times before. That the Spartans had gulled me like they gull everyone; that they exploit their chattel by permitting them the crumbs off their table, elevating one slave a fraction above another and turning each individual's miserable hunger for station into the invisible bonds which held them in chains and in thrall.
"If you hate your masters so much," I asked him, "why were you hopping like a flea during the battle, so frantic to get into the fight yourself?"
Another factor, I knew, added to Rooster's frustration. He had just got his barnfriend (as the helot boys called their illicit wenches) pregnant. Soon he would be a father. How could he flee then? He would not abandon a child, nor could he make his getaway lugging a girl and a babe.
He stomped along, cursing one of the other herd boys who had let two goats stray, chasing the urchin back after these stragglers behind the herd. "Look at me," he growled as he fell again into step beside me. "I can run as fast as any of these Spartan dick-strokers. I'm fourteen but I'll fight any twenty-year-old man-to-man and bring him down. Yet here I trudge, in this fool's nightshirt, holding the leash on a goat."
He vowed he would steal a _xyele_ and cut a Spartan's throat one day.
I told him he must not speak like this in my hearing.
"What'll you do? Report me?"
I wouldn't and he knew it.
"But by the gods," I swore to him, "raise your hand once against them, any one of them, and I'll kill you."
Rooster laughed. "Pluck a sharp stick from the roadside and drive it into your sockets, my friend. It couldn't make you any blinder than you are already."
The army reached the frontier at Oion at nightfall of the second day, and Sparta herself twelve hours later. Runners had preceded the troops; the city had known for two days the identities of the wounded and the slain. Funeral games stood already in preparation; they would be celebrated within the fortnight.
That evening and the following day were consumed in decamping the battle train: cleaning and refitting weapons and armor, reshafting spears which had been shivered in combat and rewrighting the oaken hubs of the _hoplon_ shields, disassembling and storing the riggings of the waggons, tending to the pack and draught animals, making sure each beast was properly watered and groomed and dispersed with their helot teamsters to their various _kleroi,_ the farmsteads they worked. That second night, the Peers of the train at last returned to their messes.
This was customarily a solemn evening, in the aftermath of a battle, when fallen comrades were memorialized, acts of valor recognized and dishonorable conduct censured, when errors were reviewed and turned to instruction and the grave capital of battle stored up against future need.
The messes of the Peers are customarily havens of respite and confidentiality, sanctuaries within which all converse is privileged and private. Here after the long day friends may let down their hair among friends, speak as gentlemen the truths of their hearts and even, though never to excess, embrace the mellowing comfort of a bowl or two of wine.
This night, however, was not one for ease or conviviality. The souls of the twenty-eight perished hung heavily over the city. The secret shame of the warrior, the knowledge within his own heart that he could have done better, done more, done it more swiftly or with less self-preserving hesitation; this censure, always most pitiless when directed against oneself, gnawed unspoken and unrelieved at the men's guts. No decoration or prize of valor, not victory itself, could quell it entire.
"Well," Polynikes called the youth Alexandros forward and addressed him sternly, "how did you like it?"
He meant war.
To be there, to see it raw and entire.
The evening stood now well advanced. The hour of the _epaikla_ had expired, that second course of the meal at which game meat and wheaten bread may be contributed, and now the sixteen Peers of the Deukalion mess settled, hunger satisfied, upon their hardwood couches. Now the lads who stood-to the mess for their instruction might be summoned and roasted upon the griddle.
Alexandros was made to stand forth before his elders at the position of boy's attention, hands tucked from sight beneath the folds of his cloak, eyes glued to the floor as not yet worthy of rising to meet a Peer's full in the face.
"How did you enjoy the battle?" Polynikes queried.
"It made me sick," Alexandros replied.
Under the interrogation the boy confessed that he had been unable to sleep since, neither aboard ship nor on the march home. If he closed his eyes even for a moment, he declared, he saw again with undiminished horror the scenes of slaughter, particularly the death spasm of his friend Meriones. His compassion, he acknowledged, was elicited as much by those casualties of the enemy as for the fallen heroes of his own city. Pressed hard upon this point, the boy declared the slaughter of war "barbarous and unholy."
"Barbarous and unholy, is it?" responded Polynikes, darkening with anger.
The Peers in their messes are encouraged, when they deem it useful for the instruction of youth, to single out one lad, or even another Peer, and abuse him verbally in the most stern and pitiless fashion. This is called _arosis,_ harrowing. Its purpose, much like the physical beatings, is to inure the senses to insult, to harden the will against responding with rage and fear, the twin unmanning evils of which that state called _katalepsis,_ possession, is comprised. The prized response, the one the Peers look for, is humor. Deflect defamation with a joke, the coarser the better. Laugh in its face. A mind which can maintain its lightness will not come undone in war.
But Alexandros possessed no gift for the wisecrack. It wasn't in him. All he could do was answer in his clear pure voice with the most excruciating candor. I watched him from my service station at the left of the mess entrance, beneath the carven plaque—
_Exo tes thyras ouden,_
"Out this door nothing"—meaning no word spoken within these precincts may be repeated elsewhere.
It was a form of high courage which Alexandros displayed, to stand up to the Peers' hammering without a joke or a lie. At any time during a harrowing, the object boy may signal and call a stop. This is his right under the laws of Lykurgus. Pride, however, prevented Alexandros from exercising this option, and everyone knew it.
You wanted to see war, Polynikes began. What did you imagine it would be?
Alexandros was required to answer in the Spartan style, at once, with extreme brevity.
Your eyes were horror-stricken, your heart aggrieved at the sight of the manslaughter. Answer this:
What did you think a spear was for?
A shield?
A _xiphos_ sword?
Questions of this kind would be put to the boy not in a harsh or abusive tone, which would have been easier to bear, but coldly, rationally, demanding a concisely expressed reasoned response. Alexandros was made to describe the wounds an eight-footer could produce and the types of deaths that would ensue. Should an overhand thrust be aimed at the throat or the chest? If the tendon of a foeman's calf be severed, should you pause to finish him off or press forward with the advance? If you plunge a spear into the groin above a man's testicles, should it be pulled straight out or ripped upward, blade vertical, to eviscerate the man's bowels? Alexandros' face flushed, his voice quaked and broke. Would you like to stop, boy? Is this instruction too much for you?
Answer concisely:
Can you envision a world without war?
Can you imagine clemency from an enemy?
Describe the condition of Lakedaemon without her army, without her warriors, to defend her.
Which is better, victory or defeat?
To rule or be ruled?
To make a widow of the enemy's wife or to have one's own wife widowed?
What is the supreme virtue of a man? Why? Whom of all in the city do you admire most? Why?
Define the word "mercy." Define "compassion." Are these the virtues of war or of peace? Of men or of women? Are they virtues at all?
Of the Peers who harrowed Alexandros this evening, Polynikes did not on the surface seem the most relentless or display the harshest severity. He did not lead the _arosis,_ nor was his interrogation overtly cruel or malicious. He just wouldn't let it stop. In the tone of the other men's voices, no matter how ruthlessly they grilled Alexandros, resided at bottom the unspoken fundament of inclusion. Alexandros was of their blood, he was one of them; everything they did tonight and every other night was not to break his spirit or crush him like a slave, but to make him stronger, to temper his will and render him more worthy of being called warrior, as they were, of taking his place as a Spartiate and a Peer.
Polynikes' harrowing was different. There was something personal about it. He hated the boy, though it was impossible to guess why. What made it even more painful, to watch as it must have been to endure, was Polynikes' supreme physical beauty.
In every aspect of his person, face as well as physique, the Knight was formed as flawlessly as a god. Naked in the _Gymnasion,_ even alongside scores of youths and warriors blessed in comeliness and elevated by their training to the peak of condition, Polynikes stood out, without equal, surpassing all others in symmetry of form and faultlessness of physical structure. Clothed in white robes for the Assembly, he shone like Adonis. And armed for war, with the bronze of his shield burnished, his scarlet cloak across his shoulders and the horsehair-crested helmet of a Knight pushed back upon his brow, he shone forth, peerless as Achilles.
To watch Polynikes train on the Big Ring, in preparation for the Games at Olympia or Delphi or Nemea, to behold him in the pastel light of day's end when he and the other sprinters had finished their distance work and now, under the eyes of their trainers, donned their racing armor for the final dressed sprints, even the most hardened Peers, training in the boxing oval or the wrestling pits, would pull up from their regimens and watch.
Four runners regularly trained with Polynikes: two brothers, Malineus and Gorgone, both victors at Nemea in the _diaulos_ sprint; Doreion the Knight, who could outrun a racehorse over sixty meters; and Telamonias the boxer and _enomotarch_ of the Wild Olive regiment.
The five would take their marks and a trainer would clap the start. For thirty meters, sometimes as long as fifty, the elite field remained a pack of straining bronze and flesh, laboring beneath the weight of their harness, and for a span of heartbeats the watching Peers would think, maybe this once, maybe this singular time, one will best him. Then from the fore, as the runners' accelerating power began to break the bonds of their burdens, Polynikes' churning shield would emerge, twenty pounds of oak and bronze sustained upon the pumping flesh and sinew of his left forearm; you saw his helmet flash; his polished greaves extended next, flying like the winged sandals of Hermes himself, and then, with a force and power so magnificent they stopped the heart, Polynikes would catapult out of the pack, blazing with such impossible swiftness that he seemed to be naked, even winged, and not belabored by the poundage upon his arm and across his back. Around the turning pole he flew. Daylight burst between him and his pursuers. He vaulted forward to the finish, four hundred meters total, no longer in his mind competing with these lesser fellows, these pedestrian mortals, any one of whom in another city would have been the object of adoration, mobbed by throngs of admirers, but who here, against this invincible runner, were doomed to eat dust and like it. This was Polynikes. No one could touch him. He possessed in every pore those blessings of feature and physique which the gods allow to combine in a single mortal only once in a generation.
Alexandros was beautiful too. Even with the broken nose Polynikes had gifted him with, his physical perfection approached that of the peerless runner. Perhaps this, in some way, lay at the root of the hatred the man felt for the boy. That he, Alexandros, whose joy lay in the chorus and not on the athletic field, was unworthy of this gift of beauty; that it, in him, failed to reflect the manly virtue, the _andreia,_ which it in Polynikes so infallibly proclaimed.
My own suspicion was that the runner's animus was inflamed further by the favor Alexandros had found in Dienekes' eyes. For of all the men in the city with whom Polynikes competed in virtue and excellence, he resented most my master. Not so much for the honors Dienekes had been granted by his peers in battle, for Polynikes, like my master, had been awarded the prize of valor twice, and he was ten or twelve years younger.
It was something else, some less obvious aspect of character which Dienekes possessed and which the city honored him by recognizing, instinctively, without prompting or ceremony. Polynikes saw it in the way the young boys and girls joked with Dienekes when he passed their _sphairopaedia,_ the ball-playing fields, during the noonday break. He caught it in the tilt of a smile from a matron and her maids at the springs or an old woman passing in the square. Even the helots granted my master a fondness and respect that were withheld from Polynikes, for all the heaps of honors that were his in other quarters. It galled him. Mystified him. He, Polynikes, had even produced two sons, while Dienekes' issue were all female, four daughters who, unless Arete could produce a son, would extinguish his line altogether, while Polynikes' strapping swift lads would one day be warriors and men. That Dienekes wore the respect of the city so lightly and with such self-effacing wit was even more bitter to Polynikes.
For the runner saw in Dienekes neither beauty of form nor fleetness of foot. Instead he perceived a quality of mind, a power of self-possession, which he himself, for all the gifts the gods had lavished upon him, could not call his own. Polynikes' courage was that of a lion or an eagle, something in the blood and the marrow, which summoned itself out of its own preeminence, without thought, and gloried in its instinctual supremacy.
Dienekes' courage was different. His was the virtue of a man, a fallible mortal, who brought valor forth out of the understanding of his heart, by the force of some inner integrity which was unknown to Polynikes. Was this why he hated Alexandros? Was it why he had splintered the boy's nose that evening of the eight-nighter? Polynikes sought to break more than the youth's face now. Here in the mess he wanted to crack him, to see him come apart.
"You look unhappy, _pais._ As if the prospect of battle held for you no promise of joy."
Polynikes ordered Alexandros to recite the pleasures of war, to which the boy responded by rote, citing the satisfactions of shared hardship, of triumph over adversity, of camaraderie and _philadelphia,_ love of one's comrades-in-arms.
Polynikes frowned. "Do you feel pleasure when you sing, boy?"
"Yes, lord."
"And when you flirt around with that trollop Agathe?"
"Yes, lord."
"Then imagine the pleasure that awaits you, when you clash in line of battle, shield-to-shield with an enemy burning to kill you, and you instead slay him. Can you imagine that ecstasy, you little shitworm?"
"The _pais_ is trying, lord."
"Let me assist you. Close your eyes and picture it. Obey me!"
Polynikes was keenly aware of the torment this was causing Dienekes, who held himself controlled and impassive upon his bare couch, just two places down.
"To plunge a spear, blade-deep, into a man's guts is like fucking, only better. You like to fuck, don't you?"
"The boy doesn't know, lord."
"Don't toy with me, you twittering sparrow."
Alexandros, on his feet for an hour by this time, had steeled himself utterly. He answered his tormentor's questions, frozen at attention, eyes riveted to the dirt, ready in his guts to endure anything.
"Killing a man is like fucking, boy, only instead of giving life you take it. You experience the ecstasy of penetration as your warhead enters the enemy's belly and the shaft follows. You see the whites of his eyes roll inside the sockets of his helmet. You feel his knees give way beneath him and the weight of his faltering flesh draw down the point of your spear. Are you picturing this?"
"Yes, lord."
"Is your dick hard yet?"
"No, lord."
"What? You've got your spear in a man's guts and your dog isn't stiff? What are you, a woman?"
At this point the Peers of the mess began rapping their knuckles upon the hardwood, an indication that Polynikes' instruction was going too far. The runner ignored this.
"Now picture with me, boy. You feel the foe's beating heart upon your iron and you rip it forth, twisting as you pull. A sensation of joy surges up the ash of your spear, through your hand and along your arm up into your heart. Are you enjoying this yet?"
"No, lord."
"You feel like God at that moment, exercising the right only He and the warrior in combat may experience: that of dealing death, of loosing another man's soul and sending it down to hell. You want to savor it, to twist the blade deeper and pull the man's heart and guts out upon the iron point of your spear, but you can't. Tell me why."
"Because I must move on and slay the next man."
"Are you going to weep now?"
"No, lord."
"What will you do when the Persians come?"
"Slay them, lord."
"What if you stand on my right in line of battle? Will your shield protect me?"
"Yes, lord."
"What if I advance, defended by the shadow of your shield? Will you hold it high at port before me?"
"Yes, lord."
"Will you bring down your man?"
"I will."
"And the next?"
"Yes."
"I don't believe you."
At this the Peers rapped more vigorously with their knuckles upon the tables. Dienekes spoke. "This is no longer instruction, Polynikes. This is malice."
"Is it?" the runner answered, not deigning to look in the direction of his rival. "We'll inquire of its object. Have you had enough, you psalm-singing wad of shit?"
"No, lord. The boy begs the Peer to continue."
Dienekes stepped in. Gently, with compassion, he addressed the youth, his protege. "Why do you tell the truth, Alexandros? You could lie, like every other boy, and swear you reveled in the witnessing of slaughter, you savored the sight of limbs cleaved and men maimed and murdered within the jaws of war."
"I thought of that, lord. But the company would see through me."
"You're fucking right we would," confirmed Polynikes. He heard the anger in his own voice and brought it swiftly under control. "However, out of deference to my esteemed comrade"—here he turned with a mock-courteous bow to Dienekes—"I will address my next question not to this child, but to the mess as a whole." He paused, then indicated the boy at attention before them. "Who will stand with this woman on his right in the line of battle?"
"I will," Dienekes answered without hesitation.
Polynikes snorted.
"Your mentor seeks to shield you, _paidarion._ In the pride of his own prowess he imagines he may fight for two. This is recklessness. The city cannot risk his loss, because he has eyes for the comeliness of your girlish face."
"Enough, my friend." This from Medon, senior of the mess. The Peers seconded with a chorus of knuckle raps.
Polynikes smiled. "I accede to your chastisement, gentlemen and elders. Please excuse my excess of zeal. I seek only to impart to our youthful comrade some insight into the nature of reality, the state of man as the gods have made him. May I conclude his instruction?"
"With brevity," Medon admonished.
Polynikes turned again to Alexandros. When he resumed now, his voice was gentle and without malice; if anything it seemed informed with something not unlike kindness and even, odd as it sounds, sorrow.
"Mankind as it is constituted," Polynikes said, "is a boil and a canker. Observe the specimens in any nation other than Lakedaemon. Man is weak, greedy, craven, lustful, prey to every species of vice and depravity. He will lie, steal, cheat, murder, melt down the very statues of the gods and coin their gold as money for whores. This is man. This is his nature, as all the poets attest.
"Fortunately God in his mercy has provided a counterpoise to our species' innate depravity. That gift, my young friend, is war.
"War, not peace, produces virtue. War, not peace, purges vice. War, and preparation for war, call forth all that is noble and honorable in a man. It unites him with his brothers and binds them in selfless love, eradicating in the crucible of necessity all which is base and ignoble. There in the holy mill of murder the meanest of men may seek and find that part of himself, concealed beneath the corrupt, which shines forth brilliant and virtuous, worthy of honor before the gods. Do not despise war, my young friend, nor delude yourself that mercy and compassion are virtues superior to _andreia,_ to manly valor." He finished, turning to Medon and the elders. "Forgive me for waxing long-winded."
The harrowing ended; the Peers dispersed. Outside beneath the oaks, Dienekes sought out Polynikes, addressing him by his praise-name Kallistos, which may be defined as "harmoniously beautiful" or "of perfect symmetry," though in the tone Dienekes employed, it expressed itself in the converse, as "pretty boy" or "angel face."
"Why do you hate this youth so much?" Dienekes demanded.
The runner replied without hesitation. "Because he does not love glory."
"And is love of glory the supreme virtue of a man?"
"Of a warrior."
"And of a racehorse and a hunting dog."
"It is the virtue of the gods, which they command us to emulate."
The others of the mess could overhear this exchange, though they affected not to, since, under the laws of Lykurgus, no matter discussed behind those doors may be carried over to these more public precincts. Dienekes, realizing this as well, brought himself under control and faced the Olympian Polynikes with an expression of wry amusement.
"My wish for you, Kallistos, is that you survive as many battles in the flesh as you have already fought in your imagination. Perhaps then you will acquire the humility of a man and bear yourself no longer as the demigod you presume yourself to be."
"Spare your concern for me, Dienekes, and save it for your boy friend. He has greater need of it."
That hour had arrived when the messes along the Amyklaian Way released their men, those over thirty to depart for their homes and wives, and the younger men, of the first five age-classes, to retire under arms to the porticoes of the public buildings, there to stand the night watches over the city or curl in their cloaks for sleep. Dienekes took these last moments to speak apart with Alexandros.
The man placed an arm about the boy's shoulder; they moved slowly together beneath the unlit oaks. "You know," Dienekes said, "that Polynikes would give his life for you in battle. If you fell wounded, his shield would preserve you, his spear would bring you safely back. And if death's blow did find you, he would swim without hesitation into the manslaughter and spend his last breath to retrieve your body and keep the enemy from stripping your armor. His words may be cruel, Alexandros, but you have seen war now and you know it is a hundred times crueler.
"Tonight was a lark. It was practice. Prepare your mind to endure its like again and again, until it is nothing to you, until you can laugh in Polynikes' face and return his insults with a carefree heart.
"Remember that boys of Lakedaemon have endured these harrowings for hundreds of years. We spend tears now that we may conserve blood later. Polynikes was not seeking to harm you tonight. He was trying to teach that discipline of mind which will block out fear when the trumpets sound and the battle pipers mark the beat.
"Remember what I told you about the house with many rooms. There are rooms we must not enter. Anger. Fear. Any passion which leads the mind toward that 'possession' which undoes men in war.
"Habit will be your champion. When you train the mind to think one way and one way only, when you refuse to allow it to think in another, that will produce great strength in battle."
They stopped beneath an oak and sat.
"Did I ever tell you about the goose we had on my father's _kleros?_ This bird had formed a habit, God knows why, of pecking three times at a certain patch of turf before she waddled into the water with her brothers and sisters. When I was a boy, I used to marvel at this. The goose did it every time. It was compelled to.
"One day I got it into my head to prevent her. Just to see what she would do. I took up a station on that patch of superstitious turf—I was no more than four or five years old at the time—and refused to let that goose come near it. She became frantic. She rushed at me and beat me with her wings, pecking me bloody. I fled like a rat. At once the goose recovered composure. She pecked her little spot of turf three times and slid into the water, contented as could be."
The older Peers were departing now for their homes, the younger men and boys returning to their stations.
"Habit is a mighty ally, my young friend. The habit of fear and anger, or the habit of self-composure and courage." He rapped the boy warmly upon the shoulder; they both stood.
"Go now. Get some sleep. I promise you, before you see battle again, we'll arm you with all the handiest habits."
**THIRTEEN**
**W** hen the youths began dispersing to their stations, Dienekes with his squire, Suicide, moved out to the road, joining a company of other officers assembling to proceed to the _ekklesia,_ where they were to assist in the organization of the coming funeral games. A helot boy approached Dienekes there, before the mess, dashing up with a message. I was on the point of departing with Alexandros for the open porches around the Square of Freedom to take up my berth for the night when a sharp whistle summoned me.
To my astonishment it was Dienekes.
I crossed to him swiftly, presenting myself respectfully upon his left, his shield side. "Are you acquainted with the location of my house?" he asked. These were the first words he had ever addressed directly to me. I replied that I did. "Go there now. This boy will lead you."
Dienekes said nothing more but turned and departed at once with the body of officers toward the Assembly. I had no idea what was required of me. I asked the boy if perhaps there was some mistake, was he sure it was I who was required? "It's you, all right, and we'd better make the pebbles fly."
The town house of Dienekes' family, in contradistinction to the farmstead their helot families worked three miles south along the Eurotas, stood two lanes off the Eventide Road, on the west end of the village of Pitana. It was not conjoined to other dwellings, as many in that quarter were, but isolated at the edge of a grove beneath ancient oaks and olives. It had itself been a farmhouse at some point in the past and possessed yet the unadorned utilitarian charm of a country _kleros._ The house itself was unassuming in the extreme, barely larger than a cottage, less prepossessing even than the house of my own father in Astakos, though its courtyard and grounds, nestled within a grove of myrtle and hyacinth, arose like a haven of refuge and charm. One arrived upon the site at the terminus of a series of flower-girt lanes, each seeming to draw one deeper into a space of serenity and seclusion, passing, as one went, the dappled clusters of other Peers' cottages, their hearths aglow in the evening chill, with the peal of children's laughter and the happy yapping of their hounds spilling over the founded walls. The site itself, and its bowered environs, could not have appeared farther removed from the precincts of training and of war, nor offered more contrast and comfort to those repairing from them.
Dienekes' eldest daughter, Eleiria, who was eleven at the time, let me in the gate. I perceived low white walls surrounding an immaculately swept courtyard of plain tile brick, decorated with flowers in earthen pots upon the sill. Jasmine bloomed along the unvarnished beams of an axe-hewn pergola; wisteria and oleander nestled trim upon the face; a stonework watercourse, no wider than a handbreadth, gurgled along the northern wall. A servant girl whom I did not recognize waited beside a plaited wicker garden seat in the shadows.
I was directed to a stone bowl and told to rinse my hands and feet. Several clean linen cloths hung upon a bar; I dried myself and rehung them scrupulously. My heart was hammering, though for the life of me I could not have said why. The maiden Eleiria ushered me inside to the hearth hall, the solitary room, other than Dienekes' and his wife Arete's bedchamber, of which the house was comprised.
All four of Dienekes' daughters were present, including a slumbering toddler and a newborn; the second-eldest, Alexa, now being joined by her sister, both of whom sat to the side and proceeded to card wool as if it were the normal activity for the middle of the night. These maidens were presided over by the lady Arete, who sat with the infant at her breast upon a low uncushioned stool adjacent the hearth.
I discerned at once, however, that it was not Dienekes' lady upon whom I was to attend. Instead, at her side, and more toward the meridian of the room, sat the lady Paraleia, Alexandros' mother, the wife of the _polemarch_ Olympieus.
This mistress began without ceremony to interrogate me on the harrowing her son had received not half an hour earlier in the mess. That she knew of this event at all, and so immediately, was surprise enough. Something in her eyes warned me I must choose my words with care.
The lady Paraleia declared that she was keenly cognizant of and held in profoundest respect the proscription against revealing any exchange spoken within the precincts of a Peers' mess. Nonetheless I might, without violating the sanctity of the law, yet vouchsafe to her, a mother understandably concerned about her son's welfare and future, some indication, if not of the precise words and actions of the aforesaid event, then perhaps some portion of its tone and flavor.
She inquired by way of motivation, in the identical understated tone with which the Peers of the mess had interrogated Alexandros, who it was who governed the city. The kings and the ephors, I replied at once, and of course the Laws. The lady smiled and glanced, just for a moment, toward the mistress Arete.
"Yes," she said. "Surely this must be so."
This was her way of letting me know that the women ran the show and that if I didn't want to find myself permanently back in the farmers' shitfields, I'd better start coughing up a satisfactory dose of information. Within ten minutes she had gotten everything there was to get. I sang like a bird.
She wished, the lady Paraleia began, to know everything her son had done in the hours after he had defied her wishes in the grove of the Twins and set off to follow the army to Antirhion. She grilled me as if I were a spy. The lady Arete did not interrupt. Her eldest daughters never lifted their eyes toward me nor toward the lady Paraleia, yet they remained in their modest silence riveted to every word. This was how they learned. The lesson today was how to grill a boy in service. How a lady did it. What tone she took, what questions she asked, when her voice rose with a hint of threat and when it lowered to assume a more confidential, candor-evoking tone.
What rations had Alexandros and I taken? What arms? When our food ran out, how had we acquired more? Did we encounter strangers along the way? How did her son comport himself? How did the strangers respond? Did they show him respect worthy of a Spartan? Did her son's demeanor command it?
The lady assimilated my responses, revealing nothing herself, though it was plain at certain junctures that she disapproved of her son's conduct. Only once did she permit actual anger to invest her tone, that when I acknowledged under compulsion that Alexandros had not secured the name of the boat captain who had ferried and betrayed us. The lady's voice shook. What was wrong with the boy? What had he learned all these years at his father's table and in the common mess? Didn't he see that this reptile, this fisher captain, must be punished, executed if necessary, to teach these scoundrels the price of playing perfidy with the son of a Peer of Lakedaemon? Or if prudence dictated, that he, this boatman, could be exploited to advantage? If war with the Persian came, this blackguard, turned informer, could prove an invaluable source of intelligence for the army. Even if he attempted through falsehood to play the traitor, this could be discerned and valuable knowledge acquired. Why didn't my son find out his name?
"Your servant does not know, lady. Perhaps your son did and his servant was unaware of it."
"Call yourself 'I,'" Paraleia scolded me sharply. "You're not a slave, don't talk like one."
"Yes, lady."
"The boy needs something to wet his throat, Mother." This from the maiden Eleiria, with a giggle. "Look at him. If his face gets any redder, he'll burst like a tomato."
The grilling went on for another hour. Adding to the discomfort I felt on this hot seat was the effect of the lady Paraleia's physical appearance, which bore an uncanny resemblance to that of her son. Like him, the lady was beautiful, and like him, her beauty took the unadorned, underplayed Spartan form.
The wives and maidens of my native Astakos, and those of every other city in Hellas, routinely employ cosmetics and facial paint to enhance their comeliness. These ladies are keenly aware of the effect the artificial sheen of their curls or the pink of their lips produces upon any male within range of their charms.
None of this entered into the scheme of the lady Paraleia, nor Arete either. Her _peplos_ robe was split up the side in the Spartan style, revealing her bare leg to the thigh. This in any other city would have been lewd to the point of scandalous. Yet here in Lakedaemon it was unremarkable in the extreme. This is a leg. We women possess them just like you men. For Spartan males to leer at or ogle a lady in this dress would have been unthinkable. They had beheld their mothers and sisters and daughters naked since they were old enough to open their eyes, both in the girls' and women's athletic training and in the festivals and the other women's processions.
Still these ladies, both of them, were not unaware of their personal magnetism and the effect it produced, even upon a boy in service drawn up before them. After all, wasn't Helen herself a Spartan? The wife of Menelaus, she whom Paris had carried off to Troy,
_the cause of endless suffering_
_among Trojans and Greeks, and for_
_whose peerless beauty's sake so many_
_brave Achaeans lost their lives in Troy_
_far from their native country._
Spartan women surpass for beauty all others in Hellas, and not the least of their charms is that they make so little play upon it. Aphrodite is not their goddess, but Artemis Huntress. Look at the loveliness of our hair, their bearing seems to say, which reflects the lamplight not by the artifice of the cosmetician's art, but by the sheen of health and the luster of virtue. Look in our eyes which embrace a man's, neither lowering in contrived modesty nor fluttering behind dyed lashes like Corinthian whores. Our legs we groom not in the boudoir with wax and myrtle, but under the sun in the race and upon the Ring.
They were dams, these ladies, wives and mothers whose primary calling was to produce boys who would grow to be warriors and heroes, defenders of the city. Spartan women were brood mares, the pampered damsels of other cities might scoff, but if they were mares, they were racers, Olympic champions. The athletic glow and vigor which the _gynaikagoge,_ the women's training discipline, produced in them was powerful stuff and they knew it.
Standing before these women now, my thoughts despite all efforts were wrung back into the past, to Diomache and to my mother. I saw in memory my cousin's bare legs flashing strong and well made when we raced after some hare or doe with our dogs sprinting ahead up some rock-strewn slope. I saw the smooth glowing flesh of her arm when she drew the bow, her eyes that shrank before nothing and the flush of youth and freedom that suffused the skin of her face when she smiled. I saw again my mother, who was only twenty-six at her death, and whose memory to my eyes was of surpassing gentleness and nobility. These thoughts were like a room in the house of the mind that Dienekes spoke of, a room I had sworn since the Three-Cornered Way never to permit myself to enter.
But now, finding myself here in this real room of this real house, before these womanly rustles and scents, the feminine auroras of these wives and mothers and daughters and sisters, six of them, so much female presence concentrated in so close a space, I was driven back in mind against my will. It took all my self-composure to conceal the effect of these memories and to answer the lady's continuing questions in good order. At last it seemed the inquisition was approaching its conclusion.
"Answer now one final question. Speak with candor. If you lie, I will know. Does my son possess courage? Evaluate his _andreia,_ his manly virtue, as a youth who must soon take his place as a warrior."
It took no brains to see I was treading the thinnest of ice. How could one answer a question like that? I straightened and addressed the lady directly.
"There are fourteen hundred boys in the training platoons of the _agoge._ Only one displayed the temerity to follow the army, and that in knowing defiance of his own mother's wishes, not to say full awareness of what punishment he must endure upon his return."
The lady considered this. "It is a politic answer, but a good one. I accept it."
She rose and thanked the lady Arete for arranging this interview and for providing for its confidentiality. I was told to wait outside in the courtyard. The lady Paraleia's maidservant stood there still, smirking; no doubt she had overheard every word and would blab it to all the Eurotas valley by sunrise tomorrow. In a moment the lady herself emerged, deigning neither to look at nor speak to me, and accompanied by her maid, strode off without torchlight down the dark lane.
"Are you old enough to take wine?"
The lady Arete addressed me directly, speaking from the doorway and motioning me back within the dwelling. All four daughters slept now. The lady herself prepared a bowl for me, cut six to one as for a boy. I took a grateful swallow. Clearly this night of interviews was not over.
The lady invited me to sit. She herself settled at the mistress's station beside the hearth. She placed a chunk of _alphita_ barley bread on a plate before me and brought a relish of oil, cheese and onion.
"Be patient, this night among women will soon be over. You'll be back with the men, with whom you clearly feel more comfortable."
"I am at ease, lady. Truly. It's a relief to be away from barrack life for an hour, even if it means dancing barefoot on the hot steel of the skillet."
The lady smiled at this, but it was apparent that her mind was held by a more sober subject. She drew my eyes to hers.
"Have you ever heard the name Idotychides?"
I had.
"He was a Spartiate slain in battle at Mantinea. I have seen his stone before the mess of Winged Nike on the Amyklaian Way."
"What else do you know of this man?" the lady asked. I muttered something. "What else?" she insisted.
"They say that Dekton, the helot boy called Rooster, is his bastard. By a Messenian mother, who died giving birth."
"And do you believe this?"
"I do, lady."
"Why?"
I had stuck myself in a corner now; I could see the lady perceive it. "Is it because," she answered for me, "this boy Rooster hates the Spartans so much?"
I was struck with dread that she knew this and for long moments could not find my tongue.
"Have you noticed," the lady continued in a voice that to my surprise displayed neither outrage nor anger, "that among slaves the meanest seem to bear their lot without excessive distress, while the noblest, those at the brink of freedom, chafe most bitterly? It's as if the more one in service feels himself worthy of honor, yet denied the means to achieve it, the more excruciating is the experience of subjection."
This was Rooster in a nutshell. I had never thought about it that way but, now that the lady had expressed it thus, I saw it was true.
"Your friend Rooster talks too much. And what his tongue withholds, his demeanor announces only too plainly." She quoted, virtually verbatim, several seditious statements that Dekton had spoken, in my hearing alone, I thought, on the march back from Antirhion.
I was speechless and could feel myself breaking into a sweat. The lady Arete maintained her expression inscrutable.
"Do you know what the _krypteia_ is?" she asked.
I did. "It is a secret society among the Peers. No one knows who its members are, just that they are of the youngest and strongest, and they do their work at night."
"And what work is that?"
"They make men disappear." Helots, I meant. Treasonous helots.
"Now answer this, and consider before you speak." The lady Arete paused, as if to reinforce the importance of the question she was about to put. "If you were a member of the _krypteia_ and you knew what I have just told you about this helot, Rooster, that he had expressed sentiments treasonous to the city and further declared his intention of taking action based upon them, what would you do?"
There could be only one answer.
"It would be my duty to kill him, were I a member of the _krypteia._ "
The lady absorbed this, her expression still betraying nothing. "Now answer: if you were yourself, a friend to this helot boy, Rooster, what would you do?"
I stammered something about exculpatory circumstances, that Rooster was a hothead, he often spoke without thinking, much of what he said was bluster and everyone knew it.
The lady turned toward the shadows.
"Is this boy lying?"
"Yes, Mother!"
I spun in startlement. Both older daughters were wide awake, in their shared bed, glued to every word.
"I will answer the question for you, young man," the lady said, rescuing me from my predicament. "I think you would do this. I think you would warn this boy, Rooster, to speak no more of such things within your hearing and to take no action, however slight—or you yourself would dispatch him."
I was now utterly discomfited. The lady smiled. "You are a poor liar. It is not one of your gifts. I admire that. But you tread dangerous ground. Sparta may be the greatest city in Hellas, but it is still a small town. A mouse cannot sneeze without every cat saying God bless you. The servants and helots hear everything, and their tongues can be set a-wag for the price of a honey cake."
I considered this.
"And will mine," I asked, "be loosened for the cost of a bowl of wine?"
"The boy disrespects you, Mother!" This from Alexa, who was nine. "You must have him striped!"
To my relief the lady Arete regarded me in the lamplight with neither anger nor indignation, but calmly, studying me. "A boy in your position should rightly stand in fear of the wife of a Peer of my husband's stature. Tell me: why aren't you afraid of me?"
I hadn't realized until that moment that in fact I wasn't. "I'm not sure, lady. Perhaps because you remind me of someone."
For several moments the lady did not speak, but continued regarding me with that same intense scrutiny.
"Tell me about her," she commanded.
"Who?"
"Your mother."
I flushed again. It made me squirm to think this lady divined the contents of my heart before I even spoke them.
"Go ahead, take some wine. You don't have to play tough in front of me."
What the hell. I took it. It helped. I told the lady briefly of Astakos, of its sack and of my mother and father's murder at the hands of the night-skulking warriors of Argos.
"The Argives have always been cowards," she observed, dismissing them with a snort of contempt that endeared her, more than she realized, to me. Clearly her long ears had learned my poor story already, yet she listened attentively, seeming to respond with empathy to hearing the tale from my own lips.
"You have had an unhappy life, Xeo," she said, speaking my name for the first time. To my surprise this moved me profoundly; I had to fight not to let it show.
For my part, I was summoning every ounce of self-composure I possessed, to speak correctly, in proper Greek worthy of a freeborn, and to hold myself with respect not only for her but for my own country and my own line.
"And why," the lady asked, "does a boy of no city display so much loyalty to this alien country of Lakedaemon, of which he is not, and can never be, a part?"
I knew the answer but could not judge how much I dared entrust to her. I responded obliquely, speaking briefly of Bruxieus. "My tutor instructed me that a boy must have a city or he cannot grow to be fully a man. Since I no longer possessed a city of my own, I felt free to choose any I liked."
This was a novel point of view, but I could see the lady approved of it. "Why not, then, a _polis_ of riches or opportunity? Thebes or Corinth or Athens? All that can come to you here is coarse bread and a striped back."
I replied with a proverb that Bruxieus had once quoted to Diomache and me: that other cities produce monuments and poetry, Sparta produces men.
"And is this true?" the lady inquired. "In your most candid judgment, now that you have had opportunity to study our city, its worst as well as its best?"
"It is, lady."
To my surprise these words seemed to move the lady profoundly. She averted her gaze, blinking several times. Her voice, when she summoned herself again to speak, was hoarse with affect.
"What you have heard of the Peer Idotychides is true. He was the father of your friend Rooster. He was something other as well. He was my brother."
She could see me react with surprise.
"You didn't know this?"
"No, lady."
She mastered the emotion, the grief, I now saw, that had threatened to discompose her.
"So you see," she said with a smile brought forth with effort, "that makes this young Rooster something of a nephew to me. And I an aunt to him."
I took more wine. The lady smiled.
"May I ask why the lady's family has not sponsored the boy Rooster and put him forward as a _mothax_?"
This is a special dispensation in Lakedaemon, a "stepbrother" category of youth, available to the lesser-born or bastard sons of Spartiate fathers primarily, who could despite their mean birth be sponsored and elevated, enrolled in the _agoge._ They would train alongside the sons of Peers. They could even, if they showed sufficient merit and courage in battle, become citizens.
"I have asked your friend Rooster more than once," the lady answered. "He rebuffs me."
She could see the disbelief on my face.
"With respect," she added. "Most courtly respect. But with finality."
She considered this for a moment.
"There is another curiosity of mind which one may observe among slaves, particularly those who spring from a conquered people, as this boy Rooster does, being of a Messenian mother. Those men of pride will often identify with the meaner half of their line, out of spite perhaps, or the wish not to seem to curry favor by seeking to ingratiate themselves on the better side."
This was indeed true of Rooster. He saw himself as Messenian, and fiercely so.
"I tell you this, my young friend, for your sake as well as my nephew's: the _krypteia_ knows. They have watched him since he was five. They watch you too. You speak well, you have courage, you are resourceful. None of this goes unobserved and unremarked. And I will tell you something more. There is one among the _krypteia_ who is not unknown to you. This is the Captain of Knights, Polynikes. He will not hesitate to slit a treasonous helot's throat, nor do I think that your friend Rooster, for all his strength and spirit, will outrun a champion of Olympia."
The girls by now had all succumbed to slumber. The house itself and the darkness beyond its walls seemed at last entirely, eerily still.
"War with the Persian is coming," the lady declared. "The city will need every man. Greece will need every man. But just as important, this war, which all agree will be the gravest in history, will afford a mighty stage and arena for greatness. A field upon which a man may display by his deeds the nobility denied him by his birth."
The lady's eyes met mine and held them.
"I want this boy Rooster alive when war comes. I want you to protect him. If your ear detects any hint of danger, the slightest rumor, you must come straightaway to me. Will you do this?"
I promised I would.
"You care for this boy, Xeo. Though he has scourged you, I see the friendship you share. I implore you in the name of my brother and his blood which flows in this boy Rooster's veins. Will you watch over him? Will you do this for me?"
I promised that what I could do, I would.
"Swear it."
I complied, by all the gods.
It seemed preposterous. How could I stand against the _krypteia_ or any other force that sought to murder Rooster? Still somehow my boy's promise seemed to ease the lady's distress. She studied my face for a long moment.
"Tell me, Xeo," she said softly. "Do you ever...have you ever asked anything just for yourself?"
I replied that I did not understand the lady's question.
"I command one other thing of you. Will you perform it?"
I swore I would.
"I order you one day to take an action purely for your own sake and not in service to another. You will know when the time comes. Promise me. Say it aloud."
"I promise, lady."
She rose then, with the sleeping infant in her arms, and crossed to a cradle between the beds of the other girls, laying the babe down and settling it within the soft covers. This was the signal for me to take my leave. I had risen already, as respect commanded, when the lady stood.
"May I ask one question, lady, before I go?"
Her eyes glinted teasingly. "Let me guess. Is it about a girl?"
"No, lady." Already I regretted my impulse. This question I had was impossible, absurd. No mortal could answer it.
The lady had become intrigued, however, and insisted that I continue.
"It's for a friend," I told her. "I cannot answer it myself, being too young and knowing too little of the world. Perhaps you, lady, with your wisdom may be able to. But you must promise not to laugh or take offense."
She agreed.
"Or repeat this to anyone, including your husband."
She promised.
I took a breath and plunged in.
"This friend...he believes that once, when he was a child, alone at the point of death, he was spoken to by a god."
I pulled up, minding keenly for any sign of scorn or indignation. To my relief the lady displayed none.
"This boy...my friend...he wishes to know if such a thing is possible. Could...would a being of divinity condescend to speak to a boy without city or station, a penniless child who possessed no gift to offer in sacrifice and did not even know the proper words of prayer? Or was my friend hatching phantoms, fabricating empty visions out of his own isolation and despair?"
The lady asked which god it was, who had spoken to my friend.
"The archer god. Apollo Far Striker."
I was squirming. Surely the lady will scorn such temerity and presumption. I should never have opened my cheesepipe.
But she did not mock my question nor deem it impious. "You are something of an archer yourself, I understand, and far advanced for your years. They took your bow, didn't they? It was confiscated when you first appeared in Lakedaemon?"
She declared that fortune must have guided me to her hearth this night, for yes, the goddesses of the earth flew thick and near at hand. She could feel them. Men think with their minds, the lady said; women with their blood, which is tidal and flows at the discretion of the moon.
"I am no priestess. I can respond only out of a woman's heart, which intuits and discerns truth directly, from within."
I replied that this was precisely what I wished.
"Tell your friend this," the lady said. "That which he saw was truth. His vision indeed was of the god."
Without warning, fierce tears sprung to my eyes. At once emotion overwhelmed me. I buckled and sobbed, mortified at such loss of self-command and astonished at the power of passion which had sprung seemingly from nowhere to overcome me. I buried my face in my hands and wept like a child. The lady stepped to me and held me gently, patting my shoulder like a mother and uttering kind words of assurance.
Within moments I had mastered myself. I apologized for this shameful lapse. The lady would hear none of it; she scolded me, declaring that such passion was holy, inspired by heaven, and must not be repented or apologized for.
She stood now by the open doorway, through which the starlight fell and the soft babbling of the courtyard watercourse could be heard.
"I would like to have known your mother," the lady Arete said, regarding me with kindness. "Perhaps she and I will meet someday, beyond the river. We will speak of her son, and the unhappy portion the gods have set out before him."
She touched me once upon the shoulder in dismissal.
"Go now, and tell your friend this: he may come again with his questions, if he wishes. But next time he must come in person. I wish to look upon the face of this boy who has sat and chatted with the Son of Heaven."
**FOURTEEN**
**A** lexandros and I received our whippings for Antirhion the following evening. His was administered by his father, Olympieus, before the Peers of that officer's mess; I was lashed without ceremony in the fields by a helot groundsman. Rooster helped me away afterward, alone in the darkness, down to a grove called the Anvil beside the Eurotas to bathe and dress my stripes. This was a spot sacred to Demeter of the Fields and segregated by custom to the use of Messenian helots; there had once been a smithy upon the site, hence the name.
To my relief Rooster did not treat me to his customary harangue about the life of a slave, but rather limited his diatribe to the observation that Alexandros had been whipped like a boy and I like a dog. He was kind to me and, more important, possessed expertise in cleansing and dressing that unique species of ruptured laceration which is produced by the impact of the knurled birch upon the naked flesh of the back.
First water and plenty of it, bodily immersion to the neck in the icy current. Rooster supported me from behind, elbows braced beneath my armpits, since the shock of the frigid water upon the opened weals rarely fails to knock one faint. The cold numbs the flesh swiftly, and a wash of boiled nettles and Nessos' wort may be applied and endured. This stanches the flow of blood and promotes the rapid resealing of the flesh. A dressing of wool or linen at this stage would be unendurable, even applied with the gentlest touch. But a friend's bare palm, placed lightly at first, then pressed hard into the quivering flesh and held down, brings a relief whose effects approach ecstasy. Rooster had endured his own share of thrashings and knew the drill well.
Within five minutes I could stand. In fifteen my skin could take the soft sphagnum, which Rooster pressed into the blotted mass to suck out the poison and to inject its own subtle anesthetic. "By God, there's not a virgin left," he observed, meaning a space that was still God's flesh and not ruptured and reruptured scar tissue. "You won't be humping that hymn-singer's shield across this back for a month."
He was just launching into another venomous denunciation of my boy-master when a rustle came from the bank above us. We both wheeled, ready for anything.
It was Alexandros. He stepped into view beneath the plane trees, his cloak furled forward, leaving his own throttled back bare. Rooster and I froze. Alexandros would buy himself a second whipping if he was found here at this hour, and us with him.
"Here," he said, skidding down the bank to join us, "I picked the surgeon's locker for this."
It was wax of myrrh. Two fingers' worth, wrapped in green rowan leaves. He stepped into the stream beside us.
"What have you got there on his back?" he demanded of Rooster, who stepped aside with a look of blank astonishment. Myrrh was what the Peers used on wounds of battle when they could get it, which they rarely could. They would beat Alexandros half to death if they knew he'd purloined this precious portion. "Get it on him later when you peel off the moss," Alexandros directed Rooster. "Wash it off good by dawn. If anyone smells it, it'll be all our backs and more."
He placed the wrapped leaves in Rooster's hands.
"I have to be back before count," Alexandros declared. In an instant he had melted away up the bank; we could hear his footfalls vanishing softly as he sprinted in shadow back toward the boys' stations around the Square.
"Well, bend me over and root me senseless," Rooster spoke, shaking his head. "That little lark's got bigger globes than I thought."
At dawn when we fell in before sacrifice, Rooster and I were called out from our places by Suicide, Dienekes' Scythian squire. We were white with dread. Someone had peeped on us; there would be hell to pay for sure.
"You little turdnuggets must be floating under a lucky star" was all Suicide said. He conducted us to the rear of the formation. Dienekes stood there, silent, alone in the predawn shadows. We took our stations of deference on his left, his shield side. The pipers sounded; the formation moved off. Dienekes indicated that Rooster and I were to stay put.
He held stationary before us. Suicide stood on his right, with the quiver of sawed-off javelins he called "darning needles" angled nonchalantly across his back.
"I've been examining your record," Dienekes addressed me, his first words, other than the summons two nights previous to follow the serving boy to his home, ever spoken directly to me. "The helots tell me you're worthless as a field hand. I've watched you in the sacrificial train; you can't even shave the throat of a goat correctly. And it's clear from your conduct with Alexandros that you'll follow any order, no matter how mindless or absurd." He motioned me to turn, so he could examine my back. "It seems the only talent you possess is you're a fast healer."
He bent and sniffed my back. "If I didn't know better," he observed, "I'd swear these stripes had been waxed with myrrh."
Suicide kicked me around, back to face Dienekes. "You're an unwholesome influence on Alexandros," the Peer addressed me. "A boy doesn't need another boy, and certainly not a trouble collector like you; he needs a mature man, someone with the authority to stop him when he gets some reckless stunt into his head like tracking after the army. So I'm giving him my own man." His nod indicated Suicide. "I'm sacking you," he told me. "You're through."
Oh hell. Back to the shitfields.
Dienekes turned next to Rooster. "And you. The son of a Spartiate hero and you can't even hold a sacrificial cock in your fists without strangling it. You're pathetic. You've got a mouth looser than a Corinthian's asshole and it broadcasts treason every time it yawns. I'd be doing you a favor to slit your cheesepipe right here and save the _krypteia_ the trouble."
He reminded Rooster of Meriones, the squire of Olympieus who had fallen so gallantly last week at Antirhion. Neither of us boys had any idea where this was going.
"Olympieus is past fifty, he possesses all the prudence and circumspection he needs. His next squire should balance him with youth. Somebody green and strong and reckless." He regarded Rooster with wry scorn. "God knows what folly has inspired him, but Olympieus has picked you. You will take Meriones' place. You will attend Olympieus. Report to him at once. You're his first squire now."
I could see Rooster blinking. This must be a trick.
"It's no joke," Dienekes said, "and you'd better not make it one. You're treading in the steps of a man better than half the Peers in the regiment. Screw it up and I'll spit you over the flame personally."
"I won't, lord."
Dienekes studied him a long, hard moment. "Shut up and get the fuck out of here."
Rooster took off after the formation at a run. I confess I was ill with envy. The first squire of a Peer, and not just that, but a _polemarch_ and king's tent companion. I hated Rooster for his dumb blind luck.
Or was it? As I blanched, numb with jealousy, a picture of the lady Arete shot across the eye of my mind. She was behind this. I felt even worse and regretted bitterly that I had confided to her my vision of Apollo Far Striker.
"Let me see your back," Dienekes commanded. I turned again; he whistled appreciatively. "By God, if there were an Olympic event in back-striping, you'd be the betting man's favorite." He had me face about and stand at attention before him; he regarded me thoughtfully, his gaze seeming to pierce straight through to my spine. "The qualities of a good battle squire are simple enough. He must be dumb as a mule, numb as a post and obedient as an imbecile. In these qualifications, Xeones of Astakos, I declare your credentials impeccable."
Suicide was chuckling darkly. He tugged something from behind the quiver at his back. "Go ahead, take a look," Dienekes ordered. I raised my eyes.
In the Scythian's hand stood a bow. My bow.
Dienekes commanded me to take it.
"You're not strong enough yet to be my first squire, but if you can manage to keep your head out of your ass, you might make a half-respectable second." Into my palm Suicide placed the bow, the big Thessalian cavalry weapon that had been confiscated from me at twelve, when first I crossed the frontier into Lakedaemon.
I could not stop my hands from trembling; I felt the warm ash of the bow and the living current that coursed its length and up into my palms.
"You'll pack my rations, bedding and medical kit," Dienekes instructed me. "You'll cook for the other squires and hunt for my pot, on exercises in Lakedaemon and beyond the border on campaign. Do you accept this?"
"I do, lord."
"At home you may hunt hares and keep them for yourself, but don't flaunt your good fortune."
"I won't, lord."
He regarded me with that look of wry amusement I had observed on his face before, at a distance, and which I would come to see many times more close-up.
"Who knows," my new master said, "with luck, you might even get in a potshot at the enemy."
**FIFTEEN**
**T** he army of Lakedaemon marched out in twenty-one different campaigns over the next five years, all in actions against other Hellenes. That pitch of enmity which Leonidas had sought since Antirhion to maintain focused upon the Persian now found itself of necessity directed against more immediate targets, those cities of Greece which tilted perfidiously toward playing the traitor, allying themselves in advance with the invader, to save their own skins.
Mighty Thebes, whose exiled aristocrats conspired ceaselessly with the Persian court, seeking to reclaim preeminence in their country by selling it out to the foe.
Jealous Argos, Sparta's most bitter and proximate rival, whose nobles treated openly with the agents of the Empire. Macedonia under Alexander had long since offered tokens of submission. Athens, too, had exiled aristocrats reclining within the Persian pavilions while they plotted for their own restoration as lords beneath the Persian pennant.
Sparta herself stood not immune from treason, for her deposed king, Demaratos, as well had taken up the exile's station among the sycophants surrounding His Majesty. What else could Demaratos' desire be, save reaccession to power in Lakedaemon as satrap and magistrate of the Lord of the East?
In the third year after Antirhion, Darius of Persia died. When news of this reached Greece, hope rekindled in the free cities. Perhaps now the Persian would abort his mobilization. With her King dead, would not the army of the Empire disband? Would not the Persian vow to conquer Hellas be set aside?
Then you, Your Majesty, acceded to the throne.
The army of the foe did not disband.
Her fleet did not disperse.
Instead the Empire's mobilization redoubled. The zeal of a prince freshly crowned burned within His Majesty's breast. Xerxes son of Darius would not be judged by history inferior to his father, nor to his illustrious forebears Cambyses and Cyrus the Great. These, who had vanquished and enslaved all Asia, would be joined in the pantheon of glory by Xerxes, their scion, who would now add Greece and Europe to the roll of provinces of the Empire.
Across all Hellas, _phobos_ advanced like a sapper's tunnel. One smelled the dust of its excavation in the still of morning and felt its yard-by-yard advance rumbling beneath one in his sleep. Of all the mighty cities of Greece, only Sparta, Athens and Corinth held fast. These dispatched legation after legation to the wavering _poleis,_ seeking to bind them to the Alliance. My own master was assigned in a single season to five separate overseas embassies. I puked over so many different ships' rails I couldn't recall one from the other.
Everywhere these embassies touched, _phobos_ had called first. The Fear made people reckless. Many were selling all they owned; others, more heedless, were buying. "Let Xerxes spare his sword and send his purse instead," my master observed in disgust after yet another embassy had been rebuffed. "The Greeks will trample one another's bones, racing to see who first can sell his freedom."
Always upon these legations, a part of my mind kept alert for word of my cousin. Three times in my seventeenth year the service of my master brought me through the city of the Athenians; each time I inquired after the location of the home of the gentlewoman whom Diomache and I had encountered that morning on the road to the Three Corners, when that fine lady had ordered Dio to seek her town estate and take service there. I secured at last the quarter and street but never succeeded in finding the house.
Once at a salon in the Athenian Akademe a lovely bride of twenty appeared, mistress of the household, and for a moment I was certain it was Diomache. My heart began to pound so violently that I must kneel upon one knee for fear of dropping to the floor dead faint. But the lady was not she. Nor was the bride glimpsed a year later bearing water from a spring in Naxos. Nor the physician's wife encountered under cloister in Histiaea six months thereafter.
Upon one blistering summer evening, two years before the battle at the Gates, the ship bearing my master's legation touched briefly at Phaleron, a port of Athens. Our mission completed, we had two hours before tide's turn. I was granted leave and on the run at last located the house of the family of the lady of the Three Corners. The place was shuttered; _phobos_ had driven the clan forth to landholdings in Iapygia, or so I was informed by a loitering squad of Scythian archers, those thugs whom the Athenians employ as city constabulary. Yes, the brutes remembered Diomache. Who could forget her? They took me for another of her suitors and spoke in the crude language of the street.
"The bird winged off," one said. "Too wild for the cage."
Another declared he had encountered her since, in the market with a husband, a citizen and sea officer. "The fool bitch," he laughed. "To knot with that salt-sucker, when she could have had me!"
Returning to Lakedaemon, I resolved to root this folly of longing from my heart, as a farmer burns out a stubborn stump. I told Rooster it was time I took a bride. He found one for me, his cousin Thereia, the daughter of his mother's sister. I was eighteen, she fifteen when we were joined in the Messenian fashion practiced by the helots. She bore a son within ten months and a daughter while I was away on campaign.
A husband now, I vowed to think no more of my cousin. I would eradicate my own impiety and dwell no longer upon fancies.
The years had passed swiftly. Alexandros completed his service as a youth of the _agoge;_ he was given his war shield and assumed his station among the Peers of the army. He took to wife the maiden Agathe, just as he had promised. She bore him twins, a boy and girl, before he was twenty.
Polynikes was crowned at Olympia for the second time, victor again in the sprint in armor. His wife, Altheia, bore him a third son.
The lady Arete produced for Dienekes no more children; she had come up barren after four daughters, without producing a male heir.
Rooster's wife, Harmonia, bore a second child, a boy whom he named Messenieus. The lady Arete attended the birth, providing her own midwife and assisting at the delivery with her own hands. I myself bore the torch that escorted her home. She would not speak, so torn was she between the joy of witnessing at last from her line the birth of a male, a defender for Lakedaemon, and the sorrow of knowing that this boy-child, issue of her brother's bastard, Rooster, with all his treasonous defiance of his Spartan masters, right down to the name he had chosen for his son, would face the sternest and most perilous passage to manhood.
The Persian myriads stood now in Europe. They had bridged the Hellespont and traversed all of Thrace. Still the Hellenic allies wrangled. A force of ten thousand heavy infantry, commanded by the Spartan Euanetus, was dispatched to Tempe in Thessaly, there to make a stand against the invader at the northernmost frontier of Greece. But the site, when the army got there, proved undefendable. The position could be turned by land via the pass at Gonnus and outflanked by sea through Aulis. In disgrace and mortification the force of Ten Thousand pulled out and dispersed to its constituent cities.
A desperate paralysis possessed the Congress of the Greeks. Thessaly, abandoned, had gone over to the Persian, adding her matchless cavalry to swell the squadrons of the foe. Thebes teetered at the brink of submission. Argos was sitting it out. Dread omens and prodigies abounded. The Oracle of Apollo at Delphi had counseled the Athenians,
_"Fly to the ends of the earth,"_
while the Spartan Council of Elders, notoriously slow to action, yet dithered and dawdled. A stand must be made somewhere. But where?
In the end it was their women who galvanized the Spartans into action. It came about like this.
Refugees, many brides with babes, were flooding into the last of the free cities. Young mothers took flight to Lakedaemon, islanders and relations fleeing the Persian advance across the Aegean. These brides inflamed their listeners' hatred of the foe with tales of the conquerors' atrocities in their earlier passage through the islands: how the enemy at Chios and Lesbos and Tenedos had formed dragnets at one end of the territory and advanced across each island, scouring out every hiding place, hauling forth the young boys, herding the handsomest together and castrating them for eunuchs, killing every man and raping the women, selling them forth into foreign slavery. The babies' heads these heroes of Persia dashed against the walls, splattering their brains upon the paving stones.
The wives of Sparta listened with icy fury to these tales, cradling their own infants at their breasts. The Persian hordes had swept now through Thrace and Macedonia. The baby-murderers stood upon the doorstep of Greece, and where was Sparta and her warrior defenders? Blundering homeward unblooded from the fool's errand of Tempe.
I had never seen the city in such a state as in the aftermath of that debacle. Heroes with prizes of valor skulked about, countenances downcast with shame, while their women snapped at them with scorn and held themselves aloof and disdainful. How could Tempe have happened? Any battle, even a defeat, would have been preferable to none at all. To marshal such a magnificent force, garland it before the gods, transport it all that way and not draw blood, even one's own, this was not merely disgraceful but, the wives declared, blasphemous.
The women's scorn excoriated the city. A delegation of wives and mothers presented itself to the ephors, insisting that they themselves be sent out next time, armed with hairpins and distaffs, since surely the women of Sparta could disgrace themselves no more egregiously nor accomplish less than the vaunted Ten Thousand.
In the warriors' messes the mood was even more corrosive. How much longer would the Allied Congress dither? How many more weeks would the ephors delay?
I recall vividly the morning when at last the proclamation came. The Herakles regiment trained that day in a dry watercourse called the Corridor, a blistering funnel between sand banks north of the village of Limnai. The men were running impact drills, two-on-ones and three-on-twos, when a distinguished elder named Charilaus, who had been an ephor and a priest of Apollo but now functioned primarily as a senior counselor and emissary, appeared on the crest of the bank and spoke aside to the _polemarch_ Derkylides, the regimental commander. The old man was past seventy; he had lost the lower half of a leg in battle years past. For him to have hobbled on his staff this far from the city could only mean something big had happened.
The patriarch and the _polemarch_ spoke in private. The drills went on. No one looked up, yet every man knew.
This was it.
Dienekes' men got the word from Laterides, commander of the adjacent platoon, who passed it down the line.
"It's the Gates, lads."
The Hot Gates.
Thermopylae.
No assembly was called. To the astonishment of all, the regiment was dismissed. The men were given the whole rest of the day off.
Such a holiday had only been granted half a dozen times in my memory; invariably the Peers broke up in high spirits and made for home at the trot. This time no one budged. The entire regiment stood nailed to the site, in the sweltering confines of the dry river, buzzing like a hive.
Here was the word:
Four _morai,_ five thousand men, would be mobilized for Thermopylae. The column, reinforced by four _perioikic_ regiments and packing squires and armed helots two to a man, would march out as soon as the Karneia, the festival of Apollo which prohibited taking up arms, expired. Two and half weeks.
The force would total twenty thousand men, twice the number at Tempe, concentrated in a pass ten times narrower.
Another thirty to fifty thousand allied infantry would be mobilized behind this initial force, while a main force of the allied navy, a hundred and twenty ships of war, would seal the straits at Artemisium and Andros and the narrows of the Euripus, protecting the army at the Gates from flank assault by sea.
This was a massive call-up. So massive it smelled. Dienekes knew it and so did everyone else.
My master humped back to the city accompanied by Alexandros, now a full line warrior of the platoon, his mates Bias, Black Leon and their squires. A third of the way along we overtook the elder Charilaus, shambling home with painful slowness, supported by his attendant, Sthenisthes, who was as ancient as he. Black Leon led an ass of the train on a halter; he insisted the old man ride. Charilaus declined but permitted the place to his servant.
"Cut through the shit for us, will you, old uncle?" Dienekes addressed the statesman affectionately but with a soldier's impatience for the truth.
"I relay only what I'm instructed, Dienekes."
"The Gates won't hold fifty thousand. They won't hold five."
A wry expression wizened the old-timer's face. "I see you fancy your generalship superior to Leonidas'."
One fact was self-evident even to us squires. The Persian army stood now in Thessaly. That was what, ten days to the Gates? Less? In two and a half weeks their millions would sweep through and be eighty miles beyond. They'd be parked upon our threshold.
"How many in the advance party?" Black Leon inquired of the elder.
He meant the forward force of Spartans that would, as always in advance of a mobilization, be dispatched to Thermopylae now, at once, to take possession of the pass before the Persians got there and before the main force of the allied army moved up.
"You'll hear it from Leonidas tomorrow," the old man replied. But he saw the younger men's frustration.
"Three hundred," he volunteered. "All Peers. All sires."
My master had a way of setting his jaw, a fierce clamping action of the teeth, which he employed when he was wounded on campaign and didn't want his men to know how bad. I looked. This expression stood now upon his face.
An "all-sire" unit was comprised only of men who were fathers of living sons.
This was so that, should the warriors perish, their family lines would not be extinguished.
An all-sire was a suicide unit.
A force dispatched to stand and die.
My customary duties upon return from training were to clean and stow my master's gear and look to, with the servants of the mess, the preparation of the evening meal. Instead this day Dienekes asked Black Leon for his squire to do double duty. Myself he ordered on ahead, at a run, to his own home. I was to inform the lady Arete that the regiment had been dismissed for the day and that her husband would arrive at home shortly. I was to issue an invitation to her on his behalf: would she and their daughters accompany him this afternoon for a ramble in the hills?
I raced ahead, delivered this message and was dismissed to my own pursuits. Some impulse, however, made me linger.
From the hill above my master's cottage I could see his daughters burst from the gate and dash with eager enthusiasm to greet him upon the way. Arete had prepared a basket of fruit, cheese and bread. The party was all barefoot, wearing big floppy sun hats.
I saw my master tug his wife aside beneath the oaks and there speak privately with her for several moments. Whatever he said, it prompted her tears. She embraced him fiercely, both arms flung tight about his neck. Dienekes seemed at first to resist, then in a moment yielded and clamped his wife to him, holding her tenderly.
The girls clamored, impatient to be off. Two puppies squalled underfoot. Dienekes and Arete released their embrace. I could see my master lift his youngest, Ellandra, and plant her pony style astride his shoulders. He held the maiden Alexa's hand as they set off, the girls exuberant and gay, Dienekes and Arete lagging just a little.
No main-force army would be dispatched to Thermopylae; that tale was for public consumption only, to shore up the allies' confidence and put iron in their backbones.
Only the Three Hundred would be sent, with orders to stand and die.
Dienekes would not be among them.
He had no male issue.
He could not be selected.
**SIXTEEN**
**I** must now recount an incident of battle several years previous, whose consequences at this present juncture came powerfully to affect the lives of Dienekes, Alexandros, Arete and others in this narrative. This occurred at Oenophyta against the Thebans, one year after Antirhion.
I refer to the extraordinary heroism demonstrated on that occasion by my mate Rooster. Like myself at the time, he was just fifteen and had been serving, green as grass, for less than twelve months as first squire of Alexandros' father, Olympieus.
The armies' fronts had clashed. The Menelaion, Polias and Wild Olive regiments were locked in a furious struggle with the Theban left, which was stacked twenty deep instead of the customary eight and was holding its position with terrific stubbornness. To augment this peril, the foe's wing overlapped the Spartan right an eighth of a mile; these elements now began to wheel inboard and advance, taking the Menelaion in the flank. Simultaneously the enemy's right, which was taking the most grievous casualties, lost cohesion and fell back upon the massed ranks of its rearmen. The foe's right broke in panic while his left advanced.
In the midst of this melee Olympieus received a crippling lizard-sticker wound through the arch of the foot, from the butt-spike of an enemy spear. This came, as I said, at a moment of extreme dislocation upon the field, with the enemy right collapsing and the Spartans surging into the pursuit, while the foe's left wheeled in attack, supported by numbers of their cavalry coursing uncontested across the broken field.
Olympieus found himself alone upon the open "gleaning ground" to the rear of the onrolling battle, with his foot wound rendering him crippled, while his cross-crested officer's helmet provided an irresistible target for any would-be hero of the enemy's ranging horse.
Three Theban cavalrymen went after him.
Rooster, unarmed and unarmored, sprinted headlong into the fray, snatching a spear from the ground as he ran. Dashing up to Olympieus, he not only employed his master's shield to protect him from the missile weapons of the enemy but took on the attacking horsemen single-handedly, wounding and driving off two with spear thrusts and caving in the skull of the third with the man's own helmet, which he, Rooster, in the madness of the moment, had torn off the fellow's head with his bare hands as he simultaneously ripped him out of his seat. Rooster even succeeded in capturing the handsomest of the three horses, a magnificent battle mount which he used in the aftermath to draw the litter which evacuated Olympieus safely from the field.
When the army returned to Lakedaemon after this campaign, Rooster's exploit was the talk of the city. Among the Peers his prospects were debated at length. What should be done with this boy? All recalled that though his mother was a Messenian helot, his father had been the Spartiate Idotychides, Arete's brother, a hero slain in battle at Mantinea when Rooster was two.
The Spartans, as I have noted, have a grade of warrior youth, a "stepbrother" class called a _mothax._ Bastards like Rooster and even legitimate sons of Peers who through misfortune or poverty have lost their citizenship may be, if deemed worthy, plucked from their straits and elevated to this station.
This honor was now proffered to Rooster.
He turned it down.
His stated reason was that he was already fifteen. It was too late for him; he preferred to remain in service as a squire.
This rejection of their generous offer enraged the Peers of Olympieus' mess and created an outrage, as much as the affair of a helot bastard could, within the city at large. Assertions were made to the point that this headstrong ingrate was notorious for his disloyal sentiments. He was a type not uncommon among slaves, prideful and stubborn. He sees himself as Messenian. He must either be eliminated, and his family with him, or secured beyond doubt of betrayal to the Spartan cause.
Rooster eluded assassination at the hands of the _krypteia_ that time, largely due to his youth and to Olympieus' intercession, man-to-man among the Peers. The affair faded for the moment, rekindling itself, however, upon subsequent campaigns when Rooster again and again proved himself the boldest and most valorous of the young squires, surpassing all in the army save Suicide, Cyclops, main man of the Olympic pentathlete Alpheus, and Polynikes' squire, Akanthus.
Now the Persians stood at the threshold of Greece. Now the Three Hundred were being selected for Thermopylae. Olympieus would be prominent among them, with Rooster at his shoulder in his service. Could this treasonous youth be trusted? With a blade in his fist and himself a handbreadth from the _polemarch_ 's back?
The last thing Sparta needed at this desperate hour was trouble at home with the helots. The city could not stand a revolt, even an abortive one. Rooster by this time, aged twenty, had become a force among the Messenian laborers, farmers and vineyardmen. He was a hero to them, a youth whose courage in battle could have been exploited by him as a ticket out of his servitude. He could be wearing Spartan scarlet and lording it over his mean-birthed brothers. But this he had disdained. He had declared himself Messenian, and his fellows never forgot. Who knows how many of them followed Rooster in their hearts? How many absolutely vital craftsmen and support personnel, armorers and litter bearers, squires and victualry men? It is an ill wind, they say, that blows no one good, and this Persian invasion could be the best thing that ever happened to the helots. It could spell deliverance. Freedom. Would they stand loyal? Like the gate of a mighty citadel which turns upon a single tempered hinge, much of the Messenian sentiment focused its attention upon Rooster and stood ready to take its cue from him.
It was now the night before the proclamation of the Three Hundred. Rooster was summoned to stand-to before Olympieus' mess, the Bellerophon. There, officially and with the goodwill of all, the honor of Spartan scarlet was again offered to the youth.
Again he spurned it.
I loitered deliberately in that hour outside the Bellerophon, to see which way the issue would go. It took no imagination, hearing the murmur of outrage within and beholding Rooster's swift and silent exit, to read the gravity of the issue, and its peril. An assignment for my master detained me for the bulk of an hour. At last I found opportunity to scamper free.
Beside the Little Ring where the starter's box stands is a grove with a dry course branching in three directions. There Rooster and I and other boys used to meet and even bring girls, because if you were found, you could dash away easily in the dark down one of the three dry riverbeds. I knew he would be there now, and he was. To my amazement Alexandros was with him. They were arguing. It took only moments to see it was the clash of one who wishes to be another's friend and the other who rejects him. What was startling was that it was Alexandros who wanted to be friend to Rooster. He would be in calamitous trouble if he was caught, so immediately subsequent to his initiation as a warrior. As I skittered down into the shadows of the dry course, Alexandros was cursing Rooster and declaring him a fool.
"They'll kill you now, don't you know that?"
"Fuck them. Fuck them all."
"Stop this!" I burst down between them. I recited what all three of us knew: that Rooster's prestige among the lower orders precluded him from acting for himself alone; what he did bore repercussions for his wife, his son and daughter, his family. He had cooked himself and them with him. The _krypteia_ would finish him this very night, and nothing would suit Polynikes more.
"He won't catch me if I'm not here."
Rooster had set his mind to flee, this night, to the Temple of Poseidon at Tainaron, where a helot could be granted sanctuary.
He wanted me to come. I told him he was insane. "What were you thinking when you turned them down? What they offered you is an honor."
"Fuck their honors. The _krypteia_ hunts me now, in darkness, faceless as cowards. Is that honor?"
I told him his slave's pride had bought his own ticket to hell.
"Shut up, both of you!"
Alexandros ordered Rooster to his shell, that term the Spartans use to describe the mean huts of the helots. "If you're going to run, run now!"
We sprinted away down the dark watercourse. Harmonia had both children, Rooster's daughter and infant son, packed and ready. In the smoky confines of the helot's shell, Alexandros pressed into Rooster's hand a clutch of Aeginetan obols, not much, but all he had, enough to aid a runaway.
This gesture struck Rooster speechless.
"I know you don't respect me," Alexandros told him. "You think yourself my better in skill at arms, in strength and in valor. Well, you are. I have tried, as the gods are my witness, with every fiber of my being and still I'm not half the fighter you are. I never will be. You should stand in my place and I in yours. It is the gods' injustice that makes you a slave and me free."
This from Alexandros utterly disarmed Rooster. You could see the combativeness in his eyes relent and his proud defiance slacken and abate.
"You own more of valor than I ever will," the bastard replied, "for you manufacture it out of a tender heart, while the gods sat me up punching and kicking from the cradle. And you do yourself honor to speak with such candor. You're right, I did despise you. Until this moment."
Rooster glanced at me then; I could see confusion in his aspect. He was moved by Alexandros' integrity, which pulled his heart strongly to remain and even to yield. Then with an effort he broke the spell. "But you won't influence me, Alexandros. Let the Persian come. Let him grind all Lakedaemon into dust. I'll jig on its grave."
We heard Harmonia gasp. Outside, torches flared. Shadows surrounded the shell. Its blanket flap was torn open. There in the rude doorway stood Polynikes, armed and backed by four assassins of the _krypteia._ They were all young, athletes nearly on a par with the Olympian, and pitiless as iron.
They burst in and bound Rooster with cord. The infant boy wailed in Harmonia's arms; the poor girl was barely seventeen; she shuddered and wept, pulling her daughter in terror to her side. Polynikes absorbed the sight with contempt. His glance flicked over Rooster, his wife and babes and myself, to settle with scorn upon the person of Alexandros.
"I might have known we'd find you here."
"And I you," the youth responded.
On his face was written plain his hatred of the _krypteia._
Polynikes regarded Alexandros, and his sentiments, with barely contained outrage. "Your presence here in these precincts constitutes treason. You know it and so do these others. Out of respect to your father only, I will say this once: leave now. Depart at once and nothing more will be said. The dawn will find four helots missing."
"I will not," Alexandros answered.
Rooster spat. "Kill us all, then!" he demanded of Polynikes. "Show us Spartan valor, you night-skulking cowards."
A fist smashed his teeth, silencing him.
I saw hands seize Alexandros and felt others clamp me; thongs of hide bound my wrists, a gag of linen stoppered my throat. The _krypteis_ snatched Harmonia and her babes.
"Bring them all," Polynikes ordered.
**SEVENTEEN**
**T** here stands a grove upslope behind the Deukalion mess, where the men and hounds customarily muster before setting off on a hunt. There within minutes a rump court stood assembled.
The site is a grisly one. Rude kennels extend beneath the oaks, with their game nets and chase harnesses hanging beneath the eaves of the feeding stations. The mess kitchen stores its slaughtering implements in several double-locked outbuildings; upon the inner doors hang hatchets and gutting knives, cleavers and bonebreakers; a blood-black chopping board for game fowl and poultry extends along the wall, where the birds' heads are whacked off and topple to the dirt for the hounds to scrap over. Piles of plucked feathers collect as high as a man's calf, rendered sodden by the blood drippings of the next luckless fowl to stretch its gullet beneath the chopper. Above these along the runway stand the bars of the butchery with their heavy iron hooks for the hanging, gutting and bleeding of game.
It was a foregone conclusion that Rooster must die, and his infant son with him. What remained yet at issue was the fate of Alexandros, and his treason which, if published throughout the city, would work grievous harm at this most peril-fraught hour, not only to himself and his station as a newly initiated warrior but to the prestige of his entire clan, his wife, Agathe, his mother, Paraleia, his father, the _polemarch_ Olympieus, and, not least of all, his mentor, Dienekes. This latter pair now took their place in the shadows, along with the other sixteen Peers of the Duekalion mess. Rooster's wife wept silently, her daughter beside her; the baby squalled, muffled, in her arms. Rooster knelt in his cord bonds, on his knees in the dry high-summer dust.
Polynikes paced impatiently, wanting a decision.
"May I speak?" Rooster croaked in a throat hoarse from having been throttled on the way to this summary arraignment.
"What has scum like you to say?" Polynikes demanded.
Rooster indicated Alexandros. "This man your thugs think they 'captured'...they should be declaring him a hero. He took me captive, he and Xeones. That's why they were in my shell. To arrest me and bring me in."
"Of course," Polynikes replied sarcastically. "That's why they had you bound so tightly."
Olympieus addressed Alexandros. "Is this true, son? Did you indeed place the youth Rooster in custody?"
"No, Father. I did not."
All knew that this "trial" would not last long. Discovery was inevitable, even here in the shadows, by the _agoge_ youths who stood sentry over the night city, their patrols doubled now for wartime. The assembly had perhaps five minutes, no more.
In two brief exchanges, as if the Peers couldn't divine it themselves, it became clear that Alexandros had at the eleventh hour attempted to persuade Rooster into rescinding his defiance and accepting the city's honor, that he had failed and that still he had taken no action against him.
This was treason pure and simple, Polynikes declared. Yet, he said, he personally had no wish to defame and punish the son of Olympieus, nor even myself, the squire of Dienekes. Let it end here. You gentlemen retire. Leave this helot and his brat to me.
Dienekes now spoke. He expressed his gratitude to Polynikes for this offer of clemency. There remained, however, an aspect of half-exoneration to the Knight's suggestion. Let us not leave it at that, but clear Alexandros' name entire. May he, Dienekes requested, speak on the young man's behalf?
The senior Medon assented, the Peers seconding him.
Dienekes spoke. "You gentlemen all know my feelings for Alexandros. All of you are aware that I have counseled and mentored him since he was a child. He is like a son to me, and a friend and brother as well. But I will not defend him out of these sentiments. Rather, my friends, consider these points.
"What Alexandros was attempting this night is nothing other than that which his father has been trying since Oenophyta, that is, to influence informally, by reason and persuasion, and out of friendly feeling, this boy Dekton called Rooster. To soften the bitterness he bears against us Spartans, who, he feels, have enslaved his countrymen, and to bring him around to the greater cause of Lakedaemon.
"In this endeavor, Alexandros has not this night and never has sought any advantage for himself. What good could come to him from enlisting this renegade beneath Spartan scarlet? His thought was alone for the good of the city, to harness to its use a young man of clearly demonstrated vigor and courage, the bastard son of a Peer and hero, my own wife's brother, Idotychides. In fact, you may hold me to blame along with Alexandros, for I more than once have referred to this boy Rooster as my by-blow nephew."
"Yes," Polynikes put in swiftly, "as a joke and term of derision."
"We do not joke here tonight, Polynikes."
There was a rustle among the leaves, and suddenly, to the astonishment of all, there into the slaughtering space advanced the lady Arete. I glimpsed a pair of barn urchins escaping into shadow; clearly these spies had witnessed the scene at Rooster's shell and dashed at once to relay it to the lady.
Now she came forward. Wearing a plain _peplos_ robe, with her hair down, summoned no doubt from bedtime lullabies just moments previous. The Peers parted before her, taken so by surprise that none could momentarily find voice to protest.
"What is this," she demanded with scorn, "a skull court beneath the oaks? What august verdict will you brave warriors pronounce tonight? To murder a maiden or slit the throat of an infant?"
Dienekes sought to silence her, and the others did as well, with declamations to the effect that a woman had no business here, she must depart at once, they would hear of nothing else. Arete, however, ignored these utterly, stepping without hesitation to the side of the girl Harmonia, and there seizing Rooster's infant and taking him into her arms.
"You say my presence here can serve no purpose. On the contrary," she declared to the Peers, "I can offer most apposite assistance. See? I can tilt this child's jaw back, to make his assassination easier. Which of you sons of Herakles will slice this infant's throat? You, Polynikes? You, my husband?"
More declarations of outrage ensued, insisting that the lady vacate at once. Dienekes himself voiced this in the most emphatic terms. Arete would not budge.
"If this young man's life were all that were at stake"—her gesture indicated Rooster—"I would obey my husband and you other Peers without hesitation. But who else will you heroes be compelled to murder in addition? The boy's half brothers? His uncles and cousins and their wives and children, all of them innocent and all assets which the city needs desperately in this hour of peril?"
It was reasserted that these issues were none of the lady's concern.
Actaeon the boxer addressed her directly. "With respect, lady, none can but see that your intention is to shield from extinction your honored brother's line," and he gestured to the squalling boy-child, "even in this, its bastard form."
"My brother has already achieved imperishable fame," the lady responded with heat, "which is more than can be said for any of you. No, it is simple justice I seek. This child you stand ready to murder is not the issue of this boy, Rooster."
This statement appeared so irrelevant as to border upon the preposterous.
"Then whose is he?" Actaeon demanded impatiently.
The lady hesitated not a moment.
"My husband's," she replied.
Snorts of incredulity greeted this. "Truth is an immortal goddess, lady," the senior Medon spoke sternly. "One would be wise to consider before defaming her."
"If you don't believe me, ask this girl, the child's mother."
The Peers plainly granted no credence whatever to the lady's outrageous assertion. Yet all eyes now centered upon the poor young housewoman, Harmonia.
"He is my child," Rooster broke in with vehemence, "and no one else's."
"Let the mother speak," Arete cut him off. Then to Harmonia: "Whose son is he?"
The hapless girl sputtered in consternation. Arete held the infant up before the Peers. "Let all see, the babe is well made, strong of limb and voice, with the cradled vigor which precedes strength in youth and valor in manhood."
She turned to the girl. "Tell these men. Did my husband lie with you? Is this child his?"
"No...yes...I don't..."
"Speak!"
"Lady, you terrorize the girl."
"Speak!"
"He is your husband's," the girl blurted, and began to sob.
"She lies!" Rooster shouted. He received a vicious cuff for his efforts; blood sprung from his lip, now split.
"Of course she would not tell you, her husband," the lady addressed Rooster. "No woman would. But that does not alter the facts."
With a gesture Polynikes indicated Rooster. "For the only time in his life, this villain speaks the truth. He has sired this whelp, as he says."
This opinion was seconded vigorously by the others.
Medon now addressed Arete. "I would sooner go up bare-handed against a lioness in her den than face your wrath, lady. Nor can any but commend your motive, as a wife and mother, in seeking to shield the life of an innocent. Nonetheless we of this mess have known your husband since he was no bigger than this babe here. None in the city surpasses him in honor and fidelity. We have been with him, more than once on campaign, when he has had opportunity, ample and tempting opportunity, to be faithless. Never has he so much as wavered."
This was corroborated with emphasis by the others.
"Then ask him," Arete demanded.
"We will do no such thing," Medon replied. "Even to call his honor into question would be infamous."
The Peers of the mess faced Arete, solid as a phalanx. Yet far from being intimidated, she confronted the line boldly, in a tone of order and command.
"I will tell you what you will do," Arete declared, stepping squarely before Medon, senior of the mess, and addressing him like a commander. "You will recognize this child as the issue of my husband. You, Olympieus, and you, Medon, and you, Polynikes, will then sponsor the boy and enroll him in the _agoge._ You will pay his dues. He will be given a schooling name, and that name will be Idotychides."
This was too much for the Peers to endure. The boxer Actaeon now spoke. "You dishonor your husband, and your brother's memory, even to propose such a course, lady."
"If the child were my husband's, would my argument find favor?"
"But he is not your husband's."
"If he were?"
Medon cut her short. "The lady knows full well that if a man, like this youth called Rooster, is found guilty of treason and executed, his male issue may not be allowed to live, for these, if they possess any honor whatever, will seek vengeance when they reach manhood. This is the law not merely of Lykurgus but of every city in Hellas and holds true without exception even among the barbarians."
"If you believe that, then slit the babe's throat now."
Arete stepped directly before Polynikes. Before the runner could react, her grasp sprung to his hip and snatched forth his _xiphos._ Maintaining her own hand upon the hilt, she thrust the weapon into Polynikes' hand and held the infant up, exposing its throat beneath the whetted steel.
"Honor the law, sons of Herakles. But do it here in the light where all may see, not in the darkness so beloved of the _krypteia._ "
Polynikes froze. His hand sought to tug the blade back and away, but the lady's grip would not release it.
"Can't do it?" she hissed. "Let me help. Here, I'll plunge it with you..."
A dozen voices, led by her husband's, implored Arete to hold. Harmonia sobbed uncontrollably. Rooster looked on, still bound, paralyzed with horror.
Such a fierceness stood now in the lady's eye as must have informed Medea herself as she poised the steel of slaughter above her own babes.
"Ask my husband if this child is his," Arete demanded again. "Ask him!"
A chorus of refusal greeted this. Yet what alternative did the Peers possess? Each eye now swung to Dienekes, not so much in demand that he respond to this ridiculous accusation, as simply because they were flummoxed by the lady's temerity and did not know what else to do.
"Tell them, my husband," Arete spoke softly. "Before the gods, is this child yours?"
Arete released her hand upon the blade. She swung the babe away from Polynikes' sword and held him out before her husband.
The Peers knew the lady's assertion could not be true. Yet, if Dienekes so testified, and under oath as Arete demanded, it must be accepted by all, and by the city as well, or his holy honor would be forfeit. Dienekes understood this too. He peered for a long moment into his wife's eyes, which met his, as Medon's image had so aptly suggested, like those of a lioness.
"By all the gods," Dienekes swore, "the child is mine."
Tears welled in the lady Arete's eyes, which she at once quelled.
The Peers murmured at this defilement of the oath of honor.
Medon spoke. "Consider what you are saying, Dienekes. You defame your wife by attesting to this 'truth' and yourself by swearing to this falsehood."
"I have considered, my friend," Dienekes responded.
He restated that the child was his.
"Take him, then," Arete directed at once, advancing the final pace before her husband and placing the babe gently into his grasp. Dienekes accepted the bundle as if he'd been handed a litter of serpents.
He glanced again, for a long moment, into the eyes of his wife, then turned and addressed the Peers.
"Which of you, friends and comrades, will sponsor my son and enroll him before the ephors?"
Not a peep. It was a dreadful oath to which their brother-in-arms had sworn; would they, seconding him, be impeached by it as well?
"It will be my privilege to stand up for the child," Medon spoke. "We will present him tomorrow. His name as the lady wishes shall be Idotychides, as was her brother's."
Harmonia wept with relief.
Rooster glared at the assembly with helpless rage.
"Then it is settled," said Arete. "The child will be raised by his mother within the walls of my husband's home. At seven years he will enter the Upbringing as a _mothax_ and be trained as any other blood issue of a citizen. If he proves worthy in virtue and discipline, he will when he reaches manhood receive his initiation and take his place as a warrior and defender of Lakedaemon."
"So be it," assented Medon, and the others of the mess, however reluctantly, agreed.
It was not yet over.
"This one," Polynikes indicated Rooster. "This one dies."
The warriors of the _krypteia_ now hauled Rooster to his feet. None of the mess raised a hand in his defense. The assassins commenced to drag their captive toward the shadows. In five minutes he would be dead. His body would never be found.
"May I speak?"
This from Alexandros, advancing to intercept the executioners. "May I address the Peers of the mess?"
Medon, the eldest, nodded his assent.
Alexandros indicated Rooster. "There is another way to deal with this renegade which may, I suggest, prove of greater utility to the city than summarily to dispatch him. Consider: Many among the helots honor this man. His death by assassination will make him in their eyes a martyr. Those who call him friend may for the moment be cowed by the terror of his execution but later, in the field against the Persian, their sense of injustice may find an outlet opposed to the interests of Hellas and of Lakedaemon. They may prove traitor under fire, or work harm to our warriors when they are most vulnerable."
Polynikes interrupted with anger. "Why do you defend this scum, son of Olympieus?"
"He is nothing to me," Alexandros replied. "You know he holds me in contempt and considers himself a braver man than I. In this judgment he is doubtless correct."
The Peers were abashed by this candor, expressed so openly by the young man. Alexandros continued.
"Here is what I propose: Let this helot live, but go over to the Persian. Have him escorted to the frontier and cut loose. Nothing could suit his seditious purposes more; he will embrace the prospect of dealing harm to us whom he hates. The enemy will welcome a runaway slave. Them he will provide with all the intelligence he wishes about the Spartans; they may even arm him and allow him to march beneath their banner against us. But nothing he says can injure our cause, since Xerxes already has among his courtiers Demaratos, and who can give better intelligence of the Lakedaemonians than their own deposed king?
"The defection of this youth will work no harm to us, but it will accomplish something of inestimable value: it will prevent him from being viewed by his fellows in our midst as a martyr and a hero. He will be seen by them for what he is, an ingrate who was offered a chance to wear the scarlet of Lakedaemon and who spurned it out of pride and vainglory.
"Let him go, Polynikes, and I promise you this: if the gods grant that this villain come before us again on the field of battle, then you will have no need to slay him, for I will do it myself."
Alexandros finished. He stepped back. I glanced to Olympieus; his eyes glistened with pride at the case so concisely and emphatically put forward by his son.
The _polemarch_ addressed Polynikes. "See to it."
The _krypteis_ hauled Rooster away.
Medon broke up the assembly with orders to the Peers to disperse at once to their berths or homes and repeat nothing of what had transpired here, until tomorrow at the proper hour before the ephors. He upbraided the lady Arete sternly, admonishing her that she had tempted the gods sorely this evening. Arete, now chastened and beginning to experience that quaking of the limbs which all warriors know in the aftermath of battle, accepted the elder's chastisement without protest. As she turned her path toward home, her knees failed. She stumbled, faint, and had to be braced up by her husband, who stood at her side.
Dienekes wrapped his cloak about his wife's shoulders. I could see him regard her keenly while she struggled to reclaim her self-command. A portion of him still burned, furious at her for what she had forced him to do tonight. But another part stood in awe of her, at her compassion and audacity and even, if the word may be applied, her generalship.
The lady's equilibrium returned; she glanced up to discover her husband studying her. She smiled for him. "Whatever deeds of virtue you have performed or may yet perform, my husband, none will exceed that which you have done this night."
Dienekes appeared less than convinced.
"I hope you're right," he said.
The Peers had now departed, leaving Dienekes beneath the oaks with the babe still in the crook of his arm, about to hand it back to its mother.
Medon spoke. "Let's have a look at this little bundle."
In the starlight the elder advanced to my master's shoulder. He took the infant and passed it gently across to Harmonia. Medon examined the little fellow, extending a war-scarred forefinger, which the boy clasped in his strong infant's fist and tugged upon with vigor and pleasure. The elder nodded, approving. He caressed the babe's crown once in tender benediction, then turned back with satisfaction toward the lady Arete and her husband.
"You have a son now, Dienekes," he said. "Now you may be chosen."
My master regarded the elder quizzically, uncertain of his meaning.
"For the Three Hundred," Medon said. "For Thermopylae."
**EIGHTEEN**
**H** _is Majesty read with great interest these words of the Greek Xeones which I, His historian, placed before Him in their transcribed form. The army of Persia had advanced by this date deep into Attika and made camp at that crossroads called by the Hellenes the Three-Cornered Way, two hours' march northwest of Athens. There His Majesty made sacrifice to God Ahura Mazda and distributed decorations for valor to the leading men among the Empire's forces. His Majesty had not for the preceding several days summoned into His presence the captive Xeones to hear from him in person the continuance of his tale, so consumed was He with the myriad affairs of the army and navy in the advance. Yet did His Majesty not fail to follow the narrative in His spare hours, studying it in this, the transcribed form in which His historian daily submitted it._
_In fact His Majesty had not been well for the previous several nights. His sleep had been troubled; the attendance of the Royal Surgeon had been summoned. His Majesty's rest was disturbed by untoward dreams whose content He divulged to no one, save the Magi and the circle of His most trusted counselors: the general Hydarnes, commander of the Immortals and victor at Thermopylae; Mardonius, field marshal of His Majesty's land forces; Demaratos, the deposed Spartan king and now guest-friend; and the warrioress Artemisia, queen of Halicarnassus, whose wisdom in counsel His Majesty esteemed beyond all others'._
_The incubus of these troublous dreams, His Majesty now confided, appeared to be His own remorse over the desecration, following the victory at the Hot Gates, of the body of the Spartan Leonidas. His Majesty reiterated his regret at the defilement of the corpse of this warrior who was, before all, a king._
_The general Mardonius beseeched His Majesty to recall that He had scrupulously observed all sacred ritual prescribed to expiate the lingering vapors of blood guilt, if in fact any such had been incurred. Had not His_ _Majesty subsequently ordered the execution of all those of the royal party, including His own son, the prince Rheodones, who had participated in the event? What more needed doing? Yet despite all this, His Majesty declared, the royal slumber remained restless and unsound. His Majesty in wistful tone expressed the fancy that He, perhaps in induced visions or seantic trance, might acquaint Himself personally with the shade of the man Leonidas and share with him a cup of wine._
_A silence of no short duration followed. "This fever," the general Hydarnes ventured at last, "has dulled Your Majesty's edge of command and compromised its keenness. I beg Your Majesty speak no more in this manner."_
_"Yes, yes, you're right, my friend," His Majesty replied. "As you are always."_
_The commanders turned their attention to matters military and diplomatic. Reports were delivered. The advance force of Persian infantry and cavalry, fifty thousand strong, had entered Athens and taken possession of the city. The Athenian citizenry had abandoned the place utterly, betaking themselves, with only those goods which they could bear upon their persons, by sail across the strait to Troezen and the island of Salamis, where they now held themselves as refugees, huddling about fires upon the hillsides and bewailing their sorrows._
_The city itself had offered no resistance, save that of a small band of fanatics who occupied the High City, the Acropolis, whose precincts in ancient times had been bounded by a wooden palisade. These desperate defenders had fortified themselves in this site, placing, it seems, their faith in the oracle of Apollo which some weeks previous had declared,_
"...the wooden wall alone shall not fail you."
_These lamentable remnants were routed easily by imperial archers, who slew them at a distance. So much, Mardonius decreed, for the prophecy. The bivouac fires of the Empire now burned upon the Athenian acropolis. Tomorrow His Majesty Himself would enter the city. Plans were approved for the razing of all temples and sanctuaries of the Hellenic gods and the torching of the remainder of the city. The smoke and flames, it was reported by the intelligence officer, would be visible across the strait to the_ _Athenian populace now cowering in the high goat pastures upon the island of Salamis. "They will have a front-row seat," the lieutenant said with a smile, "at the annihilation of their universe."_
_The hour had now grown late, and His Majesty had begun to display indications of fatigue. The Magus, observing, suggested that the evening might now profitably be brought to a close. All rose from their couches, prostrated themselves and made their exit, save the general Mardonius and Artemisia, whom by subtle gesture of His Majesty's hand were bade to stay. His Majesty indicated that His historian would remain as well, to record the proceedings. Clearly His Majesty's peace was troubled._
_Now alone in the tent with His two closest confidants, He spoke, relaying a dream._
_"I was on a battlefield, which seemed to extend to infinity, and over which the corpses of the slain spread beyond sight. Cries of victory filled the air; generals and men were vaunting triumphantly. Abruptly I espied the corpse of Leonidas, decapitated, with its head impaled upon a spike, as we had done at Thermopylae, the body itself nailed as a trophy to a single barren tree in the midst of the plain. I was seized with grief and shame. I raced toward the tree, shouting to my men to cut the Spartan down. In the dream it seemed that, if I could only reaffix the king's head, all would be well. He would revive, and even befriend me, which outcome I dearly desired. I reached to the spike, upon which the severed head sat impaled..."_
_"And the head was His Majesty's own," the lady Artemisia broke in._
_"Is the dream that transparent?" His Majesty inquired._
_"It is nothing and signifies nothing," the warrioress declared emphatically, continuing in a tone that deliberately made light of the matter and urged His Majesty with all speed to put it from his mind. "It means only that His Majesty, who is a king, recognizes the mortality of all kings, Himself included. This is wisdom, as Cyrus the Great Himself expressed when he spared the life of Croesus of Lydia."_
_His Majesty considered Artemisia's words for long moments. He wished by them to be convinced, yet, it was apparent, they had not succeeded entirely in stanching His concern._
_"Victory is yours, Your Majesty, and nothing can take it from you," the general Mardonius now spoke. "Tomorrow we will burn Athens,_ _which was the goal of your father, Darius, and your own and the reason you have assembled this magnificent army and navy and have toiled and struggled for so long and overcome so many obstacles. Rejoice, my lord! All Greece lies prostrate before you. You have defeated the Spartans and slain their king. The Athenians you have driven before you like cattle, compelling them to abandon the temples of their gods and all their lands and possessions. You stand triumphant, Sire, with the sole of your slipper upon the throat of Greece."_
_So complete was His Majesty's victory, Mardonius declared, that the Royal Person need detain Itself not one hour longer here in these hellish precincts at the antipodes of the earth. "Leave the dirty work to me, Your Majesty. You yourself take ship home for Susa, tomorrow, there to receive the worship and adulation of your subjects, and to attend to the far more pressing matters of the Empire, which have been in favor of this Hellenic nuisance too long neglected. I will mop up for you. What your forces do in your name is done by you."_
_"And the Peloponnese?" the warrioress Artemisia put in, citing the southern peninsula of Greece, which alone of the whole country remained unsubdued. "What would you do with it, Mardonius?"_
_"The Peloponnese is a goat pasture," the general responded. "A desert of rocks and sheep dung, with neither riches nor spoil, nor a single port possessed of haven for more than a dozen garbage scows. It is nothing and contains nothing which His Majesty needs."_
_"Except Sparta."_
_"Sparta?" Mardonius replied contemptuously, and not without heat. "Sparta is a village. The whole stinking place would fit, with room to spare, within His Majesty's strolling garden at Persepolis. It is an up-country burg, a pile of stones. It contains no temples or treasures of note, no gold; it is a barnyard of leeks and onions, with soil so thin a man may kick through it with one strike of his foot."_
_"It contains the Spartans," the lady Artemisia spoke._
_"Whom we have crushed," Mardonius replied, "and whose king His Majesty's forces have slain."_
_"We slew three hundred of them," replied Artemisia, "and it took two million of us to do it."_
_These words so incensed Mardonius that he seemed upon the point of rising from his couch to confront Artemisia physically._
_"My friends, my friends." His Majesty's conciliating tone made to quell the momentary upset. "We are here to take counsel, not brawl with each other like schoolchildren."_
_Yet the lady's fervor still burned. "What is that between your legs, Mardonius, a turnip? You speak like a man with balls the size of chickpeas."_
_She addressed Mardonius directly, controlling her anger and speaking with precision and clarity. "His Majesty's forces have not even sighted, let alone confronted or defeated, the main force of the Spartan army, which remains intact within the Peloponnese and no doubt in full preparedness, and eagerness, for war. Yes, we have killed a Spartan king. But they, as you know, have two; Leotychides now reigns, and Leonidas' son, the boy Pleistarchus; his uncle and regent, Pausanias, who will lead the army and whom I know, is every inch the equal of Leonidas in courage and sagacity. So the loss of a king means nothing to them, other than to harden their resolve and inspire them to yet greater prodigies of valor as they seek to emulate his glory._
_"Now consider their numbers. The Spartiate Peers alone comprise eight thousand heavy infantry. Add the Gentleman-Rankers and the_ perioikoi _and the tally multiplies by five. Arm their helots, which they most certainly will do, and the total swells by another forty thousand. To this stew toss in the Corinthians, Tegeates, Eleans, Mantineans, Plataeans and Megarians, and the Argives, whom these others will compel into alliance if they have not done so already, not to mention the Athenians, whose backs we have driven to the wall and whose hearts are primed with the valor of desperation."_
_"The Athenians are ashes," Mardonius broke in. "As will be their city before tomorrow's sun sets."_
_His Majesty appeared of two minds, torn between the prudence of his general's counsel and the passion of the warrioress's advice. He turned to Artemisia. "Tell me, my lady, is Mardonius right? Ought I to settle myself upon pillows and take ship for home?"_
_"Nothing could be more disgraceful, Your Majesty," the lady replied_ _without hesitation, "nor more unworthy of your own greatness." She rose to her feet now and spoke, pacing before His Majesty beneath the arcing linen of His pavilion._
_"Mardonius has recited the names of the Hellenic cities which have offered tokens of submission, and these I admit are not inconsiderable. But the flower of Hellas remains unplucked. The Spartans' nose we have barely bloodied, and the Athenians, though we have driven them from their lands, remain an intact_ polis _and a formidable one. Their navy is two hundred warships, by far the greatest in Hellas, and every vessel is manned by crack citizen crews. These may bear the Athenians anywhere in the world, where they may reestablish themselves undiminished, as potent a threat to Your Majesty's peace as ever. Nor have we depleted their manpower. Their hoplite army remains untouched, and their leaders enjoy the full respect and support of the city. We delude ourselves to underestimate these men, whom His Majesty may not know but whom I do. Themistokles, Aristides, Xanthippus the son of Ariphron; these are names of proven greatness, fired and ardent to earn more._
_"As for the poverty of Greece, what Mardonius says cannot be controverted. There is neither gold nor treasure upon these hardscrabble shores, no rich lands nor fat flocks to plunder. But are these why we came? Are these the reason Your Majesty levied and marshaled this army, the mightiest the world has ever seen? No! Your Majesty came to bring these Greeks to their knees, to compel them to offer earth and water, and this, these last defiant cities have refused and yet refuse to do._
_"Put this fatigue-spawned dream from your mind, Your Majesty. It is a false dream, a phantasm. Let the Greeks degrade themselves by resort to superstition. We must be men and commanders, exploiting oracles and portents when they suit the purposes of reason and dismissing them when they do not._
_"Consider the oracle which the Spartans were given, which all Hellas knows, and which they know we know. That either Sparta would lose a king in battle, which calamity had never in six hundred years befallen them, or the city herself would fall._
_"Well, they have lost a king. What will their seers make of this, Your Majesty? Clearly that the city now cannot fall._
_"If you retire now, Lord, the Greeks will say it was because you feared a dream and an oracle."_
_She drew up then, before His Majesty, and addressed these words directly to Him. "Contrary to what our friend Mardonius says, His Majesty has not yet claimed His victory. It dangles before Him, a ripe fruit waiting to be plucked. If His Majesty retires now to palaced luxury and leaves this prize to be taken by others, even those whom He most honors and holds dearest to Him, the glory of this triumph is tarnished and defamed. Victory cannot simply be declared, it must be won. And won, if I may say so, in person._
_"Then, and only then, may His Majesty with honor take ship and return home."_
_The warrior queen finished and resumed her position upon her couch. Mardonius offered no rebuttal. His Majesty looked from one to the other._
_"It seems my women have become men, and my men women."_
_His Majesty spoke not in rancor or disapprobation, but stretching His right hand across, He settled it with affection upon the shoulder of His friend and kinsman Mardonius, as if to reassure the general that His confidence in him remained steadfast and undiminished._
_His Majesty then straightened and with forcefulness of voice and demeanor reassumed His kingly tone._
_"Tomorrow," He vowed, "we will burn Athens to the ground and, following that, march upon the Peloponnese, there to overthrow the very foundation stones of Sparta, not ceasing until we have ground them, everlastingly, into dust."_
**NINETEEN**
**H** _is Majesty did not sleep that night. Instead He ordered the Greek Xeones summoned to Him at once, intending even at this advanced hour to interrogate the man personally, seeking further intelligence of the Spartans, who now, more so even than the Athenians, had become the focus of His Majesty's fever and obsession. The warrioress Artemisia had along with Mardonius been dismissed and was at that moment taking her leave; upon hearing these orders of His Majesty she turned back and spoke with concern for Him._
_"Sire, please, for the sake of the army and of those who love you, I beg you preserve the Royal Person, for godly though Your Majesty's spirit may be, yet it is contained within a mortal vessel. Get some sleep. Do not torment yourself with these cares, which are mere phantoms."_
_The general Mardonius seconded this with vehemence. "Why distress yourself, Lord, with this tale told by a slave? What bearing can the story of obscure officers and their petty internecine wars have upon the events of supreme moment to which we now are committed? Trouble yourself no more with this whimsy woven by a savage, who hates you and Persia with every element of his being. His story is all lies anyway, if you ask me."_
_His Majesty smiled at these words of his general. "On the contrary, my friend, I believe this fellow's tale is true in every regard and, though you may not yet grant it, very much to the point of matters with which we now grapple."_
_His Majesty indicated His campaign throne, which stood in the lamplight beneath the pinnacle of the tent. "Do you see that chair, my friends? No mortal can be lonelier or more isolated than He who sits upon it. You cannot appreciate this, Mardonius. None can who has not sat there._
_"Consider: Whom can a king trust who comes into His hearing? What man enters before Him but with some secret desire, passion, grievance or_ _claim, which he employs all his artifice and guile to conceal? Who speaks the truth before a king? A man addresses Him either in fear for that which He may seize or in avarice for that which He may bestow. None comes before Him but as a suppliant. His heart's business the flatterer speaks not aloud, but all he obscures beneath the cloak of dissemblance and dissimulation._
_"Each voice vowing allegiance, each heart declaring love, the Royal Listener must probe and examine as if He were a vendor in a bazaar, seeking the subtle indices of betrayal and deceit. How tiresome this becomes. A king's own wives whisper sweetly to Him in the darkness of the royal bedchamber. Do they love Him? How can He know, when He perceives their true passion spent in scheming and intriguing for their children's advantage or their own private gain. None speaks the truth whole to a king, not His own brother, not even you, my friend and kinsman."_
_Mardonius hastened to deny this, but His Majesty cut him short with a smile. "Of all those who come before me, only one man, I believe, speaks without desire for private profit. That is this Greek. You do not understand him, Mardonius. His heart yearns for one thing only: to be reunited with his brothers-in-arms beneath the earth. Even his passion to tell their story is secondary, an obligation imposed upon him by one of his gods, which is to him a burden and a curse. He seeks nothing from me. No, my friends, the Greek's words do not trouble or distress. They please. They restore."_
_His Majesty, standing then at the threshold of the pavilion, gestured past the guard of the Immortals to the watch fires glowing without._
_"Consider the crossing at which we now stand encamped, that site the Hellenes call the Three-Cornered Way. It would be nothing to us, mere dirt beneath our feet. Yet is not this humble plot given meaning, and even charm, to recall from the prisoner's tale that he, as a child, parted here from the maiden Diomache, his cousin whom he loved?"_
_Artemisia exchanged a glance with Mardonius._
_"His Majesty yields to sentiment," the lady addressed her King, "and fatuous sentiment at that."_
_At this moment the service portal of the pavilion parted and permission to enter was asked by the detention officers. The Greek was borne in, yet upon his litter, eyes cloth-bound as ever, by two subalterns of the Immortals preceded by Orontes, their captain._
_"Let us see the man's face," His Majesty commanded, "and may his eyes behold ours."_
_Orontes obeyed. The cloth was removed._
_The captive Xeones blinked several times in the lamplight, then looked for the first time upon His Majesty. So striking was the expression which then appeared upon the man's face that the captain remarked angrily upon it and demanded to know what arrogance possessed the fellow to stare so boldly at the Royal Person._
_"I have looked upon His Majesty's face before," the man replied._
_"Above the battle, as all the foe have."_
_"No, Captain. Here, in this tent. On the night of the fifth day."_
_"You are a liar!" Orontes struck the man in anger. For the breach to which the captive referred had in fact occurred, on the penultimate eve of battle at the Hot Gates, when a night raid of the Spartans bore a handful of their warriors within a spear's thrust of the Royal Presence, inside this very pavilion, before the intruders were driven back by the Immortals and Egyptian marines swarming to His Majesty's defense._
_"I was here," the Greek responded calmly, "and would have had my skull split apart by an axe, hurled at me by a noble, had it not struck first a ridgepole of the tent and embedded itself there."_
_At this, the general Mardonius' face lost all color. In the west portal of the chamber, precisely where the Spartan raid had penetrated, was lodged yet an axehead, driven so deep into the cedar that it could not be extracted without splitting the pole, and so had been left in place by the carpenters, sawn off at the shaft, with the pole repaired and rewound about it with cord._
_The Hellene's gaze now centered directly upon Mardonius. "This lord here threw that axe. I recognize his face as well."_
_The general's expression, for the moment struck dumb, betrayed the truth of this._
_"His sword," the Greek continued, "severed the wrist of a Spartiate warrior, at the moment of drawing back his spear to thrust at His Majesty."_
_His Majesty inquired of Mardonius if this indeed was true. The general confirmed that he had in fact inflicted such a wound upon an advancing_ _Spartan, among numerous others delivered in those moments of confusion and peril._
_"That warrior," the man Xeones declared, "was Alexandros, the son of Olympieus, of whom I spoke."_
_"The boy who followed the Spartan army? Who swam the channel before Antirhion?" Artemisia asked._
_"Grown to manhood," the Greek confirmed. "Those officers who bore him from this tent protected by the shadows of their shields, those were the Knight Polynikes and my master, Dienekes."_
_All paused for several moments, absorbing this._
_His Majesty spoke: "These truly were the men who penetrated here, into this tent?"_
_"They and others, Lord. As His Majesty saw."_
_The general Mardonius received this intelligence with skepticism bordering on outrage. He accused the prisoner of fabricating this tale from snatches he had overheard from the cooks or medical personnel who attended him. The captive denied this respectfully but with vehemence._
_Orontes, responding to Mardonius in his capacity as Commander of the Guard, proclaimed it inconceivable that the Greek could have acquired knowledge of these events in the manner the general suggested. The captain himself had personally overseen the prisoner's sequestration. No one, either of the commissariat or of the Royal Surgeon's staff, had been allowed alone with the man, even for a moment, without the immediate supervision of His Majesty's Immortals, and these were, as all knew, without peer in scruple and attendance._
_"Then he has this tale from the rumor mill of the battle," Mardonius rejoined, "from the Spartan warriors who did in fact breach His Majesty's line."_
_All attention now swung to the captive Xeones, who, quite undistressed by these accusations which could have produced his death upon the spot, regarded Mardonius with level gaze and addressed him without fear._
_"I might have learned of these events, lord, in the manner which you suggest. But how, sir, would I know to recognize you, of all these others, as the man who hurled the axe?"_
_His Majesty had now crossed to the spot where the axehead was embed_ _ded and with His dagger cut through the enwrapping cord to expose the weapon. Upon the steel of the axehead, His Majesty identified the double-headed griffin of Ephesus, whose corps of armorers' privilege it was to provide all edged blades and lances for Mardonius and his commanders._
_"Tell me now," His Majesty addressed the general, "that no god's hand is at work here."_
_His Majesty declared that He and His counselors had already from the captive's tale gleaned much that was as instructive as it was unanticipated. "How much more of value may we yet learn?"_
_With a gesture of warmth, His Majesty motioned the man Xeones closer to Him and had the fellow's still gravely ill form propped upon a settle._
_"Please, my friend, continue your tale. Tell it as you wish, in whatever manner the god instructs you."_
**TWENTY**
**I** had watched the army marshal on the plain beneath Athena of the Brazen House perhaps fifty times over the previous nine years, in various strengths of call-up as it prepared to march out to one campaign or another. This one, the corps dispatched to the Hot Gates, was the slenderest ever. Not a two-thirds call-up as before Oenophyta, when nearly six thousand warriors, squires and their battle train had filled the plain, nor a half-mobilization, forty-five hundred, as before Achilleion, nor even a two- _mora,_ twenty-five hundred, when Leonidas led the force to Antirhion which Alexandros and I had followed as boys.
Three hundred.
The meager tally seemed to rattle about the plain like peas in a jar. Just three dozen pack animals stood in the fore along the roadway. There were only eight waggons; the sacrificial herd was marshaled by two scared-looking goat boys. Supply trains had already been dispatched and dumps set up along the six-day route. In addition, it was anticipated, the allied cities would provide provisions along the way, as the Spartan forerunners picked up the various contingents which would complete the force and bring it to its full complement of four thousand.
An august silence pervaded the valedictory sacrifices performed by Leonidas in his role as chief priest, assisted by Olympieus and Megistias, the Theban seer who had come to Lakedaemon of his own volition, with his son, out of love not of his native city alone, but of all Hellas, to contribute without fee or reward his art in divination.
The entire army, all twelve _lochoi,_ had been drawn up, not under arms because of the Karneian prohibition but yet in their scarlet cloaks, to witness the march-out. Each warrior of the Three Hundred stood garlanded, _xiphos_ -armed with shield at the carry, scarlet cloak draped across his shoulders, while his squire stood at his side holding the spear until the sacrifices were completed. It was the month, as I said, of Karneius, the new year having begun at midsummer as it does in the Greek calendar, and each man was due to receive his new cloak for the year, replacing the now-threadbare one he had worn for the previous four seasons. Leonidas ordered the issue discontinued for the Three Hundred. It would be an improvident use of the city's resources, he declared, to provide new garments for men who would have use of them for so brief a time.
As Medon had predicted, Dienekes was chosen as one of the Three Hundred. Medon himself was selected. At fifty-six he was the fourth oldest, behind only Leonidas himself, who was past sixty, Olympieus and Megistias the seer. Dienekes would command the _enomotia_ from the Herakles regiment. The brothers and champions of Olympia, Alpheus and Maron, were likewise selected; they would join the platoon representing the Oleaster, the Wild Olive, whose position would be to the right of the Knights, in the center of the line. Fighting as a _dyas,_ the paired pentathlete and wrestler towered invincible; their inclusion greatly heartened one and all. Aristodemos the envoy was also selected. But most startling and controversial was the election of Alexandros.
At twenty he would be the youngest line warrior and one of only a dozen, including his _agoge_ -mate Ariston (of Polynikes' "broken noses"), without experience of battle. There is a proverb in Lakedaemon, "the reed beside the staff," whose meaning is that a chain is made stronger by its possession of one unproven link. The tender hamstring that drives the wrestler to compensate with skill and cunning, the lisp that the orator extends his brilliance to overcome. The Three Hundred, Leonidas felt, would fight best not as a company of individual champions, but as a sort of army in miniature, of young and old, green and seasoned. Alexandros would join the platoon of the Herakles commanded by Dienekes; he and his mentor would fight as a _dyas._
Alexandros and Olympieus were the only father and son selected for the Three Hundred. Alexandros' infant boy, also named Olympieus, would be their survivor and maintain the line. It was a sight of extreme poignancy, there along the Aphetaid, the Going-Away Street, to watch Alexandros' bride, Agathe, only nineteen years old, hold up this babe for the final farewell. Alexandros' mother, Paraleia, she who had interrogated me so masterfully after Antirhion, stood beside the girl beneath the same myrtle grove from which Alexandros and I had departed that night years ago to follow the army.
Good-byes were said on the march as the formation trooped solemnly past the rubble-walled assembly platform called the Forts, beneath the hero shrines of Lelex and Amphiareus, to the road's turn at the Running Course, above which the boys' platoons clustered at Axiopoinos, the Temple of Athena of Just Requital, Athena Tit for Tat. I watched Polynikes bid his three lads farewell; the eldest at eleven and nine stood already in the _agoge._ They straightened within their black cloaks with the gravest dignity; each would have cut off his right arm for the chance to march now with his father.
Dienekes paused before Arete on the roadside adjacent the Hellenion, whose porches stood bedecked in laurel with ribands of yellow and blue for the Karneia; she held out Rooster's boy, now named Idotychides. My master took each of his daughters in turn into his arms, lifting the younger two and kissing them with tenderness. Arete he embraced one time, setting his cheek against her neck, to smell the scent of her hair for the last time.
Two days previous to this gentle moment, the lady had summoned me in private, as she always did before a march-out. It is the Spartans' custom during the week preceding a departure for war for the Peers to pass a day neither in training nor drill, but at their ease upon the _kleroi,_ the farmsteads each warrior holds under the laws of Lykurgus and from which he draws the produce which supports himself and his family as a citizen and a Peer. These "county days," as they are called, comprise a homely tradition deriving, reason must surmise, from the warrior's natural wish to revisit prior to battle, and in a sense bid farewell to, the happy scenes of his childhood. That and the more practical purpose, in the ancient days at least, of outfitting and provisioning himself from the _kleros_ 's stores. A county day is a fair, one of the rare occasions when a Peer and those who serve his land may congregate as fellows and stuff their bellies with a carefree heart. In any event this was where we headed, to the farmstead called Daphneion, several mornings before the march-out to the Gates.
Two families of Messenian helots worked this land, twenty-three in all, including a pair of grandmothers, twins, so ancient they could not recall which of them was which, plus the only slightly less dotty stump-leg Kamerion, who had lost his right foot in service as Dienekes' father's squire. This toothless gaffer could outswear the foulest-tongued sailor and presided at his own insistence, and to the delight of all, as master of ceremonies for the day.
My own wife and children served this farm as well. Neighbors visited from the adjoining landholdings. Prizes were awarded in whimsical categories; there was a country dance, outdoors on the threshing floor beside the laurel grove from which the farm derived its name, and various children's games were held, before the party settled in late afternoon to a communal feast beneath the trees, at which Dienekes himself and the lady Arete and their daughters did the serving. Gifts were exchanged, quarrels and grievances patched up, claims pressed and complaints aired. If a lad of the _kleros_ sought betrothal to his sweetheart from the overhill farm, he might approach Dienekes now and claim his blessing.
Invariably two or three of the sturdier helot youths and men would be slated to accompany the army, as craftsmen or armorers, battle squires or javelineers. Far from resenting or seeking to shirk these perils, the young bucks reveled in the manly attention; their sweethearts clung to them throughout the day, and many a proposal of marriage was spawned in the wine-merry amorousness of these bright country afternoons.
By the time the merry party had "put aside all desire for food and drink," as Homer says, more grain and fruit, wine and cakes and cheeses had been heaped at Dienekes' feet than he could carry into a hundred battles. He now retired to the courtyard table, with the elders of the farm, to conclude whatever details remained to set the affairs of the _kleros_ in order before the march-out.
It was when the men had turned themselves to this business that the lady Arete motioned me to join her in private. We sat before a table in the farm kitchen. It was a cheerful spot, warm in the late sun that flooded through the courtyard doorway. The lad Idotychides, Rooster's boy, played outside with two other naked urchins, including my own son, Skamandridas. The lady's eyes rested for a moment, with sorrow it seemed, upon the roughhousing little fellows.
"The gods remain always a jump ahead of us, don't they, Xeo?"
This was the first hint I had received from her lips confirming that which none possessed the courage to ask: that the lady had indeed not foreseen the consequences of her action, that night of the _krypteia,_ when she had saved the babe's life.
She cleared a space upon the table. Into my care the lady placed, as ever, those articles of her husband's kit which it was a wife's responsibility to provide. The surgeon's packet, bound in the thick oxhide roll that doubled as a wrap for a splint or, bound flat atop the flesh, as seal for a puncture. The three curved needles of Egyptian gold, called by the Spartans "fishhooks," with their spool of catgut twine and steel lancet, for use in the tailor's art of sewing flesh. The compresses of bleached linen, the tourniquet binders of leather, the copper "dog bites," the needle-nosed grippers for extracting arrowheads or, more often, the shards and slivers that fly from the clash of steel upon iron and iron upon bronze.
Next, money. A cache of Aeginetan obols that, as all coin or currency, the warriors were forbidden to carry but which, discovered serendipitously within a squire's pack, would come in handy at some on-route market or beside the sutler's waggon, to procure forgotten necessaries or to purchase a treat to lift the heart.
Finally those articles of purely personal significance, the little surprises and charms, items of superstition, the private talismans of love. A girl's sketch in colored wax, a riband from a daughter's hair, a charm in amber carved by a child's untutored hand. Into my care the lady placed a packet of sweets and trifles, sesame cakes and candied figs. "You may rifle your share," she said, smiling, "but save a few for my husband."
There was always something for me. This day it was a pouch of coins of the Athenians, twenty in all, tetradrachms, nearly three months' pay for a skilled oarsman or hoplite of their army. I was astonished that the lady possessed such a sum, even of her own purse, and struck dumb at her extravagant generosity. These "owls," as they were called from the image on their obverse, were good not just in the city of Athena but anywhere in Greece.
"When you accompanied my husband on embassy to Athens last month," the lady broke my dumbstruck silence, "did you find occasion to visit your cousin? Diomache. That is her name, isn't it?"
I had and she knew it. This wish of mine, long-sought, had indeed at last been fulfilled. Dienekes had dispatched me upon the errand himself. Now I glimpsed a hint of the lady's pot-stirring. I asked if it was she, Arete, who had contrived it all.
"We wives of Lakedaemon are forbidden fine gowns or jewelry or cosmetics. It would be heartless in the extreme, don't you think, to ban as well a little innocent intrigue?"
She smiled at me, waiting.
"Well?" she asked.
"Well, what?"
My wife, Thereia, was gossiping with the other farm women, out in the courtyard. I squirmed. "My cousin is a married woman, lady. As I am a married man."
The lady's eyes threw sparks of mischief. "You would not be the first husband bound by love to someone other than his wife. Nor she the first wife."
At once all teasing gaiety fled from the lady's glance. Her features became grave and shadowed, it seemed, with sorrow.
"The gods played the same trick on my husband and me." She rose, indicating the door and the courtyard beyond. "Come, let us take a walk."
The lady led barefoot up the slope to a shady spot beneath the oaks. In what country other than Lakedaemon would a noblewoman's soles be so thick with callus that they may tread upon the spiky leaves of oak and not feel their spiny barbs?
"You know, Xeo, that I was wife to my husband's brother before I was married to him."
This I did know, having learned it, as I said, from Dienekes himself.
"Iatrokles was his name, I know you have heard the story. He was killed at Pellene, a hero's death, at thirty-one. He was the noblest of his generation, a Knight and a victor at Olympia, gifted by the gods with virtue and beauty much like Polynikes in this generation. He pursued me passionately, with such impetuousness that he called me from my father's house when I was still a girl. All this the Spartans know. But I will tell you something now which no one, except my husband, knows."
The lady had reached a low bole of oak, a natural bench within the shade of the grove. She sat and indicated that I should take the place beside her.
"Down there," she said, gesturing to the open space between two outbuildings and the track that led to the threshing floor. "Right there where the path turns was where I first saw Dienekes. It was on a county day just like this. The occasion was Iatrokles' first march-out. He was twenty. My father had brought me and my brother and sisters over from our own _kleros_ with gifts of fruit and a yearling goat. The boys of the farm were playing, right there, when I came, holding my father's hand, over this knoll where you and I now sit."
The lady drew up. For a moment she searched my eyes, as if to make certain of their attention and understanding.
"I saw Dienekes first from behind. Just his bare shoulders and the back of his head. I knew in an instant that I would love him and only him all my life."
Her expression grew sober before this mystery, the summons of Eros and the unknowable workings of the heart.
"I remember waiting for him to turn, so I could see him, see his face. It was so odd. In a way it was like an arranged match, where you wait with your heart fluttering to behold the face you will and must love.
"At last he turned. He was wrestling another boy. Even then, Xeo, Dienekes was unhandsome. You could hardly believe he was his brother's brother. But to my eyes he appeared _eueidestatos,_ the soul of beauty. The gods could not have crafted a face more open or touching to my heart. He was thirteen then. I was nine."
The lady paused for a moment, gazing solemnly down at the spot of which she spoke. The occasion did not present itself, she declared, throughout her whole girlhood when she could speak in private with Dienekes. She observed him often on the running courses and in the exercises with his _agoge_ platoon. But never did one share a moment with the other. She had no idea if he even knew who she was.
She knew, however, that his brother had chosen her and had been speaking with the elders of her family.
"I wept when my father told me I had been given to Iatrokles. I cursed myself for the heartlessness of my ingratitude. What more could a girl ask than this noble, virtuous man? But I could not master my own heart. I loved the brother of this man, this fine brave man I was to marry.
"When Iatrokles was killed, I grieved inconsolably. But the cause of my distress was not what people thought. I feared that the gods had answered by his death the self-interested prayer of my heart. I waited for Dienekes to choose a new husband for me as was his obligation under the laws, and when he didn't, I went to him, shamelessly, in the dust of the _palaistra,_ and compelled him to take me himself as his bride.
"My husband embraced this love and returned it in kind, both of us over the still-warm bones of his brother. The delight was so keen between us, our secret joy in the marriage bed, that this love itself became a curse to us. My own guilt I could requite; it is easy for a woman because she can feel the new life growing inside her, that her husband has planted.
"But when each child was born and each a female, four daughters, and then I lost the gift to conceive, I felt, and my husband did too, that this was a curse from the gods for our passion."
The lady paused and glanced again down the slope. The boys, including my son and little Idotychides, had dashed out from the courtyard and now played their carefree sport directly below the site where we sat.
"Then came the summons of the Three Hundred to Thermopylae. At last, I thought, I perceive the true perversity of the gods' plan. Without a son, my husband cannot be called. He will be denied this greatest of honors. But in my heart I didn't care. All that mattered was that he would live. Perhaps for only another week or month, until the next battle. But still he would live. I would still hold him. He would still be mine."
Now Dienekes himself, his farm business completed within, emerged onto the flat below. There he joined playfully with the roughhousing boys, already obeying in their blood the instincts of battle and of war.
"The gods make us love whom we will not," the lady declared, "and disrequite whom we will. They slay those who should live and spare those who deserve to die. They give with one hand and take with the other, answerable only to their own unknowable laws."
Dienekes had now spotted Arete, watching him from above. He lifted the boy Idotychides playfully and made the lad's little arm wave up the slope. Arete compelled her own to answer.
"Now, inspired by blind impulse," she spoke toward me, "I have saved the life of this boy, my brother's bastard's son, and lost my husband's in the process."
She spoke these words so softly and with such sorrow that I felt my own throat catch and the burning begin in my eyes.
"The wives of other cities marvel at the women of Lakedaemon," the lady said. "How, they ask, can these Spartan wives stand erect and unblinking as their husbands' broken bodies are borne home to a grave or, worse, interred beneath some foreign dirt with nothing save cold memory to clutch to their hearts? These women think we are made of stauncher stuff than they. I will tell you, Xeo. We are not.
"Do they think we of Lakedaemon love our husbands less than they? Are our hearts made of stone and steel? Do they imagine that our grief is less because we choke it down in our guts?"
She blinked once, dry-eyed, then turned her glance to mine.
"The gods have played a game with you too, Xeo. But it may not be too late to steal a roll of the dice. This is why I have given you this pouch of 'owls.'"
Already I knew what her heart intended.
"You are not Spartan. Why should you bind yourself by her cruel laws? Haven't the gods stolen enough from you already?"
I begged her to speak of this no more.
"This girl you love, I can have her brought here. Just ask it."
"No! Please."
"Then run. Get out tonight. Go there."
I replied at once that I could not.
"My husband will find another to serve him. Let another die instead of you."
"Please, lady. This would be dishonor."
I felt my cheek sting and realized the lady had struck me. "Dishonor?" She spat the word with revulsion and contempt.
Down the slope the boys and Dienekes had been joined by the other lads of the farm. A game of ball had started. The boys' cries of _agon,_ of contention and competition, pealed brightly up the slope to where the lady sat.
One could feel only gratitude for that which had sprung so nobly from her heart: the wish to grant to me that clemency which she felt _moira,_ fate, had denied her. To grant to me and her whom I loved a chance to slip the bonds she felt herself and her husband imprisoned in.
I could offer nothing save that which she already knew. I could not go. "Besides, the gods would be there already. As ever, one jump ahead."
I saw her shoulders straighten then, as her will brought to heel the gallant but impossible impulse of her heart.
"Your cousin will learn where your body lies, and with what honor you perished. By Helen and the Twins, I swear this."
The lady rose from her bench of oak. The interview was over. She had become again a Spartan.
Now here on the morn of the march-out I beheld upon her face that same austere mask. The lady released her husband's embrace and gathered her children to her, resuming that posture, erect and solemn, replicated by the line of other Spartan wives extending fore and rear beneath the oaks.
I saw Leonidas embrace his wife, Gorgo, "Bright Eyes," their daughters, and his son, the boy Pleistarchus, who would one day take his place as king.
My own wife, Thereia, held me hard, grinding against me beneath her Messenian white robe, while she held our infants crooked in one arm. She would not be husbandless for long. "Wait at least until I'm out of sight," I joked, and held my children, whom I hardly knew. Their mother was a good woman. I wish I could have loved her as she deserved.
The final sacrifices were over, omens taken and recorded. The Three Hundred formed up, each Peer with a single squire, in the long shadows cast by distant Parnon, with the entire army in witness upon the shield-side slope. Leonidas assumed his place before them, beside the stone altar, garlanded as they. The remainder of the whole city, old men and boys, wives and mothers, helots and craftsmen, stood drawn up upon the spear-side rise. It was not yet daybreak; the sun still had not peeked above Parnon's crest.
"Death stands close upon us now," the king spoke. "Can you feel him, brothers? I do. I am human and I fear him. My eyes cast about for a sight to fortify the heart for that moment when I come to look him in the face."
Leonidas began softly, his voice carrying in the dawn stillness, heard with ease by all.
"Shall I tell you where I find this strength, friends? In the eyes of our sons in scarlet before us, yes. And in the countenances of their comrades who will follow in battles to come. But more than that, my heart finds courage from these, our women, who watch in tearless silence as we go."
He gestured to the assembled dames and ladies, singling out two matriarchs, Pyrrho and Alkmene, and citing them by name. "How many times have these twain stood here in the chill shade of Parnon and watched those they love march out to war? Pyrrho, you have seen grandfathers and father troop away down the Aphetaid, never to return. Alkmene, your eyes have held themselves unweeping as husband and brothers have departed to their deaths. Now here you stand again, with no few others who have borne as much and more, watching sons and grandsons march off to hell."
This was true. The matriarch Pyrrho's son Doreion stood garlanded now among the Knights; Alkmene's grandsons were the champions Alpheus and Maron.
"Men's pain is lightly borne and swiftly over. Our wounds are of the flesh, which is nothing; women's is of the heart—sorrow unending, far more bitter to bear."
Leonidas gestured to the wives and mothers assembled along the still-shadowed slopes.
"Learn from them, brothers, from their pain in childbirth which the gods have ordained immutable. Bear witness to that lesson they teach: nothing good in life comes but at a price. Sweetest of all is liberty. This we have chosen and this we pay for. We have embraced the laws of Lykurgus, and they are stern laws. They have schooled us to scorn the life of leisure, which this rich land of ours would bestow upon us if we wished, and instead to enroll ourselves in the academy of discipline and sacrifice. Guided by these laws, our fathers for twenty generations have breathed the blessed air of freedom and have paid the bill in full when it was presented. We, their sons, can do no less."
Into each warrior's hand was placed by his squire a cup of wine, his own ritual chalice, presented to him on the day he became a Peer and brought forth only for ceremonies of the gravest solemnity. Leonidas held his own aloft with a prayer to Zeus All-Conquering and Helen and the Twins. He poured the libation.
"In years six hundred, so the poets say, no Spartan woman has beheld the smoke of the enemy's fires."
Leonidas lifted both arms and straightened, garlanded, raising his countenance to the gods.
"By Zeus and Eros, by Athena Protectress and Artemis Upright, by the Muses and all the gods and heroes who defend Lakedaemon and by the blood of my own flesh, I swear that our wives and daughters, our sisters and mothers, will not behold those fires now."
He drank, and the men followed him.
**TWENTY-ONE**
**H** is Majesty is well familiar with the topography of the approaches, defiles and the compressed battle plain wherein his armies engaged the Spartans and their allies at the Hot Gates. I will pass over this, addressing instead the composition of the Greek forces and the state of discord and disorder which prevailed as these arrived and took station, preparatory to defending the pass.
When the Three Hundred—now reinforced by five hundred heavy infantry from Tegea and a matching number from Mantinea, along with two thousand combined from Orchomenos and the rest of Arkadia, Corinth, Phlius and Mycenae, plus seven hundred from Thespiae and four hundred from Thebes—arrived at Opountian Lokris, ten miles from the Hot Gates, there to be joined by a thousand heavy infantry from Phokis and Lokris, they found instead the entire countryside deserted.
Only a few boys and young men of the neighborhood remained, and these occupied themselves in looting the abandoned homes of their neighbors and appropriating whatever stores of wine they could disinter from their compatriots' caches. They took to their heels at the sight of the Spartans, but the rangers ran them down. The army and populace of Lokris, the pimply pillagers reported, had taken to the hills, while the locals' chieftains were scurrying north toward the Persians as fast as their spindly shanks would bear them. In fact, the urchins claimed, their headmen had already capitulated.
Leonidas was furious. It was determined, however, in a hasty and decidedly ungentle interrogation of these farmyard freebooters, that the Lokrians of Opus had gotten the day of assembly wrong. Apparently the month called Karneius in Sparta is named Lemendieon in Lokris. Further its start is counted backward from the full moon, rather than forward from the new. The Lokrians had expected the Spartans two days earlier and, when these failed to appear, determined that they had been left in the lurch. They bolted amid bitter oaths and maledictions, which the gales of rumor scattered swiftly into neighboring Phokis, in which country the Gates themselves are located and whose inhabitants were already in terror of being overrun. The Phokians had hightailed it too.
All along the march north, the allied column had encountered country tribes and villagers fleeing, streaming south along the military road, or what had now become a military road. Tattered clan groups fled before the Persian advance, bearing their pitiful possessions in shoulder sacks contrived from bedcovers or bundled cloaks, balancing their ragged parcels like water vessels atop their heads. Sunken-cheeked husbandmen wheeled handbarrows whose cargoes were more often flesh than furniture, children whose legs had given out from the tramp or bundled ancients hobbled with age. A few had oxcarts and pack asses. Pets and farm stock jostled underfoot, gaunt hounds cadging for a handout, doleful-looking swine being kicked along as if they knew they would be supper in a night or two. The main of the refugees were female; they trudged barefoot, shoes slung about their necks to save the leather.
When the women descried the allied column approaching, they vacated the road in terror, scrambling up the hillsides, clutching their infants and spilling possessions as they fled. There came always that moment when it broke upon these dames that the advancing warriors were their own countrymen. Then the alteration which overtook their hearts bordered upon the ecstatic. The women skittered back down the hardscrabble slopes, pressing tight about the column, some numb with wonder, others with tears coursing down their road-begrimed faces. Grandmothers crowded forward to kiss the young men's hands; farm matrons threw their arms about the necks of the warriors, embracing them in moments that were simultaneously poignant and preposterous. "Are you Spartans?" they inquired of the sun-blackened infantrymen, the Tegeates and Mycenaeans and Corinthians, Thebans, Philiasians and Arkadians, and many of these lied and said they were. When the women heard that Leonidas in person led the column, many refused to believe it, so accustomed had they become to betrayal and abandonment. When the Spartan king was pointed out to them and they saw the corps of Knights about him and at last believed, many could not bear the relief. They buried their faces in their hands and sank upon the roadside, overcome.
As the allies beheld this scene repeated, eight, ten, a dozen times a day, a grim urgency took possession of their hearts. All haste must be made; the defenders must at all costs reach and fortify the pass before the arrival of the enemy. Unordered, each man lengthened stride. The pace of the column soon outstripped the capacity of the train to keep up. The waggons and pack asses were simply left behind, to catch up as best they could, their necessaries transferred to the marching men's backs. For myself, I had stripped shoeless and rolled my cloak into a shoulder pad; my master's shield I bore in its hide case, along with his greaves and breastplate weighing something over sixty-five pounds, plus both our bedding and field kit, my own weapons, three quivers of ironheads wrapped in oiled goatskin and sundry other necessaries and indispensables: "fishhooks" and catgut; bags of medicinal herbs, hellebore and foxglove, euphorbia and sorrel and marjoram and pine resin; arterial straps, bindings for the hand, compresses of linen, the bronze "dogs" to heat and jam into puncture wounds to cauterize the flesh, "irons" to do the same for surface lacerations; soap, footpads, moleskins, sewing kit; then the cooking gear, a spit, a pot, a handmill, flint and firesticks; grit-and-oil for polishing bronze, oilcloth fly for rain, the combination pick and shovel called from its shape a _hyssax,_ the soldier's crude term for the female orifice. This in addition to rations: unmilled barley, onions and cheese, garlic, figs, smoked goat meat; plus money, charms, talismans.
My master himself bore the spare shield chassis with double bronze facings, both our shoes and strapping, rivets and instrument kit, his leather corselet, two ash and two cornelwood spears with spare ironheads, helmet, and three _xiphe,_ one on his hip and the other two lashed to the forty-pound rucksack stuffed with more rations and unmilled meal, two skins of wine and one of water, plus the "goodie bag" of sweetcakes prepared by Arete and his daughters, double-wrapped in oiled linen to keep the onion stink of the rucksack from invading it. All up and down the column, squire and man humped loads of two hundred to two hundred and twenty pounds between them.
The column had acquired a nonrostered volunteer. This was a roan-colored hunting bitch named Styx who belonged to Pereinthos, a Skirite ranger who was one of Leonidas' "king's selections." The dog had followed its master down to Sparta from the hills and now, having no home to return to, continued her attendance upon him. For an hour at a crack she would patrol the length of the column, all business, memorizing by scent the position of each member of the march, then returning to her Skirite master, who had now been nicknamed Hound, there to resume her tireless trot at his heel. There was no doubt that in the bitch's mind, all these men belonged to her. She was herding us, Dienekes observed, and doing a hell of a job.
With each passing mile, the countryside grew thinner. Everyone was gone. At last in the nation of the Phokians, nearing the Gates, the column entered country utterly denuded and abandoned. Leonidas dispatched runners into the mountain fastnesses to which the army of the locals had withdrawn, informing them in the name of the Hellenic Congress that the allies stood indeed upon the site and that it was their intention to defend the Phokians' and Lokrians' country whether these themselves showed their faces or not. The king's message was not inscribed upon the customary military dispatch roll, but scribbled on that kind of linen wrapper used to invite family and friends to a dance. The final sentence read: "Come as you are."
The allies themselves reached the village of Alpenoi that afternoon, the sixth after the march-out from Lakedaemon, and Thermopylae itself half an hour later. Unlike the countryside, the battlefield, or what would become the battlefield, stood far from deserted. A number of denizens of Alpenoi and Anthela, the north-end village that fronts the stream called Phoenix, had erected makeshift commercial ventures. Several had barley and wheat bread baking. One fellow had set up a grogshop. A pair of enterprising trollops had even established a two-woman brothel in one of the abandoned bathhouses. This became known at once as the Sanctuary of Aphrodite Fallen, or the "two-holer," depending upon who was seeking directions and who proffering them.
The Persians, the rangers reported, had not yet reached Trachis either by land or sea. The plain to the north lay yet unpopulated by the enemy camp. The fleet of the Empire, it was reported, had put out from Therma in Macedonia either yesterday or the day before. Their thousand warships were even now traversing the Magnesia coast, the advance elements expected to reach the landing beaches at Aphetae, eight miles north, within twenty-four hours.
The land forces of the enemy had departed Therma ten days earlier; their columns were advancing, so fugitives from the north reported, via the coastal and inland routes, cutting through forests as they came. Their rangers were anticipated at the same time as the fleet.
Now Leonidas emerged to the fore.
Before the allied camps were even staked out, the king dispatched raiding parties into the country of Trachis, immediately north of the Gates. These were to torch every stalk of grain and capture or drive off every piece of livestock, down to hedgehogs and barn cats, which might make a meal for the enemy.
In the wake of these raiders, reconnaissance parties were sent out, surveyors and engineers from each allied detachment, with orders to proceed as far north as the landing beaches likely to be appropriated by the Persians. These men were to map the area, as best they could in the gathering darkness, concentrating upon the roads and trails available to the Persian in his advance to the Narrows. Although the allied force possessed no cavalry, Leonidas made certain to include skilled horsemen in this party; though afoot, they could best assess how enemy cavalry might operate. Could Xerxes get horsemen up the trail? How many? How fast? How could the allies best counter this?
Further the reconnaissance parties were to apprehend and detain any locals whose topographical knowledge could be of assistance to the allies. Leonidas wanted yard-by-yard intelligence of the immediate northern approaches and, most crucial recalling Tempe, an ironclad assessment of the mountain defiles south and west, seeking any undiscovered track by which the Greek position could be outflanked and enveloped.
At this point a prodigy occurred which nearly broke the allies' will before they had even unshouldered their kits. An infantryman of the Thebans trod accidentally into a nest of baby snakes and received upon the bare ankle the full poison of half a dozen infant serpents, whose venom, all hunters know, is more to be feared than a full grown's because the young ones have not yet learned how to deal it out in doses, but instead inject it into the flesh in full measure. The infantryman died within the hour amid horrible sufferings, despite being bled white by the surgeons.
Megistias the seer was summoned while the stricken Theban writhed yet in the throes. The remainder of the army, ordered by Leonidas to assess at once the extension and reinforcement of the ancient Phokian Wall across the Gates, nonetheless amid their labors loitered with cold dread as the snakebitten man's life, emblematic all felt of their own, ebbed rapidly and agonizingly.
It was Megistias' son, at last, who thought to inquire the man's name.
This was, his mates reported, Perses.
At once all omen-spawned gloom dispelled as Megistias stated the prodigy's meaning, which could not have been plainer: this man, ill-starred in his mother's election of name, represented the enemy, who had in invading Greece stepped into a litter of serpents. Unfledged though these were and disunited, the fanged babes yet stood capable of delivering their venom into the foe's vital stream and bringing him low.
Night had fallen when this fortune-crossed fellow expired. Leonidas had him interred immediately with honor, then returned the men's hands at once to work. Orders were issued for every stonemason within the allied ranks to present himself, regardless of unit. Chisels, picks and levers were collected and more sent for from Alpenoi village and the surrounding countryside. The party set forth down the track to Trachis. The masons were ordered to destroy as much of the trail as possible, and also to chisel into the stone in plainest view the following message:
_Greeks conscripted by Xerxes:_
_If under compulsion you must fight us your brothers,_
_fight badly._
Simultaneously work was begun on rebuilding the ancient Phokian Wall which blocked the Narrows. This fortification, when the allies arrived, was little more than a pile of rubble. Leonidas demanded a proper battle wall.
A wry scene ensued as various engineers and draughtsmen of the allied militias assembled in solemn council to survey the site and propose architectural alternatives. Torches had been positioned to light the Narrows, diagrams were sketched in the dirt; one of the captains of the Corinthians produced an actual drawn-to-scale blueprint. Now the commanders began wrangling. The wall should be erected right at the Narrows, blocking the pass. No, suggested another, better it should be set back fifty meters, creating a "triangle of death" between the cliffs and the battle wall. A third captain urged a setback distance of twice that, giving the allied infantry room to mass and maneuver. Meanwhile the troops loitered about, as Hellenes will, offering their own sage counsel and wisdom.
Leonidas simply picked up a boulder and marched to a spot. There he set the stone in place. He lifted a second and placed it beside the first. The men looked on dumbly as their commander in chief, whom all could see was well past sixty, stooped to seize a third boulder. Someone barked: "How long do you imbeciles intend to stand by, gaping? Will you wait all night while the king builds the wall himself?"
With a cheer the troops fell to. Nor did Leonidas cease from his exertions when he saw other hands joined to labor, but continued alongside the men as the pile of stones began to rise into a legitimate fortress. "Nothing fancy, brothers," the king guided the construction. "For a wall of stone will not preserve Hellas, but a wall of men."
As he had done at every engagement at which it had been my privilege to observe him, the king stripped and worked alongside his warriors, shirking nothing, but pausing to address individuals, calling by name those he knew, committing to memory the names and even nicknames of others heretofore unknown to him, often clapping these new mates upon the back in the manner of a comrade and friend. It was astonishing with what celerity these intimate words, spoken only to one man or two, were relayed warrior-to-warrior down the line, filling the hearts of all with courage.
It was now the changing of the first watch.
"Bring me the villain."
With these words Leonidas summoned an outlaw of the region who had fallen in with the column along the route and enlisted for pay to aid in reconnaissance. Two Skiritai brought the man forward. To my astonishment, I knew him.
This was the youth of my own country who called himself Sphaireus, "Ball Player," the wild boy who had taken to the hills following my city's destruction and had kicked about a man's stuffed skull as his sign of outlaw princeship. Now this criminal advanced into the margins of the king's fire, no longer a smooth-cheeked boy but a scarred and bearded man grown.
I approached him. The fellow recognized me. He was delighted to resume our acquaintance and vastly amused at the fate which had brought us, two orphans of fire and sword, to this, the very epicenter of Hellas' peril.
The outlaw stood in sizzling high spirits over the prospect of war. He would haunt its margins and prey upon the broken and the vanquished. War to him was big business; it was clear without words that he thought me a dunce for electing to serve, and for not a penny's pay or profit. "Whatever happened to that tasty bit of steam you used to tramp with?" he asked me. "What was her name—your cousin?" "Steam" was the salacious slang of my country for a female of fair and tender years.
"She's dead," I lied, "and you will be too for the price of another word."
"Easy, countryman! Back your oars. I'm only fanning the breeze."
The king's officers summoned the brigand away before he and I could speak further. Leonidas needed a buck whose soles knew how to grip the hardscrabble track of a goat trail, some stoutheart to scramble up the sheer three-thousand-foot face of Kallidromos which towered above the Narrows. He wanted to know what was up top and how dangerous it was to get there. Once the enemy took possession of the Trachinian plain and the northern approaches, could the allies get a party, even a single man, across the shoulder of the mountain and into their rear?
Ball Player appeared decidedly unenthusiastic about his participation in this hazardous venture. "I'll go with him." This from the Skirite Hound, a mountaineer himself. "Anything to get off building this miserable wall." Leonidas accepted this offer with alacrity. He instructed his paymaster to compensate the outlaw handsomely enough to get him to go, but poorly enough to make sure he came back.
Around midnight the Phokians and Lokrians of Opus began arriving from the mountains. The king welcomed the fresh allies warmly, making no mention of their near desertion but instead guiding them at once to that section of the camp which had been assigned to their use and in which hot broth and freshly baked loaves awaited them.
A terrific storm had sprung up, north along the coast. Bolts resounded furiously in the distance; though the sky above the Gates stood yet clear and brilliant, the men were getting spooked. They were tired. The six days' hump had taken the starch out of them; fears unspoken and demons unseen began to prey upon their hearts. Nor could the newly arrived Phokians and Lokrians fail to discern the slender, not to say suicidally small, numbers of the force which proposed to hold off the myriads of the enemy.
The native vendors, even the whores, had vanished, like rats evacuating to their holes presaging a quake.
There was a man among the loitering locals, a merchantman's mate, he said, who had sailed for years out of Sidon and Tyre. I chanced to be present, around a fire of the Arkadians, when this fellow began to fan the flames of terror. He had seen the Persian fleet firsthand and had the following tale to tell.
"I was on a grain galley out of Mytilene last year. We got taken by Phoenicians, part of the Great King's fleet. They confiscated our cargo. We had to trail them in under escort and unload it at one of his supply magazines. This was at Strymon on the Thracian coast. The sight I beheld there numbed the senses with awe."
More men began to cluster about the circle, listening gravely. "The dump was big as a city. One thought, coming in, that a range of hills stood beyond it. But when you got close, the hills turned out to be salt meat, towering in hogsheads of brine, stacked to the heavens.
"I saw weapons, brothers. Stands of arms by the tens of thousands. Grain and oil, bakers' tents the size of stadiums. Every article of war matériel the mind could imagine. Sling bullets. Lead sling bullets stacked a foot high, covering an acre. The trough of oats for the King's horses was a mile long. And in the middle of all rose one oilcloth-shrouded pyramid, big as a mountain. What in heaven could be under that? I asked the officer of marines guarding us. 'Come on,' he says, 'I'll show you.' Can you guess, my friends, what rose there, stacked to the sky beneath those covers?
"Paper," the ship's mate declared.
None of the Arkadians grasped the significance.
"Paper!" the Trachinian repeated, as if to drum the meaning into his hearers' thick skulls. "Paper for scribes to take inventory. Inventory of men. Horses. Arms. Grain. Orders for troops and more orders, papers for reports and requisitions, muster rolls and dispatches, courts-martial and decorations for valor. Paper to keep track of every supply the Great King is bringing, and every item of loot he plans on taking back. Paper to write down countries burned and cities sacked, prisoners taken, slaves in chains..."
At this moment my master chanced to arrive at the gathering's margins. He discerned at once the terror graven upon the listeners' faces; without a word he pressed forward into the firelight. At the sight of a Spartiate officer among his listeners, the ship's mate redoubled his fervor. He was enjoying the current of dread his tale had spawned.
"But the most fearsome remains yet to be told, brothers," the Trachinian continued. "That same day, as our gaolers marched us to supper, we passed the Persian archers in their practice. Not the Olympian gods themselves could have assembled such myriads! I swear to you, mates, so numerous were the multitudes of bowmen that when they fired their volleys, the mass of arrows blocked out the sun!"
The rumormonger's eyes burned with pleasure. He turned to my master, as if to savor the flame of dread his tale had ignited even in a Spartan. To his disappointment Dienekes regarded him with a cool, almost bored detachment.
"Good," he said. "Then we'll have our battle in the shade."
In the middle of the second watch came the first panic. I was still awake, securing my master's covered shield against the rain which threatened, when I heard the telltale rustle of bodies shifting, the alteration in the rhythm of men's voices. A terror-swept camp sounds completely different from a confident one. Dienekes rose out of a sound sleep, like a sheepdog sensing murmurs of disquiet among his fold. "Mother of bitches," he grunted, "it's starting already."
The first raiding parties had returned to camp. They had seen torches, cavalry brands of the Persians' mounted rangers, and had made their own prudent withdrawal before getting cut off. You could see the foe plainly now, they reported, from the shoulder of the mountain, two miles or less down the trail. Some of the forward sentries had made sorties on their own as well, and these had now returned to camp to confirm the report.
Beyond the shoulder of Kallidromos, upon the sprawling plain of Trachis, the advance units of the Persians were arriving.
**TWENTY-TWO**
**W** ithin minutes of the sighting of the enemy forerunners, Leonidas had the entire Spartan contingent on its feet and armed, with orders to the allies to marshal in succession and be ready to move forward. The remainder of that night, and all the next day, were consumed with ravaging in earnest the plain of Trachis and the hillsides above, penetrating along the coast as far north as the Spercheios and inland to the citadel and the Trachinian cliffs. Watch fires were set across the entire plain, not little rabbit-roasters as customary, but roaring bonfires, to create the impression of vast numbers of men. The allied units shouted insults and imprecations to one another across the darkness, trying to sound as cheerful and confident as possible. By morning the plain was blanketed end-to-end in fire smoke and sea fog, exactly as Leonidas wanted. I was among the final four parties, stoking bonfires as the murky dawn came up over the gulf. We could see the Persians, mounted reconnaissance units and marine archers of the foe's fast scout corvettes, upon the far bank of the Spercheios. We shouted insults and they shouted back.
The day passed, and another. Now the main-force units of the foe began streaming in. The plain commenced to fill with the enemy. All Greek parties withdrew before the Median tide. The scouts could see the King's officers claiming the prime sites for His Majesty's pavilions and staking out the lushest pasturage for his horses.
They knew the Greeks were here, and the Greeks knew they were.
That night Leonidas summoned my master and the other _enomotarchai,_ the platoon leaders, to the low knoll behind the Phokian Wall upon which he had established his command post. Here the king began to address the Spartan officers. Meanwhile the commanders from the other allied cities, also summoned to council, began arriving. The timing of this was as the king intended. He wanted the allied officers to overhear the words he spoke seemingly for Spartan ears alone.
"Brothers and comrades," Leonidas addressed the Lakedaemonians clustered about him, "it appears that the Persian, despite our impressive showmanship, remains unconvinced of the prudence of packing his kit and embarking for home. It looks like we're going to have to fight him, after all. Hear, then, what I expect from each of you.
"You are the elect of Hellas, officers and commanders of the nation of Lakedaemon, chosen by the Isthmaian Congress to strike the first blow in defense of our homeland. Remember that our allies will take their cue from you. If you show fear, they will be afraid. If you project courage, they will match it in kind. Our deportment here must not differ from any other campaign. On the one hand, no extraordinary precautions; on the other, no unwonted recklessness. Above all, the little things. Maintain your men's training schedule without alteration. Omit no sacrifice to the gods. Continue your gymnastics and drills-at-arms. Take time to dress your hair, as always. If anything, take more time."
By now the allied officers had arrived at the council fire and were assuming their stations amid the already assembled Spartans. Leonidas continued as if to his own countrymen, but with an ear to the new arrivals as well.
"Remember that these our allies have not trained their whole lives for war, as we have. They are farmers and merchants, citizen-soldiers of their cities' militias. Nonetheless they are not unmindful of valor or they would not be here. For the Phokians and Lokrians of Opus, this is their country; they fight to defend home and family. As for the men of the other cities, Thebans and Corinthians and Tegeates, Orchomenians and Arkadians, Philiasians, Thespaians, Mantineans and the men of Mycenae, these display to my mind even nobler _andreia,_ for they come uncompelled, not to defend their own hearths, but all Greece."
He motioned the new arrivals forward.
"Welcome, brothers. Since I find myself among allies, I am making a long-winded speech."
The officers settled in with an anxious chuckle. "I am telling the Spartans," Leonidas resumed, "what I now tell you. You are the commanders, your men will look to you and act as you do. Let no officer keep to himself or his brother officers, but circulate daylong among his men. Let them see you and see you unafraid. Where there is work to do, turn your hand to it first; the men will follow. Some of you, I see, have erected tents. Strike them at once. We will all sleep as I do, in the open. Keep your men busy. If there is no work, make it up, for when soldiers have time to talk, their talk turns to fear. Action, on the other hand, produces the appetite for more action.
"Exercise campaign discipline at all times. Let no man heed nature's call without spear and shield at his side.
"Remember that the Persian's most formidable weapons, his cavalry and his multitudes of archers and slingers, are rendered impotent here by the terrain. That is why we chose this site. The enemy can get no more than a dozen men at a time through the Narrows and mass no more than a thousand before the Wall. We are four thousand. We outnumber him four to one."
This produced the first genuine laughter. Leonidas sought to instill courage not by his words alone but by the calm and professional manner with which he spoke them. War is work, not mystery. The king confined his instructions to the practical, prescribing actions which could be taken physically, rather than seeking to produce a state of mind, which he knew would evaporate as soon as the commanders dispersed beyond the fortifying light of the king's fire.
"Look to your grooming, gentlemen. Keep your hair, hands and feet clean. Eat, if you have to choke it down. Sleep, or pretend to. Don't let your men see you toss. If bad news comes, relay it first to those in grade above you, never directly to your men. Instruct your squires to buff each man's _aspis_ to its most brilliant sheen. I want to see shields flashing like mirrors, for this sight strikes terror into the enemy. Leave time for your men to sharpen their spears, for he who whets his steel whets his courage.
"As for your men's understandable anxiety concerning the immediate hours, tell them this: I anticipate action neither tonight nor tomorrow, nor even the day after. The Persian needs time to marshal his men, and the more myriads he is burdened with, the longer this will take. He must wait upon the arrival of his fleet. Beaching grounds are scarce and slender upon this inhospitable coast; it will take the Persian days to lay out roadsteads and secure at anchor his thousands of warships and transports.
"Our own fleet, as you know, holds the strait at Artemisium. Breaking through will require of the enemy a full-scale sea battle; preparing for this will consume even more of his time. As for assaulting us here in the pass, the foe must reconnoiter our position, then deliberate how best to attack it. No doubt he will send emissaries first, seeking to achieve by diplomacy what he hesitates to hazard at the cost of blood. This you need not concern yourselves with, for all treating with the enemy will be done by me." Here Leonidas bent to the earth and lifted a stone thrice the size of a man's fist. "Believe me, comrades, when Xerxes addresses me, he might as well be talking to this."
He spat upon the rock and slung it away into the dark.
"Another thing. You have all heard the oracle declaring that Sparta will either lose a king in battle or the city herself be extinguished. I have taken the omens and the god has answered that I am that king and that this site will be my grave. Be assured, however, that this foreknowledge will nowise render me reckless with other men's lives. I swear to you now, by all the gods and by the souls of my children, that I will do everything in my power to spare you and your men, as many as I possibly can, and still defend the pass effectively.
"Finally this, brothers and allies. Wherever the fighting is bloodiest, you may expect to discover the Lakedaemonians in the forefront. But convey this, above all, to your men: let them not yield preeminence in valor to the Spartans, rather strive to outdo them. Remember, in warfare practice of arms counts for little. Courage tells all, and we Spartans have no monopoly on that. Lead your men with this in mind and all will be well."
**TWENTY-THREE**
**I** t was the standing order of my master on campaign that he be woken two hours before dawn, an hour prior to the men of his platoon. He insisted that these never behold him prone upon the earth, but awake always to the sight of their _enomotarch_ on his feet and armed.
This night Dienekes slept even less. I felt him stir and roused myself. "Lie still," he commanded. His hand pressed me back down. "It's not even past second watch." He had dozed without removing his corselet and now creaked to his feet, all his scarred joints groaning. I could hear him crack the bones of his neck and hawk dry phlegm from the lungs he had seared at Oinoe, inhaling fire, which wound like the others had never truly healed.
"Let me help you, sir."
"Sleep. Don't make me tell you twice."
He snatched one of his spears from the stacked arms and shouldered his _aspis_ by its sling cord. He took his helmet, seating it by its nasal into the warpack he now slung across his shoulders. He gimped off on his bad ankle. He was making for Leonidas' cluster among the Knights, where the king would be awake and perhaps wanting company.
Across the cramped confines the camp slumbered. A waxing moon stood above the strait, the air unseasonably chill for summer, dank as it is by the sea and made more raw by the recent storms; you could hear the breakers clearly, combing in at the base of the cliffs. I glanced across at Alexandros, pillowed upon his shield beside the snoring form of Suicide. Watch fires had banked down; across the camp the sleeping warriors' forms had stilled into lumpy piles of cloaks and sleeping capes that looked more like sacks of discarded laundry than men.
Toward the Middle Gate, I could see the bathhouses of the spa. These were cheerful structures of unmilled lumber, their stone thresholds worn smooth by the tread of bathers and summer visitors dating from centuries. The oiled paths meandered prettily under the oaks, lit by the olivewood lamps of the spa. A burnished wood plaque hung beneath each lamp, a snatch of verse carved upon it. I recall one:
_As at birth the soul_
_steps into the liquid body_
_So step you now, friend, into these baths,_
_releasing flesh into soul,_
_reunited, divine._
I remembered something my master had once said about battlefields. This was at Tritaea, when the army met the Achaians in a field of seedling barley. The climactic slaughter had taken place opposite a temple to which in time of peace the deranged and god-possessed were conveyed by their families, to pray and offer sacrifice to Demeter Merciful and Persephone. "No surveyor marks out a tract and declares, 'Here we shall have a battle.' The ground is often consecrated to a peaceful purpose, frequently one of succor and compassion. The irony can get pretty thick sometimes."
And yet within Hellas' mountainous and topographically hostile confines, there existed those sites hospitable to war—Oenophyta, Tanagra, Koroneia, Marathon, Chaeronea, Leuktra—those plains and defiles upon and within which armies had clashed for generations.
This pass of the Hot Gates was such a site. Here in these precipitous straits, contending forces had slugged it out as far back as Jason and Herakles. Hill tribes had fought here, savage clans and seaborne raiders, migrating hordes, barbarians and invaders of the north and west. The tides of war and peace had alternated in this site for centuries, bathers and warriors, one come for the waters, the other for blood.
The battle wall had now been completed. One end abutted the sheer face of the cliff, with a stout tower flush to the stone, the other tailing off at an angle across the slope to the cliffs and the sea. It was a good-looking wall. Two spear-lengths thick at the base and twice the height of a man. The face toward the enemy had not been erected sheer in the manner of a city battlement, but left deliberately sloped, right up to the actual sallyports at the crest, where the final four feet rose vertical as a fortress. This was so the warriors of the allies could scamper rearward to safety if they had to, and not find themselves pinned and crushed against their own wall.
The rear face sloped up in stacked steps for the defenders to mount to the battlements, atop which had been anchored a stout timber palisade sheathed in hides which the standing watches could cast loose so that the tow arrows of the enemy would not set the palisade alight. The masonry was ragged stuff but sturdy. Towers stood at intervals, reinforcing redoubts right, left and center and secondary walls behind these. These strongpoints had been built solid to the height of the primary wall, then stacked with heavy stones to a man's height beyond. These loose boulders could be tumbled, should necessity dictate, into the breaches of the lower sallyports. I could see the sentries now atop the Wall and the three ready platoons, two Arkadian and one Spartan, in full _panoplia,_ at each redoubt.
Leonidas was in fact awake. His long steel-colored hair could be distinguished clearly beside the commanders' fire. Dienekes attended him there among a knot of officers. I could make out Dithyrambos, the Thespaian captain; Leontiades, the Theban commander; Polynikes; the brothers Alpheus and Maron, and several other Spartan Knights.
The sky had begun to lighten; I became aware of forms stirring beside me. Alexandros and Ariston had come awake as well and now roused themselves and took station beside me. These young warriors, like myself, found their gaze drawn irresistibly to the officers and champions surrounding the king. The veterans, all knew, would acquit themselves with honor. "How will we do?" Alexandros put into words the anxiety that loitered unspoken in his youthful mates' hearts. "Will we find the answer to Dienekes' question? Will we discover within ourselves 'the opposite of fear'?"
Three days before the march-out from Sparta, my master had assembled the warriors and squires of his platoon and outfitted a hunt at his own expense. This was in the form of a farewell, not to each other, but to the hills of their native country. None spoke a word of the Gates or of the trials to come. It was a grand outing, blessed by the gods with several excellent kills including a fine boar brought down in its charge by Suicide and Ariston with the javelin and the foot-braced pike.
At dusk the hunters, beyond a dozen with twice that number of squires and helots serving as beaters, settled in high spirits about several fires among the hills above Therai. _Phobos_ took a seat as well. As the other huntsmen made merry around their separate blazes, diverting themselves with lies of the chase and good-fellow jesting, Dienekes cleared space beside his own station for Alexandros and Ariston and bade them sit. I discerned then my master's subtle intent. He was going to speak of fear, for these unblooded youths whom he knew despite their silence, or perhaps because of it, had begun in their hearts to dwell upon the trials to come.
"All my life," Dienekes began, "one question has haunted me. What is the opposite of fear?"
Down the slope the boar flesh was coming ready; portions were being shared out to eager hands. Suicide came up, with bowls for Dienekes, Alexandros and Ariston, and one apiece for himself, Ariston's squire Demades and me. He settled on the earth across from Dienekes, flanked by two of the hounds who had noses for the scraps and knew Suicide as a notorious soft touch.
"To call it _aphobia,_ fearlessness, is without meaning. This is just a name, thesis expressed as antithesis. To call the opposite of fear fearlessness is to say nothing. I want to know its true obverse, as day of night and heaven of earth."
"Expressed as a positive," Ariston ventured.
"Exactly!" Dienekes met the young man's eyes in approval. He paused to study both youths' expressions. Would they listen? Did they care? Were they, like him, true students of this subject?
"How does one conquer fear of death, that most primordial of terrors, which resides in our very blood, as in all life, beasts as well as men?" He indicated the hounds flanking Suicide. "Dogs in a pack find courage to take on a lion. Each hound knows his place. He fears the dog ranked above and feeds off the fear of the dog below. Fear conquers fear. This is how we Spartans do it, counterpoising to fear of death a greater fear: that of dishonor. Of exclusion from the pack."
Suicide took this moment to toss several scraps to the dogs. Furiously their jaws snapped these remnants from the turf, the stronger of the two seizing the lion's share.
Dienekes smiled darkly.
"But is that courage? Is not acting out of fear of dishonor still, in essence, acting out of fear?"
Alexandros asked what he was seeking.
"Something nobler. A higher form of the mystery. Pure. Infallible."
He declared that in all other questions one may look for wisdom to the gods. "But not in matters of courage. What have the immortals to teach us? They cannot die. Their spirits are not housed, as ours, in this." Here he indicated the body, the flesh. "The factory of fear."
Dienekes glanced again to Suicide, then back to Alexandros, Ariston and me. "You young men imagine that we veterans, with our long experience of war, have mastered fear. But we feel it as keenly as you. More keenly, for we have more intimate experience of it. Fear lives within us twenty-four hours a day, in our sinews and our bones. Do I speak the truth, my friend?"
Suicide grinned darkly in reply.
My master grinned back. "We cobble our courage together on the spot, of rags and remnants. The main we summon out of that which is base. Fear of disgracing the city, the king, the heroes of our lines. Fear of proving ourselves unworthy of our wives and children, our brothers, our comrades-in-arms. For myself I know all the tricks of the breath and of song, the pillars of the _tetrathesis,_ the teachings of the _phobologia._ I know how to close with my man, how to convince myself that his terror is greater than my own. Perhaps it is. I employ care for the men-at-arms serving beneath me and seek to forget my own fear in concern for their survival. But it's always there. The closest I've come is to act despite terror. But that's not it either. Not the kind of courage I'm talking about. Nor is beastlike fury or panic-spawned self-preservation. These are _katalepsis,_ possession. A rat owns as much of them as a man."
He observed that often those who seek to overcome fear of death preach that the soul does not expire with the body. "To my mind this is fatuousness. Wishful thinking. Others, barbarians primarily, say that when we die we pass on to paradise. I ask them all: if you really believe this, why not make away with yourself at once and speed the trip?
"Achilles, Homer tells us, possessed true _andreia._ But did he? Scion of an immortal mother, dipped as a babe in the waters of Styx, knowing himself to be save his heel invulnerable? Cowards would be rarer than feathers on fish if we all knew that."
Alexandros inquired if any of the city, in Dienekes' opinion, possessed this true _andreia._
"Of all in Lakedaemon, our friend Polynikes comes closest. But even his valor I find unsatisfactory. He fights not out of fear of dishonor, but greed for glory. This may be noble, or at least unbase, but is it true _andreia?_ "
Ariston asked if this higher courage in fact existed.
"It is no phantom," Dienekes declared with conviction. "I have seen it. My brother Iatrokles possessed it in moments. When I beheld its grace upon him, I stood in awe. It radiated, sublime. In those hours he fought not like a man but a god. Leonidas has it on occasion. Olympieus doesn't. I don't. None of us here does." He smiled. "Do you know who owns it, this pure form of courage, more than any other I have known?"
None around the fire answered.
"My wife," Dienekes said. He turned to Alexandros. "And your mother, the lady Paraleia." He smiled again. "There is a clue here. The seat of this higher valor, I suspect, lies in that which is female. The words themselves for courage, _andreia_ and _aphobia,_ are female, whereas _phobos_ and _tromos,_ terror, are masculine. Perhaps the god we seek is not a god at all, but a goddess. I don't know."
You could see it did Dienekes good to speak of this. He thanked his listeners for sitting still for it. "The Spartans have no patience for such inquiries of the salon. I remember asking my brother once, on campaign, a day when he had fought like an immortal. I was mad to know what he had felt in those moments, what was the essence experienced within? He looked at me as if I had taken leave of sanity. 'Less philosophy, Dienekes, and more virtue.'"
He laughed. "So much for that."
My master turned sidelong then, as if to draw this inquiry to a close. Yet some impulse drew him back, to Ariston, upon whose features stood that expression of one of youthful years nerving himself to venture speech before his elders. "Spit it out, my friend," Dienekes urged him.
"I was thinking of women's courage. I believe it is different from men's."
The youth hesitated. Perhaps, his expression clearly bespoke, it smacked of immodesty or presumptuousness to speculate upon matters of which he possessed no experience.
Dienekes pressed him nonetheless. "Different, how?"
Ariston glanced to Alexandros, who with a grin reinforced his friend's resolve. The youth took a breath and began: "Man's courage, to give his life for his country, is great but unextraordinary. Is it not intrinsic to the nature of the male, beasts as well as men, to fight and to contend? It's what we were born to do, it's in our blood. Watch any boy. Before he can even speak, he reaches, impelled by instinct, for the staff and the sword—while his sisters unprompted shun these implements of contention and instead cuddle to their bosom the kitten and the doll.
"What is more natural to a man than to fight, or a woman to love? Is this not the imperative of a mother's blood, to give and to nurture, above all the produce of her own womb, the children she has borne in pain? We know that a lioness or she-wolf will cast away her life without hesitation to preserve her cubs or pups. Women the same. Now consider, friends, that which we call women's courage:
"What could be more contrary to female nature, to motherhood, than to stand unmoved and unmoving as her sons march off to death? Must not every sinew of the mother's flesh call out in agony and affront at such an outrage? Must not her heart seek to cry in its passion, 'No! Not my son! Spare him!' That women, from some source unknown to us, summon the will to conquer this their own deepest nature is, I believe, the reason we stand in awe of our mothers and sisters and wives. This, I believe, Dienekes, is the essence of women's courage and why it, as you suggested, is superior to men's."
My master acknowledged these observations with approval. At his side Alexandros shifted, however. You could see the young man was not satisfied.
"What you say is true, Ariston. I had never thought of it in that way before. Yet something must be added. If women's victory were simply to stand dry-eyed as their sons march off to death, this would not alone be unnatural, but inhuman, grotesque and even monstrous. What elevates such an act to the stature of nobility is, I believe, that it is performed in the service of a higher and more selfless cause.
"These women of whom we stand in awe donate their sons' lives to their country, to the people as a whole, that the nation may survive even as their own dear children perish. Like the mother whose story we have heard from childhood who, on learning that all five of her sons had been killed in the same battle, asked only, 'Was our nation victorious?' and, being told that it was, turned for home without a tear, saying only, 'Then I am happy.' Is it not this element—the nobility of setting the whole above the part—that moves us about women's sacrifice?"
"Such wisdom from the mouths of babes!" Dienekes laughed and rapped both lads affectionately upon the shoulder. "But you have not yet answered my question. What is the opposite of fear?
"I will tell you a story, my young friends, but not here or now. At the Gates you shall hear it. A story of our king, Leonidas, and a secret he confided to Alexandros' mother, Paraleia. This tale will advance our inquiry into courage—and will tell as well how Leonidas came to select those he did for the Three Hundred. But for this hour we must put a period to our salon or the Spartans, overhearing, will declare us effeminate. And they will be right!"
Now in camp at the Gates we three youths could see our _enomotarch,_ responding to dawn's first glimmer, take leave of the king's council and return to his platoon, stripping his cloak to call the men to gymnastics. "On our feet, then." Ariston sprung up, snapping Alexandros and me from our preoccupations. "The opposite of fear must be work."
Drill-at-arms had barely begun when a sharp whistle from the Wall summoned every man to alert.
A herald of the enemy was advancing into view at the throat of the Narrows.
This messenger drew up at a distance, calling out a name in Greek, that of Alexandros' father, the _polemarch_ Olympieus. When the herald was motioned forward, escorting a single officer of the enemy embassy and a boy, he cried further by name after three other Spartan officers, Aristodemos, Polynikes and Dienekes.
These four were summoned at once by the officer of the watch, he and all others in hearing astonished and by no means uncurious about the specificity of the enemy's request.
The sun was full up now; scores of allied infantrymen stood watching upon the Wall. Forward advanced the Persian embassy. Dienekes recognized its principal at once. This was the captain Ptammitechus, "Tommie," the Egyptian marine we had encountered and exchanged gifts with four years previous at Rhodes. The boy, it turned out, was his son. The lad spoke excellent Attic Greek and served as interpreter.
A scene of warm reacquaintance ensued, with abundant clapping of backs and clasping of hands. Surprise was expressed by the Spartans that the Egyptian was not with the fleet; he was, after all, a marine, a sea fighter. Tommie responded that he only, and his immediate platoon, had been detached to duty with the land armies, seconded to the Imperial Command at his own request for this specific purpose: to act as an informal ambassador to the Spartans, whose acquaintance he recalled with such warmth and whose welfare he wished above all to succor.
By now the crowd surrounding the marine had swelled to above a hundred. The Egyptian towered half a head over even the tallest Hellene, his tiara of pressed linen adding further to his stature. His smile flashed brilliant as ever. He bore a message, he declared, from King Xerxes himself, which he had been charged to deliver to the Spartans alone.
Olympieus, who had been senior envoy during the Rhodian embassy, now assumed that position in this parley. He informed the Egyptian that no treating would be done on a nation-by-nation basis. It was one for all among the Greeks, and that was that.
The marine's cheerful demeanor did not falter. At that moment the main body of Spartans, led by Alpheus and Maron, was running shield drills immediately before the Wall, working with and instructing two platoons of the Thespaians. Tommie observed the brothers for a number of moments, impressed. "I will alter my request, then," he said, smiling, to Olympieus. "If you, sir, will escort me to your king, Leonidas, I will deliver my message to him as commander of the Hellenic allies as a whole."
My master was plainly fond of this personable fellow and delighted to see him again. "Still wearing steel underpants?" he inquired through the boy interpreter.
Tommie laughed and displayed, to the further amusement of the assembly, an undergarment of white Nile linen. Then, with a gesture friendly and informal, he seemed to set aside his role as envoy and speak, for the moment, man-to-man.
"I pray that armor of mail need never be employed between us, brothers." He indicated the camp, the Narrows, the sea, seeming to include the defense as a whole in the sweep of his arm. "Who knows how this may turn out? It may all blow over, as it did for your force of Ten Thousand at Tempe. But if I may speak as a friend, to you four only, I would urge you thus: do not let hunger for glory, nor your own pride in arms, blinder you to the reality your forces now confront.
"Death alone awaits you here. The defenders cannot hope to stand, even for a day, in the face of the multitudes His Majesty brings against you. Nor will all the armies of Hellas prevail in the battles yet to come. Surely you know this, as does your king." He paused to let his son deliver the translation and to study the response upon the faces of the Spartans. "I beg you hearken to this counsel, friends, offered from my own heart as one who bears the most profound respect for you as individuals and for your city and its wide and well-deserved fame. Accept the inevitable, and be ruled with honor and respect—"
"You may stop there, friend," Aristodemos cut him off.
Polynikes put in with heat: "If that's all you came to tell us, brother, stick it between the creases."
The Egyptian maintained his level and amiable demeanor. "You have my word and His Majesty's upon it: if the Spartans will yield now and surrender their arms, none will exceed them in honor beneath the King's banner. No Persian foot will tread the soil of Lakedaemon now or forever, this His Majesty swears. Your country will be granted dominion over all Greece. Your forces will take their place as the foremost unit in His Majesty's army, with all the fortune and glory such prominence commands. Your nation has but to name its desires. His Majesty will grant them and, if I may claim to know his heart, will shower further gifts upon his new friends, in scale and costliness beyond imagining."
At this, the breath of every allied listener stoppered in his throat. Each eye stood fearfully upon the Spartans. If the Egyptian's offer was bona fide, and there was no reason to believe it wasn't, it meant deliverance for Lakedaemon. All she need do was forsake the Hellenic cause. What now would be these officers' response? Would they at once convey the envoy to their king? Leonidas' word would be tantamount to law, so preeminent stood his stature among the Peers and ephors.
Out of the blue, the fate of Hellas suddenly teetered upon the precipice. The allied listeners stood nailed to the site, awaiting breathlessly the response of these four warriors of Lakedaemon.
"It seems to me," Olympieus addressed the Egyptian with barely a moment's hesitation, "that if His Majesty truly wished to make the Spartans his friends, he would find them of far greater service with their arms than without."
"Further, experience has taught us," Aristodemos added, "that honor and glory are boons which cannot be granted by the pen but must be earned by the spear."
My glance scanned in this moment the faces of the allies. Tears stood in the eyes of not a few; others seemed so undone with relief that their knees threatened to give way beneath them. The Egyptian clearly discerned this. He smiled, gracious and patient, not abashed in the least.
"Gentlemen, gentlemen. I trouble you with matters which should and must be debated, not here in the marketplace so to speak, but in private before your king. Please, if you will, conduct me to him."
"He'll tell you the same, brother," Dienekes declared.
"And in far cruder language," put in another Spartan among the crowd.
Tommie waited for the laughter to subside.
"May I hear this response, then, from the king's own lips?"
"He'd have us whipped, Tommie," Dienekes put in with a smile.
"He'd tear the hide off our backs," spoke the same man who had interposed a moment earlier, "even to propose such a course of dishonor."
The Egyptian's eyes swung now to this speaker, whom he perceived to be an older Spartan, clad in tunic and homespun cloak, who now stepped into the second rank, at the shoulder of Aristodemos. For a moment the marine was taken aback to discover this graybeard, who clearly bore the weight of more than sixty summers, yet stood in infantryman's raiment among the other, far younger warriors.
"Please, my friends," the Egyptian continued, "do not respond out of pride or the passion of the moment but permit me to place before your king the wider consequences of such a decision. Let me set the Persian Majesty's ambitions in perspective.
"Greece is just the jumping-off point. The Great King already rules all Asia; Europe now is his goal. From Hellas His Majesty's army moves on to conquer Sikelia and Italia, from there to Helvetia, Germania, Gallia, Iberia. With you on our side, what force can stand against us? We will advance in triumph to the Pillars of Herakles themselves and beyond, to the very walls of Oceanus!
"Please, brothers, consider the alternatives. Stand now in pride of arms and be crushed, your country overrun, wives and children enslaved, the glory of Lakedaemon, not to say her very existence, effaced forever from the earth. Or elect, as I urge, the course of prudence. Assume with honor your rightful station in the forefront of the invincible tide of history. The lands you rule now will be as nothing beside the domains the Great King will bestow upon you. Join us, brothers. Conquer with us all the world! Xerxes son of Darius swears this: no nation or army will surpass you in honor among all His Majesty's forces! And if, my Spartan friends, the act of abandoning your Hellene brothers strikes you as dishonorable, King Xerxes extends his offer further, to all Greeks. All Hellenic allies, regardless of nation, will he set in freedom at your shoulder and honor second only to yourselves among his minions!"
Neither Olympieus nor Aristodemos nor Dienekes nor Polynikes lifted voice in response. Instead the Egyptian saw them defer to the older man in the homespun cloak.
"Among the Spartans any may speak, not just these ambassadors, as we are all accounted Peers and equals before the law." The elder now stepped forward. "May I take the liberty to suggest, sir, an alternative course, which I feel certain will find favor, not among the Lakedaemonians alone, but with all the Greek allies?"
"Please do," responded the Egyptian.
All eyes centered upon the veteran.
"Let Xerxes surrender to us," he proposed. "We will not fail to match his generosity, but set him and his forces foremost among our allies and grant to him all the honors which he so munificently proposes to shower upon us."
A laugh burst from the Egyptian.
"Please, gentlemen, we squander precious time." He turned away from the older man, not without a hint of impatience, and pressed his request again to Olympieus. "Conduct me at once to your king."
"No use, friend," answered Polynikes.
"The king is a crusty old bugger," Dienekes added.
"Indeed," put in the older man. "He is a foul-tempered and irascible fellow, barely literate, in his cups most days before noon, they say."
A smile now spread across the features of the Egyptian. He glanced to my master and to Olympieus.
"I see," said Tommie.
His look returned to the older man, who, as the Egyptian now discerned, was none other than Leonidas himself.
"Well then, venerable sir," Tommie addressed the Spartan king directly, dipping his brow in a gesture of respect, "since it seems I am to be frustrated in my desire to speak in person with Leonidas, perhaps, in deference to the gray I behold in your beard and the many wounds my eyes espy upon your body, you yourself, sir, will accept this gift from Xerxes son of Darius in your king's stead."
From a pouch the Egyptian produced a double-handled goblet of gold, magnificent in craftsmanship and encrusted with precious gems. He declared that the engravings thereupon represented the hero Amphiktyon, to whom the precinct of Thermopylae was sacred, along with Herakles and Hyllus, his son, from whom the race of the Spartans, and Leonidas himself, was descended. The cup was so heavy that the Egyptian had to hold it out with both hands.
"If I accept this generous gift," Leonidas addressed him, "it must go into the war treasury of the allies."
"As you wish." The Egyptian bowed.
"Then convey the Hellenes' gratitude to your King. And tell him my offer will remain open, should God grant him the wisdom to embrace it."
Tommie passed the goblet to Aristodemos, who accepted it for the king. A moment passed, in which the Egyptian's eyes met first Olympieus', then settled gravely upon my master's. An expression of solemnity, sober to the point of sorrow, shrouded the marine's eyes. Clearly he discerned now the inevitability of that which he had sought with such charity and concern to avert.
"If you fall in capture," he addressed the Spartans, "call my name. I will exert every measure of influence to see that you are spared."
"You do that, brother," Polynikes answered, hard as steel.
The Egyptian recoiled, stung. Dienekes stepped in swiftly, clasping the marine's hand warmly in his own.
"Till we meet," Dienekes said.
"Till then," Tommie replied.
**TWENTY-FOUR**
**T** hey wore trousers.
Pantaloons of purple, bloused below the knee, topping calf-length boots of doeskin or some other precious product of the tannery. Their tunics were sleeved and embroidered, beneath mail jackets of armor shaped like fish scales; their helmets open-faced and brilliantly plumed, of hammered iron shaped like domes. Their cheeks they wore rouged and their ears and throats bedecked with ornament. They looked like women and yet the effect of their raiment, surreal to Hellene eyes, was not that which evoked contempt, but terror. One felt as if he were facing men from the underworld, from some impossible country beyond Oceanus where up was down and night day. Did they know something the Greeks didn't? Were their light skirmisher shields, which seemed almost ludicrously flimsy contrasted to the massive twenty-pound oak and bronze, shoulder-to-knee _aspides_ of the Hellenes, somehow, in some undivinable way, superior? Their lances were not the stout ash and cornelwood eight-footers of the Greeks but lighter, slender, almost javelin-like weapons. How would they strike with these? Would they hurl them or thrust them underhand? Was this somehow more lethal than the overhand employed by the Greeks?
They were Medes, the vanguard division of the troops who would first assault the allies, though none among the defenders knew this for certain at the time. The Greeks could not distinguish among Persians, Medes, Assyrians, Babylonians, Arabians, Phrygians, Karians, Armenians, Cissians, Cappadocians, Paphlagonians, Bactrians nor any of the other five score Asiatic nations save the Ionian Hellenes and Lydians, the Indians and Ethiopians and Egyptians who stood out by their distinctive arms and armor. Common sense and sound generalship dictated that the commanders of the Empire grant to one nation among their forces the honor of drawing first blood. It made further sense, so the Greeks surmised, that when making trial of an enemy for the first time, a prudent general would not commit the flower of his troops—in His Majesty's case his own Ten Thousand, the Persian household guard known as the Immortals—but rather hold these elite in reserve against the unexpected.
In fact this was the selfsame strategy adopted by Leonidas and the allied commanders. These kept the Spartans back, choosing to honor, after much debate and discussion, the warriors of Thespiae. These were granted first position and now, on the morning of the fifth day, stood formed in their ranks, sixty-four shields across, upon the "dance floor" formed by the Narrows at the apex, the mountain wall on one side, the cliffs dropping to the gulf on the other and the reconstructed Phokian Wall at the rear.
This, the field of slaughter, comprised an obtuse triangle whose greatest depth lay along the southern flank, that which was anchored by the mountain wall. At this end the Thespaians were drawn up eighteen deep. At the opposing end, alongside the drop-off to the sea, their shields were staggered to a depth of ten. This force of the men of Thespiae totaled approximately seven hundred.
Immediately to their rear, atop the Wall, stood the Spartans, Philiasians and Mycenaeans, to a total of six hundred. Behind these every other allied contingent was likewise drawn up, all in full _panoplia._
Two hours had elapsed since the enemy had first been sighted, half a mile down the track to Trachis, and still no motion had come. The morning was hot. Down the track, the roadway widened into an open area about the size of the agora of a small city. There, just after dawn, the lookouts had espied the Medians assembling. Their numbers were about four thousand. These, however, were only the foe who could be seen; the shoulder of the mountain hid the trail and the marshaling stations beyond. One could hear the enemy trumpets and the shouted orders of their officers moving more and more men into position beyond the shoulder. How many more thousands massed there out of sight?
The quarter hours crawled by. The Medes continued marshaling, but did not advance. The Hellenic lookouts began shouting insults down at them. Back in the Narrows, the heat and other exigencies had begun to work on the chafing, impatient Greeks. It made no sense to sweat longer under the burdens of full armor. "Dump 'em but be ready to hump 'em!" Dithyrambos, the Thespaian captain, called out to his countrymen in the coarse slang of his city. Squires and servants dashed forward among the ranks, each assisting his man in disencumbering himself of breastplate and helmet. Corselets were loosened. Shields rested already against knees. The felt undercaps which the men wore beneath their helmets came off and were wrung like bath linens, saturated with sweat. Spears were plunged at the position of rest, butt-spike-first, into the hard dirt, where they stood now in their numbers like an iron-tipped forest. The troops were permitted to kneel. Squires with skins of water circulated, replenishing the parched warriors. It was a safe bet that many skins contained refreshment more potent than that scooped from a spring.
As the delay grew longer, the sense of unreality heightened. Was this another false alarm, like the previous four days? Would the Persian attack at all?
"Snap out of those daydreams!" an officer barked.
The troops, bleary-eyed and sun-scorched, continued eyeing Leonidas on the Wall with the commanders. What were they talking about? Would the order come to stand down?
Even Dienekes grew impatient. "Why is it in war you can't fall asleep when you want to and can't stay awake when you have to?" He was just stepping forward to address a steadying word to his platoon when from out front among the foreranks rose a shout of such intensity that it cut his words off in midbreath. Every eye swung skyward.
The Greeks now saw what had caused the delay.
There, several hundred feet above and one ridgeline removed, a party of Persian servants escorted by a company of their Immortals was erecting a platform and a throne.
"Mother of bitches." Dienekes grinned. "It's young Purple Balls himself."
High above the armies, a man of between thirty and forty years could be descried plainly, in robes of purple fringed with gold, mounting the platform and assuming his station upon the throne. The distance was perhaps eight hundred feet, up and back, but even at that range it was impossible to mistake the Persian monarch's surpassing handsomeness and nobility of stature. Nor could the supreme self-assurance of his carriage be misread even at this distance. He looked like a man come to watch an entertainment. A pleasantly diverting show, one whose outcome was foreordained and yet which promised a certain level of amusement. He took his seat. A sunshade was adjusted by his servants. We could see a table of refreshments placed at his side and, upon his left, several writing desks set into place, each manned by a secretary.
Obscene gestures and shouted insults rose from four thousand Greek throats.
His Majesty rose with aplomb in response to the jeers. He gestured elegantly and, it seemed, with humor, as if acknowledging the adulation of his subjects. He bowed with a flourish. It seemed, though the distance was too great to be sure, as if he were smiling. He saluted his own captains and settled regally upon his throne.
My place was on the Wall, thirty stations in from the left flank anchored by the mountain. I could see, as could all the Thespaians before the Wall and every Lakedaemonian, Mycenaean and Philiasian atop it, the captains of the enemy, advancing now to the sound of their trumpets, in the van before the massed ranks of their infantry. My God, they looked handsome. Six division commanders, each, it seemed, taller and nobler than the next. We learned later that these were not merely the flower of the Median aristocracy, but that their ranks were reinforced by the sons and brothers of those who had been slain ten years earlier by the Greeks at Marathon. But what froze the blood was their demeanor. Their carriage shone forth, bold to the point of contemptuous. They would brush the defenders aside, that's what they thought. The meat of their lunch was already roasting, back down in camp. They would polish us off without raising a sweat, then return to dine at their leisure.
I glanced to Alexandros; his brow glistened, pale as a winding sheet; his wind came in strangled, wheezing gasps. My master stood at his shoulder, one pace to the fore. Dienekes' attention held riveted to the Medes, whose massed ranks now filled the Narrows and seemed to extend endlessly beyond, out of sight along the track. But no emotion disclarified his reason. He was gauging them strategically, coolly assessing their armament and the bearing of their officers, the dress and interval of their ranks. They were mortal men like us; was their vision struck, like ours, with awe of the force which stood now opposed to them? Leonidas had stressed again and again to the officers of the Thespaians that their men's shields, greaves and helmets must be bossed to the most brilliant sheen possible. These now shone like mirrors. Above the rims of the bronze-faced _aspides,_ each helmet blazed magnificently, overtopped with a lofty horsehair crest, which as it trembled and quavered in the breeze not only created the impression of daunting height and stature but lent an aspect of dread which cannot be communicated in words but must be beheld to be understood.
Adding further to the theater of terror presented by the Hellenic phalanx and, to my mind most frightful of all, were the blank, expressionless facings of the Greek helmets, with their bronze nasals thick as a man's thumb, their flaring cheekpieces and the unholy hollows of their eye slits, covering the entire face and projecting to the enemy the sensation that he was facing not creatures of flesh like himself, but some ghastly invulnerable machine, pitiless and unquenchable. I had laughed with Alexandros not two hours earlier as he seated the helmet over his felt undercap; how sweet and boyish he appeared in one instant, with the helmet cocked harmlessly back upon his brow and the youthful, almost feminine features of his face exposed. Then with one undramatic motion, his right hand clasped the flare of the cheekpiece and tugged the ghastly mask down; in an instant the humanity of his face vanished, his gentle expressive eyes became unseeable pools of blackness chasmed within the fierce eye sockets of bronze; all compassion fled in an instant from his aspect, replaced with the blank mask of murder. "Push it back," I cried. "You're scaring the hell out of me." It was no joke.
This now Dienekes was assessing, the effect of Hellenic armor upon the enemy. My master's eyes scanned the foe's ranks; you could see piss stains darkening the trouser fronts of more than one man. Spear tips shivered here and there. Now the Medes formed up. Each rank found its mark, each commander his station.
More endless moments passed. Tedium stood displaced by terror. Now the nerves began to scream; the blood pounded within the recesses of the ears. The hands went numb; all sensation fled the limbs. One's body seemed to treble in weight, all of it cold as stone. One heard one's own voice calling upon the gods and could not tell if the sound was in his head or if he was shamefully crying aloud.
His Majesty's vantage may have been too elevated upon the overstanding mountain to descry what happened next, what stroke of heaven immediately precipitated the clash. It was this. Of a sudden a hare started from the cliffside, dashing out directly between the two armies, no more than thirty feet from the Thespaian commander, Xenocratides, who stood foremost in advance of his troops, flanked by his captains, Dithyrambos and Protokreon, all of them garlanded, with their helmets tucked under their arms. At the sight of this wildly sprinting prey, the roan bitch Styx, who had been already barking furiously, loose at the right flank of the Greek formation, now bolted like a shot into the open. The effect would have been comical had not every Hellene's eye seized upon the event at once as a sign from heaven and attended breathlessly upon its outcome.
The hymn to Artemis, which the troops were singing, faltered in midbreath. The hare fled straight for the Median front-rankers, with Styx hot on its heels and mad with pursuit. Both beasts appeared as screaming blurs, the puffs of dust from their churning feet hanging motionless in the air while their bodies, stretched to the full in the race, streaked on before them. The hare sped straight toward the mass of the Medians, at the approach to which it panicked and tore into a tumble, end over end, as it attempted a right-angle turn at top speed. In a flash Styx was on it; the hound's jaws seemed to snap the prey in two, but, to the astonishment of all, the hare burst free, unscathed, and in an eyeblink had regained full velocity in flight.
A zigzag chase ensued, in duration fewer than a dozen heartbeats, in which hare and hound traversed thrice the _oudenos chorion,_ the no-man's-land, between the armies. A hare will always flee uphill; its forelegs are shorter than its rear. The speedster sprung now for the mountain wall, attempting to scamper to salvation. But the face was too sheer; the fugitive's feet skidded out from under; it tumbled, fell back. In an instant its form hung limp and broken within the Stygian jaws.
A cheer rose from the throats of four thousand Greeks, certain that this was an omen of victory, the answer to the hymn it had so serendipitously interrupted. But now from the ranks of the Medes stepped forth two archers. As Styx turned, seeking his master to show off the prize, a pair of cane arrows, launched from no farther than twenty yards and striking simultaneously, slammed into the beast's flank and throat, tumbling him head over heels into the dust.
A cry of anguish erupted from the Skirite whom all had come to call "Hound." For agonizing moments his dog flopped and writhed, pinioned mortally by the enemy's shafts. We heard the enemy commander cry an order in his tongue. At once a thousand Median archers elevated their bows. "Here it comes!" someone cried from the Wall. Every Hellene's shield was snatched at once to high port. That sound which is not a sound but a silence, a rip like that of fabric torn in the wind, now keened from the fisted grips of the enemy's massed bowmen as their string hands released and their triple-pointed bronzeheads sprung as one into the air, shafts singing, driving them forward.
While these missiles arced yet through the aether, the Thespaian commander, Xenocratides, seized the instant. "Zeus Thunderer and Victory!" he cried, tearing the garland from his brow and jerking his helmet down into position of combat, covering, save the eye slits, his entire face. In an instant every man of the Hellenes followed suit. A thousand arrows rained on them in homicidal deluge. The _sarpinx_ bellowed. "Thespiae!"
From where I stood atop the Wall, it seemed as if the Thespaians closed to the foe within the space of two heartbeats. Their front ranks hit the Medes not with that sound of thunder, bronze upon bronze, which the Hellenes knew from collisions with their own kind, but with a less dramatic, almost sickening crunch, like ten thousand fistfuls of kindling stalks snapped in the vineyardman's fists, as the metallic facings of the Greeks' shields collided with the wall of wicker thrown up by the Medes. The enemy reeled and staggered. The Thespaians' spears rose and plunged. In an instant the killing zone was obscured within a maelstrom of churning dust.
The Spartans atop the wall held motionless as that peculiar bellowslike compression of ranks unfolded before their sight; the first three ranks of the Thespaians compacted against the foe and churned like a movable wall upon them; now the succeeding ranks, fourth, fifth, sixth, seventh, eighth and more, between whom an interval had opened in the rush, caught up, wave succeeding wave and compressing one upon the other, as each man elevated his shield to high port and planted it as squarely as his terror-unstrung limbs would permit into the back of the comrade before him, seating his left shoulder beneath the upper rim, and, digging his soles and toes into the earth for purchase, hurled himself with all his force into the melee. The heart stopped with the awe of it, as each warrior of the Thespaians cried out to his gods, to the souls of his children, to his mother, to every entity, noble or absurd, which he could imagine of aid, and, forgetting his own life, waded with impossible courage into the mob of murder.
What had been a moment earlier a formation of troops, discernible as ranks and files, even as individuals, transformed in the space of a heartbeat into a roiling mass of manslaughter. The Thespaian reserves could not contain themselves; they, too, hurled themselves forward, pressing the weight of their ranks into the backs of their brothers, heaving against the compacted mass of the enemy.
Behind these the Thespaians' squires danced like ants on a skillet, unranked and unarmored, some backpedaling in terror, others dashing forward, crying out to each other to remember their courage and not fail the men they served. Toward these servants of the train now sailed a second and third rainbow of arrows, loosed by the massed enemy archers stationed to the rear of their lancers and fired in arching fusillades directly over their comrades' plumed heads. The bronzeheads struck the earth in a ragged but discernible front, like a squall line at sea. One could see this curtain of death withdraw rearward as the Median archers fell back behind their lancers, maintaining an interval so they could concentrate their fire upon the mass of the Greeks assaulting them and not squander it, lobbing shafts over their heads. One Thespaian squire dashed recklessly forward to the squall line. A bronzehead nailed him right through the foot. He cavorted off, howling in pain and cursing himself for an idiot.
"Forward to Lion Stone!"
With a cry, Leonidas dismounted his post atop the Wall and advanced down the stone slope, which had been erected deliberately with a descendible incline, into the open before the Spartans, Mycenaeans and Philiasians. These now followed, as the "beaten zone" of the enemy's bronzeheads retreated under the furious push of the Thespaians, maintaining the dress of their lines, as they had rehearsed half a hundred times in the preceding four days, forming up in ready position on the level ground before the Wall.
Along the mountain face to the left, three stones, each at twice the height of a man so they could be seen above the dust of battle, had been selected as benchmarks.
Lizard Stone, so named for a particularly fearless fellow of that species who took his sun thereupon, stood farthest forward of the Phokian Wall, closest to the Narrows, perhaps a hundred and fifty feet from the actual mouth of the pass. This was the line to which the enemy would be permitted to advance. It had been determined by trial with our own men that a thousand of the foe, densely packed, could fit between this demarcation and the Narrows. A thousand, Leonidas had ordered, will be invited to the dance. There, at Lizard Stone, they will be engaged and their advance checked.
Crown Stone, second of the three and another hundred feet rearward of Lizard, defined the line at which each relief detachment would marshal, immediately before being hurled into the fray.
Lion Stone, rearmost of the three and directly in front of the Wall, marked the waiting line—the runners' chute, at which each relief unit would marshal, leaving enough space between itself and those actually fighting for the rear ranks of the combatants to maneuver, to give ground if necessary, to rally, for one flank to support another and for the wounded to be withdrawn.
Along this demarcation the Spartans, Mycenaeans and Philiasians now took their stations. "Dress the line!" the _polemarch_ Olympieus bellowed. "Close up your interval!" He prowled before the front, disdaining the drizzle of arrows, shouting to his platoon commanders, who relayed the orders to their men.
Leonidas, out farther still before Olympieus, surveyed the roiling, dust-choked struggle ahead at the Narrows. The sound, if anything, had increased. The clash of sword and spear upon shield, the ringing bell-like toll of the bowl-shaped bronze, the cries of the men, the sharp cracking explosions as lances shivered under impact and snapped in two; all echoed and reverberated between the mountain face and the Narrows like some _theatron_ of death circumvallated within its own stone amphitheater. Leonidas, still garlanded, with his helmet up, turned and signaled to the _polemarch._
"Shields to rest!" Olympieus' voice boomed. Along the Spartan line, _aspides_ were lowered and set upright upon the earth, top rims balanced against each man's thigh, with the shield's forearm sheath and gripcord ready to hand. All helmets were up, each man's face still exposed. Beside Dienekes, his captain-of-eight, Bias, was hopping like a flea. "This is it, this is it, this is it."
"Steady, gentlemen." Dienekes stepped forward to let his men see him. "Rest those cheeseplates." In the third rank Ariston, beside himself with agitation, yet clutched his shield at port. Dienekes reached through and whacked him with the flat of his lizard-sticker. "Are you showing off?" The youth snapped to, blinking like a boy awoken from a nightmare. For a full heartbeat you could see he had no idea who Dienekes was or what he wanted. Then, with a start and a sheepish expression, he recovered himself and lowered his shield to position of rest against his knee.
Dienekes prowled before the men. "All eyes on me! Here, brothers!" His voice penetrated, hard and throaty, carrying with the hoarse bark all combatants know when their tongue turns to leather. "Look at me, don't look at the fighting!"
The men tore their eyes from the flood and ebb of murder which was taking place a stone's lob in front of them. Dienekes stood before them, his back to the enemy. "This is what's happening, a blind man could tell just from the sound." Dienekes' voice carried despite the din from the Narrows. "The enemy's shields are too small and too light. They can't protect themselves. The Thespaians are carving them up." The men's glances kept tearing away toward the struggle. "Look at me! Put your lamps here, goddamn you! The enemy hasn't broken yet. They feel their King's eyes upon them. They're falling like wheat but their courage hasn't failed. I said, look at me! In the killing zone, you see our allies' helmets now, rising out of the slaughter; it seems as if the Thespaians are mounting a wall. They are. A wall of Persian bodies."
This was true. Distinctly could be beheld a rise of men, a wave of its own within the boiling melee. "The Thespaians will only last a few more minutes. They're exhausted from killing. It's a grouse shoot. Fish in a net. Listen to me! When our turn comes, the enemy will be ready to cave. I can hear him cracking now. Remember: we're going in for a boxer's round. In and out. Nobody dies. No heroes. Get in, kill all you can, then get out when the trumpets sound."
Behind the Spartans, on the Wall, which had been filled with the third wave of Tegeates and Opountian Lokrians twelve hundred strong, the wail of the _sarpinx_ cut the din. Out front, Leonidas raised his spear and tugged his helmet down. You could see Polynikes and the Knights advance to envelop him. The Thespaians' round was over. "Hats down!" Dienekes bellowed. "Cheeseplates up!"
The Spartans came in frontally, eight deep, at a double interval, allowing the Thespaian rearmen to withdraw between their files, man by man, one rank at a time. There was no order to it; the Thespaians just dropped from exhaustion; the Lakedaemonian tread rolled over them. When the Spartan _promachoi,_ the forerankers, got within three shields of the front, their spears began plunging at the foe over the allies' shoulders. Many of the Thespaians just dropped and let themselves be trampled; their mates pulled them to their feet once the line had passed over them.
Everything Dienekes had said proved true. The Medes' shields were not only too light and too small, but their lack of mass prevented them from gaining purchase against the Hellenes' wide and weighty, bowl-shaped _aspides._ The enemy's targeteer shields slid off the convex fronts of the Greeks', deflecting up and down, left and right, exposing their bearers' necks and thighs, throats and groins. The Spartans struck overhand with their spears, again and again into the faces and gorges of the enemy. The Medes' armament was that of skirmishers, of lightly armed warriors of the plains, whose role was to strike swiftly, from beyond range of spear thrust, dealing death at a distance. This dense-packed phalanx warfare was hell on them.
And yet they stood. Their valor was breathtaking, beyond reckless to the point of madness. It became sacrifice, pure and simple; the Medes gave up their bodies as if flesh itself were a weapon. In minutes the Spartans, and no doubt the Mycenaeans and Philiasians as well, though I couldn't see them, were beyond exhaustion. Simply from killing. Simply from the arm's thrust of the spear, the shoulder's heave of the shield, the thunder of blood through the veins and the hammering of the heart within the breast. The earth grew, not littered with enemy bodies, but piled with them. Stacked with them. Mounded with them.
At the heels of the Spartans, their squires abandoned all thought of inflicting casualties with their own missile weapons, turning to nothing but dragging out trampled corpses of the foe to help their men maintain footing. I saw Demades, Ariston's squire, slit three wounded Medes' throats in fifteen seconds, slinging their carcasses back onto a mound already writhing with groaning men.
Discipline had broken among the Median forerankers; officers' bawled orders could not be heard amid the din, and even if they could, the men were so overwhelmed in the crush they could not respond to them. Still the rank and file had not panicked. In desperation they cast aside bows, lances and shields and simply grappled with bare hands onto the weapons of the Spartans. They clutched at spears, hanging on with both hands and struggling to wrest them from the Spartans' grips. Others of the foe flung themselves bodily onto the Lakedaemonians' shields, clasping the top rim and pulling the bowls of the _aspides_ down, scratching and clawing at the Spartans with fingers and fingernails.
Now the slaughter in the forefront became man-to-man, with only the wildest semblance of rank and formation. The Spartans slew belly-to-belly with the murderously efficient thrust-and-draw of their short _xiphos_ swords. I saw Alexandros, his shield torn from his grip, plunging his _xiphos_ into the face of a Mede whose hands clawed and pounded at Alexandros' groin.
The middle-rankers of the Lakedaemonians surged into this bedlam, spears and shields still intact. But the Medes' capacity for reinforcement seemed limitless; above the fray, one could glimpse the next thousand reinforcements thundering into the Narrows like a flood, with more myriads behind, and yet more after that. Despite the catastrophic magnitude of their casualties, the tide began to flow in the enemy's favor. The weight of their masses alone began to buckle the Spartan line. The only thing that stopped the foe from swamping the Hellenes outright was that they couldn't get enough men through the Narrows quickly enough; that, and the wall of Median bodies that now obstructed the confines like a landslide.
The Spartans fought from behind this wall of flesh as if it were a battlement of stone. The enemy swarmed atop it. Now we in the rear could see them; they became targets. Twice Suicide drilled javelins right over Alexandros' shoulder into Medes lunging at the youth from atop the mound of corpses. Bodies were underfoot everywhere. I mounted atop what I thought was a stone, only to feel it writhe and wriggle beneath me. It was a Mede, alive. He plunged the stub end of a shattered _machaera_ scimitar three inches into my calf; I bellowed in terror and toppled into the tangle of other gore-splattered limbs. The foe came at me with his teeth. He seized my arm as if to tear it from its socket; I punched him in the face with my bow still in my grip. Suddenly a foot planted itself massively upon my back. A battle-axe fell with a grisly swoosh; the enemy's skull split like a melon. "What are you looking for down there?" a voice bellowed. It was Akanthus, Polynikes' squire, spray-blasted with blood and grinning like a madman.
The enemy flooded over the wall of bodies. By the time I got to my feet I had lost sight of Dienekes; I couldn't tell which platoon was which or where my proper station was. I had no idea how long we had been in the fight. Was it two minutes or twenty? I had two spears, spares, lashed to my back, their iron sheathed in leather so that, should I tumble accidentally, the spearpoints would work no harm to our comrades. Every other squire bore the same burden; they were all as scrambled as I was.
Up front you could hear the Median lancers' shafts snapping as they clashed and shivered against the Spartan bronze. The Spartans' eight-footers made a different sound than the shorter, lighter lances of the foe. The flood was working against the Lakedaemonians, not from want of valor, but simply in consequence of the overwhelming masses of men which the enemy flung into the teeth of the line. I was frantic to locate Dienekes and deliver my spares. The scene was chaos. I could hear breakdowns right and left and see the rear-rankers of the Spartans buckling as the files before them gave way beneath the weight of the Median onslaught. I had to forget my master and serve where I could.
I dashed to a point where the line was thinnest, only three deep and beginning to swell into the desperate inverse bulge that precedes an out-and-out break. A Spartan fell backward amid the maw of slaughter; I saw a Mede lop the warrior's head clean off with a thunderous slash of a scimitar. The skull toppled, helmet and all, severed from its torso and rolling in the dust, with the marrow gushing and the bone of the spine showing grayish white and ghastly. Helmet and head vanished amid a storm of churning greaves and shod and unshod feet. The murderer loosed a cry of triumph, raising his blade to heaven; half an instant later a crimson-clad warrior buried an eight-footer so deep in the foeman's guts that its killing steel burst free, clear out the man's back. I saw another Mede pass out in terror. The Spartan couldn't haul the weapon back out, so he broke it right off, planting his foot on the still-living enemy's belly and snapping the ash shaft in two. I had no idea who this hero was, and never did find out.
"Spear!" I heard him bellow, the hellish eye sockets of his helmet spinning to the rear for relief, for a spare, for anything to call to hand. I tore both eight-footers off my back and thrust them into the unknown warrior's hands. Backward. He seized one and whirled, planting it with both hands into another Mede's throat, butt-spike-first. His shield's gripcord had been severed or snapped from within; the _aspis_ itself had fallen to the dirt. There was no room to retrieve it. Two Medes lunged toward the Spartan with lances leveled, only to be intercepted by the massive bowl of his rankmate's shield, dropping into place to defend him. Both enemy lances snapped as their heads drove against the bronze facing and oak bowlwork of the shield. In the rush, their momentum carried them forward, sprawling onto the ground atop and tangled with the first Spartan. He drove his _xiphos_ into the first Mede's belly, rose with a cry of homicide and slashed the second hilt-deep across both eyes. The enemy clutched his face in horror, blood gushing between the fingers of his clenched and clawing hands. The Spartan seized with both hands his own fallen shield and brought its rim down like an onion chopper, with such force upon the enemy's throat that it nearly decapitated him.
"Re-form! Re-form!" I heard an officer shouting. Someone shoved me aside from behind. In an instant other Spartans, from another platoon, surged forward, reinforcing the membrane-thin front which teetered at the brink of buckling. This was fighting "scrambled." It stopped the heart to behold the gallantry of it. In moments, what had been a situation at the brink of catastrophe was transformed by the discipline and order of the reinforcing ranks into a strongpoint, a fulcrum of vantage. Each man who found himself in the fore, no matter what rank he had held in formation, now assumed the role of officer. These closed ranks and lapped shields, shadow-to-shadow. A wall of bronze rose before the scrambled mass, buying precious instants for those who found themselves in the rear to re-form and remarshal, surging into position in second, third, fourth ranks, and take on that station's role and rally to it.
Nothing fires the warrior's heart more with courage than to find himself and his comrades at the point of annihilation, at the brink of being routed and overrun, and then to dredge not merely from one's own bowels or guts but from one's own discipline and training the presence of mind not to panic, not to yield to the possession of despair, but instead to complete those homely acts of order which Dienekes had ever declared the supreme accomplishment of the warrior: to perform the commonplace under far-from-commonplace conditions. Not only to achieve this for oneself alone, as Achilles or the solo champions of yore, but to do it as part of a unit, to feel about oneself one's brothers-in-arms, in an instance like this of chaos and disorder, comrades whom one doesn't even know, with whom one has never trained; to feel them filling the spaces alongside him, from spear side and shield side, fore and rear, to behold one's comrades likewise rallying, not in a frenzy of mad possession-driven abandon, but with order and self-composure, each man knowing his role and rising to it, drawing strength from him as he draws it from them; the warrior in these moments finds himself lifted as if by the hand of a god. He cannot tell where his being leaves off and that of the comrade beside him begins. In that moment the phalanx forms a unity so dense and all-divining that it performs not merely at the level of a machine or engine of war but, surpassing that, to the state of a single organism, a beast of one blood and heart.
The foemen's arrows rained upon the Spartan line. From where I found myself, just behind the rear-rankers, I could see the warriors' feet, at first churning in disarray for purchase on the blood and gore-beslimed earth, now settle into a unison, a grinding relentless cadence. The pipers' wail pierced the din of bronze and fury, sounding the beat which was part music and part pulse of the heart. With a heave, the warriors' shield-side foot pressed forward, bows-on to the enemy; now the spear-side foot, planted at a ninety-degree angle, dug into the mud; the arch sank as every stone of the man's weight found purchase upon the insole, and, with left shoulder planted into the inner bowl of the shield whose broad outer surface was pressed into the back of the comrade before him, he summoned all force of tissue and tendon to surge and heave upon the beat. Like ranked oarsmen straining upon the shaft of a single oar, the unified push of the men's exertions propelled the ship of the phalanx forward into the tide of the enemy.
Up front the eight-footers of the Spartans thrust downward upon the foe, driven by each man's spear arm in an overhand strike, across the upper rim of his shield, toward the enemy's face, throat and shoulders. The sound of shield against shield was no longer the clash and clang of initial impact, but deeper and more terrifying, a grinding metallic mechanism like the jaws of some unholy mill of murder. Nor did the men's cries, Spartans and Medes, rise any longer in the mad chorus of rage and terror. Instead each warrior's lungs pumped only for breath; chests heaved like foundry bellows, sweat coursed onto the ground in runnels, while the sound which arose from the throats of the contending masses was like nothing so much as a myriad quarrymen, each harnessed to the twined rope of the sled, groaning and straining to drag some massive stone across the resisting earth.
War is work, Dienekes had always taught, seeking to strip it of its mystery. The Medes, for all their valor, all their numbers and all the skill they doubtless possessed in the type of open-plain warfare with which they had conquered all Asia, had not served their apprenticeship in this, Hellene-style heavy-infantry combat. Their files had not trained to hold line of thrust and gather themselves to heave in unison; the ranks had not drilled endlessly as the Spartans had in maintaining dress and interval, cover and shadow. Amid the manslaughter the Medes became a mob. They shoved at the Lakedaemonians like sheep fleeing a fire in a shearing pen, without cadence or cohesion, fueled only by courage, which, glorious though it was, could not prevail against the disciplined and cohesive assault which now pressed upon them.
The luckless foemen in front had nowhere to hide. They found themselves pinned between the mob of their own fellows trampling them from behind and the Spartan spears plunging upon them from the fore. Men expired simply from want of breath. Their hearts gave out under the extremity. I glimpsed Alpheus and Maron; like a pair of yoked oxen the brothers, fighting shoulder-to-shoulder, formed the tempered steel point of a twelve-deep thrust that drove into and split the Median ranks a hundred feet out from the mountain wall.
The Knights, to the twins' right, drove into this breach with Leonidas fighting in the van; they turned the enemy line into a flank and pressed furiously upon the foemen's unshielded right. God help the sons of the Empire seeking to stand against these, Polynikes and Doreion, Terkleius and Patrokles, Nikolaus and the two Agises, all matchless athletes in the prime of young manhood, fighting alongside their king and mad to seize the glory that now quavered within their grasp.
For myself, I confess the horror of it nearly overcame me. Though I had loaded up double with two packed quivers, twenty-four ironheads, the demands of fire had come so fierce and furious that I was down to nothing before I could spit. I was firing between the helmets of the warriors, point-blank into the faces and throats of the foe. This was not archery, it was slaughter. I was pulling ironheads from the bowels of still-living men to reload and replenish my spent stock. The ash of a shaft drawn across my bow hand slipped from its notch, slimy with gore and tissue; warheads dripped blood before they were even fired. Overwhelmed by horror, my eyes clamped shut of their own will; I had to tear at my face with both hands to drive them open. Had I gone mad?
I was desperate to find Dienekes, to get to my station covering him, but the part of my mind which still owned its wits ordered me to rally myself here, contribute here.
In the crush of the phalanx each man could sense the sea change as the rush of emergency passed like a wave, replaced by the steadying, settling sensation of fear passing over, composure returning and the drill settling to the murderous work of war. Who can say by what unspoken timbre the tidal flow of the fight is communicated within the massed ranks? Somehow the warriors sensed that the Spartan left, along the mountain face, had broken the Medes. A cheer swept laterally like a storm front, rising and multiplying from the throats of the Lakedaemonians. The enemy knew it too. They could feel their line caving in.
Now at last I found my master. With a cry of joy I spotted his cross-crested officer's helmet, in the fore, pressing murderously upon a knot of Median lancers who no longer offered attack but only stumbled rearward in terror, casting away their shields as they fouled upon the desperate press of men behind them. I sprinted toward his position, across the open space immediately to the rear of the grinding, gnashing, advancing Spartan line. This strip of hinter ground comprised the only corridor of haven upon the entire field, in the overshot gap between the hand-to-hand slaughter of the line and the "beaten zone" of the Median archers' arrows, which they flung from the rear of their own lines over the clashing armies toward the Hellenic formations waiting in reserve.
The Median wounded had dragged themselves into this pocket of sanctuary, they and the terror-stricken, the possum players and the exhausted. Enemy bodies were everywhere, the dead and the dying, the trampled and the overrun, the maimed and the massacred. I saw a Mede with a magnificent beard sitting sheepishly upon the ground, cradling his intestines in his hands. As I dashed past, one of his own kinsmen's arrows rained from above, nailing his thigh to the turf. His eyes met mine with the most piteous expression; I don't know why, but I dragged him a half dozen strides, into the mainland of the pocket of illusory safety. I looked behind. The Tegeates and the Opountian Lokrians, our allies next up into the fray, knelt in their ranks, massed along the line below Lion Stone with their shields interleaved and elevated to deflect the deluge of enemy shafts. The expanse of earth before them bristled like a pincushion, as dense with enemy arrows as the quills of a hedgehog's spine. The palisade of the Wall was afire, blazing with the tow bolts of the enemy by the hundred.
Now the Median lancers cracked. Like a child's game of bowls, their stacked files toppled rearward; bodies fell and tumbled upon one another as those in the fore attempted to flee and those in the rear became entangled pell-mell with their flight. The ground before the Spartan advance became a sea of limbs and torsos, trousered thighs and bellies, the backs of men crawling hand over hand across their fallen comrades, while others, pinned upon their backs, writhed and cried out in their tongue, hands upraised, pleading for quarter.
The slaughter surpassed the mind's capacity to assimilate it. I saw Olympieus thrashing rearward, treading not upon ground, but upon the flesh of the fallen foe, across a carpet of bodies, the wounded as well as the dead, while his squire, Abattus, flanked him, sinking his lizard-sticker, punching the spiked shaft downward like a boatman poling a punt, into the bellies of the yet-unslain enemy as they passed. Olympieus advanced into plain view of the allied reserves in position along the Wall. He stripped his helmet so the commanders could see his face, then pumped thrice with his horizontally held spear. "Advance! Advance!"
With a cry that curdled the blood, they did.
I saw Olympieus pause bareheaded and stare at the foe-strewn earth about him, himself overcome by the scale of the carnage. Then he reseated his helmet; his face vanished beneath the blood-blasted bronze and, summoning his squire, he strode back to the slaughter.
To the rear of the routed lancers stood their brothers, the Median archers. These were drawn up in still-ordered ranks, twenty deep, each bowman in station behind a body-height shield of wicker, its base anchored to the earth with spikes of iron. A no-man's-land of a hundred feet separated the Spartans from this wall of bowmen. The foe now began firing directly into their own lancers, the last pockets of the valiant who yet grappled with the Lakedaemonian advance.
The Medes were shooting their own men in the back.
They didn't care if they slew ten of their brothers, if one lucky bolt could nail a Spartan.
Of all the moments of supreme valor which unfolded throughout this long grisly day, that which the allies upon the Wall now beheld surpassed all, nor could any who witnessed it place any sight beneath heaven alongside it as equal. As the Spartan front routed the last remaining lancers, its forerankers emerged into the open, exposed to what was now the nearly point-blank fire of the Median archers. Leonidas himself, at his age having survived a melee of murder whose physical expenditure alone would have pressed beyond the limits of endurance even the stoutest youth in his prime, yet summoned the steel to stride to the fore, shouting the order to form up and advance. This command the Lakedaemonians obeyed, if not with the precision of the parade ground, then with a discipline and order beyond imagining under the circumstances. Before the Medes had time to loose their second broadside, they found themselves face-to-face with a front of sixty-plus shields, the _lambdas_ of Lakedaemon obscured beneath horrific layers of mud, gore and blood which ran in rivers down the bronze and dripped from the leather aprons pended beneath the _aspides,_ the oxhide skirts which protected the warriors' legs from precisely the fusillade into which they now advanced. Heavy bronze greaves defended the calves; above each shield rim extended only the armored crowns of the helmets, eye slits alone exposed, while overtopping these waved the front-to-back horsehair plumes of the warriors and the transverse crests of the officers.
The wall of bronze and crimson advanced into the Median fire. Cane arrows ripped with murderous velocity into the Spartan lines. Possessed by terror, an archer will always shoot high; you could hear these overshot shafts hailing and clattering as they ripped at crown height past the Spartan foreranks and tore into the forest of spears held at the vertical; then the missiles tumbled, spent, among the armored ranks. Bronzehead bolts caromed off bronze-faced shields with a sound like a hammer on an anvil, their furious drumming punctuated by the concussive _thwock_ of a dead-on shot penetrating metal and oak so the head lanced through the shield like a nail piercing a board.
I myself had planted shoulder and spine into the back of Medon, senior of the Deukalion mess, whose station of honor stood rearmost of the first file in Dienekes' platoon. The pipers were hunkered immediately in the lee of the formation, unarmed and unarmored, crouching for cover as close to the heels of the rear-rankers as they could without tripping them, all the while summoning breath to skirl out the shrill _aulos'_ s beat. The densely packed ranks advanced not in a mobbed disordered charge shouting like savages, but dead silent, sober, almost stately, with a dread deliberateness in time to the pipers' keening wail. Between the fighting fronts, the hundred-foot gap had narrowed to sixty. Now the Medes' fire redoubled. You could hear the orders bawled by their officers and feel the air itself vibrate as the ranks of the foe loosed their fusillades in ever more furious succession.
A single arrow blazing past one's ear can turn the knees to jelly; the honed warhead seems to scream with malevolence, the hurdling weight of the shaft driving its death-dealing cargo; then come the fletched feathers communicating by their silent shriek the homicidal intent of the enemy. A hundred arrows make a different sound. Now the air seems to thicken, to become dense, incandescent; it vibrates like a solid. The warrior feels encapsulated as in a corridor of living steel; reality shrinks to the zone of murder in which he finds himself imprisoned; the sky itself cannot be glimpsed nor even remembered.
Now come a thousand arrows. The sound is like a wall. There is no space within, no interval of haven. Solid as a mountain, impenetrable; it sings with death. And when those arrows are launched not skyward in long-range arcing trajectory to beat upon the target driven by the weight of their own fall, but instead are fired point-blank, dead flush from the chute of the bowman's grip, so that their flight is level, flat, loosed at such velocity and at such close range that the archer does not trouble even to calculate drop into his targeting equation; this is the rain of iron, hellfire at its purest.
Into this the Spartans advanced. They were told later by the allies observing from the Wall that at this instant, as the spears of the Spartans' front ranks lowered in unison from the vertical plane of advance into the leveled position of attack and the serried phalanx lengthened stride to assault the foe at the double; at this moment, His Majesty, looking on, had leapt to his feet in terror for his army.
The Spartans knew how to attack wicker. They had practiced against it beneath the oaks on the field of Otona, in the countless repetitions when we squires and helots took station with practice shields, planted our heels and braced with all our strength, awaiting the massed shock of their assault. The Spartans knew the spear was worthless against the interlatticed staves; its shaft penetrated the wicker only to become imprisoned and impossible to extract. Likewise the thrust or slash of the _xiphos,_ which caromed off as if striking iron. The enemy line must be struck, shock troop style, and overwhelmed, bowled over; it had to be hit so hard and with such concentrated force that its front-rankers caved and toppled, one rank backward upon another, like plateware in a cabinet when the earth quakes.
This is precisely what happened. The Median archers were drawn up not in a massed square front-to-back with each warrior reinforcing his comrades against the shock of assault, but honeycombed in alternating fronts, each rank at the shoulder of the one before it, so that the bowmen in the second could fire in the gaps left by the first, and on in this fashion rearward throughout the formation.
Moreover the enemy ranks were not stacked with the massed compaction of the Spartan phalanx. There was a void, an interval between ranks dictated by the physical demands of the bow. The result of this was precisely what the Lakedaemonians expected: the forerank of the enemy collapsed immediately as the first shock hit it; the body-length shields seemed to implode rearward, their anchoring spikes rooted slinging from the earth like tent pins in a gale. The forerank archers were literally bowled off their feet, their wall-like shields caving in upon them like fortress redoubts under the assault of the ram. The Spartan advance ran right over them, and the second rank, and the third. The mob of enemy mid-rankers, urged on by their officers, sought desperately to dig in and hold. Closed breast-to-breast with the Spartan shock troops, the foe's bows were useless. They flung them aside, fighting with their belt scimitars. I saw an entire front of them, shieldless, slashing wildly with a blade in each hand. The valor of individual Medes was beyond question, but their light hacking blades were harmless as toys; against the massed wall of Spartan armor, they might as well have been defending themselves with reeds or fennel stalks.
We learned that evening, from Hellenic deserters who had fled in the confusion, that the foe's rearmost ranks, thirty and forty back from the front, had been pressed rearward so resistlessly by the collapse of the men up front that they began tumbling off the Trachinian track into the sea. Pandemonium had apparently reigned along a section several hundred yards long, beyond the Narrows, where the trail ran flush against the mountain wall, with the gulf yawning eighty feet below. Over this brink, the deserters reported, hapless lancers and archers had toppled by the score, clinging to the men before them and pulling these down with them to their deaths. His Majesty, we heard, was forced to witness this, as his vantage lay almost directly above the site. This was the second moment, so the observers reported, when His Majesty sprung to his feet in dread for the fate of his warriors.
The ground immediately to the rear of the Spartan advance, as expected, was littered with the trampled forms of the enemy dead and wounded. But there was a new wrinkle. The Medes had been overrun with such speed and force that numbers of them, far from inconsiderable, had survived intact. These now rose and attempted to rally, only to find themselves assaulted almost at once by the massed ranks of the allied reserves who were already advancing in formation to reinforce and relieve the Spartans. A second slaughter now ensued, as the Tegeates and Opountian Lokrians fell upon this yet-unreaped harvest. Tegea lies immediately adjacent to the territory of Lakedaemon. For centuries the Spartans and Tegeates had battled over the border plains before, in the previous three generations, becoming fast allies and comrades. Of all the Peloponnesians save the Spartans, the warriors of Tegea are the fiercest and most skilled. As for the Lokrians of Opus, this was their country they were fighting for; their homes and temples, fields and sanctuaries, lay within an hour's march of the Hot Gates. Quarter, they knew, stood not within the invader's lexicon; neither would it be found in theirs.
I was dragging a wounded Knight, Polynikes' friend Doreion, to the safety of the field's shoulder when my foot slipped in an ankle-deep stream. Twice I tried to regain balance and twice fell. I was cursing the earth. What perverse spring had suddenly burst forth from the mountainside when none had shown itself in this place before? I looked down. A river of blood covered both feet, draining across a gouge in the dirt like the gutter of an abattoir.
The Medes had cracked. The Tegeates and Opountian Lokrians surged in reinforcement through the ranks of the spent Spartans, pressing the assault upon the reeling enemy. It was the allies' turn now. "Put the steel to 'em, boys!" one among the Spartans cried as the wave of allied ranks advanced ten deep from the rear and both flanks and closed into a massed phalanx before the warriors of Sparta, who at last drew up, limbs quaking with fatigue, and collapsed against one another and upon the earth.
At last I found my master. He was on one knee, shattered with exhaustion, clinging with both fists to his shivered blade-bereft spear which was driven butt-spike-first into the earth and from which he hung like a broken marionette upon a stick. The weight of his helmet bore his head groundward; he possessed strength neither to lift it nor to pull it off. Alexandros collapsed beside him, on all fours with the crown of his helmet, crest-first, mashed with exhaustion into the dirt. His ribcage heaved like a hound's, while spittle, phlegm and blood dripped from the bronze of his cheekpieces in a frothing lather.
Here came the Tegeates and Lokrians, surging past us.
There they went, driving the enemy before them.
For the first interval in what seemed an eternity, the dread of imminent extinction lifted. The Lakedaemonians dropped to the earth where they stood, on knees first, then knees and elbows, then simply sprawling, on sides and on backs, collapsing against one another, sucking breath in gasping labored need. Eyes stared vacantly, as if blind. None could summon strength to speak. Weapons drooped of their own weight, in fists so cramped that the will could not compel the muscles to release their frozen grasp. Shields toppled to earth, bowl-down and defamed; exhausted men collapsed into them face-first and could not find strength even to turn their faces to the side to breathe.
A fistful of teeth spit from Alexandros' mouth. When he recovered strength sufficient to prise his helmet off, his long hair came out at the roots in wads, a tangled mass of salt sweat and matted blood. His eyes stared, blank as stones. He collapsed like a child, burying his face in my master's lap, weeping the dry tears of those whose shattered substance has no more fluid to spend.
Suicide came up, shot through both shoulders and oblivious with elation. He stood above the collapsed ranks of men, fearless, peering out to where the allies had now closed with the last of the Medes and were hacking them to pieces with such a grisly din that it seemed the slaughter was taking place ten paces away instead of a hundred.
I could see my master's eyes, pools of black behind the hooded eye slits of his helmet. His hand gestured feebly to the empty spear sheath across my back. "What happened to my spares?" his throat croaked hoarsely.
"I gave them away."
A moment passed while he waited for breath. "To our own men, I hope."
I helped him off with his helmet. It seemed to take minutes, so swollen with sweat and blood was his felt undercap and the tangled clotted mass of his hair. The water bearers had arrived. None among the warriors possessed the strength even to cup his hands, so the liquid was simply splashed upon rags and blouses which the men pressed to their lips and sucked. Dienekes swabbed the tangled hair off his face. His left eye was gone. Sliced through, leaving a ghoulish socket of tissue and blood.
"I know" was all he said.
Aristomenes and Bias and others of the platoon, Black Leon and Leon Donkeydick, now surfaced into view, gasping upon the earth, their arms and legs sliced and lacerated with innumerable slashes, glistening with mud and blood. They and other scrambled men from other scrambled units lay heaped upon one another like a frieze on a temple wall.
I knelt now at my master's side, pressing the water rag as a compress into the hollow where his eye had been. The fabric welled with fluid like a sponge.
Out front, where the enemy were falling back in wild disorder, the victors of the moment could see Polynikes, on his feet, alone, with his arms raised toward the fleeing foe. He wrenched his helmet from his skull, dripping blood and sweat, and flung it in triumph upon the earth.
"Not today, you sons of whores!" he bellowed at the foe in flight. "Not today!"
**TWENTY-FIVE**
**I** cannot state with certainty how many times on that first day each allied contingent took its turn upon the triangle bounded by the Narrows and the mountain face, the sea cliffs and the Phokian Wall. I can declare with conviction only that my master went through four shields, two whose oak underchassis were shattered by repeated blows, one whose bronze plexus was staved in and a fourth whose gripcord and forearm sheath were ripped right out of their sockets. Replacements were not hard to find. One had only to stoop, so many were the discards littered upon the field, with their owners dead and dying beside them.
Of the sixteen in my master's _enomotia_ were slain on that first day Lampitos, Söobiades, Telemon, Sthlenelaides, Ariston and grievously wounded Nikandros, Myron, Charillon and Bias.
Ariston fell in the fourth and final siege, that against His Majesty's Immortals. Ariston was that youth of twenty years, one of Polynikes' "broken noses," whose sister Agathe had been given as a bride to Alexandros. That made them brothers-in-law.
The retrieval party found Ariston's body around midnight, along the mountain wall. His squire Demades' form law sprawled atop Ariston's with his shield still in place seeking to protect his master, both of whose shins had been shattered by the blows of a _sagaris_ battle-axe. The shaft of an enemy lance was broken off just beneath Demades' left nipple. Although Ariston had sustained more than twenty wounds upon his own body, it was a single blow to the head, apparently delivered with some kind of mace or battle sledge, which had ultimately slain him, crushing both helmet and skull directly above the eyeline.
The tickets of the dead were customarily held and distributed by the chief battle priest, in this case Alexandros' father, the _polemarch_ Olympieus. He himself had been killed, however, slain by a Persian arrow an hour before nightfall, just prior to the final clash with the Persian Immortals. Olympieus had taken shelter with his men on the rampart of the Wall, in the lee of the palisade, preparatory to arming himself for the day's final siege. Of all things, he was writing in his journal. The unburned timbers of the palisade protected him, he thought; he had stripped helmet and cuirass. But the arrow, guided by some perverse fate, pierced the single opening available to it, a space no wider than a man's hand. It struck Olympieus in the cervical spine, severing the spinal cord. He died minutes later, without regaining speech or consciousness, in his son's arms.
With that, Alexandros had lost father and brother-in-law in a single afternoon.
Among the Spartans, the most grievous casualties of the first day were suffered by the Knights. Of thirty, seventeen were either killed or incapacitated too severely to fight. Leonidas was wounded six times but walked off the field under his own power. Astonishingly Polynikes, fighting all day in the forefront of the bloodiest action, had sustained no more than the slashes and lacerations incidental to action, a number of them doubtless inflicted by his own errant steel and that of his mates. He had, however, severely strained both hamstrings and pulled his left shoulder, simply from exertion and the excessive demands made upon the flesh in moments of supreme necessity. His squire, Akanthus, had been killed defending him, luck-lessly like Olympieus, just minutes before the cessation of the day's slaughter.
The second attack had commenced at noon. These were the mountain warriors of Cissia. None among the allies even knew where the hell this place was, but wherever it was, it bred men of ungodly valor. Cissia, the allies learned later, is a country of stern and hostile highlands not far from Babylon, dense with ravines and defiles. This contingent of the enemy, far from being daunted by the cliff wall of Kallidromos, took this obstacle in stride, clambering up and along its face, rolling stones down indiscriminately upon their own troops as well as the allies. I myself could not view this struggle directly, being stationed during that interval behind the Wall, all efforts consumed with tending my master's wounds and those of our platoon and looking to their and my own necessaries. But I could hear it. It sounded like the whole mountain coming down. At one point, from where Dienekes and Alexandros were, in the Spartan camp a hundred feet rearward of the Wall, we could see the ready platoons, in this rotation the Mantinean and the Arkadians, pouring up to the battlements of the Wall and there hurling javelins, spears and even dismantled boulders down upon the attackers, who, in the elation of the triumph they thought at hand, were keening a bloodcurdling wail which I can only replicate as "Elelelelele."
The Thebans, we learned that evening, were the ones who threw back the Cissian assault. These warriors of Thebes held the right flank, as the allies saw it, alongside the sea cliffs. Their commander, Leontiades, and the picked champions fighting alongside him managed to secure a breach in the mob of the enemy, about forty feet out from the cliffs. The Thebans poured into this break and began shoving the cutoff ranks of the foe, about twenty files in breadth, toward the cliffs. Again the weight of the allied armor proved irresistible. The enemy right were rooted backward by the press of their own failing comrades. They toppled into the sea, as before in the rout of the Medes, clutching at the trousers, sword belts and finally the ankles of their fellows, pulling them over with them. The scale and celerity of the slaughter had clearly been massive, made more so by the ghastly manner in which the slain perished, that is, tumbling eighty and a hundred feet to have their bodies broken upon the rocks below or, escaping that, to drown in armor in the sea. Even from our position an eighth of a mile away and above the din of battle, we could hear plainly the cries of the falling men.
The Sacae were the next nation elected by Xerxes to assault the allies. These massed below the Narrows around midafternoon. They were plainsmen and mountain men, warriors of the eastern empire, and the bravest of all the troops the allies faced. They fought with battle-axes and inflicted, for a time, the most grisly casualties upon the Greeks. Yet in the end their own courage was their undoing. They did not break or panic; they simply came on in wave after wave, clawing over the fallen bodies of their brothers to hurl themselves as if seeking their own slaughter upon the shields and iron spearpoints of the Greeks. Against these Sacae were arrayed at first the Mycenaeans, the Corinthians and the Philiasians, with the Spartans, Tegeates and Thespaians in ready reserve. These last were flung into battle almost at once, as the Mycenaeans and Corinthians spent themselves in the mill of murder and became too exhausted to continue. The reserves likewise became shattered with fatigue and themselves had to be relieved by the third rotation of Orchomenians and other Arkadians, these last having barely gotten out of the previous melee and had time to gnaw a hard biscuit and gulp down a snootful of wine.
By the time the Sacae broke, the sun was well over the mountain. The "dance floor," now in full shadow, looked like a field ploughed by the oxen of hell. Not an inch remained unchurned and unriven. The rock-hard earth, sodden now with blood and piss and the unholy fluids which had spilled from the entrails of the slain and the butchered, lay churned in places to the depth of a man's calf. There is a spring sacred to Persephone, behind the sallyport adjacent to the Lakedaemonian camp, where in the morning, immediately following the repulsion of the Median assault, the Spartans and Thespaians had collapsed in exhaustion and triumph. In that initial instant of salvation, however temporary all knew it must be, a flush of supreme joy had flooded over the entire allied camp. Panoplied men faced one another and slammed shields together, just for the joy of it, like boys rejoicing in the clamor of bronze upon bronze. I saw two warriors of the Arkadians standing face-to-face, pounding each other with fists upon the shoulders of their leathers, tears of joy streaming down their faces. Others whooped and danced. One warrior of the Philiasians grasped the corner of the redoubt with both hands and pounded his helmeted brow against the stone, bang bang bang, like a lunatic. Others writhed upon the ground, as horses will do sand-bathing, so overflushed with joy that they could discharge its excess in no other way.
Simultaneously a second wave of emotion coursed through the camp. This was of piety. Men embraced one another, weeping in awe before the gods. Prayers of thanksgiving were sung from fervent hearts, and none took shame to voice them. Across the expanse of the camp, one saw knots of warriors kneeling in invocation, circles of a dozen with clasped hands, knots of three and four with arms around each other's shoulders, pairs crouching knee-to-knee and everywhere individuals upon the earth in prayer.
Now, seven hours into the slaughter, all such observance of piety had fled. Men stared with hollow eyes upon the riven plain. Across this farmer's field of death lay sown such a crop of corpses and shields, hacked-up armor and shattered weapons, that the mind could not assimilate its scale nor the senses give it compass. The wounded, in numbers uncountable, groaned and cried out, writhing amid piles of limbs and severed body parts so intertangled one could not distinguish individual men, but the whole seemed a Gorgon-like beast of ten thousand limbs, some ghastly monster spawned by the cloven earth and now draining itself, fluid by fluid, back into that chthonic cleft which had given it birth. Along the face of the mountain the stone glistened scarlet to the height of a man's knee.
The faces of the allied warriors had by this hour clotted into featureless masks of death. Blank eyes stared from sunken sockets as if the divine force, the _daimon,_ had been extinguished like a lamp, replaced by a weariness beyond description, a stare without affect, the hollow gaze of hell itself. I turned to Alexandros; he looked fifty years old. In the mirror of his eyes I beheld my own face and could no longer recognize it.
A temper toward the enemy now arose which had not been present before. This was not hatred but rather a refusal to reckon quarter. A reign of savagery began. Acts of barbarity which had been hitherto unthinkable now presented themselves to the mind and were embraced without a quibble. The theater of war, the stink and spectacle of carnage on such a scale, had so overwhelmed the senses with horror that the mind had grown numb and insensate. With perverse wit, it actually sought these and sought to intensify them.
All knew that the next attack would be the day's last; nightfall's curtain would adjourn the slaughter until tomorrow. It was also clear that whichever force the foe next threw into the line would be his finest, the cream saved for this hour when the Hellenes labored in exhaustion and stood the likeliest chance of being overthrown by fresh troops. Leonidas, who had not slept now in more than forty hours, yet prowled the lines of defenders, assembling each allied unit and addressing it in person. "Remember, brothers: the final fight is everything. All we have achieved so far this day is lost if we do not prevail now, at the end. Fight as you have never fought before."
In the intervals between the first three assaults, each warrior readying for the next engagement had striven to scour clean the face of his shield and helmet, to present again to the foe the gleaming terror-inspiring surface of bronze. As the threshing mill of murder progressed throughout the day, however, this housekeeping became honored increasingly in the breach, as each knurl and inlay on the shield acquired a grisly encrustment of blood and dirt, mud and excrement, fragments of tissue, flesh, hair and gore of every description. Besides, the men were too tired. They didn't care anymore. Now Dithyrambos, the Thespaian captain, sought to make a virtue of necessity. He ordered his men to cease from burnishing their shields, and instead to paint and streak them, and the men's own body armor, with yet more blood and gore.
This Dithyrambos, by trade an architect and by no means a professional soldier, had already distinguished himself with such magnificent courage throughout the day that the prize of valor, it was a foregone conclusion, would be his by acclamation. His gallantry had elevated him second only to Leonidas in prestige among the men. Dithyrambos now, stationing himself in the open in full view of all the men, proceeded to smear his own shield, which was already nearly black with dried blood, with yet more gore and guts and fresh dripping fluid. The allies in line, the Thespaians, Tegeates and Mantineans, ghoulishly followed suit. The Spartans alone abstained, not out of delicacy or decorum, but simply in obedience to their own laws of campaign, which command them to adhere without alteration to their customary disciplines and practices of arms.
Dithyrambos now ordered the squires and servants to hold their places, to refrain from sweeping the advance ground of enemy bodies. Instead, he sent his own men out onto the arena with orders to heap the corpses in display in the most ghoulish manner possible, so as to present to the next wave of the foe, whose marshaling trumpets could already be heard around the shoulder of the Narrows, the most ghastly and terrifying spectacle possible.
"Brothers and allies, my own beautiful dogs from hell!" he addressed the warriors, striding helmetless before the lines, his voice carrying powerfully even to those upon the Wall and marshaling in the ready-ground behind. "This next wave will be the day's last. Cinch up your balls, men, for one final surpassing effort. The enemy believes us exhausted and anticipates dispatching us to the underworld beneath his onslaught of fresh, rested troops. What he doesn't know is we're already there. We crossed the line hours ago." He gestured to the Narrows and its carpeting of horror. "We stand already in hell. It is our home!"
A cheer rose from the line, overtopped by wild profane shouts and whoops of hellish laughter.
"Remember, men," Dithyrambos' voice rose yet more powerfully, "that this next wave of Asiatic ass-fuckers has not seen us yet. Consider what they have seen. They know only that three of their mightiest nations have advanced against us wearing their testicles and come back without them.
"And I promise you: they are _not_ fresh. They've been sitting on their dogblossoms all day, watching their allies carried and dragged back, hacked to pieces by us. Believe me, their imaginations have not been idle. Each man has conjured his own head cleaved at the neck, his own guts spilling into the dirt and his own cock and balls brandished before him on the point of a Greek spear! We're not the ones who are worn out, they are!"
Fresh shouting and tumult erupted from the allies, save the Spartans, while the Thespaians on the field continued their butchery. I glanced to Dienekes, who observed this all with a grim twist upon his features.
"By the gods," he declared, "it's getting ugly out there."
We could see the Spartan Knights, led by Polynikes and Doreion, taking their stations about Leonidas in the forefront of the line. Now a lookout came running back in from the forwardmost post. This was Hound, the Spartan Skirite; he sprinted straight to Leonidas and made his report. The news spread swiftly: the next wave would be Xerxes' own household guard, the Immortals. The Greeks knew that these comprised His Majesty's picked champions, the flower of Persian nobility, princes schooled from birth "to draw the bow and speak the truth."
More to the point, their numbers were ten thousand, while the Greeks had fewer than three thousand still fit to fight. The Immortals, all knew, derived their name from the custom of the Persians that replaced at once each royal guardsman who died or retired, thus keeping the number of Xerxes' finest always at ten thousand.
This corps of champions now advanced into view at the neck of the Narrows. They wore not helmets, but tiaras, soft felt caps topped with skull-crowns of metal glistening like gold. These half-helmets possessed no cover for the ears, neck or jaw and left the face and throat entirely exposed. The warriors wore earrings; some of their faces were painted with eye kohl and rouged like women. Nonetheless they were magnificent specimens, selected it seemed not merely, as the Hellenes well knew, for valor and nobility of family but for height and handsomeness of person as well. Each man looked more dashing than the fellow at his shoulder. They wore sleeved tunics of silk, purple rimmed with scarlet, protected by a sleeveless coat of mail in the shape of fish scales, and trousers atop calf-height doeskin boots. Their weapons were the bow, belt scimitar and short Persian lance, and their shields, like the Medes' and the Cissians', were shoulder-to-groiners made of wicker. Most astonishing of all, however, was the quantity of gold ornament each Immortal wore upon his person in the form of brooches and bracelets, amulets and adornments. Their commander, Hydarnes, advanced to the fore, the only mounted antagonist the allies had so far beheld. His tiara was peaked like a monarch's crown and his eyes shone brilliantly beneath kohled lashes. His horse was spooking, refusing to advance across the charnel sward of corpses. The foe drew up in ranks on the flat beyond the Narrows. Their discipline was impeccable. They were spotless.
Leonidas now strode forth to address the allies. He confirmed what each Hellenic warrior presumed by sight, that the division of the enemy now advancing into view was indeed Xerxes' own Immortals and that the number of their company, as nearly as could be estimated by eye, was the full ten thousand.
"It would appear, gentlemen," Leonidas' voice ascended powerfully, "that the prospect of facing the picked champions of all Asia should daunt us. But I swear to you, this battle will prove the most dustless of all."
The king used the Greek word _akoniti,_ whose application is customarily to wrestling, boxing and the pankration. When a victor overthrows his opponent so swiftly that the bout fails even to raise the dust of the arena, he is said to have triumphed _akoniti,_ in a "no-duster."
"Listen," Leonidas proceeded, "and I will tell you why. The troops Xerxes throws at us now are, for the first time, of actual Persian blood. Their commanders are the King's own kinsmen; he has brothers out there, and cousins and uncles and lovers, officers of his own line whose lives are precious to him beyond price. Do you see him up there, upon his throne? The nations he has sent against us so far have been mere vassal states, spear fodder to such a despot, who squanders their lives without counting the cost. These"—Leonidas gestured across the Narrows to the space where Hydarnes and the Immortals now marshaled—"these he treasures. These he loves. Their murder he will feel like an eight-footer in the guts.
"Remember that this battle at the Hot Gates is not the one Xerxes came here to fight. He anticipates far more momentous struggles to come, in the heartland of Hellas against the main force of our armies, and for these clashes he wishes to preserve the flower of his army, the men you see before you now. He will be frugal with their lives today, I promise you.
"As to their numbers: they are ten thousand, we are four. But each man we slay will sting like a regiment to their King. These warriors are to him like miser's gold, which he hoards and covets beyond all else in his treasury.
"Kill one thousand and the rest will crack. One thousand and their master will pull the remainder out. Can you do that for me, men? Can four of you kill one of them? Can you give me one thousand?"
**TWENTY-SIX**
**H** is Majesty himself may best judge the precision of Leonidas' forecast. Suffice it to note, for this record, that darkness found the Immortals in shattered retreat, under His Majesty's orders as Leonidas had predicted, leaving the broken and dying upon the _orchestra,_ the dance floor, of the Narrows.
Behind the allied Wall the spectacle was one of corresponding horror. A downpour had drenched the camp shortly after nightfall, drowning what few fires remained with none to tend them, all effort of squires, attendants and mates being required to succor the wounded and the maimed. Slides toppled from the wall of Kallidromos, sluicing the upper camp with rivers of mud and stone. Across this sodden expanse, slain and spared sprawled limb upon limb, many still in armor, the slumber of the exhausted so profound that one could not distinguish the living from the dead. Everything was soaked and muck-begrimed. Stores of dressings for the wounded had long since been depleted; the spa-goers' tents requisitioned by the Skiritai rangers as shelter now found their linen called to duty a second time, as battle compresses. The stink of blood and death rose with such palpable horror that the asses of the supply train bawled all night and could not be quietened.
There was a third unrostered member of the allied contingent, a volunteer other than the outlaw Ball Player and the roan bitch Styx. This was an _emporos,_ a merchant of Miletus, Elephantinos by name, whose disabled waggon the allied column had chanced upon during its march through Doris, a day prior to arrival at the Gates. This fellow despite his misfortune of the road maintained the merriest of spirits, sharing a lunch of green apples with his hobbled ass. Upon the brow of his waggon rose a hand-painted standard, an advertisement as it were of his congeniality and eagerness of custom. The sign intended to declare, "The best service only for you, my friend." The tinker had misspelled, however, several words, chiefly "friend," _philos,_ which his hand had inscribed _phimos,_ the term in Doric for a contraction of the flesh which covers the male member. The waggon's banner declared roughly thus:
"The best service only for you, my foreskin."
The luster of this poesy rendered the fellow an instant celebrity. Several squires were detached to assist him, for which courtesy the tradesman expressed abounding gratitude. "And where, if one may inquire, is this magnificent army bound?"
"To die for Hellas," someone answered.
"How delightful!" Toward midnight the tinker appeared in camp, having tracked the column all the way to the Gates. He was welcomed with enthusiasm. His specialty lay in applying an edge to steel, and at this, he testified, he stood without peer. He had been sharpening farmers' scythes and housewives' cleavers for decades. He knew how to make even the meanest untempered trowel hold an edge, and moreover, he vowed, he would donate his services to the army in repayment of their kindness upon the highway.
The fellow employed an expression with which he spiked his conversation whenever he wished to emphasize a point. "Wake up to this!" he would say, though in his dense Ionian accent it came out as "Weck up to thees!"
This phrase was immediately and with high glee adopted by the entire army.
"Cheese and onions again, weck up to thees!"
"Double drill all day, weck up to thees!"
One of the two Leons in Dienekes' platoon, Donkeydick, rousted the merchant that succeeding dawn by brandishing before his slumber-dazed eyes a prodigious erection. "They call this a _phimos,_ weck up to thees!"
The tradesman became a kind of mascot or talisman to the troops. His presence was welcomed at every fire, his company embraced by youths as well as veterans; he was considered a raconteur and boon companion, a jester and a friend.
Now in the wake of this first day's slaughter, the merchant appointed himself as well unofficial chaplain and confessor to the young warriors whom he had over the past days come to care for more intimately than sons. He passed all night among the wounded, bearing wine, water and a consoling hand. His accustomed cheerfulness he contrived to redouble; he diverted the maimed and mutilated with profane tales of his travels and misadventures, seductions of housewives, robberies and thrashings sustained upon the road. He had armed himself as well, from the discards; he would fill a gap tomorrow. Many of the squires, uncompelled by their masters, had taken upon themselves the same role.
All night the forges roared. The hammers of the smiths and foundrymen rang without ceasing, repairing spear and sword blades, beating out the bronze for fresh shield facings, while wrights and carpenters manned spokeshaves limning fresh spearshafts and shield carriages for the morrow. The allies cooked their meals over fires made from the spent arrows and shivered spearshafts of the enemy. The natives of Alpenoi village who a day earlier had peddled their produce for profit, now, beholding the sacrifice of the defenders, donated their goods and foodstuffs and hastened off with shuttles and handbarrows to bear up more.
Where were the reinforcements? Were any coming at all? Leonidas, sensing the preoccupation of the army, eschewed all assembly and councils of war, circulating instead in person among the men, transacting the business of the commanders as he went. He was dispatching more runners to the cities, with more appeals for aid. Nor was it lost upon the warriors that he selected always the youngest. Was this for speed of foot, or the king's wish to spare those whose share of remaining years was the greatest?
Each soldier's thoughts turned now toward his family, to those at home whom his heart loved. Shivering, exhausted men scribbled letters to wives and children, mothers and fathers, many of these missives little more than scratches upon cloth or leather, fragments of ceramic or wood. The letters were wills and testaments, final words of farewell. I saw the dispatch pouch of one runner preparing to depart; it was a jumble of paper rolls, wax tablets, potshards, even felt scraps torn from helmet undercaps. Many of the warriors simply sent amulets which their loved ones would recognize, a charm that had pended from the chassis of a shield, a good-luck coin drilled through for a neckband. Some of these bore salutations—"Beloved Amaris"..."Delia from Theagones, love." Others bore no name at all. Perhaps the runners of each city knew the addressees personally and could take it upon themselves to ensure delivery. If not, the contents of the pouch would be displayed in the public square or the agora, perhaps set out before the temple of the city's Protectress. There the anxious families would congregate in hope and trepidation, awaiting their turn to pore through the precious cargo, desperate for any message, wordless or otherwise, from those whom they loved and feared to behold again only in death.
Two messengers came in from the allied fleet, from the Athenian corvette assigned as courier between the navy below and the army up top. The allies had engaged the Persian fleet this day, inconclusively, but without buckling. Our ships must hold the straits or Xerxes could land his army in the defenders' rear and cut them off; the troops must hold the pass or the Persian could advance by land to the narrows of the Euripus and trap the fleet. So far, neither had cracked.
Polynikes came and sat for a few minutes beside the fires around which the remains of our platoon had gathered. He had located a renowned _gymnastes,_ an athletic trainer named Milon, whom he knew from the Games at Olympia. This fellow had wrapped Polynikes' hamstrings and given him a _pharmakon_ to kill the pain.
"Have you had enough of glory, Kallistos?" Dienekes inquired of the Knight.
Polynikes answered only with a look of surpassing grimness. He seemed chastened, out of himself for once.
"Sit down," my master said, indicating a dry space beside him.
Polynikes settled gratefully. Around the circle the platoon slumbered like dead men, heads pillowed upon each other and their yet-gore-encrusted shields. Directly across from Polynikes, Alexandros stared with awful blankness into the fire. His jaw had been broken; the entire right side of his face glistened purple; the bone itself was cinched shut with a leather strap.
"Let's have a look at you." Polynikes craned forward. He located among the trainer's kit a waxed wad of euphorbia and amber called a "boxer's lunch," the kind pugilists employ between matches to immobilize broken bones and teeth. This Polynikes kneaded warm until it became pliable. He turned to the trainer. "You better do this, Milon." Polynikes took Alexandros' right hand in his own, for the pain. "Hang on. Squeeze till you break my fingers."
The trainer spit from his own mouth into Alexandros' a purge of uncut wine to cleanse the clotted blood, then with his fingers extracted a grotesque gob of spittle, mucus and phlegm. I held Alexandros' head; the youth's fist clamped Polynikes'. Dienekes watched as the trainer inserted the sticky amber wad between Alexandros' jaws, then gently clamped the shattered bone down tight upon it. "Count slowly," he instructed the patient. "When you hit fifty, you won't be able to prise that jaw apart with a crowbar."
Alexandros released the Knight's hand. Polynikes regarded him with sorrow.
"Forgive me, Alexandros."
"For what?"
"For breaking your nose."
Alexandros laughed, his broken jaw making him grimace.
"It's your best feature now."
Alexandros winced again. "I'm sorry about your father," Polynikes said. "And Ariston."
He rose to move on to the next fire, glancing once to my master, then returning his gaze to Alexandros.
"There is something I must tell you. When Leonidas selected you for the Three Hundred, I went to him in private and argued strenuously against your inclusion. I thought you would not fight."
"I know," Alexandros' voice ground through his cinched jaw.
Polynikes studied him a long moment.
"I was wrong," he said.
He moved on.
Another round of orders came, assigning parties to retrieve corpses from no-man's-land. Suicide's name was among those detailed. Both his shot shoulders had seized up; Alexandros insisted on taking his place.
"By now the king will know about the deaths of my father and Ariston." He addressed Dienekes, who as his platoon commander could forbid him to participate in the retrieval detail. "Leonidas will try to spare me for my family's sake; he'll send me home with some errand or dispatch. I don't wish to disrespect him by refusing."
I had never beheld such an expression of balefulness as that which now framed itself upon my master's face. He gestured to a flat of sodden earth beside him in the firelight.
"I've been watching these little myrmidons."
There in the dirt, a war of the ants was raging.
"Look at these champions." Dienekes indicated the massed battalions of insects grappling with impossible valor atop a pile of their own fellows' fallen forms, battling over the desiccated corpse of a beetle.
"This one here, this would be Achilles. And there. That must be Hektor. Our bravery is nothing alongside these heroes'. See? They even drag their comrades' bodies from the field, as we do."
His voice was dense with disgust and stinking with irony. "Do you think the gods look down on us as we do upon these insects? Do the immortals mourn our deaths as keenly as we feel the loss of these?"
"Get some sleep, Dienekes," Alexandros said gently.
"Yes, that's what I need. My beauty rest."
He lifted his remaining eye toward Alexandros. Out beyond the redoubts of the Wall, the second watch of sentries was receiving their orders, preparatory to relieving the first. "Your father was my mentor, Alexandros. I bore the chalice the night you were born. I remember Olympieus presenting your infant form to the elders, for the 'ten, ten and one' test, to see if you were deemed healthy enough to be allowed to live. The magistrate bathed you in wine and you came up squawling, with your infant's voice strong and your little fists clenched and waving. 'Hand the boy to Dienekes,' your father instructed Paraleia. 'My son will be your protege,' Olympieus told me. 'You will teach him, as I have taught you.'"
Dienekes' right hand plunged the blade of his _xiphos_ into the dirt, annihilating the Iliad of the ants.
"Now sleep, all of you!" he barked to the men yet surviving of his platoon and, himself rising, despite all protests that he, too, embrace the boon of slumber, strode off alone toward Leonidas' command post, where the king and the other commanders yet stood to their posts, awake and planning the morrow's action.
I saw Dienekes' hip give way beneath him as he moved; not the bad leg, but the sound. He was concealing from his men's sight yet another wound—from the cast of his gait, deep and crippling. I rose at once and hastened to his aid.
**TWENTY-SEVEN**
**T** hat spring called the Skyllian, sacred to Demeter and Perseph one, welled from the base of the wall of Kallidromos just to the rear of Leonidas' command post. Upon its stone-founded approach my master drew up, and I hurrying in his wake overhauled him. No curses or commands to withdraw rebuffed me. I draped his arm about my neck and took his weight upon my shoulder. "I'll get water," I said.
An agitated knot of warriors had clustered about the spring; Megistias the seer was there. Something was amiss. I pressed closer. This spring, renowned for its alternating flows of cold and hot, had gushed since the allies' arrival with naught but sweet cold water, a boon from the goddesses to the warriors' thirst. Now suddenly the fount had gone hot and stinking. A steaming sulphurous brew spewed forth from the underworld like a river of hell. The men trembled before this prodigy. Prayers to Demeter and the _Kore_ were being sung. I begged a half-helmetful of water from the Knight Doreion's skin and returned to my master, steeling myself to mention nothing.
"The spring's gone sulphurous, hasn't it?"
"It presages the enemy's death, sir, not ours."
"You're as full of shit as the priests."
I could see he was all right now.
"The allies need your cousin upon this site," he observed, settling in pain upon the earth, "to intercede with the goddess on their behalf."
He meant Diomache.
"Here," he said. "Sit beside me."
This was the first time I had heard my master refer aloud to Diomache, or even acknowledge his awareness of her existence. Though I had never, in our years, presumed to burden him with details of my own history prior to entering his service, I knew he knew it all, through Alexandros and the lady Arete.
"This is a goddess I have always felt pity for—Persephone," my master declared. "Six months of the year she rules as Hades' bride, mistress of the underworld. Yet hers is a reign bereft of joy. She sits her throne as a prisoner, carried off for her beauty by the lord of hell, who releases his queen under Zeus' compulsion for half the year only, when she comes back to us, bringing spring and the rebirth of the land. Have you looked closely at statues of her, Xeo? She appears grave, even in the midst of the harvest's joy. Does she, like us, recall the terms of her sentence—to retire again untimely beneath the earth? This is the sorrow of Persephone. Alone among the immortals the _Kore_ is bound by necessity to shuttle from death to life and back again, intimate of both faces of the coin. No wonder this fount whose twin sources are heaven and hell is sacred to her."
I had settled now upon the ground beside my master. He regarded me gravely.
"It's too late, don't you think," he pronounced, "for you and I to keep secrets from one another?"
I agreed the hour was far advanced.
"Yet you preserve one from me."
He would have me speak of Athens, I could see, and the evening barely a month previous wherein I had at last—through his intercession—met again my cousin.
"Why didn't you run?" Dienekes asked me. "I wanted you to, you know."
"I tried. She wouldn't let me."
I knew my master would not compel me to speak. He would never presume to tread where his presence sowed distress. Yet instinct told me the hour to break silence had come. At worst my report would divert his preoccupation from the day's horror and at best turn it, perhaps, to more propitious imaginings.
"Shall I tell you of that night in Athens, sir?"
"Only if you wish."
It was upon an embassy, I reminded him. He, Polynikes and Aristodemos had traveled on foot from Sparta then, without escort, accompanied only by their squires. The party had covered the distance of 140 miles in four days and remained there in the city of the Athenians for four more, at the home of the _proxenos_ Kleinias the son of Alkibiades. The object of the legation was to finalize the eleventh-hour details of coordinating land and sea forces at Thermopylae and Artemisium: times of arrival for army and fleet, modes of dispatch between them, courier encryptions, passwords and the like. Unspoken but no less significant, Spartans and Athenians wished to look each other in the eye one last time, to make sure both forces would be there, in their places, at the appointed hour.
On the evening of the third day, a salon was held in honor of the embassy at the home of Xanthippus, a prominent Athenian. I loved to listen at these affairs, where debate and discourse were always spirited and often brilliant. To my great disappointment, my master summoned me alone before table and informed me of an urgent errand I must run. "Sorry," he said, "you'll miss the party." He placed into my hands a sealed letter, with instructions to deliver it in person to a certain residence in the seaport town of Phaleron. A boy servant of the house awaited without, to serve as guide through the nighttime streets. No particulars were given beyond the addressee's name. I assumed the communication to be a naval dispatch of some urgency and so traveled armed.
It took the time of an entire watch to traverse that labyrinth of quarters and precincts which comprises the city of the Athenians. Everywhere men-at-arms, sailors and marines were mobilizing; chandlery waggons rumbled under armed escorts, bearing the rations and supplies of the fleet. The squadrons under Themistokles were readying for embarkation to Skiathos and Artemisium. Simultaneously families by the hundreds were crating their valuables and fleeing the city. As numerous as were the warcraft moored in lines across the harbor, their ranks were eclipsed by the ragtag fleet of merchantmen, ferries, fishing smacks, pleasure boats and excursion craft evacuating the citizenry to Troezen and Salamis. Some of the families were fleeing for points as distant as Italia. As the boy and I approached Phaleron port, so many torches filled the streets that the passage was lit bright as noon.
Lanes became crookeder as we approached the harbor. The stink of low tide choked one's nostrils; gutters ran with filth, backed up into a malodorous stew of fish guts, leek shavings and garlic. I never saw so many cats in my life. Grogshops and houses of ill fame lined streets so narrow that daylight's cleansing beams, I was certain, never penetrated to the floors of their canyons to dry the slime and muck of the night's commerce in depravity. The whores called out boldly as the boy and I passed, advertising their wares in coarse but good-humored tongue. The man to whom we were to deliver the letter was named Terrentaius. I asked the lad if he had any idea who the fellow was or what station he held. He said he had been given the house name alone and nothing more.
At last the boy and I located it, an apartment structure of three stories called the Griddle after the slop shop and inn which occupied its street-level floor. I inquired within for the man Terrentaius. He was absent, the publican declared, with the fleet. I asked after the man's ship. Which vessel was he officer of? A round of hilarity greeted this query. "He's a lieutenant of the ash," one of the tippling seamen declared, meaning the only thing he commanded was the oar he pulled. Further inquiries failed to elicit any additional intelligence.
"Then, sir," the boy guide addressed me, "we are instructed to deliver the letter to his wife."
I rejected this as nonsense.
"No, sir," replied the lad with conviction, "I have it from your master himself. We are to place the letter in the hands of the man's mistress, by name Diomache."
With but a moment's consideration I perceived in this event the hand, not to say the long arm, of the lady Arete. How had she tracked down and located, from the remove of Lakedaemon, this house and this woman? There must be a hundred Diomaches in a city the size of Athens. No doubt the lady Arete had maintained her intentions secret, anticipating that I, made aware of them in advance, would have found excuse to evade their obligation. In this, she was doubtless correct.
In any event my cousin, it was discovered, was not present in the apartments, nor could any of the seamen inform us as to her whereabouts. My guide, a lad of resourcefulness, simply stepped into the alley and bellowed her name. In moments the grizzled heads of half a dozen backstreet dames appeared above, among the hanging laundry of the lane-facing windows. The name and site of a harbor-town temple were shouted down to us.
"She'll be there, boy. Just follow the shore."
My guide set out again in the lead. We traversed more stinking sea-town streets, more alleys choked with traffic of the natives clearing out. The boy informed me that many of the temples in this quarter functioned less as sanctuaries of the gods and more as asylums for the cast-out and the penniless, particularly, he said, wives "put by" by their husbands. Meaning those deemed unfit, unwilling, or even insane. The boy pressed ahead in merry spirits. It was all a grand adventure to him.
At last we stood before the temple. It was nothing but a common house, perhaps home in former times to a middling-prosperous trader or merchant, sited upon a surprisingly cheerful slope two streets above the water. A copse of olives sheltered a walled enclave whose inner precinct could not be glimpsed from the street. I rapped at the gate; after an interval a priestess, if such a lofty title may be applied to a gowned and masked housewoman of fifty years, responded. She informed us the sanctuary was that of Demeter and Covert _Kore,_ Persephone of the Veil. None but females might enter. Behind the shroud which concealed her face, the priestess was clearly frightened, nor could one blame her, the streets running with whoremongers and cutpurses. She would not let us in. No avenue of appeal proved of avail; the woman would neither confirm my cousin's presence nor agree to convey a message within. Again my boy guide took the bull by the horns. He opened his cheesepipe and bawled Diomache's name.
We were admitted at last to a rear courtyard, the lad and I. The house upon entry proved far more capacious, and a good deal cheerier, than it had appeared from the street. We were not permitted passage through the interior but escorted along an outer path. The dame, our chaperon, confirmed that a matron by name Diomache was indeed among those novices currently resident within the sanctuary. She was at this hour attending to duties in the kitchen; an interview, however, of a few minutes' duration might be granted, with permission from the asylum mother. My guide, the boy, was offered refreshment; the dame took him away for a feed.
I was standing, alone in the courtyard, when my cousin entered. Her children, both girls, one perhaps five, the other a year or two older, clung fearfully to her skirts; they would not come forward when I knelt and held out my hand. "Forgive them," my cousin said. "They are shy of men." The dame led the girls away to the interior, leaving me at last alone with Diomache.
How many times in imagination had I rehearsed this moment. Always in conjured scenarios, my cousin was young and beautiful; I ran to her arms and she to mine. Nothing of the sort now occurred. Diomache stepped into view in the lamplight, garbed in black, with the entire breadth of the court dividing us. The shock of her appearance unstrung me. She was unveiled and unhooded. Her hair was cropped short. Her years were no more than twenty-four yet she looked forty, and a hard-used forty at that.
"Can it really be you, cousin?" she inquired in the same teasing voice she had taken with me since we were children. "You are a man, as you were always so impatient to become."
Her lightness of tone served only to compound the despair which now seized my heart. The picture I had held so long before the eye of the mind was of her in the bloom of youth, womanly and strong, exactly as she had been the morning we parted at the Three-Cornered Way. What terrible hardships had been visited upon her by the intervening years? The vision of the whore-infested streets was fresh upon me, the crude seamen and the mean existence of these refuse-choked lanes. I sank, overcome with grief and regret, upon a bench along the wall.
"I should never have left you," I said, and meant it with all my heart. "Everything that has happened is my fault for not being at your side to defend you."
I cannot recall a word of what was spoken over the next several minutes. I remember my cousin moving to the bench beside me. She did not embrace me, but touched me with tender clemency upon the shoulder.
"Do you remember that morning, Xeo, when we set out for market with Stumblefoot and your little clutch of ptarmigan eggs?" Her lips declined into a sorrowful smile. "The gods set our lives upon their courses that day. Courses from which neither of us has had the option to stray."
She asked if I would take wine. A bowl was brought. I recalled the letter I bore and delivered it now to my cousin. Beneath its wrapper, it was addressed to her, not her husband; she opened and read it. Its contents were in the lady Arete's hand. When Diomache finished she did not show it to me, but tucked it away without a word beneath her robe.
My eyes, adjusted now to the lamplight of the court, studied my cousin's face. Her beauty remained, I saw, but altered in a manner both grave and austere. The age in her eyes, which had at first shocked and repelled me, I now perceived as compassion and even wisdom. Her silence was profound as the lady Arete's; her bearing spartan beyond spartan. I was daunted and even in awe. She seemed, like the goddess she served, a maiden hauled off untimely by the dark forces of the underworld and now, restored by some covenant with those pitiless gods, bearing in her eyes that primal female wisdom which is simultaneously human and inhuman, personal and impersonal. Love for her flooded my heart. Yet did she appear, inches from my grasp, as august as an immortal and as impossible to hold.
"Do you feel the city about us now?" she asked. Outside the walls, the rumble of evacuees and their baggage trains could clearly be heard. "It's like that morning at Astakos, isn't it? Perhaps within weeks this mighty city will be fired and razed, as our own was on that day."
I begged her to tell me how she was. Truly.
She laughed.
"I've changed, haven't I? Not the husband bait you always took me for. I was foolish then too; I thought as highly of my prospects. But this is not a woman's world, cousin. It never was and never will be."
From my lips blurted a course of passionate impulse. She must come with me. Now. To the hills, where we had flown once and been happy once. I would be her husband. She would be my wife. Nothing would ever harm her again.
"My sweet cousin," she replied with tender resignation, "I have a husband." She indicated the letter. "As you have a wife."
Her seemingly passive acceptance of fate infuriated me. What husband is he who abandons his wife? What wife is she taken without love? The gods demand of us action and the use of our free will! That is piety, not to buckle beneath necessity's yoke like dumb beasts!
"This is Lord Apollo talking." My cousin smiled and touched me again with patient gentleness.
She asked if she could tell me a story. Would I listen? It was a tale she had confided to no one, save her sisters of the sanctuary and our dearest friend Bruxieus. Only a few minutes remained to us. I must be patient and attend.
"Do you remember that day when the Argive soldiers shamed me? You knew I turned the hands of murder upon the issue of that violation. I aborted myself. But what you didn't know was that I hemorrhaged one night and nearly died. Bruxieus saved me as you slept. I bound him by oath never to tell you."
She regarded me with the same self-consecrated gaze I had observed upon the features of the lady Arete, that expression born of feminine wisdom which apprehends truth directly, through the blood, unobscured by the cruder faculty of reason.
"Like you, cousin, I hated life then. I wanted to die, and nearly did. That night in fevered sleep, feeling the blood draining from me like oil from an overturned lamp, I had a dream.
"A goddess stood above me, veiled and cowled. I could see nothing but her eyes, yet so vivid was her presence that I felt certain she was real. More real than real, as if life itself were the dream and this, the dream, life in its profoundest essence. The goddess spoke no words but merely looked upon me with eyes of supreme wisdom and compassion.
"My soul ached with the desire to behold her face. I was consumed with this need and implored her, in words that were not words but only the fervent appeal of my heart, to loose her veil and let me see the whole of her. I knew without thought that what would be revealed would be of supreme consequence. I was terrified and at the same time trembling with anticipation.
"The goddess unbound her veil and let it fall. Will you understand, Xeo, if I say that what was revealed, the face beyond the veil, was nothing less than that reality which exists beneath the world of flesh? That higher, nobler creation which the gods know and we mortals are permitted to glimpse only in visions and transports.
"Her face was beauty beyond beauty. The embodiment of truth as beauty. And it was human. So human it made the heart break with love and reverence and awe. I perceived without words that this alone was real which I beheld now, not the world we see beneath the sun. And more: that this beauty existed here, about us at every hour. Our eyes were just too blind to see it.
"I understood that our role as humans was to embody here, upon this shadowed and sorrow-bound side of the Veil, those qualities which arise from beyond and are the same on both sides, ever-sustaining, eternal and divine. Do you understand, Xeo? Courage, self-lessness, compassion and love."
She drew up and smiled.
"You think I'm loony, don't you? I've gone cracked with religion. Like a woman."
I didn't. I told her briefly of my own glimpse beyond the veil, that night within the grove of snow. Diomache acknowledged gravely.
"Did you forget your vision, Xeo? I forgot mine. I lived a life of hell here in this city. Until one day the goddess's hand guided me within these walls."
She indicated a modestly scaled but superb statue in an alcove of the court. I looked. It was a bronze of Veiled Persephone.
"This," my cousin declared, "is the goddess whose mystery I serve. She who passes from life to death and back again. The _Kore_ has preserved me, as the Lord of the Bow has protected you."
She placed her hands atop mine and drew my eyes to hers.
"So you see, Xeo, nothing has transpired amiss. You think you have failed to defend me. But everything you've done has defended me. As you defend me now."
She reached within the folds of her garment and produced the letter written in the lady Arete's hand.
"Do you know what this is? A promise to me that your death will be honored, as you and I honored Bruxieus and we three sought to honor our parents."
The housewoman appeared again from the kitchen. Diomache's children awaited within; my boy guide had finished his feed and stood impatient to depart. Diomache rose and held out both hands to me. The lamplight fell kindly upon her; in its gentle glow her face appeared as beautiful as it had to my eyes of love, those short years that seemed so long past. I stood too and embraced her. She tugged the cowl atop her cropped hair and slid the veil in place across her face.
"Let neither of us pity the other," my cousin spoke in parting. "We are where we must be, and we will do what we must."
**TWENTY-EIGHT**
**S** uicide shook me awake two hours before dawn. "Look what crawled in through the bunghole."
He was pointing to the knoll behind the Arkadian camp, where deserters from the Persian lines were being interrogated beside the watch fires. I squinted but my eyes refused to focus. "Look again," he said. "It's your seditious mate, Rooster. He's asking for you."
Alexandros and I went over together. It was Rooster, all right. He had crossed from the Persian lines with a party of other deserters; the Skiritai had him bound, naked, to a post. They were going to execute him; he had asked for a moment alone with me before they opened his throat.
On all sides the camp was rousing; half the army stood already on station, the other half arming. Down the track toward Trachis you could hear the enemy trumpets, forming up for Day Two. We found Rooster next to a pair of Median informers who had talked a good-enough game that they were actually being given breakfast. Not Rooster. The Skiritai had worked him over so hard that he had to be propped up, slumped against the post where his throat would be slit.
"Is that you, Xeo?" He squinted through eyeholes battered purple as a boxer's.
"I've brought Alexandros."
We managed to dribble some wine down Rooster's throat.
"I'm sorry about your father" were his first words to Alexandros. He, Rooster, had served six years as squire to Olympieus and saved his life at Oenophyta, when the Theban cavalry had ranged down upon him. "He was the noblest man of the city, not excepting Leonidas."
"How can we help you?" Alexandros asked.
Rooster wished to know first who else was still alive. I told him Dienekes, Polynikes and some others and recited the names of the dead whom he knew. "And you're alive too, Xeo?" His features twisted into a grin. "Your crony Apollo must be saving you for something extraordinary."
Rooster had a simple request: that I arrange to have delivered to his wife an ancient coin of his nation, Messenia. This thumb-worn obol, he told us, he had carried in secret his whole life. He placed it into my care; I vowed to send it with the next dispatch runner. He clasped my hand in gratitude, then, lowering his voice in exigency, tugged me and Alexandros near.
"Listen closely. This is what I came to tell you."
Rooster spit it out quickly. The Hellenes defending the pass had another day, no more. His Majesty even now was offering the wealth of a province to any guide who could inform him of a track through the mountains by which the Hot Gates could be encircled. "God made no rock so steep that men couldn't climb it, particularly driven by gold and glory. The Persians will find a way around to your rear, and even if they don't, their fleet will break the Athenian sea line within another day. No reinforcements are coming from Sparta; the ephors know they'd only be enveloped too. And Leonidas will never pull himself or any of you out, dead or alive."
"You took that beating just to deliver this news?"
"Listen to me. When I went over to the Persians, I told them I was a helot fresh from Sparta. The King's own officers interrogated me. I was right there, two squares from Xerxes' tent. I know where the Great King sleeps and how to deliver men right to his doorstep."
Alexandros laughed out loud. "You mean attack him in his tent?"
"When the head dies, the snake dies. Pay attention. The King's pavilion stands just beneath the cliffs at the top of the plain, right by the river, so his horses can water before the rest of the army fouls the stream. The gorge produces a torrent coming out of the mountains; the Persians think it impassable, they have less than a company on guard. A party of half a dozen could get in, in darkness, and maybe even get out."
"Yes. We'll flap our wings and fly right over."
The camp had come fully awake now. At the Wall the Spartans were already massing, if so grand a term may be applied to so meager a force. Rooster told us that he had offered to guide a party of raiders into the Persian camp in return for freedom for his wife and children in Lakedaemon. This was why the Skiritai had beaten him; they thought it a trick designed to deliver brave men into the enemy's hands for torture or worse. "They won't even relay my words to their own officers. I beg you: inform someone of rank. Even without me it can work. By all the gods I swear it!"
I laughed at this reborn Rooster. "So you've acquired piety as well as patriotism."
The Skiritai called to us sharply. They wanted to finish Rooster and get themselves into armor. Two rangers jerked him to his feet, to lash him upright to the post, when a clamor interrupted from the rear of the camp. We all turned and stared back down the slope.
Forty men of the Thebans had deserted during the night. A half dozen had been slain by sentries, but the others had made good their escape. All save three, who had just now been discovered, attempting to conceal themselves among the mounds of the dead.
This luckless trio was now hauled forth by a squad of Thespaian sentries and dumped into the open to the rear of the Wall, smack amid the marshaling army. Blood was in the air. The Thespaian Dithyrambos strode to the breach and took charge.
"What punishment for these?" he shouted to the encircling throng.
At this moment Dienekes appeared at Alexandros' shoulder, summoned by the commotion. I seized the instant to plead for Rooster's life, but my master made no answer, his attention held by the scene playing out below.
A dozen mortal punishments had been shouted out by the thronging warriors. Blows of homicidal intent were struck at the terrified captives; it took Dithyrambos himself, wading into the fray with his sword, to drive the men back.
"The allies are possessed," Alexandros observed with dismay. "Again."
Dienekes looked on coldly. "I will not witness this a second time."
He strode forward, parting the mob before him, and thrust himself to the fore beside the Thespaian Dithyrambos.
"These dogs must receive no mercy!" Dienekes stood over the bound and blindered captives. "They must suffer the most hateful penalty imaginable, so that no other will be tempted to emulate their cowardice."
Cries of assent rose from the army. Dienekes' raised hand quelled the tumult.
"You men know me. Will you accept the punishment I propose?"
A thousand voices shouted aye.
"Without protest? Without a quibble?"
All swore to abide by Dienekes' sentence.
From the knoll behind the Wall, Leonidas and the Knights, including Polynikes, Alpheus and Maron, looked on. All sound stilled save the wind. Dienekes stepped to the kneeling captives and snatched off their blinders.
His blade cut the prisoners loose.
Bellows of outrage thundered from every quarter. Desertion in the face of the enemy was punishable by death. How many more would flee if these traitors walked off with their lives? The whole army will fall apart!
Dithyrambos, alone among the allies, seemed to divine Dienekes' subtler intent. He stepped forward beside the Spartan, his raised sword silencing the men so that Dienekes could speak.
"I despise that seizure of self-preservation which unmanned these cravens last night," Dienekes addressed the thronging allies, "but far more I hate that passion, comrades, which deranges you now."
He gestured to the captives on their knees before him. "These men you call coward today fought shoulder-to-shoulder beside you yesterday. Perhaps with greater valor than you."
"I doubt it!" came a shouted cry, succeeded by waves of scorn and cries for blood pelting down upon the fugitives.
Dienekes waited for the tumult to subside. "In Lakedaemon we have a name for that state of mind which holds you now, brothers. We call it 'possession.' It means that yielding to fear or anger which robs an army of order and reduces it to a rabble."
He stepped back; his sword gestured to the captives upon the ground.
"Yes, these men ran last night. But what did you do? I'll tell you. Every one of you lay awake. And what were the covert petitions of your hearts? The same as these." The blade of his _xiphos_ indicated the pitiful wretches at his feet. "Like these, you yearned for wives and children. Like these, you burned to save your own skin. Like these, you laid plans to fly and live!"
Cries of denial struggled to find voice, only to sputter and fail before Dienekes' fierce gaze and the truth it embodied.
"I thought those thoughts too. All night I dreamt of running. So did every officer and every Lakedaemonian here, including Leonidas."
A chastened silence held the mob.
"Yes!" a voice cried. "But we didn't do it!"
More murmurs of assent, mounting.
"That's right," Dienekes spoke softly, his glance no longer lifted in address to the army but turned now, hard as flint, upon the trio of captives. "We didn't do it."
He regarded the fugitives for one pitiless moment, then stepped back so the army could behold the three, bound and held at swordpoint, in their midst.
"Let these men live out their days, cursed by that knowledge. Let them wake each dawn to that infamy and lie down each night with that shame. That will be their sentence of death, a living extinction far more bitter than that trifle the rest of us will bear before the sun sets tomorrow."
He stepped beyond the felons, toward that margin of the throng which led away to safety. "Clear a runway!"
Now the fugitives began to beg. The first, a beardless youth barely past twenty, declared that his poor farmstead lay less than half a week from here; he had feared for his new bride and infant daughter, for his infirm mother and father. The darkness had unmanned him, he confessed, but he repented now. Clasping his bound hands in supplication, he lifted his gaze toward Dienekes and the Thespaian. Please, sirs, my crime was of the moment. It is passed. I will fight today and none will fault my courage.
Now the other two chimed in, both men past forty, vowing mighty oaths that they, too, would serve with honor.
Dienekes stood over them. "Clear a runway!"
The crush of men parted to open a lane down which the trio might pass in safety out of the camp.
"Anyone else?" Dithyrambos' voice ascended in challenge to the army. "Who else feels like a stroll? Let him take the back door now, or shut his cheesepipe from here to hell."
Surely no sight under heaven could have been more baleful or infamous, so pitiful were the postures of the wretches and the slouching increments of their gait as they passed out along the avenue of shame between the ranks of their silent comrades.
I looked down into the faces of the army. Fled was the self-serving fury which had cried in false righteousness for blood. Instead in each chastened countenance stood graven a purged and pitiless shame. The cheap and hypocritical rage which had sought to vent itself upon the runaways had been turned inward by the intervention of Dienekes. And that rage, refired within the forge of each man's secret heart, now hardened into a resolution of such blistering infamy that death itself seemed a trifle alongside it.
Dienekes turned and stalked back up the knoll. Nearing myself and Alexandros, he was intercepted by an officer of the Skiritai, who clasped his hand in both his own. "That was brilliant, Dienekes. You shamed the whole army. Not one will dare budge from this dirt now."
My master's face, far from displaying satisfaction, instead stood darkened into a mask of grief. He glanced back toward the three miscreants, slouching miserably off with their lives. "Those poor bastards served their turn in the line all day yesterday. I pity them with all my heart."
The criminals had now emerged at the far end of the gauntlet of infamy. There the second man, the one who had groveled most shamelessly, turned and shouted back at the army. "Fools! You're all going to die! Fuck you all, and damn you to hell!"
With a cackle of doom he vanished over the brow of the slope, followed by his scampering mates, who cast glances back over their shoulders like curs.
At once Leonidas passed an order to the _polemarch_ Derkylides, who relayed it to the officer of the watch: from here on, no sentries would be posted to the rear, no precautions taken to prevent further desertions.
With a shout the men broke up and marshaled to their ready stations.
Dienekes had now reached the compound where Alexandros and I waited with Rooster. The officer of the Skiritai was a man named Lachides, brother of the ranger called Hound.
"Give this villain to me, will you, friend?" Dienekes' weary gesture indicated Rooster. "He's my bastard nephew. I'll slit his throat myself."
**TWENTY-NINE**
**H** is Majesty knows far better than I the details of the intrigue by which the ultimate betrayal of the allies was effected; that is, who the traitor was of the Trachinian natives who came forward to inform His Majesty's commanders of the existence of the mountain track by which the Hot Gates could be encircled, and what reward was paid this criminal from the treasury of Persia.
The Greeks drew hints of this calamitous intelligence first from the omens taken on the morning of the second day's fighting, corroborated further by rumors and reports of deserters throughout the day, and ultimately confirmed by eyewitness testimony upon that evening, the end of the allies' sixth in possession of the pass of Thermopylae.
A nobleman of the enemy had come over to the Greek lines at the time of the changing of the first watch, approximately two hours after the cessation of the day's hostilities. He identified himself as Tyrrhastiadas of Kyme, a captain-of-a-thousand in the conscripted forces of that nation. This prince was the tallest, best-looking and most magnificently appareled personage of the enemy who had thus far deserted. He addressed the assembly in errorless Greek. His wife was a Hellene of Hallicarnassus, he declared; that, and the compulsion of honor, had impelled him to cross over to the allied lines. He informed the Spartan king that he had been present before Xerxes' pavilion this very evening when the traitor, whose name I have learned but here and evermore refuse to repeat, had come forward to claim the reward offered by His Majesty and to volunteer his services in guiding the forces of Persia along the secret track.
The noble Tyrrhastiadas went on to report that he had personally observed the issuance by His Majesty of the orders of march and the marshaling of the Persian battalions. The Immortals, their losses replaced and now numbering again their customary ten thousand, had set out at nightfall under command of their general, Hydarnes. They were on the march at this very moment, led by their traitor guide. They would be in the allied rear, in position to attack, by dawn.
His Majesty, cognizant of the catastrophic consequence for the Greeks of this betrayal, may marvel at their response in assembly to the timely and fortuitous warning delivered by the noble Tyrrhastiadas.
They didn't believe him.
They thought it was a trick.
Such an irrational and self-deluding response may be understood only in the light not alone of the exhaustion and despair which had by that hour overwhelmed the allies' hearts but by the corresponding exaltation and contempt of death, which are, like the mated faces of a coin, their obverse and concomitant.
The first day's fighting had produced acts of extraordinary valor and heroism.
The second began to spawn marvels and prodigies.
Most compelling of all was the simple fact of survival. How many times amid the manslaughter of the preceding forty-eight hours had each warrior stood upon the instant of his own extinction? Yet still he lived. How many times had the masses of the foe in numbers overwhelming assaulted the allies with unstoppable might and valor? Yet still the front had held.
Three times on that second day the lines of the defenders teetered upon the point of buckling. His Majesty beheld the moment, immediately before nightfall, when the Wall itself stood breached and the massed myriads of the Empire clambered upon and over the stones, vaunting their victory cry. Yet somehow the Wall stood; the pass did not fall.
All day long, that second of battle, the fleets had clashed off Skiathos in mirrored reflection of the armies at the Gates. Beneath the bluffs of Artemisium the navies hammered each other, driving bronze ram against sheathed timber as their brothers contended steel against steel upon land. The defenders of the pass beheld the burning hulks, smudges against the horizon, and closer in, the flotsam of staved-in beams and spars, shivered oars and sailors' bodies facedown in the shore current. It seemed that Greek and Persian contended no longer as antagonists, but rather had entered, both sides, into some perverse pact whose aim was neither victory nor salvation, but merely to incarnadine earth and ocean with their intermingled blood. The very heavens appeared that day not as a peopled realm, assigning by their witness meaning to events below, but rather as a blank unholy face of slate, compassionless and indifferent. The mountain wall of Kallidromos overstanding the carnage seemed beyond all to embody this bereavement of pity in the featureless face of its silent stone. All creatures of the air had fled. No sign of green shoot lingered upon the earth nor within the clefts of rock.
Only the dirt itself possessed clemency. Alone the stinking soup beneath the warriors' tread proffered surcease and succor. The men's feet churned it into broth ankle-deep; their driving legs furrowed it to the depth of the calf, then they themselves fell upon it on their knees and fought from there. Fingers clawed at the blood-blackened muck, toes strained against it for purchase, the teeth of dying men bit into it as if to excavate their own graves with the clamp of their jaws. Farmers whose hands had taken up with pleasure the dark clods of their native fields, crumbling between their fingers the rich earth which brings forth the harvest, now crawled on their bellies in this sterner soil, clawed at it with the nubs of their busted fingers and writhed without shame, seeking to immure themselves within earth's mantle and preserve their backs from the pitiless steel.
In the _palaistrai_ of Hellas, the Greeks love to wrestle. From the time a boy can stand, he grapples with his mates, dusted with grit in pits of sand or oiled with ooze in rings of mud. Now the Hellenes wrestled in less holy precincts, where the sluice pail held not water but blood, where the prize was death and the umpire spurned all calls for stay or quarter. One witnessed again and again in the battles of the second day a Hellenic warrior fight for two hours straight, retire for ten minutes, without taking food and gulping only a cupped handful of water, then return to the fray for another two-hour round. Again and again one saw a man receive a blow that shattered the teeth within his jaw or split the bone of his shoulder yet did not make him fall.
On the second day I saw Alpheus and Maron take out six men of the foe so fast that the last two were dead before the first pair hit the ground. How many did the brothers slay that day? Fifty? A hundred? It would have taken more than an Achilles among the foe to bring them down, not solely in consequence of their strength and skill but because they were two who fought with a single heart.
All day His Majesty's champions came on, advancing in wave after wave with no interval to distinguish between nations or contingents. The rotation of forces which the allies had employed on the first day became impossible. Companies of their own will refused to forsake the line. Squires and servants took up the arms of the fallen and assumed their places in the breach.
No longer did men waste breath to cheer or rally one another to pride or valor. No more did warriors exult or vaunt their hearts in triumph. Now in the intervals of respite these simply fell, wordless and numb, into heaps of the unstrung and the undone. In the lee of the Wall, upon every hollow of sundered earth, one beheld knots of warriors shattered by fatigue and despair, eight or ten, twelve or twenty, dropped where they fell, in unmoving postures of horror and grief. None spoke or stirred. Instead the eyes of each stared without sight into inexpressible realms of private horror.
Existence had become a tunnel whose walls were death and within which prevailed no hope of rescue or deliverance. The sky had ceased to be, and the sun and stars. All that remained was the earth, the churned riven dirt which seemed to wait at each man's feet to receive his spilling guts, his shattered bones, his blood, his life. The earth coated every part of him. It was in his ears and nostrils, in his eyes and throat, under his nails and in the crease of his backside. It coated the sweat and salt of his hair; he spat it from his lungs and blew it slick with snot from his nose.
There is a secret all warriors share, so private that none dare give it voice, save only to those mates drawn dearer than brothers by the shared ordeal of arms. This is the knowledge of the hundred acts of his own cowardice. The little things that no one sees. The comrade who fell and cried for aid. Did I pass him by? Choose my skin over his? That was my crime, of which I accuse myself in the tribunal of my heart and there condemn myself as guilty.
All a man wants is to live. This before all: to cling to breath. To survive.
Yet even this most primal of instincts, self-preservation, even this necessity of the blood shared by all beneath heaven, beasts as well as man, even this may be worn down by fatigue and excess of horror. A form of courage enters the heart which is not courage but despair and not despair but exaltation. On that second day, men passed beyond themselves. Feats of heart-stopping valor fell from the sky like rain, and those who performed them could not even recall, nor state with certainty, that the actors had been themselves.
I saw a squire of the Philiasians, no more than a boy, take up his master's armor and wade into the manslaughter. Before he could strike a blow, a Persian javelin shattered his shin, driving straight through the bone. One of his mates rushed to the lad to bind his gushing artery and drag him to safety. The youth beat back his savior with the flat of his sword. He hobbled upon his spear used as a crutch, then on his knees, into the fray, still hacking at the foe from the earth where he perished.
Other squires and servants seized iron pegs and, themselves unshod and unarmored, scaled the mountain face above the Narrows, hammering the pins into cracks of rock to secure themselves, from these exposed perches hurling stones and boulders down upon the foe. The Persian archers turned these boys into pincushions; their bodies dangled crucified from pitons or tumbled from their fingerholds to crash upon the roiling slaughter below.
The merchant Elephantinos dashed into the open to save one of these lads yet living, hung up on a ledge above the rear of battle. A Persian arrow tore the old man's throat out; he fell so fast he seemed to vanish straight into the earth. Fierce fighting broke out over his corpse. Why? He was no king or officer, only a stranger who tended the young men's wounds and made them laugh with "Weck up to thees!"
Night had nearly fallen. The Hellenes were reeling from casualties and exhaustion, while the Persians continued pouring fresh champions into the fray. Those in the foe's rear were being driven onward by the whips of their own officers; these pressed with zeal upon their fellows, driving them forward into the Greeks.
Does His Majesty remember? A violent squall had broken then over the sea; rain began sheeting in torrents. By this point most of the allies' weapons had been spent or broken. The warriors had gone through a dozen spears apiece; none yet bore his own shield, which had been staved in long since; he defended himself with the eighth and tenth he had snatched from the ground. Even the Spartans' short _xiphos_ swords had been sundered from excess of blows. The steel blades held, but the hafts and grips had come undone. Men were fighting with stubs of iron, thrusting with shivered half-spears bereft of warhead and butt-spike.
The host of the foe had hacked their way forward, within a dozen paces of the Wall. Only the Spartans and Thespaians remained before this battlement, all others of the allies having been beaten back behind or upon it. The massed myriads of the enemy extended all the way from the Narrows, flooding at will across the hundred-yard triangle before the Wall.
The Spartans fell back. I found myself beside Alexandros atop the Wall, hauling one man after another up and over, while the allies rained javelins and shivered spears, stones and boulders and even helmets and shields down upon the onpressing foe.
The allies cracked and reeled. Back they fell in a disordered mass, fifty feet, a hundred, beyond the Wall. Even the Spartans withdrew in disorder, my master, Polynikes, Alpheus and Maron themselves, shattered by wounds and exhaustion.
The enemy literally tore the stones from the face of the Wall. Now the tide of their multitude flooded over the toppled ruins, skidding down the stadium steps of the Wall's rear onto the open earth before the unprotected camps of the allies. Vanquishment was moments away when for cause inexplicable, the foe, with victory before him in his palm, pulled up in fear and could not find courage to press home the kill.
The enemy drew up, seized by a terror without source or signature.
What force had unmanned their hearts and robbed them of valor, no faculty of reason may divine. It may have been that the warriors of the Empire could not credit the imminence of their own triumph. Perhaps they had been fighting for so long on the foreside of the Wall that their senses could not embrace the reality of at last achieving the breach.
Whatever it was, the foe's momentum faltered. A moment of unearthly stillness seized the field.
Suddenly from the heavens a bellow of unearthly power, as that from the throats of fifty thousand men, pealed through the aether. The hair stood straight up on my neck; I spun toward Alexandros; he, too, held rooted, paralyzed in awe and terror, as every other man upon the field.
A bolt of almighty magnitude slammed overhead into the wall of Kallidromos. Thunder boomed, great stones blew from the cliff face; smoke and sulphur rent the air. On rolled that unearthly cry, nailing all in place with terror save Leonidas, who now strode to the fore with upraised spear.
"Zeus Savior!" the king's voice rose into the thunder. "Hellas and freedom!"
He cried the _paean_ and rushed forward upon the foe. Fresh courage flooded the allies' hearts; they roared into the counterattack. Back over the Wall the enemy tumbled in panic at this prodigy of heaven. I found myself again atop its slick and sundered stones, firing shaft after shaft into the mass of Persians and Bactrians, Medes and Illyrians, Lydians and Egyptians, stampeding in flight below.
The ghastliness of the carnage that followed, His Majesty's own eyes may testify to. As the foreranks of the Persians fled in terror, the whips of their rearmen drove their reinforcing fellows forward. As when two waves, one crashing shoreward before the storm, the other returning seaward down the steep slope of the strand, collide and annihilate one another in spray and foam, so did the crash and wheel of the Empire's armies turn force upon force to trample by thousands those trapped within the riptide of its vortex.
Leonidas had earlier called upon the allies to build a second wall, a wall of Persian bodies. Precisely this now eventuated. The foe fell in such numbers that no warrior of the allies planted sole upon the earth. One trod upon bodies. On bodies atop bodies.
Ahead the Hellenic warriors could see the enemy stampeding into the whips of their own rearmen, charging them, slaying with spear and sword their own fellows in blood madness to escape. Scores and hundreds toppled into the sea. I saw the Spartan front ranks literally scaling the wall of Persian bodies, needing assistance from the second-rankers just to propel themselves over.
Suddenly the piled mass of the dead gave way. An avalanche of bodies began. In the Narrows the allies scrambled rearward toward safety atop a landslide of corpses, which fed upon itself, gaining momentum from its own weight as it tumbled with onrolling might upon the Persians, back down the track toward Trachis. So grotesque was this sight that the Hellenic warriors, unordered by command, but of their own instinct, pulled up where they stood and discontinued the press of their advance, looking on in awe as the enemy perished in numbers uncountable, swallowed and effaced beneath this grisly avalanche of flesh.
Now, in the night assembly of the allies, this prodigy was recalled and cited as evidence of the intercession of the gods. The nobleman Tyrrhastiadas stood beside Leonidas, before the assembled Greeks, urging them with what was clearly the passionate beneficence of his heart to retreat, withdraw, get out. The noble repeated his report of the ten thousand Immortals, even now advancing upon the mountain track to encircle the allies. Less than a thousand Hellenes remained still capable of resistance. What could these hope to effect against ten times their number striking from the undefended rear, while a thousand times their total compounded the assault from the fore?
Yet such was the exaltation produced by that final prodigy that the allies would neither listen nor pay heed. Men came forward in assembly, skeptics and agnostics, those who acknowledged their doubt and even disdain of the gods; these same men now swore mighty oaths and declared that this bolt of heaven and the unearthly bellow which had accompanied it had been none other than the war cry of Zeus Himself.
More heartening news had come in from the fleet. A storm, unseasonably spawned this prior night, had wrecked two hundred of the enemy's warships on the far shore of Euboea. One fifth of His Majesty's navy, the Athenian corvette captain Habroniches reported with exultation, had been lost with all hands; he had beheld the wreckage this day with his own eyes. Might not this, too, be the work of God?
Leontiades, the Theban commander, stepped forward, seconding and inflaming the derangement. What force of man, he demanded, may stand up before the rage of heaven? "Bear this in mind, brothers and allies, that nine-tenths of the Persian's army are conscripted nations, drafted against their will at the point of a sword. How will Xerxes continue to hold them in line? Like cattle as today, driven onward with whips? Believe me, men, the Persian's allies are cracking. Discontent and disaffection are spreading like pestilence through their camp; desertion and mutiny lie one more defeat away. If we can hold tomorrow, brothers, Xerxes' predicament will compel him to force the issue at sea. Poseidon who shakes the earth has already wreaked havoc once upon the Persian's pride. Perhaps the god may cut him down to size again."
The Greeks, inflamed by the Theban commander's passion, hurled harsh words at the Kymean Tyrrhastiadas. The allies swore it was not they who stood now in peril, but Xerxes himself and his overweening pride which had called forth the wrath of the Almighty.
I did not need to glance to my master to read his heart. This derangement of the allies was _katalepsis,_ possession. It was madness, as surely the speakers themselves knew even as they spewed their grief- and horror-spawned rage at the convenient target of the Kymean noble. The prince himself bore this abuse in silence, sorrow darkening his already grave features.
Leonidas dismissed the assembly, instructing each contingent to turn its attention to the repair and refitting of weapons. He dispatched the Athenian captain, Habroniches, back to the fleet, with orders to inform the naval commanders Eurybiades and Themistokles of all he had heard and seen here tonight.
The allies dispersed, leaving only the Spartans and the nobleman Tyrrhastiadas beside the commander's fire.
"A most impressive testimony of faith, my lord," the prince spoke after some moments. "Such devout orations cannot fail to sustain your men's courage. For an hour. Until darkness and fatigue efface the passion of the moment, and fear for themselves and their families resurfaces, as it must, within their hearts."
The noble repeated with emphasis his report of the mountain track and the Ten Thousand. He declared that if the hand of the gods was at all present in this day's events, it was not their benevolence seeking to preserve the Hellenic defenders but their perverse and unknowable will acting to detach them from their reason. Surely a commander of Leonidas' sagacity perceived this, as clearly as he, lifting his glance to the cliff of Kallidromos, could behold there upon the rock the scores of lightning scars where over decades and centuries numerous other random bolts had in the natural course of coastal storms struck here upon this, the loftiest and most proximate promontory.
Tyrrhastiadas again pressed Leonidas and the officers to credit his report. The _demos_ in assembly may elect to disbelieve him; they may denounce and even execute him as a spy; their reason may deceive itself and embrace a propitious prospect for the morrow. Their king and commander, however, cannot permit himself such luxury.
"Say," the Persian pressed, "that I am an agent of intrigue. Believe I have been sent by Xerxes. Say that my intention is in his interest, to influence you by guile and artifice to quit the pass. Say and believe all this. Yet still my report is true.
"The Immortals are coming.
"They will appear by morning, ten thousand strong, in the allied rear."
With a step the noble moved before the Spartan king, addressing him with passion, man-to-man.
"This struggle at the Hot Gates will not be the decisive one, my lord. That battle will come later, deeper into Greece, perhaps before the walls of Athens, perhaps at the Isthmus, perhaps within the Peloponnese, beneath the peaks of Sparta herself. You know this. Any commander who can read terrain and topography knows it.
"Your nation needs you, sir. You are the soul of her army. You may say that a king of Lakedaemon never retreats. But valor must be tempered with wisdom or it is merely recklessness.
"Consider what you and your men have accomplished at the Hot Gates already. The fame you have won in these six days will live forever. Do not seek death for death's sake, nor to fulfill a vain prophecy. Live, sir, and fight another day. Another day with your whole army at your back. Another day when victory, decisive victory, may be yours."
The Persian gestured to the Spartan officers clustered in the light of the council fire. The _polemarch_ Derkylides, the Knights Polynikes and Doreion, the platoon commanders and the warriors, Alpheus and Maron and my master. "I beg you, sir. Conserve these, the flower of Lakedaemon, to give their lives another day. Spare yourself for that hour.
"You have proved your valor, my lord. Now, I beseech you, demonstrate your wisdom.
"Withdraw now.
"Get yourself and your men out while you still can."
**THIRTY**
**T** here would be eleven in the party to raid His Majesty's tent.
Leonidas refused to hazard a greater number; he begrudged even this many, of the hundred and eight who remained of the Three Hundred yet in condition to fight, countenancing the inclusion of five Peers only, and that purely to give the party credibility among the allies.
Dienekes would lead, as the ablest small-unit commander. The Knights Polynikes and Doreion were included for their speed and prowess and Alexandros, over Leonidas' objections seeking to spare him, to fight beside my master as a _dyas._ The Skiritai Hound and Lachides would go. They were mountaineers; they knew how to scale sheer faces. The outlaw Ball Player would serve as guide up the cliff face of Kallidromos, and Rooster would take the company into the enemy camp. Suicide and I were included to support Dienekes and Alexandros and to augment with javelin and bow the party's striking power. The final Spartiate was Telamonias, a boxer of the Wild Olive regiment; after Polynikes and Doreion he was the fastest of the Three Hundred and the only one of the raiders unhampered by wounds.
The Thespaian Dithyrambos had been the force behind the adoption of the plan, conceiving of it on his own without prompting from Rooster, whom my master had not executed, after all, but instead ordered detained in camp throughout the second day with instructions to look to the wounded and the repair and replacement of weapons. Dithyrambos had lobbied strenuously with Leonidas in favor of the raid, and now, disappointed as he was not to be included himself, he stood to hand to wish the party well.
Night's chill had descended upon the camp; as the nobleman Tyrrhastiadas had predicted, fear now stood hard upon the allies; they were one rumor away from terror and one perceived prodigy from panic. Dithyrambos understood the militiamen's hearts. These needed some prospect to fix their hopes upon this night, some expectation to hold them steadfast till morning. Let the raid succeed or fail, it did not matter. Just send men out. And if indeed the gods have taken our part in this cause, well...Dithyrambos grinned and clasped my master's hand in farewell.
Dienekes divided the party into two units, one of five under Polynikes, the other of six under his own command. Each squad was to scale the cliff face independently, advancing across Kallidromos on its own to the rendezvous point beneath the cliffs of Trachis. This to increase the likelihood, in the event of ambush or capture, of at least one party getting through to strike.
When the men were armed and ready to move out, both parties presented themselves for final orders before Leonidas. The king spoke to them alone, without the allies or even the Spartan officers present. A cold wind had gotten up. The sky rumbled above Euboea. The mountain face loomed overhead; the moon, as yet only partially shrouded, could be glimpsed above the shreds of wind-torn fog.
Leonidas offered the parties wine from his personal store and poured the libations from his own plain cup. He addressed each man, squires included, not by his name, but by his nickname, and even the diminutive of that. He called Doreion "Little Hare," the Knight's play name from childhood. Dekton he addressed not as Rooster, but "Roo," and touched him with tenderness upon the shoulder.
"I've had your papers of manumission drawn," the king informed the helot. "They'll be in the courier's pouch for Lakedaemon tonight. They emancipate you as well as your family, and they free your infant son."
This was the babe whose life the lady Arete had saved, that night before the _krypteia;_ the child whose being had made Dienekes under Lakedaemonian law the father of a living son and thus eligible for inclusion among the Three Hundred.
It was this infant whose life would mean Dienekes' death, and Alexandros' and Suicide's by their association with him. And mine as well.
"If you wish"—Leonidas' eyes met Rooster's in the gust-driven firelight—"you may change the name Idotychides, by which the babe is now called. It is a Spartan name, and we all know you bear scant affection for our race."
This name Idotychides, one may recall, was that of Rooster's father, Arete's brother, who had fallen in battle years before. The name the lady had insisted upon giving the babe, that night of the rump court behind the mess.
"You're free to call your son by his Messenian name," Leonidas continued to Rooster, "but you must tell me now, before I seal the papers and dispatch them."
I had seen Rooster whipped and beaten any number of times upon our chores and details in Lakedaemon. But never till this moment had I seen his eyes well and fill.
"I am struck with shame, sir," he addressed Leonidas, "to have extracted this kindness by extortion." Rooster straightened before the king. He declared the name Idotychides a noble one, which his son would be proud to bear.
The king nodded and placed his hand, warm as a father's, upon Dekton's shoulder. "Come back alive this night, Roo. I'll get you out to safety in the morning."
Before Dienekes' party had climbed half a mile above Alpenoi, heavy pellets of rain began to strike. The gentle slope had turned to cliff wall, whose composition was maritime conglomerate, chalky and rotten. When the downpour hit, the surface turned to soup.
Ball Player took the lead past the initial ascent, but it soon became apparent that he had lost his way in the dark; we were off the main track and into the bewildering network of goat trails that crisscrossed the steepening face. The party made up the trace as it went along, groping in the dark with one man taking his turn in the lead, unburdened, while the others followed bearing the shields and weapons. None wore helmets, just undercaps of felt. These became drenched and sodden, spilling cascades from their brimless fronts into the men's eyes. The climb became out-and-out mountaineering, traversing from toehold to handhold with each man's cheek mashed flat against the rotten face, while icy torrents sluiced upon him, accompanied by landslides of mud as the boulders and stones of the face released their hold, and all this in the dark.
For myself, my shot calf had cramped up and now burned as if a poker had been buried molten within the flesh. Each upward heave compelled exertion of this muscle; the pain nearly knocked me faint. Dienekes was laboring even more miserably. His old wound from Achilleion prevented him from raising his left arm above his shoulder; his right ankle was incapable of flexion. To top it off, the socket of his gouged eye had begun to bleed afresh; rainwater mingled with the dark blood, runneling through his beard and down onto the leathers of his corselet. He squinted across to Suicide, whose pair of shot shoulders made him slither like a snake, arms held low to his side as he writhed up the crumbling, mud-slick, rotting slope.
"By the gods," Dienekes muttered, "this outfit is a mess."
The party reached the first crest after an hour. We were above the fog now; the rain ceased; at once the night became clear, windy and cold. The sea rumbled a thousand feet below, blanketed an eighth of a mile deep in a marine fog whose cottony peaks shone brilliant white beneath a moon only one night shy of full. Suddenly Ball Player signed for silence; the party dropped for cover. The outlaw pointed out across a chasm.
Upon the opposite ridge, a third of a mile away, could be descried the tented throne of His Majesty, the one from upon which he had observed the first two days of battle. Servants were dismantling the platform and pavilion.
"They're packing up. For where?"
"Maybe they've had enough. They're heading home."
The party skittered down off the skyline to a shadowed ledge where it could not be seen. Everything the men bore was soaked. I wrung a compress and wound it fresh for my master's eye. "My brains must be leaking along with the blood," he said. "I can't think of another explanation for why I'm out here on this ass-fucked errand."
He had the men take more wine, for the warmth and to deaden the pain of their various wounds. Suicide continued squinting across to the far ridge and the Persian servants striking their master's theater seats. "Xerxes thinks tomorrow will be the end. Bet on it: we'll see him on horseback at dawn, in the Narrows, to savor his triumph at close hand."
The ridge saddle was broad and level; with Ball Player in the lead, the party made good time for the next hour, following game trails that wove among the scrub sumac and fireweed. The track ran inland now, the sea no longer in sight. We crossed two more ridges, then struck a wild watercourse, one of the torrents that fed the Asopus. At least that's what our outlaw guide guessed. Dienekes touched my shoulder, indicating a peak to the north.
"That's Oita. Where Herakles died."
"Do you think he'll help us tonight?"
The party reached a wooded upslope that had to be climbed hand over hand. Suddenly a swift crashing burst from the thicket above. Forms shot forth, invisible. Every hand flew to a weapon.
"Men?"
The sound receded swiftly above.
"Deer."
In a heartbeat the beasts were a hundred feet gone. Silence. Just the wind, tearing the treetops above us.
For some reason, this serendipitous find heartened the party tremendously. Alexandros pushed forward into the thicket. The earth where the deer had taken shelter was dry, crushed and matted where the herd had lain, flank-to-flank. "Feel the grass. It's still warm."
Ball Player assumed a stance to urinate. "Don't," Alexandros nudged him. "Or the deer will never use this nest again."
"What's that to you?"
"Piss down the slope," Dienekes commanded.
Odd as it sounds, the feeling within that cozy copse evoked a hearth of home, a haven. One could still smell that deery smell, the gamy scent of their coats. None of the party spoke, yet each, I will wager, was thinking the same thought: how sweet it would be, right now, to lie down here like the deer and close one's eyes. To allow all fear to depart one's limbs. To be, just for a moment, innocent of terror.
"It's good hunting country," I observed. "Those were boar runs we passed through. I'll bet there are bear up here, and even lion."
Dienekes' glance met Alexandros' with a glint. "We'll have ourselves a hunt here. Next fall. What do you say?"
The youth's broken face contorted into a grin.
"You'll join us, Rooster," Dienekes proposed. "We'll take a week and make an event of it. No horses or beaters, just two dogs per man. We'll live off the hunt and come home draped in lionskins like Herakles. We'll even invite our dear friend Polynikes."
Rooster regarded Dienekes as if he had gone mad. Then a wry grin settled into place upon his features.
"Then it's settled," my master said. "Next fall."
From the succeeding crest the party followed the watercourse down. The torrent was loud and discipline got careless. From out of nowhere arose voices.
Every man froze.
Rooster crouched in the lead; the party was strung out in column, the worst possible lineup to fight from. "Are they speaking Persian?" Alexandros whispered, straining his ears toward the sound.
Suddenly the voices froze too.
They had heard us.
I could see Suicide, two steps below me, silently stretch behind his shoulder, slipping a pair of "darning needles" from his quiver. Dienekes, Alexandros and Rooster all clutched eight-footers; Ball Player readied a throwing axe.
"Hey, fuckers. Is that you?"
Out of the darkness stepped Hound, the Skirite, with a sword in one hand and a dagger in the other.
"By the gods, you scared the shit out of us!"
It was Polynikes' party, pausing to gnaw a heel of dry bread.
"What is this, a picnic?" Dienekes slid down among them. We all clapped our mates in relief. Polynikes reported that the outroute his party had taken, the lower track, had been fast and easy. They had been in this clearing a quarter of an hour.
"Come down here." The Knight motioned to my master. "Take a look at this."
The whole party followed. On the opposite bank of the watercourse, ten feet up the slope, stretched a track wide enough for two men to pass abreast. Even in the deep shadow of the gorge, you could see the churned-up earth.
"It's the mountain track, the one the Immortals are taking. What else can it be?"
Dienekes knelt to feel the earth. It was freshly trodden, the passage no more than two hours old. You could glimpse on the uphill side the ridges where the marching soles of the Ten Thousand had caved the hill in, and the slides on the downslope from the weight of their passage.
Dienekes chose one of Polynikes' men, Telamonias the boxer, to retrace the track their party had taken and inform Leonidas. The man groaned with disappointment. "None of that," Dienekes snapped. "You're the fastest who knows the trail. It has to be you."
The boxer sprinted off.
Another of Polynikes' party was absent. "Where's Doreion?"
"Down the track. Taking a snoop."
A moment later the Knight, whose sister Altheia was Polynikes' wife, came loping into view from below. He was _gymnos,_ naked for speed.
"What happened to your dog?" Polynikes greeted him merrily. "The little fellow has shriveled into an acorn."
The Knight grinned and snatched his cloak from where it hung upon a tree. He reported that the track ended about a quarter mile down. There an entire forest had been felled, probably this very evening, immediately after the Persians had learned of the track. The Immortals had no doubt marshaled there, on the freshly cleared ground, before setting out.
"What's there now?"
"Cavalry. Three, maybe four squadrons."
These were Thessalians, the Knight reported. Greeks whose country had gone over to the enemy.
"They're snoring like farmers. The fog is soup. Every nose is buried in a cloak, sentries too."
"Can we go around?"
Doreion nodded. "It's all pine. A carpet of soft needles. You can cross on a dead run and not make a sound."
Dienekes indicated the clearing in which the parties now stood. "This will be our rally point. We'll assemble here after. You'll guide us back from this point, Doreion, or one of your party, by the way you came, the fast way."
Dienekes had Rooster rebrief both parties on the layout of the enemy camp, in case something happened to him on the way down. The last of the wine was shared out. The skin in its sequence chanced to pass from Polynikes' hand to Rooster's. The helot seized this moment of intimacy before action. "Tell me the truth. Would you have killed my son that night with the _krypteia?"_
"I'll kill him yet," the runner answered, "if you fuck us up tonight."
"In that case," the helot said, "I look forward with even greater anticipation to your death."
It was time for Ball Player to depart. He had agreed to guide the party this far and no farther. To the surprise of all, the outlaw seemed torn. "Look," he offered haltingly, "I want to keep on with you, you're good men, I admire you. But I can't in good conscience without being compensated."
This struck the entire party as hilarious.
"Your scruples are stern, outlaw," Dienekes observed.
"You want compensation?" Polynikes clutched his own privates. "I'll save this for you."
Ball Player alone did not laugh.
"Goddamn you," he muttered, more to himself than to the others. With further grumbled curses, he took his place in the undermanned column. He was staying.
The party would no longer be divided; from here it would advance in teams of five, Ball Player attached to Polynikes' four to make up for Telamonias, but in tandem, each unit supporting the other.
The squads ghosted without incident past the snoozing Thessalians. The presence of this Greek cavalry was extremely good fortune. The way back, if there was one, would inevitably be in disorder; it would be of no small advantage to have a landmark as conspicuous in the dark as an acres-wide swath of felled forest. The Thessalians' horses could be stampeded to create confusion, and, if the party had to flee under fire through their camp, its shouts to one another in Greek would not betray it among the Greek-speaking Thessalians.
Another half hour brought the squads to the edge of a wood directly above the citadel of Trachis. The channel of the Asopus thundered beneath the city walls. It roared in torrent, deafening, with a sharp cold wind keening down the throat of the gorge.
We could see the enemy camp now.
Surely no sight beneath heaven, not Troy under siege, nor the war of the gods and Titans itself, could have equaled in scale that which now spread before our vision.
As far as the eye could see, three miles of plain extending to the sea, five miles across, with plain and more plain extending beyond sight around the shoulders of the Trachinian cliffs, thousands of acres square and all of it incandescent with the mist-magnified fires of the enemy.
"So much for them packing up."
Dienekes motioned Rooster to him. The helot laid it out as he remembered:
Xerxes' horses drink upstream of all, before the rest of the camp. Rivers are sacred to the Persians and must be preserved unprofaned. The whole upper valley is staked out as pasturage. The Great King's pavilion, so Rooster swore, stood at the head of the plain, within bowshot of the river.
The party dropped down, directly beneath the citadel walls, and entered the current. The Eurotas in Lakedaemon is mountain-fed; even in summer its snowmelt is bone-numbingly cold. The Asopus was worse. One's limbs went to ice within moments. It was so cold we feared for our safety; if you had to get out and run, you couldn't feel your own legs and feet.
Mercifully the torrent lessened a few hundred yards down. The party rolled its cloaks into bundles and floated them on shields turned bowls-up. Dams had been erected by the enemy to abate the torrent and facilitate the watering of horses and men. Pickets had been stationed atop these, but the fog and wind made conditions so inhospitable, the hour was so late and the sentries so complacent, deeming infiltration unthinkable, that the party was able to steal past, bellying over the spillways, then coasting swiftly into the shadows along the bank.
The moon had set. Rooster could not pick out His Majesty's pavilion. "It was here, I swear it!" He pointed across to a rise of land, upon which stood nothing but a street of grooms' tents snapping in the wind and a rope picket line shoulder-to-shoulder with horses standing miserably in the gale. "They must have moved it."
Dienekes himself drew his blade. He was going to open Rooster's throat on the spot as a traitor. Rooster swore by every god he could think of; he wasn't lying. "Things look different in the dark," he offered lamely.
Polynikes saved him. "I believe him, Dienekes. He's so fucking stupid, this is just the way he would screw it up."
The party slithered on, neck-deep in marrow-numbing rapids. At one point Dienekes' leg became snarled in a tangle of reeds; he had to submerge with his _xiphos_ to cut himself free. He came up snorting.
I asked what he was laughing at.
"I was just wondering if it was possible to get any more miserable." He chuckled darkly. "I suppose if a river snake crawled up my ass and gave birth to quintuplets..."
Suddenly Rooster's hand nudged my master's shoulder. A hundred paces ahead stood another dam and spillway. Three linen pavilions abutted a pleasant beach; a lantern-lit walkway snaked up the slope, past a hide corral in which were confined a dozen blanket-draped war mounts of such magnificence that the worth of each alone must have equaled the produce of a small city.
Directly above rose a copse of oak, lit by iron cressets howling in the gale, and beyond, past a single picket line of Egyptian marines, could be glimpsed the pennanted kingposts of a pavilion so vast it looked like it housed a battalion.
"That's it." Rooster pointed. "That's Xerxes' tent."
**THIRTY-ONE**
**T** he warrior's thoughts at the brink of action, my master had often observed (as the student of fear he ever declared himself to be), follow a pattern unvarying and ineluctable. There appears always an interval, often brief as a heartbeat, wherein the inward eye summons the following tripartite vision, often in the selfsame order:
First to the inmost heart appear the faces of those he loves who do not share his immediate peril: his wife and mother, his children, particularly if they are female, particularly if they are young. These who will remain beneath the sun and preserve within their hearts the memory of his passage, the warrior greets with fondness and compassion. To them he bequeaths his love and to them bids farewell.
Next arise before the inward eye the shades of those already across the river, they who stand awaiting upon the distant shore of death. For my master these comprised his brother, Iatrokles, his father and mother and Arete's brother, Idotychides. These, too, the warrior's heart greets in silent vision, summons their aid and then releases.
Lastly advance the gods, whichever a man feels have favored him most, whichever he feels himself most to have favored. Into their care he releases his spirit, if he can.
Only when this triple obligation has been requited does the warrior revert to the present and turn, as if arising from a dream, to those at his shoulder, they who in a moment will undergo with him the trial of death. Here, Dienekes often observed, is where the Spartans most hold advantage over all who face them in battle. Beneath what alien banner could one discover at his shoulder such men as Leonidas, Alpheus, Maron, or here in this dirt Doreion, Polynikes and my master, Dienekes, himself? These who will share the ferry with him, the warrior's heart embraces with a love surpassing all others granted by the gods to humankind, save only that of a mother for her babe. To them he commits all, as they all to him.
My own eyes now glanced to Dienekes, crouched upon the riverbank helmetless in his scarlet cloak which showed dead black in the darkness. His right hand was kneading the joints of his immobile ankle seeking to restore flexion, as he in compact phrase issued the instructions which would drive the men he commanded into action. At his shoulder Alexandros had scraped a fistful of sand from the bank and was scoring it along the haft of his eight-footer, abrading the surface for a grip. Polynikes with a curse worked his forearm into the sodden bronze and leather sleeve of his shield, seeking the point of balance and proper hold upon the gripcord. Hound and Lachides, Ball Player, Rooster and Doreion likewise completed their preparations. I glanced to Suicide. He was sorting swiftly through his darning needles, like a surgeon selecting his instrument, picking those three, one for his throwing hand, two for his free, whose heft and balance promised the truest flight. I moved in a crouch beside the Scythian, with whom I was paired in the assault. "See you in the ferry," he said, and tugged me with him toward the flank from which we would attack.
Would his be the last face I would see? This Scythian, mentor and instructor to me since I was fourteen. He had taught me cover and interval, dress and shadow; how to stanch a puncture wound, set a broken collarbone; how to take down a horse upon the open field, drag a wounded warrior from battle using his cloak. This man with his skill and fearlessness could have hired himself out as a mercenary to any army in the world. To the Persians if he wished. He would have been appointed captain-of-a-hundred, achieved fame and glory, women and wealth. Yet he chose to remain in the harsh academy of Lakedaemon, in service for no pay.
I thought of the merchant Elephantinos. Of all in camp, Suicide had taken most to this gay, ebullient fellow; the pair had become fast friends. On the evening before the first battle, when my master's platoon had settled, preparing the evening meal, this Elephantinos had appeared upon his rounds. He had traded away all his wares, bartered his waggon and ass, sold even his own cloak and shoes. Now on this night he circulated with a basket of pears and sweetmeats, distributing these treats to the warriors as they sat to their suppers. He stopped beside our fire. My master often sacrificed in the evenings; nothing much, just a crust of barley loaf and a libation, not praying aloud, just offering within his heart a few silent words to the gods. He would never reveal the contents of his prayer, but I could read it upon his lips and overhear the odd mumble. He was praying for Arete and his daughters.
"It is these young boys who should practice such piety," the merchant observed, "not you grisly veterans!"
Dienekes greeted the _emporos_ warmly. "You mean 'grizzled,' my friend."
"I mean grisly, weck up to thees!"
He was invited to sit. Bias was still alive then; he joked with the merchant about his want of forethought. How will the old-timer get away now, without his ass and waggon?
Elephantinos made no reply.
"Our friend will not be leaving," Dienekes spoke softly, his gaze upon the earth.
Alexandros and Ariston arrived with a hare they had traded for with some boys from Alpenoi village. The old man smiled at the comradely ragging they endured from their mates over this prize. It was a "winter hare," so scrawny it wouldn't flavor a stew for two men, let alone sixteen. The merchant regarded my master.
"To see you veterans with gray in your beards, it is only right that you should stand here at the Gates. But these boys." His gesture indicated Alexandros and Ariston, including in its sweep myself and several other squires barely out of their teens. "How may I leave, when these babes remain?"
"I envy you comrades," the merchant continued when the emotion had cleared from his throat. "I have searched all my life for that which you have possessed from birth, a noble city to belong to." His smithy-scarred hand indicated the fires springing to life across the camp and the warriors, old and young, now settling beside them. "This will be my city. I will be her magistrate and her physician, her orphans' father and her fool."
He handed out his pears and moved on. One could hear the laughter he brought to the next fire, and the one after that.
The allies had been on station at the Gates for four nights then. They had observed the scale of the Persian host, on land and sea, and knew well the odds insuperable that faced them. Yet it was not until that moment, I felt, at least for my master's platoon, that the reality of the peril to Hellas and the imminence of the defenders' own extinction truly struck home. A profound soberness settled with the vanishing sun.
For long moments no one spoke. Alexandros was skinning the hare, I was grinding barley meal in a handmill; Medon prepared the ground oven, Black Leon was chopping onions. Bias reclined against the stump of an oak felled for firewood, with Leon Donkeydick upon his left. To the startlement of all, Suicide began to speak.
"There is a goddess in my country called Na'an," the Scythian broke the silence. "My mother was a priestess of this cult, if such a grand title may be applied to an illiterate countrywoman who lived all her life out of the back of a waggon. My mind is recalled to this by our friend the merchant and the two-wheeled cart he calls his home."
This was as much speech at one time as I, or any other, had heard Suicide give voice to. All expected him to halt right there. To their astonishment, the Scythian continued.
His priestess mother taught him, Suicide said, that nothing beneath the sun is real. The earth and everything upon it is but a forestander, the material embodiment of a finer and more profound reality which exists immediately behind it, invisible to mortal sense. Everything we call real is sustained by this subtler fundament which underlies it, indestructible, unglimpsed beyond the curtain.
"My mother's religion teaches that those things alone are real which cannot be perceived by the senses. The soul. Mother love. Courage. These are closer to God, she taught, because they alone are the same on both sides of death, in front of the curtain and behind.
"When I first came to Lakedaemon and beheld the phalanx," Suicide went on, "I thought it the most ludicrous form of warfare I had ever seen. In my country we fight on horseback. This to me was the only way, grand and glorious, a spectacle that stirs the soul. The phalanx looked like a joke to me. But I admired the men, their virtue, which was so clearly superior to that of every other nation I had observed and studied. It was a puzzle to me."
I glanced to Dienekes across the fire, to see if he had previously heard these thoughts articulated by Suicide, perhaps in the years before I had entered his service, when the Scythian alone stood as his squire. Upon my master's face was written rapt attention. Clearly this bounty from Suicide's lips was as novel to him as to the others.
"Do you remember, Dienekes, when we fought the Thebans at Erythrae? When they broke and ran? This was the first rout I had witnessed. I was appalled by it. Can there exist a baser, more degrading sight beneath the sun than a phalanx breaking apart in fear? It makes one ashamed to be mortal, to behold such ignobility even in an enemy. It violates the higher laws of God." Suicide's face, which had been a grimace of disdain, now brightened into a cheerier mode. "Ah, but the opposite: a line that holds! What can be more grand, more noble?
"One night I dreamt I marched within the phalanx. We were advancing across a plain to meet the foe. Terror froze my heart. My fellow warriors strode all around me, in front, behind, to all sides. They were all me. Myself old, myself young. I became even more terrified, as if I were coming apart into pieces. Then all began to sing. All the 'me's,' all the 'myself's.' As their voices rose in sweet concord, all fear fled my heart. I woke with a still breast and knew this was a dream straight from God.
"I understood then that it was the glue that made the phalanx great. The unseen glue that bound it together. I realized that all the drill and discipline you Spartans love to pound into each other's skulls were really not to inculcate skill or art, but only to produce this glue."
Medon laughed. "And what glue have you dissolved, Suicide, that finally allows your jaws to flap with such un-Scythian immoderation?"
Suicide grinned across the fire. Medon was the one, it was said, who had originally given the Scythian his nickname, when he, guilty of a murder in his country, had fled to Sparta, where he asked again and again for death.
"When I first came to Lakedaemon and they called me 'Suicide,' I hated it. But in time I came to see its wisdom, unintentional as it was. For what can be more noble than to slay oneself? Not literally. Not with a blade in the guts. But to extinguish the selfish self within, that part which looks only to its own preservation, to save its own skin. That, I saw, was the victory you Spartans had gained over yourselves. That was the glue. It was what you had learned and it made me stay, to learn it too.
"When a warrior fights not for himself, but for his brothers, when his most passionately sought goal is neither glory nor his own life's preservation, but to spend his substance for them, his comrades, not to abandon them, not to prove unworthy of them, then his heart truly has achieved contempt for death, and with that he transcends himself and his actions touch the sublime. This is why the true warrior cannot speak of battle save to his brothers who have been there with him. This truth is too holy, too sacred, for words. I myself would not presume to give it speech, save here now, with you."
Black Leon had been listening attentively. "What you say is true, Suicide, if you will forgive me for calling you that. But not everything unseen is noble. Base emotions are invisible as well. Fear and greed and lust. What do you say about them?"
"Yes," Suicide acknowledged, "but don't they feel base? They stink to heaven, they make one sick within the heart. The noble invisible things feel different. They are like music, in which the higher notes are the finer.
"This was another thing that puzzled me when I arrived in Lakedaemon. Your music. How much of it there was, not alone the martial odes or war songs you sing as you advance upon the foe, but in the dances and the choruses, the festivals and the sacrifices. Why do these consummate warriors honor music so, when they forbid all theater and art? I believe they sense that the virtues are like music. They vibrate at a higher, nobler pitch."
He turned to Alexandros. "That is why Leonidas chose you for the Three Hundred, my young master, though he knew you had never before stood among the trumpets. He believes you will sing here at the Gates in that sublime register, not with this"—he indicated the throat—"but with this." And his hand touched his heart.
Suicide drew up, suddenly awkward and abashed. Around the fire each face regarded him soberly and with respect. Dienekes broke the silence with a laugh.
"You're a philosopher, Suicide."
The Scythian grinned back. "Yes," he nodded, "weck up to thees!"
A messenger appeared, summoning Dienekes to Leonidas' council. My master motioned me to accompany him. Something had changed within him; I could sense it as we picked our way among the network of trails that crisscrossed the camps of the allies.
"Do you remember the night, Xeo, when we sat with Ariston and Alexandros and spoke of fear and its opposite?"
I said I did.
"I have the answer to my question. Our friends the merchant and the Scythian have given it to me."
His glance took in the fires of the camp, the nations of the allies clustered in their units, and their officers, whom we could see, like us approaching from all quarters the king's fire, ready to respond to his needs and receive his instructions.
"The opposite of fear," Dienekes said, "is love."
**THIRTY-TWO**
**T** wo sentries covered the west, the rear of His Majesty's pavilion. Dienekes chose this side to attack because it was the dreariest and least prominent, the flank most exposed to the gale. Of all the fragmented images that remain from this brawl which was over no more than fifty heartbeats after it began, the most vivid is of the first sentinel, an Egyptian marine, a six-footer with a helmet the color of gold, decorated with stubby silver griffin's wings. These marines, as His Majesty knows, wear as a badge of pride brightly colored regimental sashes of wool. It is their custom on station to drape these pennants crosswise over the chest and belt them at the waist. This night this sentry had wound his over his nose and mouth to protect against the gale and the scoring of the driven dust, enwrapping ears and brow as well, with the merest slitted sliver held open for the eyes. His body-length wicker shield he bore before him at port, wrestling its unwieldly mass in the blow. It took little imagination to perceive his misery, alone in the cold beside a single cresset howling in the blast.
Suicide advanced undetected to within thirty feet of the fellow, snaking on his belly past the buttoned-up tents of His Majesty's grooms and the loudly snapping windbreaks of linen which shielded the horses from the gale. I was half a length behind him; I could see him mutter the two-word prayer—"Deliver him," meaning the foe—to his savage gods.
Blearily the sentry blinked up. Out of the darkness, tearing directly for him, he beheld the hurtling form of the Scythian clutching in his left fist a pair of dart-length javelins, with the bronze-sheathed killing point of a third poised in throwing position beside his right ear. So bizarre and unexpected must this sight have been that the marine did not even react with alarm. With his spear hand he tugged nonchalantly at the sash that shielded his eyes, as if muttering to himself at the obligation to respond to this sudden and unwonted irritation.
Suicide's first javelin drove so powerfully through the apple of the man's throat that its point burst all the way through the neck and out the spine, its ash extending crimson, half an arm's length beyond. The man dropped like a rock. In an instant Suicide was upon him, tearing the darning needle out with such a savage wrench that it brought half the man's windpipe with it.
The second sentry, ten feet to the left of the first, was just turning in bewilderment, clearly disbelieving yet the evidence of his senses, when Polynikes blindsided him on a dead sprint, slamming the man on his unshielded right a blow of such ferocious impact with his own shoulder-driven shield that the fellow was catapulted off his feet and hurled bodily through the air. The breath expelled from the guardsman's lungs, his spine crashed into the dirt; Polynikes' lizard-sticker punched through his breast so hard you could hear the bone shiver and crack even over the gale.
The raiders dashed to the tent wall. Alexandros' blade slashed a diagonal in the bucking linen. Dienekes, Doreion, Polynikes, Lachides, then Alexandros, Hound, Rooster and Ball Player blasted through. We had been seen. The sentries on either side bawled the alarm. It had all happened so swiftly, however, that the pickets could not at first credit the substance their eyes beheld. Clearly they had orders to remain at their posts and this they half did, at least the nearest two, advancing toward Suicide and me (the only ones yet outside the pavilion) with an abashed and befuddled tentativeness. I had an arrow nocked in my bow, with three more clutched in my left fist around the grip, and was raising to fire. "Hold!" Suicide shouted into my ear in the gale. "Give 'em a grin."
I thought he was mad. But that's just what he did. Gesturing like a crony, calling to the sentries in his tongue, the Scythian put on a performance, acting as if this were just some kind of drill which perhaps these sentries had missed at the briefing. It held them for about two heartbeats. Then another dozen marines roared from the pavilion's front. We turned and plunged into the tent.
The interior was pitch-black and filled with shrieking women. The rest of our party was nowhere to be seen. We saw lamplight flare across the chamber. It was Hound. A naked woman had him about one leg, burying her teeth into the meat of his calf. The lamplight from the next chamber illuminated the Skirite's blade as he drove it like a cleaver, slicing through the gristle of her cervical spine. Hound gestured to the chamber. "Torch it!"
We were in some kind of concubines' seraglio. The pavilion as a whole must have had twenty chambers. Who the hell knew which was the King's? I dashed for the single lit lamp and jammed its flame into a closet of women's undergarments; in an instant the whole brothel was howling.
Marines were pouring in behind us, among the shrieking whores. We raced after Hound, in the direction he had taken down the corridor. Clearly we were all the way at the pavilion's rear. The next chamber must have been the eunuchs'; I saw Dienekes and Alexandros, shield by shield, blast through a pair of skull-shaved titans, not even pausing to strike but just bowling them over. Rooster disemboweled one with a swing of his _xiphos;_ Ball Player chopped another down with his axe. Polynikes, Doreion and Lachides emerged ahead, from some kind of bedchamber, spearpoints dripping blood. "Fucking priests!" Doreion shouted in frustration. A Magus staggered forth, gutted, and dropped.
Doreion and Polynikes were in the lead when the party hit His Majesty's chamber. The space was vast, big as a barn and studded with so many ridgepoles of ebony and cedar that it looked like a forest. Lamps and cressets lit the vault like noon. The ministers of the Persians were awake and assembled in council. Perhaps they had risen early for the morrow, perhaps they had never gone to bed. I turned the corner into this chamber just as Dienekes, Alexandros, Hound and Lachides caught up with Polynikes and Doreion and formed in line, shield by shield, to attack. We could see the generals and ministers of His Majesty, thirty feet away across the floor, which was not dirt but platformed wood, stout and level as a temple, and carpeted so thick with rugs that it muffled all sound of onrushing feet.
It was impossible to tell which of the Persians was His Majesty, all were so magnificently appareled and all of such surpassing height and handsomeness. Their numbers were a dozen, excluding scribes, guards and servants, and every man was armed. Clearly they had learned of the attack only moments earlier; they clutched scimitars, bows and axes and seemed by their expressions not yet to believe the evidence of their eyes. Without a word the Spartans charged.
Suddenly there were birds. Exotic species by the dozen and the score, apparently brought from Persia for His Majesty's amusement, now clattered into flight at the feet of the onsurging Spartans. Some array of cages had either been spilled or trampled open, who knows by whom, perhaps one of the Spartans in the confusion, perhaps a quick-thinking servant of His Majesty, but at once and in the midst of the attack, a hundred or more shrieking harpies erupted into the interior of the pavilion, flying creatures of every hue, howling and churning the space to madness with the wild clatter and frenzy of their wings.
Those birds saved His Majesty. They and the ridgepoles which supported the vault of the pavilion like the hundred columns of a temple. These in combination, and their unexpectedness, threw off the rush of the attackers just enough for His Majesty's marines and those remaining household guards of the Immortals to secure with their swarming bodies the space before His Majesty's person.
The Persians within the tent fought just as their fellows had in the pass and at the Narrows. Their accustomed weapons were of the missile type, javelins, lances and arrows, and they sought space, an interval of distance from which to launch them. The Spartans on the other hand were trained to close breast-to-breast with the foe. Before one could draw breath, the locked shields of the Lakedaemonians were pincushioned with arrow shafts and lanceheads. One heartbeat more and their bronze facings slammed into the frantically massing bodies of the foe. For an instant it seemed as if they would utterly trample the Persians. I saw Polynikes bury his eight-footer overhand in the face of one nobleman, jerk its gore-dripping point free and plunge it into the breast of another. Dienekes, with Alexandros on his left, slew three so quickly the eye could barely assimilate it. On the right Ball Player was hacking like a madman with his throwing axe, directly into a shrieking knot of priests and secretaries cowering upon the floor.
The servants of His Majesty sacrificed themselves with stupefying valor. Two directly ahead of me, youths without even the start of a beard, tore in tandem a carpet from the floor, thick as a shepherd's winter coat, and, employing it as a shield, flung themselves upon Rooster and Doreion. If one had had time to laugh, the sight of Rooster's fury as he plunged his _xiphos_ in frustration into that rug would have prompted gales of hilarity. He tore the first servant's throat out with his bare hands and caved in the second's skull with a lamp still aflame.
For myself, I had loosed with such furious speed all four of the arrows I clutched ready in my left hand that I was empty and groping to the quiver before I could spit. There was no time even to follow the shafts' flight to see if they had found their marks. My right hand was just clutching a fistful more from the sleeve at my shoulder when I raised my eyes and saw the burnished steel head of a hurled battle-axe pinwheeling straight for my skull. Instinct jerked my legs from beneath me; it seemed an eternity before my weight began to make me fall. The axehead was so close I could hear its whirling thrum and see the purple ostrich plume on its flank and the double-headed griffin imprinted on the steel. The killing edge was half an arm's length from the space between my eyes when a ridgepole of cedar, whose presence I had not even been aware of, intercepted the homicidal rush of its flight. The axehead buried palm-deep in the wood. I had half an instant to glimpse the face of the man who had flung the blade and then the whole wall of the chamber blew apart.
Egyptian marines poured through, twenty of them followed at once by twenty more. The whole side of the tent was now open to the gale. I saw the captain Tommie clash shield-to-shield with Polynikes. Those lunatic birds thrashed everywhere. Hound went down. A two-handed axe tore open his guts. An arrow shaft ripped through Doreion's throat; he reeled backward with blood spewing from his teeth. Dienekes was hit; he buckled rearward onto Suicide. In the fore remained only Alexandros, Polynikes, Lachides, Ball Player and Rooster. I saw the outlaw stagger. Polynikes and Rooster were swamped by inrushing marines.
Alexandros was alone. He had singled out the person of His Majesty or some nobleman he took for him and now, with his eight-footer cocked overhand above his right ear, prepared to hurl the spear across the wall of enemy defenders. I could see his right foot plant, concentrating all force of leg and limb behind the blow. Just as his shoulder started forward, arm extended in the throw, a noble of the Persians, the general Mardonius I later learned, delivered with his scimitar a blow of such force and precision that it took Alexandros' hand off right at the wrist.
As in moments of extreme emergency time seems to slow, permitting the vision to perceive instant by instant that which unfolds before the eyes, I could see Alexandros' hand, its fingers still gripping the spear, hang momentarily in midair, then plummet, yet clutching the ashen shaft. His right arm and shoulder continued forward with all their force, the stump at the wrist now spraying bright blood. For an instant Alexandros did not realize what had happened. Discomfiture and disbelief flooded his eyes; he couldn't understand why his spear was not flying forward. A blow of a battle-axe thundered upon his shield, driving him to his knees. I was in too tight to use my bow to defend him; I dove for the fallen shaft of his eight-footer, hoping to thrust it back at the Persian noble before his scimitar could find the mark to decapitate my friend.
Before I could move, Dienekes was there, the huge bronze bowl of his shield covering Alexandros. "Get out!" he bellowed to all above the din. He hauled Alexandros to his feet the way a countryman yanks a lamb out of a torrent.
We were outside, in the gale.
I saw Dienekes cry an order from no farther than two arm's lengths and could not hear a word of it. He had Alexandros on his feet and was pointing up the slope past the citadel. We would not flee by the river, there was no time. "Cover them!" Suicide shouted into my ear. I felt scarlet-cloaked forms flee past me and could not tell who was who. Two were being carried. Doreion staggered from the pavilion, mortally wounded, amid a swarm of Egyptian marines. Suicide slung darning needles into the first three so fast, each seemed to sprout a lance in the belly as if by magic. I was shooting too. I saw a marine hack Doreion's head off. Behind him, Ball Player plunged from the tent, burying his axe in the man's back; then he, too, fell beneath a hail of pike and sword blows. I was empty. So was Suicide. He made to rush the enemy bare-handed; I clutched his belt and dragged him back screaming. Doreion, Hound and Ball Player were dead; the living would need us more.
**THIRTY-THREE**
**T** he space immediately east of the pavilion stood occupied exclusively by the picketed mounts of His Majesty's personal riding stock and the service tents of their grooms. Through this open-air paddock the raiding party now fled. Linen windbreaks had been erected, dividing the enclosure into squares. It was like racing through the hanging laundry of a city's humble quarter. As Suicide and I overtook our comrades among the wind-numbed mounts, on a dead run and with the blood of terror pounding within our temples, we encountered Rooster at the party's rear, gesturing urgently to us to slow, to stop. Walk.
The party emerged into the open. Armored men advanced toward us by the hundreds. But these, as fortune or a god's hand would have it, had not been summoned to arms in response to the attack upon their King, but stood in fact in total ignorance of it. They were simply rising to the call of reveille, groggy yet and grumbling in the gale-pounded dark, to arm for the morning's resumption of battle. The marines' shouts of alarm from the pavilion were shredded in the teeth of the gale; their foot pursuit lost its way at once among the myriads in the dark.
The flight from the Persian camp became attended, as are so many moments in war, by a sense of reality so dislocated as to border upon, and even surpass, the bizarre. The party made good its escape neither sprinting nor flying, but limping and hobbling. The raiders trudged in the open, making no attempt to conceal themselves from the enemy but in fact approaching and even engaging him in converse. Irony compounded, the party itself helped spread the alarm of attack, helmetless as it was and bloodied, bearing shields from which the _lambda_ of Lakedaemon had been effaced and carrying across its shoulders one desperately wounded, Alexandros, and one already dead, Lachides. For all the world, the group appeared like a squad of overwhelmed pickets. Dienekes speaking in Boeotian Greek, or as near as he could come to the accent, and Suicide in his own Scythian dialect, addressed those officers whose arming men we passed through, spreading the word "mutiny" and gesturing back, not wildly but wearily, toward the pavilion of His Majesty.
Nobody seemed to give a damn. The great bulk of the army, it was clear, were grudging draftees whose nations had been conscripted into service against their will. These now in the dank and gale-torn dawn sought only to warm their own backsides, fill their bellies and get through the day's fighting with their heads still attached.
The raiding party even received unwitting aid for Alexandros from a squad of Trachinian cavalrymen, struggling to ignite a fire for their breakfast. These took us for Thebans, the faction of that nation who had gone over to the Persian, whose turn it was that night to provide inner-perimeter security. The cavalrymen provided us with light, water and bandages while Suicide, with the hands of experience surer than any battlefield surgeon's, secured the hemorrhaging artery with a copper "dog bite." Already he, Alexandros, was deep in shock.
"Am I dying?" he asked Dienekes in that sad detached tone so like a child's, the voice of one who seems to stand already at his own shoulder.
"You'll die when I say you can," Dienekes answered gently.
The blood was coming in surges from Alexandros' severed wrist despite the arterial clamp, sheeting from the hacked-off veins and the hundred vessels and capillaries within the pulpy tissue. With the flat of a _xiphos_ gray-hot from the fire, Suicide cauterized and bound the stump, lashing a tourniquet about the pinion point beneath the biceps. What none was aware of in the dark and the confusion, not even Alexandros himself, was the puncture wound of a lancepoint beneath his second rib and the blood pooling internally at the base of his lungs.
Dienekes himself had been wounded in the leg, his bad leg with the shattered ankle, and had lost his own share of blood. He no longer had the strength to carry Alexandros. Polynikes took over, slinging the yet-conscious warrior over his right shoulder, loosening the gripcord of Alexandros' shield to hang it as protection across his back.
Suicide collapsed halfway up the slope before the citadel. He had been shot in the groin, sometime back in the pavilion, and didn't even know it. I took him; Rooster carried Lachides' body. Dienekes' leg was coming unstrung; he needed bearing himself. In the starlight I could see the look of despair in his eyes.
We all felt the dishonor of leaving Doreion's body and Hound's, and even the outlaw's, among the foe. The shame drove the party like a lash, impelling each exhaustion-shattered limb one pace more up the brutal, steepening slope.
We were past the citadel now, skirting the felled wood where the Thessalian cavalry were picketed. These were all awake now and armed, moving out for the day's battle. A few minutes later we reached the grove where earlier we had startled the slumbering deer.
A Doric voice hailed us. It was Telamonias the boxer, the man of our party whom Dienekes had dispatched back to Leonidas with word of the mountain track and the Ten Thousand. He had returned with help. Three Spartan squires and half a dozen Thespaians. Our party dropped in exhaustion. "We've roped the trail back," Telamonias informed Dienekes. "The climbing's not bad."
"What about the Persian Immortals? The Ten Thousand."
"No sign when we left. But Leonidas is withdrawing the allies. They're all pulling out, everyone but the Spartans."
Polynikes set Alexandros gently down upon the matted grass within the grove. You could still smell the deer. I saw Dienekes feel for Alexandros' breath, then flatten his ear, listening, to the youth's chest. "Shut up!" he barked at the party. "Shut the fuck up!"
Dienekes pressed his ear tighter to the flat of Alexandros' sternum. Could he distinguish the sound of his own heart, hammering now in his chest, from that beat which he sought so desperately within the breast of his protege? Long moments passed. At last Dienekes straightened and sat up, his back seeming to bear the weight of every wound and every death across all his years.
He lifted the young man's head, tenderly, with a hand beneath the back of his neck. A cry of such grief as I had never heard tore from my master's breast. His back heaved; his shoulders shuddered. He lifted Alexandros' bloodless form into his embrace and held it, the young man's arms hanging limp as a doll's. Polynikes knelt at my master's side, draped a cloak about his shoulders and held him as he sobbed.
Never in battle or elsewhere had I, nor any of the men there present beneath the oaks, beheld Dienekes loose the reins of self-command with which he maintained so steadfast a hold upon his heart. You could see him summon now every reserve of will to draw himself back to the rigor of a Spartan and an officer. With an expulsion of breath that was not a sigh but something deeper, like the whistle of death the _daimon_ makes escaping within the avenue of the throat, he released Alexandros' life-fled form and settled it gently upon the scarlet cloak spread beneath it on the earth. With his right hand he clasped that of the youth who had been his charge and protege since the morn of his birth.
"You forgot about our hunt, Alexandros."
Eos, pallid dawn, bore now her light to the barren heavens without the thicket. Game trails and deer-trodden traces could be discerned. The eye began to make out the wild, torrent-cut slopes so like those of Therai on Taygetos, the oak groves and shaded runs that, it was certain, teemed with deer and boar and even, perhaps, a lion.
"We would have had such a grand hunt here next fall."
**THIRTY-FOUR**
**T** _he preceding pages were the last delivered to His Majesty prior to the burning of Athens._
_The Army of the Empire stood at that time, two hours prior to sunset, some six weeks after the victory at Thermopylae, drawn up on line within the western walls of the city of Athena. An incendiary brigade of 120,000 men there dressed at a double-arms interval and advanced across the capital, putting all temples and shrines, magistracies and public buildings, gymnasia, houses, factories, schools and warehouses to the torch._
_At that time the man Xeones, who had hitherto been recovering steadily from his wounds sustained at the battle for the Hot Gates, suffered a reverse. Clearly the witnessing of the immolation of Athens had distressed the man profoundly. In fever he inquired repeatedly after the fate of the seaport Phaleron wherein, he had told us, lay the temple of Persephone of the Veil, that sanctuary in which his cousin, the girl Diomache, had taken refuge. None could provide intelligence of the fate of this precinct. The captive began to fail further; the Royal Surgeon was summoned. It was determined that several punctures of the thoracic organs had reopened; internal bleeding had become severe._
_At this point His Majesty stood unavailable, being on station with the fleet, which was drawing up in preparation for imminent engagement with the navy of the Hellenes, expected to commence with the dawn. The morrow's fight, it was anticipated eagerly by His Majesty's admirals, would eliminate all resistance of the enemy at sea and leave the unconquered remainder of Greece, Sparta and the Peloponnese, helpless before the final assault of His Majesty's sea and land forces._
_I, His Majesty's historian, received at this hour orders summoning me to establish a secretaries' station to observe the sea battle at His Majesty's side and note, as they occurred, all actions of the Empire's officers deserving of valorous commendation. I was able, however, before repairing to_ _this post, to remain at the Greek's side for most of the evening. The night grew more apocalyptic with each hour. The smoke of the burning city rose thick and sulphurous across the plain; the flames from the Acropolis and the merchant and residential quarters lit the sky bright as noon. In addition, a violent quake had struck the coast, toppling numerous structures and even portions of the city walls. The atmosphere bordered upon the primordial, as if heaven and earth, as well as men, had harnessed themselves to the engines of war._
_The man Xeones remained lucid and calm throughout this interval. Intelligence requested by the captain Orontes had reached the medical pavilions to the effect that the priestesses of Persephone, presumably including the captive Xeones' cousin, had evacuated themselves to Troezen across the bay. This seemed to steady the man profoundly. He appeared convinced that he would not survive the night and was distressed only insomuch as this would cut short the telling of his tale. He wished, he said, to have recorded in what hours remained as much of the conclusion of the actual battle as he could dictate. He began at once, returning in memory to the site of the Hot Gates._
**T** he upper rim of the sun had just pierced the horizon when the party began the descent of the final cliff above the Hellenes' camp. Alexandros' and Lachides' bodies were lowered by rope, along with Suicide, whose wound in the groin had robbed him of the use of his lower limbs. Dienekes needed a rope too. We crabbed down backward. Over my shoulder I could see men packing up below, the Arkadians and Orchomenians and Mycenaeans. For a moment I thought I saw the Spartans moving out too. Could it be that Leonidas, acknowledging the futility of defense, had given the order for all to withdraw? Then my glance, instinctively turning to the man beside me on the face, met the eyes of Polynikes. He could read the wish for deliverance so transparent upon my features. He just grinned.
At the base of the Phokian Wall, what remained of the Spartans, barely above a hundred Peers yet able to fight, had already completed their gymnastics and had themselves in arms. They were dressing their hair, preparing to die.
We buried Alexandros and Lachides in the Spartan precinct beside the West Gate. Both their breastplates and helmets, Alexandros' and Lachides', were preserved aside for use; their shields Rooster and I had already stacked among the arms at the camp. No coin for the ferryman could be located among Alexandros' kit, nor did my master or I possess a surrogate. Somehow I had lost them all, that purse which the lady Arete had placed into my safekeeping upon the evening of that final county day in Lakedaemon.
"Here," Polynikes offered.
He held out, still folded in a wrap of oiled linen, the coin his wife had burnished for himself, a silver tetradrachm minted by the citizens of Elis in his honor, to commemorate his second victory at Olympia. Upon one face was stamped the image of Zeus Lord of the Thunder, with winged Nike above his right shoulder. The obverse bore a crescent of wild olive in which was centered the club and lionskin of Herakles, in honor of Sparta and Lakedaemon.
Polynikes set the coin in place himself. He had to prise Alexandros' jaws apart, on the side opposite the "boxer's lunch" of amber and euphorbia which with steadfast loyalty yet held the fractured bone immobilized. Dienekes chanted the Prayer for the Fallen; he and Polynikes slid the body, wrapped in its scarlet cloak, into the shallow trench. It took no time to cover it with dirt. Both Spartans stood.
"He was the best of us all," Polynikes said.
Lookouts were hastening in from the western peak. The Ten Thousand had been spotted; they had completed their all-night encirclement and stood now in full force six miles in the Hellenes' rear. They had already routed the Phokian defenders on the summit. The Greeks at the Gates had perhaps three hours before the Persian Immortals could complete the descent and be in position to attack.
Other messengers were arriving from the Trachis side. His Majesty's lookout throne, as the raiding party had observed last night, had been dismantled. Xerxes upon his royal chariot was advancing in person, with fresh myriads at his back, to resume the assault on the Hellenes from the fore.
The burial ground stood a considerable distance, above half a mile, from the Spartan assembly point by the Wall. As my master and Polynikes returned, the contingents of the allies tramped past, withdrawing to safety. True to his word, Leonidas had released them, all save the Spartans.
We watched the allies as they passed. First came the Mantineans, in nothing resembling order; they seemed to slouch as if all strength had left their knees and hams. No one spoke. The men were so filthy they looked like they were made of dirt. Grit caked every pore and cavity of flesh, including the creases at the pockets of their eyes and the glue of sputum that collected uncleared in the corners of their mouths. Their teeth were black; they spit, it seemed, with every fourth step and the gobs landed black upon the black earth. Some had stuck their helmets upon their heads, cocked back without thought, as if their skulls were just convenient knobs to hang the bowls upon. Most had slung theirs, nasal foremost, across the bundles of their rolled cloaks which they bore as packs across their shoulders against the biting gripcords of their slung shields. Though the dawn was still chill, the men trudged in sweat. I never saw soldiers so exhausted.
The Corinthians came next, then the Tegeates and the Opountian Lokrians, the Philiasians and the Orchomenians, intermixed with the other Arkadians and what was left of the Mycenaeans. Of eighty original hoplites of that city, eleven remained yet able to walk, with another two dozen borne prone upon litters or strapped to pole-drags drawn by the pack animals. Man leaned upon man and beast upon beast. You could not tell the concussed and the skull-fractured, those who no longer possessed the sense of who or where they were, from their fellows stricken to numbness by the horrors and strain of the past six days. Nearly every man had sustained multiple wounds, most in the legs and head; a number had been blinded; these shuffled at the sides of their brothers, hands tucked in the crook of a friend's elbow, or else trailed alongside the baggage animals, holding the end of a tether attached to the pack frame.
Past the avenues of the fallen trudged the spared, each bearing himself neither with shame nor guilt, but with that silent awe and thanksgiving of which Leonidas had spoken in the assembly following the battle at Antirhion. That these warriors yet drew breath was not their own doing and they knew it; they were no more nor less brave or virtuous than their fallen fellows, just luckier. This knowledge expressed itself with a poet's eloquence in the blank and sanctified weariness inscribed upon their features.
"I hope we don't look as bad as you," Dienekes grunted to a captain of the Philiasians as he passed.
"You look worse, brothers."
Someone had set the bathhouses and the spa compound on fire. The air had stilled and the wet wood burned with acrid sullenness. The smoke and stink of these blazes now added their cheerless component to the already baleful scene. The column of warriors emerged out of smoke and sank again within it. Men threw the rags of their discarded kit, blood-begrimed cloaks and tunics, used-up packs and gear bundles; everything that would burn was flung willy-nilly upon the flame. It was as if the allies withdrawing intended to abandon not so much as a scrap to the enemy's use. They lightened their loads and marched out.
Men held out their hands to the Spartans as they strode by, touching palm to palm, fingers to fingers. A warrior of the Corinthians gave Polynikes his spear. Another handed Dienekes his sword. "Give them hell, fuckers."
Passing the spring, we came upon Rooster. He was pulling out too. Dienekes drew up and stopped to take his hand. No shame stood upon Rooster's face. Clearly he felt he had discharged his duty and more, and the liberty with which Leonidas had gifted him was in his eyes no more than his birthright, which had been denied him all his life and now, long overdue, had been fairly and honorably won by his own hand. He clasped Dienekes' hand and promised to speak with Agathe and Paraleia when he reached Lakedaemon. He would inform them of the valor with which Alexandros and Olympieus had fought and with what honor they had fallen. Rooster would make report to the lady Arete too. "If I may," he requested, "I would like to honor Alexandros before I go."
Dienekes thanked him and told him where the grave lay. To my surprise, Polynikes took Rooster's hand too. "The gods love a bastard," he said.
Rooster informed us that Leonidas had freed with honor all the helots of the battle train. We could see a group of a dozen now, passing out among the warriors of Tegea. "Leonidas has released the squires as well," Rooster declared, "and all the foreigners who serve the army." He addressed my master. "That means Suicide—and Xeo too."
Behind Rooster the train of allied contingents continued their march-out.
"Will you hold him now, Dienekes?" Rooster asked.
He meant me.
My master did not look in my direction but spoke in reply toward Rooster. "I have never compelled Xeo's service. Nor do I now."
He drew up and turned to me. The sun had fully risen; east, by the Wall, the trumpets were sounding. "One of us," he said, "should crawl out of this hole alive." He ordered me to depart with Rooster.
I refused.
"You have a wife and children!" Rooster seized my shoulders, gesturing with passion to Dienekes and Polynikes. "Theirs is not your city. You owe it nothing."
I told him the decision had been made years ago.
"You see?" Dienekes addressed Rooster, indicating me. "He never had good sense."
Back at the Wall we saw Dithyrambos. His Thespaians had refused Leonidas' order. To a man they disdained to withdraw, but insisted upon abiding and dying with the Spartans. There were about two hundred of them. Not a man among their squires would pull out either. Fully four score of the freed Spartan squires and helots stood fast as well. The seer Megistias had likewise scorned to retreat. Of the original three hundred Peers, all were present or dead save two. Aristodemos, who had served as envoy at Athens and Rhodes, and Eurytus, a champion wrestler, had both been stricken with an inflammation of the eyes that rendered them sightless. They had been evacuated to Alpenoi. The _katalogos,_ the muster roll, of survivors marshaling at the Wall numbered just above five hundred.
As for Suicide, my master before departing to bury Alexandros had commanded him to remain here at the Wall, upon a litter. Dienekes apparently had anticipated the squires' release; he had left orders for Suicide to be borne off with their column to safety. Now here the Scythian stood, on his feet, grinning ghoulishly as his master returned, himself armored in corselet and breastplate with his loins cinched in linen and bound with leather straps from a pack mount. "I can't shit," he pronounced, "but by hell's flame, I can still fight."
The ensuing hour was consumed with the commanders reconfiguring the contingent into a front of sufficient breadth and depth, remarshaling the disparate elements into units and assigning officers. Among the Spartans, those squires and helots remaining were simply absorbed into the platoons of the Peers they served. They would fight no longer as auxiliaries but take their places in bronze within the phalanx. There was no shortage of armor, only of weapons, so many had been shivered or smashed in the preceding forty-eight hours. Two dumps of spares were established, one at the Wall and the second a furlong to the rear, halfway to a small partially fortified hillock, the most natural site for a beleaguered force to rally upon and make its last stand. These dumps were nothing grand—just swords stuck blade-first into the dirt and eight-footers jammed beside them, lizard-stickers down.
Leonidas summoned the men to assembly. This was done without so much as a shout, so few yet stood upon the site. The camp itself seemed suddenly broad and capacious. As for the dance floor before the Wall, its sundered turf lay yet littered with Persian corpses by the thousand as the enemy had left the second day's casualties to rot upon the field. Those wounded who had survived the night now groaned with their last strength, crying for aid and water, and many for the merciful stroke of extinction. For the allies the prospect of fighting again, out there upon that farmer's field of hell, seemed more than thought could bear.
This, too, was Leonidas' decision. It had been agreed among the commanders, the king now informed the warriors, no longer to fight in sallies from behind the Wall as in the previous two days but instead to put its stones at the defenders' backs and advance in a body into the widest part of the pass, there to engage the enemy, the allied scores against the Empire's myriads. The king's intent was that each man sell his life as dearly as possible.
Just as order of battle was being assigned, a herald's trumpet of the enemy sounded from beyond the Narrows. Under a banner of parley a party of four Persian riders in their most brilliant armor picked their way across the carpet of carnage and reined in directly beneath the Wall. Leonidas had been wounded in both legs and could barely hobble. With painful effort he mounted the battlement; the troops climbed with him; the whole force, what there was of it, looked down on the horsemen from atop the Wall.
The envoy was Ptammitechus, the Egyptian marine Tommie. This time his young son did not accompany him as interpreter; that function was performed by an officer of the Persians. Both their mounts, and the two heralds', were balking violently amid the underfoot corpses. Before Tommie could commence his speech, Leonidas cut him off.
"The answer is no," he called down from the Wall.
"You haven't heard the offer."
"Fuck the offer," Leonidas cried with a grin. "And yourself, sir, along with it!"
The Egyptian laughed, his smile flashing as brilliantly as ever. He strained against the reins of his spooking horse. "Xerxes does not want your lives, sir," Tommie called. "Only your arms."
Leonidas laughed. "Tell him to come and get them."
With a wheel-about, the king terminated the interview. Despite his carved-up legs he disdained help dismounting the Wall. He whistled up the assembly. Atop the stones the Spartans and Thespaians watched the Persian envoys rein their mounts about and withdraw.
Behind the Wall, Leonidas again took station before the assembly. The triceps muscle in his left arm had been severed; he would fight today with his shield strapped with leather across his shoulder. The Spartan king's demeanor nonetheless could only be described as cheerful. His eyes shone and his voice carried easily with force and command.
"Why do we remain in this place? A man would have to be cracked not to ask that question. Is it for glory? If it were for that alone, believe me, brothers, I'd be the first to wheel my ass to the foe and trot like hell over that hill."
Laughter greeted this from the king. He let the swell subside, raising his good arm for silence.
"If we had withdrawn from these Gates today, brothers, no matter what prodigies of valor we had performed up till now, this battle would have been perceived as a defeat. A defeat which would have confirmed for all Greece that which the enemy most wishes her to believe: the futility of resistance to the Persian and his millions. If we had saved our skins today, one by one the separate cities would have caved in behind us, until the whole of Hellas had fallen."
The men listened soberly, knowing the king's assessment accurately reflected reality.
"But by our deaths here with honor, in the face of these insuperable odds, we transform vanquishment into victory. With our lives we sow courage in the hearts of our allies and the brothers of our armies left behind. They are the ones who will ultimately produce victory, not us. It was never in the stars for us. Our role today is what we all knew it was when we embraced our wives and children and turned our feet upon the march-out: to stand and die. That we have sworn and that we will perform."
The king's belly grumbled, loudly, of hunger; from the front ranks laughter broke the assembly's sober mien and rippled to the rear. Leonidas motioned with a grin to the squires preparing bread, urging them to snap it up.
"Our allied brothers are on the road to home now." The king gestured down the track, the road that ran to southern Greece and safety. "We must cover their withdrawal; otherwise the enemy's cavalry will roll unimpeded through these Gates and ride our comrades down before they've gotten ten miles. If we can hold a few hours more, our brothers will be safe."
He inquired if any among the assembly wished also to speak.
Alpheus stepped to the fore. "I'm hungry too so I'll keep it short." He drew up shyly, in the unwonted role of speaker. I realized for the first time that his brother, Maron, stood nowhere among the ranks. This hero had died during the night, I heard a man whisper, of wounds sustained the previous day.
Alpheus spoke quickly, unblessed by the orator's gift but graced simply with the sincerity of his heart. "In one way only have the gods permitted mortals to surpass them. Man may give that which the gods cannot, all he possesses, his life. My own I set down with joy, for you, friends, who have become the brother I no longer possess."
He turned abruptly and melted back into the ranks.
The men began calling for Dithyrambos. The Thespaian stepped forth with his usual profane glint. He gestured toward the pass beyond the Narrows, where the advance parties of the Persians had arrived and begun staking out the marshaling salients for the army. "Just go out there," he proclaimed, "and have fun!"
Dark laughter cut the assembly. Several others of the Thespaians spoke. They were more curt than the Spartans. When they finished, Polynikes stepped to the front.
"It is no hard thing for a man raised under the laws of Lykurgus to offer up his life for his country. For me and for these Spartans, all of whom have living sons, and who have known since boyhood that this was the end they were called to, it is an act of completion before the gods."
He turned solemnly toward the Thespaians and the freed squires and helots.
"But for you, brothers and friends...for you who will this day see all extinguished forever..."
The runner's voice cracked and broke. He choked and blew snot into his hand in lieu of the tears to whose issue his will refused to permit. For long moments he could not summon speech. He motioned for his shield; it was passed to him. He displayed it aloft.
"This _aspis_ was my father's and his father's before him. I have sworn before God to die before another man took this from my hand."
He crossed to the ranks of the Thespaians, to a man, an obscure warrior among them. Into the fellow's grasp he placed the shield.
The man accepted it, moved profoundly, and presented his own to Polynikes. Another followed, and another, until twenty, thirty shields had traded hands. Others exchanged armor and helmets with the freed squires and helots. The black cloaks of the Thespaians and the scarlet of the Lakedaemonians intermingled until all distinction between the nations had been effaced.
The men called for Dienekes. They wanted a quip, a wisecrack, something short and pithy as he was known for. He resisted. You could see he did not wish to speak.
"Brothers, I'm not a king or a general. I've never held rank beyond that of a platoon commander. So I say to you now only what I would say to my own men, knowing the fear that stands unspoken in each heart—not of death, but worse, of faltering or failing, of somehow proving unworthy in this, the ultimate hour."
These words had struck the mark; one could read it plain on the faces of the silent, raptly attending men.
"Here is what you do, friends. Forget country. Forget king. Forget wife and children and freedom. Forget every concept, however noble, that you imagine you fight for here today. Act for this alone: for the man who stands at your shoulder. He is everything, and everything is contained within him. That's all I know. That's all I can tell you."
He finished and stepped back. At the rear of the assembly a commotion was heard. The ranks rustled; into view emerged the Spartan Eurytus. This was the man, stricken with field blindness, who had been evacuated to Alpenoi village, along with the envoy Aristodemos, felled by this same inflammation. Now Eurytus returned, sightless, yet armed and in armor, led by his squire. Without a word he steered himself into place among the ranks.
The men, whose courage had already been high, felt this now refire and redouble.
Leonidas stepped forth now and reassumed the _skeptron_ of command. He proposed that the Thespaian captains take these final moments to commune in private with their own countrymen, while he spoke apart for the Spartans alone.
The men of the two cities divided, each to its own. There remained just over two hundred Peers and freedmen of Lakedaemon. These assembled, without regard to rank or station, compactly about their king. All knew Leonidas would address appeals to nothing so grand as liberty or law or the preservation of Hellas from the tyrant's yoke.
Instead he spoke, in words few and plain, of the valley of the Eurotas, of Parnon and Taygetos and the cluster of five unwalled villages which alone comprise that _polis_ and commonwealth which the world calls Sparta. A thousand years from now, Leonidas declared, two thousand, three thousand years hence, men a hundred generations yet unborn may for their private purposes make journey to our country.
"They will come, scholars perhaps, or travelers from beyond the sea, prompted by curiosity regarding the past or appetite for knowledge of the ancients. They will peer out across our plain and probe among the stone and rubble of our nation. What will they learn of us? Their shovels will unearth neither brilliant palaces nor temples; their picks will prise forth no everlasting architecture or art. What will remain of the Spartans? Not monuments of marble or bronze, but this, what we do here today."
Out beyond the Narrows the enemy trumpets sounded. Clearly now could be seen the vanguard of the Persians and the chariots and armored convoys of their King.
"Now eat a good breakfast, men. For we'll all be sharing dinner in hell."
**THIRTY-FIVE**
**H** is Majesty witnessed at close range, with His own eyes, the magnificent valor demonstrated by the Spartans, Thespaians and their emancipated squires and servants upon this, the final morning of defense of the pass. He has no need of my recounting the events of this battle. I will report only those instances and moments which may have escaped the notice of His Majesty's vantage, again, as he has requested, to shed light upon the character of the Hellenes he there called his enemy.
Foremost among all, and indisputable in claim to preeminence, may be only one man, the Spartan king, Leonidas. As His Majesty knows, the main force of the Persian army, advancing as it had on the previous two days along the track from Trachis, did not commence its assault until long after the sun was fully up. The hour of attack in fact was closer to midday than morning and came while the Ten Thousand Immortals had not yet made their appearance in the allied rear. Such was Leonidas' disdain for death that he actually slept for most of this interval. Snoozed might be a more apt description, so free from care was the posture the king assumed upon the earth, cushioned upon his cloak as a ground cloth, legs crossed at the ankles, arms folded across his breast, his eyes shaded by a straw sun hat and his head pillowed insouciantly upon the bowl of his shield. He might have been a boy, herding goats in some sleepy summer dale.
Of what does the nature of kingship consist? What are its qualities in itself; what the qualities it inspires in those who attend it? These, if one may presume to divine the meditations of His Majesty's heart, are the questions which most preoccupy his own reason and reflection.
Does His Majesty recall that moment, upon the slope beyond the Narrows, after Leonidas had fallen, struck through with half a dozen lances, blinded beneath his helmet staved in from the blow of a battle-axe, his left arm useless with its splintered shield lashed to his shoulder, when he fell at last under the crush of the enemy? Can His Majesty recall that surge within the melee of slaughter when a corps of Spartans hurled themselves into the teeth of the vaunting foe and flung them back, to retrieve the corpse of their king? I refer neither to the first time nor the second or third, but the fourth, when there stood fewer than a hundred of them, Peers and Knights and freedmen, dueling an enemy massed in their thousands.
I will tell His Majesty what a king is. A king does not abide within his tent while his men bleed and die upon the field. A king does not dine while his men go hungry, nor sleep when they stand at watch upon the wall. A king does not command his men's loyalty through fear nor purchase it with gold; he earns their love by the sweat of his own back and the pains he endures for their sake. That which comprises the harshest burden, a king lifts first and sets down last. A king does not require service of those he leads but provides it to them. He serves them, not they him.
In the final moments before the actual commencement of battle, when the lines of the Persians and Medes and Sacae, the Bactrians and Illyrians, Egyptians and Macedonians, lay so close across from the defenders that their individual faces could be seen, Leonidas moved along the Spartan and Thespaian foreranks, speaking with each platoon commander individually. When he stopped beside Dienekes, I was close enough to hear his words.
"Do you hate them, Dienekes?" the king asked in the tone of a comrade, unhurried, conversational, gesturing to those captains and officers of the Persians proximately visible across the _oudenos chorion,_ the no-man's-land.
Dienekes answered at once that he did not. "I see faces of gentle and noble bearing. More than a few, I think, whom one would welcome with a clap and a laugh to any table of friends."
Leonidas clearly approved my master's answer. His eyes seemed, however, darkened with sorrow.
"I am sorry for them," he avowed, indicating the valiant foemen who stood so proximately across. "What wouldn't they give, the noblest among them, to stand here with us now?"
That is a king, Your Majesty. A king does not expend his substance to enslave men, but by his conduct and example makes them free. His Majesty may ask, as Rooster did, and the lady Arete, why one such as I whose station could most grandly be called service and most meanly slavery, why one of such condition would die for those not of his kin and country. The answer is, they were my kin and country. I set down my life with gladness, and would do it again a hundred times, for Leonidas, for Dienekes and Alexandros and Polynikes, for Rooster and Suicide, for Arete and Diomache, Bruxieus and my own mother and father, my wife and children. I and every man there were never more free than when we gave freely obedience to those harsh laws which take life and give it back again.
Those events of the actual battle I count as nothing, for the fight was over in its profoundest sense before it began. I had slept, sitting upright against the Wall, following Leonidas' example, while we waited that hour and the hour after and the hour after that for His Majesty's army to make its move.
In my doze I discovered myself again among the hills above the city of my childhood. I was no longer a boy but myself of grown years. My cousin was there, in years still a girl, and our dogs, Lucky and Happy, exactly as they had been in the days following the sack of Astakos. Diomache had given chase to a hare and was climbing, bare-legged with extraordinary swiftness, a slope which seemed to ascend to the heavens. Bruxieus waited atop, as did my mother and father: I knew, though I could not see them. I gave chase too, seeking to overtake Diomache with all my grown strength. I could not. However swiftly I mounted, she remained ever elusive, always an interval ahead, calling to me gaily, teasingly, that I would never run fast enough to catch her.
I came awake with a start. There awaited the massed Persians, less than a bowshot away.
Leonidas stood upon his feet, out front. Dienekes as always took his stance before his platoon, which was drawn up at seven-and-three, wider and shallower than on either previous day. My place was third in the second file, for the first time in my life without my bow but clutching instead in my right hand the heavy haft of the eight-footer which had last been Doreion's. Around my left forearm, braced tight against the elbow, stood wrapped the linen-cushioned bronze sleeve bolted through the oak and the bronze facing of the _aspis_ which had been Alexandros'. The helmet I wore had belonged to Lachides and the cap beneath had been that of Ariston's squire, Demades.
"Eyes on me!" Dienekes barked, and the men as always tore their glance from the enemy, who marshaled now so near across the interval that we could see the irises beneath their lashes and the gaps between their teeth. There were ungodly numbers of them. My lungs howled for air; I could feel the blood pounding within my temples and read its pulse upon the vessels of the eyes. My limbs were stone; I could feel neither hands nor feet. I prayed with every fiber, simply for the courage not to faint. Suicide stood upon my left. Dienekes stood before.
At last came the fight, which was like a tide, and within which one felt as a wave beneath the storming whims of the gods, waiting for their fancy to prescribe the hour of his extinction. Time collapsed. Elements blurred and merged. I remember one surge carrying the Spartans forward, driving the enemy by the score into the sea, and another which propelled the phalanx rearward like boats lashed gunwale-to-gunwale driven before the irresistible storm. I recall my feet, planted solid with all my strength upon the earth slick with blood and urine, as they were driven rearward, in place before the push of the foe, like the fleece-wrapped soles of a boy playing upon the mountain ice.
I saw Alpheus take on a Persian chariot single-handedly, slaying general, henchman and both flank guardsmen. When he fell, shot through the throat by a Persian arrow, Dienekes dragged him out. He got up, still fighting. I saw Polynikes and Derkylides hauling Leonidas' corpse, each with a weaponless hand upon the shoulders of the king's shattered corselet, striking at the foe with their shields as they drew back. The Spartans re-formed and rushed, fell back and broke, then re-formed again. I killed a man of the Egyptians with the butt-spike of my shivered spear as he drove his own into the wall of my guts, then an instant later, falling under the blow of an axe, clawed free over a Spartan corpse, only to recognize, beneath its hacked-open helmet, the shattered face of Alpheus.
Suicide hauled me from the fray. At last the Ten Thousand Immortals could be seen, advancing in line of battle to complete their envelopment. What remained of the Spartans and Thespaians fell rearward from the plain, to the Narrows, pouring through the sallyports of the Wall toward the final hillock.
The allies were so few now, and their weapons so spent and broken, that the Persians made bold to attack with cavalry, as they would in a rout. Suicide fell. His right foot had been chopped off. "Put me on your back!" he commanded. I knew without more words what he meant. I could hear arrows and even javelins thrumming into his yet-living flesh, shielding me as I bore him.
I saw Dienekes yet alive, slinging away one shattered _xiphos_ and poring through the dirt for another. Polynikes churned past me, carrying Telamonias hobbled beside him. Half the runner's face had been sheared off; blood gushed in sheets from the opened bone of his cheek. "The dump!" he was calling, meaning that magazine of weapons which Leonidas had ordered placed in reserve behind the Wall. I felt the tissue of my belly tear and the intestines begin to spill. Suicide hung life-spent upon my back. I turned rearward toward the Narrows. Persian and Median archers in their thousands hailed bronzeheads down upon the retreating Spartans and Thespaians. Those who reached the dump were shredded like pennants in a gale.
The defenders staggered toward the knoll upon which the last stack of weapons had been cached. No more than sixty remained; Derkylides, astonishingly unwounded, rallied the survivors into a circular front. I found a strap and cinched my guts in. I was struck, for just a moment, with the impossible beauty of the day. For once no haze obscured the channel; one could make out individual stones upon the hills across the strait and track the game trails up the slopes, turn by turn.
I saw Dienekes reel beneath the blow of an axe, but had not myself the strength to rally to him. Medes and Persians, Bactrians and Sacae, were not merely pouring over the Wall but dismantling it with a frenzy. I could see horses beyond. The officers of the foe no longer required whips to drive their men forward. Over the broken stones of the Wall thundered the horsemen of His Majesty's cavalry, followed by the bucking chariots of his generals.
The Immortals marshaled round about the hillock now, pouring bowfire point-blank into the Spartans and Thespaians crouched beneath the slender shelter of their shattered and staved-in shields. Derkylides led the rush upon them. I saw him fall, and Dienekes, fighting beside him. Neither had shields, nor for all that could be seen, weapons of any kind. They went down not like heroes of Homer crashing clamorously within the carapaces of their armor, but like commanders completing their last and dirtiest job.
The enemy stood, invincible in the might of their missile fire, but somehow the Spartans reached them. They fought without shields, with only swords and then bare hands and teeth. Polynikes went after an officer. The runner still had his legs. So swiftly did he cross the space at the base of the hillock that his hands found the foe's throat even as a storm of Persian steel tore his back apart.
The last few dozen upon the hillock, rallied now by Dithyrambos, both of whose arms had been shredded by enemy fire and hung now useless at his sides, pincushioned with bolts, sought to form a front for a final rush. Chariots and Persian horsemen stampeded pell-mell into the Spartans. A battle waggon, afire, rolled over both my legs. Before the defenders, completely encircling the hillock, the Immortals had formed now in bowmen's ranks. Their bolts thundered upon the last unarmed and shattered warriors. From their rear, more archers hurled volleys over the heads of their comrades to rain upon the last survivors among the Hellenes. Backs and bellies bristled with the fletched spines of arrow butts; shot-to-pieces men sprawled in rag piles of bronze and scarlet.
The ear could hear His Majesty bawling orders, so near at hand ranged he upon his chariot. Was he calling in his foreign tongue for his men to cease fire, to capture the final defenders alive? Were those to whom he cried the marines of Egypt, under their captain, Ptammitechus, who spurned their monarch's order and rushed in to gift what Spartans and Thespaians they could reach with the final boon of death? It was impossible to see or hear within the tumult. The marines parted toward the flanks. The fury of the Persian archers redoubled as they sought with the numberless shafts of their fusillades to extinguish at last the stubborn foe who had made them pay so dearly for this mean measure of dirt.
As when a hailstorm descends unseasonably from the mountains and hurls from the sky its icy pellets upon the husbandman's newly sprouted crop, so did the bolts of the Persians in their myriads thunder down upon the Spartans and Thespaians. Now the farmer assumes his anxious station in the doorway, hearing the deluge upon the tiles of the roof, watching its bullets of ice clatter and rebound upon the stones of the walk. How fare the sprouts of spring barley? One here and there survives, as if by miracle, and holds yet its head aloft. But the planter knows this state of clemency cannot endure. He turns his face away, in obedience to the laws of God, while without, beneath the storm, the final shaft breaks and falls, overwhelmed by the insuperable onslaught of heaven.
**THIRTY-SIX**
**S** _uch was the end of Leonidas and the defenders of the pass at Thermopylae, as related by the Greek Xeones and compiled in transcription by His Majesty's historian Gobartes the son of Artabazos and completed the fourth day of Arahsamnu, Year Five of His Majesty's Accession._
_This date, in the bitter irony of God Ahura Mazda, was the same upon which the naval forces of the Persian Empire suffered the calamitous defeat at the hands of the Hellenic fleet, in the Straits of Salamis, off Athens, that catastrophe which sent to their deaths so many valiant sons of the East and, by its consequences for the supply and support of the army, doomed the entire campaign to disaster._
_That oracle of Apollo delivered earlier to the Athenians, which declared,_
"The wooden wall alone shall not fail you,"
_had revealed its fateful truth, the timbered stronghold manifesting itself not as that ancient palisade of the Athenian Acropolis so speedily overrun by His Majesty's forces, but as a wall of ships' hulls and the sailors and marines of Hellas who manned them so superbly, dealing the death blow to His Majesty's ambitions of conquest._
_The magnitude of the calamity effaced all consideration of the captive Xeones and his tale. Care of the man himself was forsaken amid the chaos of defeat, as every physician and tender of the Royal Surgeon's staff made haste to the shore opposite Salamis, there to minister to the myriad wounded of the imperial armada, washed up amid the charred and splintered wreckage of their vessels of war._
_When darkness at last brought surcease from the slaughter, a greater terror seized the Empire's camp. This was of the wrath of His Majesty. So many officers of the court were being put to the sword, or so my notes_ _recall, that the Historian's staff cried quits to the task of recording their names._
_Terror overran the pavilions of His Majesty, heightened not only by the great quake which shook the city precisely at the hour of sunset but also by the apocalyptic aspect of the siting of the army's bivouac, there within the razed and still-smoldering city of the Athenians. Midway through the second watch the general Mardonius sealed His Majesty's chamber and debarred entry to any further officers. His Majesty's Historian was able to procure only the scantiest of instructions as to the disposition of the day's records. Upon dismissal, I inquired purely as an afterthought for orders concerning the Greek Xeones and his papers._
_"Kill him," the general Mardonius replied without hesitation, "and burn every page of that compilation of falsehoods, whose recording has been folly from the first and the merest mention of which at this hour will serve only to drive His Majesty into further paroxysms of rage."_
_Other duties held me several hours. These at length completed, I proceeded in search of Orontes, captain of the Immortals, whose responsibility it must be to carry out these orders of Mardonius. I located the officer upon the shore. He was clearly in a state of exhaustion, overwrought both with the grief of the day's defeat and with his own frustration as a soldier at being unable, other than by pulling dying sailors out of the water, to aid the valiant mariners of the fleet. Orontes composed himself at once, however, and turned his attention to the matter at hand._
_"If you'd like to find your head still upon your shoulders tomorrow," the captain declared when he had been informed of the general's order, "you will pretend you never heard or saw Mardonius."_
_I protested that the order had been issued in the name of His Majesty. It could not be ignored._
_"It can't, can't it? And what will be the general's story tomorrow or a month hence, after his order has been carried out, when His Majesty sends for you and asks to see the Greek and his notes?_
_"I will tell you what will happen," the captain continued. "Even now in His Majesty's chambers, His chancellors and ministers press upon Him the necessity to withdraw the Royal Person, to take ship for Asia, as Mardonius has urged before. This time I think His Majesty will take heed."_
_Orontes declared his conviction that His Majesty would order the bulk of the army to remain in Hellas, under command of the general Mardonius and charged to complete the conquest of Greece in His name. That task accomplished, His Majesty will possess His victory. Today's calamity will be forgotten in that bright glow of triumph._
_"Then in the delectation of conquest," Orontes continued, "His Majesty will call for these notes of the Greek Xeones, as a sweetcake to cap the banquet of victory. If you and I stand before Him empty-handed, which of us will point the finger at Mardonius, and who will believe our declarations of innocence?"_
_I asked then what we must do._
_Orontes' heart was clearly torn. Memory recalled that he, as chief field commander of the Immortals beneath Hydarnes, their general, had strode in the van during the Ten Thousand's night envelopment of the Spartans and Thespaians at Thermopylae and had served with extraordinary valor in the final morning's assault, facing the Spartans hand-to-hand and securing for His Majesty the conquest of the foe. Orontes' own arrows had been among those fatal shafts flung at close range into the final defenders, perhaps into the flesh of the very men whose histories had been recounted within the captive Xeones' tale._
_This knowledge, one could not help but read upon the captain's countenance, increased further his reluctance to deal harm to this man with whom he so clearly identified as a fellow soldier and even, one must at this point state, a friend._
_All this notwithstanding, Orontes summoned himself to duty. He dispatched two officers of the Immortals with orders to remove the Greek from the Surgeon's tent and deliver him at once to the staff pavilion of the Immortals. After several hours attending to other more urgent business, he and I proceeded to that site. We walked in together. The man Xeones sat up conscious upon his litter, though drawn and gravely enfeebled._
_Clearly he divined our purpose. His aspect was one of good cheer. "Come, gentlemen," he spoke before I or Orontes could give voice to our mission. "How may I assist you in your task?" His dispatch would not require the blade, he adjudged. "For the stroke of a feather, I feel, will be blow enough to finish the job."_
_Orontes inquired of the Greek Xeones if he grasped fully the magnitude_ _of the victory his countrymen's navy had achieved this day. The man affirmed that he did. He expressed the opinion, however, that the war was far from over. The issue remained very much in doubt._
_Orontes imparted his extreme reluctance to carry out the sentence of execution. In light of the present disorder within the Empire's camp, he declared it a matter of slender difficulty to spirit the fellow out unobserved. Did the man Xeones, Orontes inquired, possess friends or compatriots yet within Attika to whom we could deliver him? The captive smiled. "Your army has done an admirable job of driving any such off," he observed. "And besides, His Majesty will need all His men to bear more important baggage."_
_Yet did Orontes seek any excuse to postpone the moment of execution. "Since you ask no favor of us, sir," the captain addressed the prisoner, "may I request one of you?"_
_The man replied that he would gladly grant all that remained within his power._
_"You have cheated us, my friend," Orontes declared with wry expression. "Deprived us of a tale of which your master, the Spartan Dienekes, so you said, promised to speak. This was around the fire during that last hunt of which you spoke, when he and Alexandros and Ariston addressed the subject of fear. Do you remember? Your master cut off the youths' discourse with a pledge that, when they reached the Hot Gates, he would tell them a tale of Leonidas and the lady Paraleia, on the subject of courage and of what criteria the Spartan king employed to select the Three Hundred. Or did Dienekes in fact fail to speak of this?"_
_No, the captive Xeones confirmed, his master had found occasion and did in truth impart this tale. But, the prisoner asked, absent His Majesty's imperative to continue documenting the events of this narrative, did the captain indeed wish to keep on?_
_"We whom you call foe are flesh and blood," Orontes replied, "with hearts no less capable of attachment than your own. Does it strike you as implausible that we in this tent, His Majesty's historian and myself, have come to care for you, sir, not alone as a captive relaying an account of battle but as a man and even a friend?"_
_Orontes requested, as a kindness to himself who had followed with keen interest and empathy the antecedent chapters of the Greek's tale, if the man_ _would, as a comrade and so far as his strength permitted, relate to us now this final portion._
_"What had the Spartan king to say of women's courage, and how did your master, Dienekes, in fact relate it to his young friends and proteges?"_
_The man Xeones propped himself with effort, and assistance from myself and the officers, upright upon his settle. Summoning his strength, he drew a breath and resumed:_
**I** will impart this tale to you, my friends, as my master related it to me and to Alexandros and Ariston at the Hot Gates—not in his own voice, but in that of the lady Paraleia, Alexandros' mother, who recounted it in her own words to Dienekes and the lady Arete, only hours after its occurrence.
The time of the conveying of the lady Paraleia's tale was an evening three or four days before the march-out from Lakedaemon to the Hot Gates. The lady Paraleia had betaken herself for this purpose to the home of Dienekes and Arete, bringing with her several other women, all mothers and wives of warriors selected for the Three Hundred. None of the women knew what the lady wished to say. My master stood on the moment of excusing himself, that the ladies may have their privacy. Paraleia, however, requested that he remain. He must hear this too, she said. The ladies seated themselves about Paraleia. She began:
"What I tell you now, Dienekes, you must not repeat to my son. Not until you reach the Hot Gates, and not then, until the proper moment. That hour may be, if the gods so ordain, that of your own death or his. You will know it when it comes. Now attend closely, Dienekes, and you, ladies.
"This forenoon I received a summons from the king. I went at once, presenting myself within the courtyard of his home. I was early; Leonidas had not arrived from his business of organizing the march-out. His queen, Gorgo, however, awaited upon a bench in the shade of a plane tree, apparently intentionally. She welcomed me and bade me sit. We were alone, absent all servants and attendants.
"'You are wondering, Paraleia,' she began, 'why my husband has sent for you. I will tell you. He wishes to address your heart, and what he imagines must be your feelings of injustice at being singled out, so to speak, to bear a double grief. He is keenly aware that in selecting for the Three Hundred both Olympieus and Alexandros he has robbed you twice, of son as well as husband, leaving only the babe Olympieus to carry on your line. He will speak to this when he comes. But first, I must confide in you from my own heart, woman-to-woman.'
"She is quite young, our queen, and looked tall and lovely, though in that shadowed light exceedingly grave.
"'I have been daughter of one king and now wife to another,' Gorgo said. 'Women envy my station but few grasp its stern obligations. A queen may not be a woman as others. She may not possess her husband or children as other wives and mothers, but may hold them only in stewardship to her nation. She serves them, the hearts of her countrymen, not her own or her family's. Now you too, Paraleia, are summoned to this stern sisterhood. You must take your place at my shoulder in sorrow. This is women's trial and triumph, ordained by God: to abide with pain, to endure grief, to bear up beneath sorrow's yoke and thus to endow others with courage.'
"Hearing these words of the queen, I confess to you, Dienekes, and you, ladies, that my hands trembled so that I feared I may not command them—not alone with the foreknowledge of grief but of rage as well, blind bitter fury at Leonidas and the heartlessness with which he decanted the double measure of sorrow into my cup. Why me? my heart cried in anger. I stood upon the moment of giving voice to this outrage when the sound of the gate opening came from the outer court, and in a moment Leonidas himself entered. He had just come from the marshaling ground and bore his dusty footgear in his hand. Perceiving his lady and myself in intimate converse, he divined at once the subject of our intercourse.
"With apology for his tardiness he sat, thanking me for presenting myself so punctually and inquiring after my ailing father and others of our family. Though it was plain he bore a thousand burdens of the army and the state, not excepting the prescience of his own imminent death and the bereavement of his beloved wife and children, yet as he took his bench he dismissed all from his mind and addressed himself to me alone with undiverted attention.
"'Do you hate me, lady?' These were his initial words. 'Were I you, I would. My hands would now be trembling with fury hard-suppressed.' He cleared a space upon his bench. 'Come, daughter. Sit here beside me.'
"I obeyed. The lady Gorgo moved subtly closer upon her settle. I could smell the king's sweat of his exercise and feel the warmth of his flesh beside me as, when a girl, I had known my own father's when he had called me to his counsel. Again the heart's surfeit of grief and anger threatened to take me out of hand. I fought this back with all my force.
"'The city speculates and guesses,' Leonidas resumed, 'as to why I elected those I did to the Three Hundred. Was it for their prowess as individual men-at-arms? How could this be, when among champions such as Polynikes, Dienekes, Alpheus and Maron I nominated as well unblooded youths such as Ariston and your own Alexandros? Perhaps, the city supposes, I divined some subtle alchemy of this unique aggregation. Maybe I was bribed, or paying back favors. I will never tell the city why I appointed these three hundred. I will never tell the Three Hundred themselves. But I now tell you.
"'I chose them not for their own valor, lady, but for that of their women.'
"At these words of the king a cry of anguish escaped my breast, as I understood before he spoke what further he would now say. I felt his hand about my shoulder, comforting me.
"'Greece stands now upon her most perilous hour. If she saves herself, it will not be at the Gates (death alone awaits us and our allies there) but later, in battles yet to come, by land and sea. Then Greece, if the gods will it, will preserve herself. Do you understand this, lady? Well. Now listen.
"'When the battle is over, when the Three Hundred have gone down to death, then will all Greece look to the Spartans, to see how they bear it.
"'But who, lady, who will the Spartans look to? To you. To you and the other wives and mothers, sisters and daughters of the fallen.
"'If they behold your hearts riven and broken with grief, they, too, will break. And Greece will break with them. But if you bear up, dry-eyed, not alone enduring your loss but seizing it with contempt for its agony and embracing it as the honor that it is in truth, then Sparta will stand. And all Hellas will stand behind her.
"'Why have I nominated you, lady, to bear up beneath this most terrible of trials, you and your sisters of the Three Hundred? Because you can.'
"From my lips sprang these words, reproving the king: 'And is this the reward of women's virtue, Leonidas? To be afflicted twice over, and bear a double grief?'
"On this instant the queen Gorgo reached for me, to offer succor. Leonidas held her back. Instead, yet securing my shoulder within the grasp of his warm arm, he addressed my outburst of anguish.
"'My wife reaches for you, Paraleia, to impart by her touch intelligence of the burden she has borne without plaint all her life. This has ever been denied her, to be simply bride to Leonidas, but always she must be wife to Lakedaemon. This now is your role as well, lady. No longer may you be wife to Olympieus or mother to Alexandros, but must serve as wife and mother of our nation. You and your sisters of the Three Hundred are the mothers now of all Greece, and of freedom itself. This is stern duty, Paraleia, to which I have called my own beloved wife, the mother of my children, and have now as well summoned you. Tell me, lady. Was I wrong?'
"Upon these words of the king, all self-command fled my heart. I broke down, weeping. Leonidas pulled me to him in kindness; I buried my face in his lap, as a girl does with her father, and sobbed, unable to constrain myself. The king held me firmly, his embrace neither stern nor unkind, but bearing me up with gentleness and solace.
"As when a wildfire upon a hillside at last consumes itself and flares no more, so my fit of grief burned itself out. A peace settled clemently upon me, as if gift not alone of that strong arm which clasped me yet in its embrace, but of some more profound source, ineffable and divine. Strength returned to my knees and courage to my heart. I rose before the king and wiped my eyes. These words I addressed to him, not of my own will it seemed, but prompted by some unseen goddess whose source and origin I could not name.
"'Those were the last tears of mine, my lord, that the sun will ever see.'"
**THIRTY-SEVEN**
**T** _hese were the final words spoken by the captive Xeones. The man's voice trailed off; his vital signs ebbed swiftly. Within moments he lay still and cold. His god had used him up and restored him at last to that station to which he yearned most to return, reunited with the corps of his comrades beneath the earth._
_Immediately outside the captain Orontes' tent, armored elements of His Majesty's forces were clamorously withdrawing from the city. Orontes ordered the man Xeones' body borne without upon his litter. Chaos reigned. The captain was past due at his post; each succeeding moment heightened the urgency of his departure._
_His Majesty will recall the state of anarchy which prevailed upon that morning. Numerous street youths and blackguards, the scum of the Athenian polity, that element of such mean station as not even to merit evacuation but who instead had been marooned by their betters and left to prowl the streets as predators, now made bold to penetrate the margins of His Majesty's camp. These villains were looting everything they could lay hands upon. As our party emerged onto that now-rubbled boulevard called by the Athenians the Sacred Way, a clutch of these felons chanced to be herded past by subalterns of His Majesty's military police._
_To my astonishment the captain Orontes hailed these officers. He ordered them to release the miscreants to his charge and themselves begone. The malefactors were three in number and of the scurviest disposition imaginable. They drew themselves up before Orontes and the officers of the Immortals, clearly expecting to be executed upon the spot. I was commanded by the captain to translate._
_Orontes demanded of these rogues if they were Athenians. Not citizens, they replied, but men of the city. Orontes indicated the coarsecloth wrap which draped the form of the man Xeones._
_"Do you know what this garment is?"_
_The villains' leader, a youth not yet twenty, responded that it was the scarlet cloak of Lakedaemon, that mantle worn only by a warrior of Sparta. Clearly none of the criminals could summon explanation for the presence of the body of this man, a Hellene, here now in the charge of his Persian enemy._
_Orontes interrogated the wretches further. Did they know the location, in the seaport precinct of Phaleron, of that sanctuary known as Persephone of the Veil?_
_The thugs replied in the affirmative._
_To my further astonishment, and that of the officers as well, the captain produced from his purse three gold darics, each a month's pay for an armored infantryman, and held this treasure out to the reprobates._
_"Take this man's body to that temple and remain with it until the priestesses return from their evacuation. They will know what to do with it."_
_Here one of the officers of the Immortals broke in to protest. "Look at these criminals, sir. They are swine! Place gold in their hands and they'll dump man and litter in the first ditch they come to."_
_No time remained for debate. Orontes, myself and the officers all must make haste to our stations. The captain held up, for the briefest of intervals, examining the faces of the three scoundrels before him._
_"Do you love your country?" he demanded._
_The villains' expressions of defiance answered for them._
_Orontes indicated the form upon the litter._
_"This man, with his life, has preserved it. Bear him with honor."_
_There we left him, the corpse of the Spartan Xeones, and in a moment were swept ourselves into the irresistible current of decampment and retreat._
**THIRTY-EIGHT**
**T** _here remain to be appended two final postscripts regarding the man and the manuscript which will at last round this tale into completion._
_As the captain Orontes had predicted, His Majesty took ship for Asia, leaving in Greece under command of Mardonius the elite corps of the army, some 300,000 including Orontes himself and the Ten Thousand Immortals, with orders to winter in Thessaly and resume the conflict when campaigning weather returned in the spring. Come that season, so vowed the general Mardonius, the irresistible might of His Majesty's army would once and for all deliver into subjection the whole of Hellas. I myself remained, in the capacity of historian, upon station with this corps._
_At last in the spring His Majesty's land forces faced the Hellenes in battle upon that plain adjacent to the Greek city of Plataea, a day's march northwest of Athens._
_Across from the 300,000 of Persia, Media, Bactria, India, the Sacae and the Hellenes conscripted under His Majesty's banner stood 100,000 free Greeks, the main force comprised of the full Spartan army—5000 Peers, plus the Lakedaemonian_ perioikoi, _armed squires and helots to a total of 75,000—flanked by the hoplite militia of their Peloponnesian allies, the Tegeates. The army's strength was completed by lesser-numbered contingents from a dozen other Greek states, foremost among whom stood the Athenians, to the number of 8000, upon the left._
_One need not recount the particulars of that calamitous defeat, so grimly familiar are they to His Majesty, nor the details of the appalling losses to famine and disease of the flower of the Empire upon the long retreat to Asia. It may suffice to note, from the perspective of an eyewitness, that everything the man Xeones had forecast proved true. Our warriors beheld again that line of_ lambdas _upon the interleaved shields of Lakedaemon, not this time in breadth of fifty or sixty as in the confines of the Hot Gates, but ten thousand across and eight deep, as Xeones had_ _described them, an invincible tide of bronze and scarlet. The courage of the men of Persia once again proved no match for the valor and magnificent discipline of these warriors of Lakedaemon fighting to preserve their nation's freedom. It is my belief that no force under heaven, however numerous, could have withstood their onslaught upon that day._
_In the hot-blood aftermath of the slaughter, the historian's station within the Persian palisade was overrun by two battalions of armed helots. These, under orders of the Spartan commander in chief, Pausanias, to take no prisoners, began butchering without quarter every man of Asia they could lay steel upon. In this exigency I thrust myself forward and began crying out in Greek, imploring the conquerors for mercy for our men._
_Such, however, stood the Greeks' fear of the multitudes of the East, even in disarray and defeat, that none heeded or gave pause. Hands were laid upon my own person and my throat drawn back beneath the blade. Inspired perhaps by God Ahura Mazda, or in the instance by terror alone, I found my voice crying out from memory the names of those Spartans of whom the man Xeones had spoken. Leonidas. Dienekes. Alexandros. Polynikes. Rooster. At once the helot warriors drew up their swords._
_All slaughter ceased._
_Spartiate officers appeared and restored order to the mob of their armored serfs. I was hauled forward, hands bound, and dumped upon the earth before one of the Spartans, a magnificent-looking warrior, his flesh yet steaming with the gore and tissue of conquest. The helots had informed him of the names I had cried out. The warrior stood over my kneeling form, regarding me gravely._
_"Do you know who I am?" he demanded._
_I replied that I did not._
_"I am Dekton, son of Idotychides. It was my name you called when you cried 'Rooster.'"_
_Scruple compels me here to state that what spare physical description the captive Xeones had supplied of this man failed in all ways to do him justice. The warrior who stood above me was a splendid specimen in the prime of youth and vigor, six feet and more in stature, possessed of a comeliness of person and nobility of bearing that belied utterly the mean birth and station from which, it was clear, he had in the interval arisen._
_I now knelt within this man's power, pleading for mercy. I told him of_ _his comrade Xeones' survival following the battle at Thermopylae, his resuscitation by the Royal Surgeon's staff and his dictation of the document by which I, its transcriber, had acquired knowledge of those names of the Spartans which I had, seeking pity, cried out._
_By now a dozen other Spartiate warriors had clustered, encircling my kneeling form. As one, they scorned the document unseen and denounced me for a liar._
_"What fiction of Persian heroism is this you have concocted of your own fancy, scribe?" one among them demanded. "Some carpet of lies woven to flatter your King?"_
_Others declared that they knew well the man Xeones, squire of Dienekes. How dare I cite his name, and that of his noble master, in craven endeavor to save my own skin?_
_Throughout this, the man Dekton called Rooster held silent. When the others' fury had at last spent itself, he put to me one question only, with Spartan brevity: where had the man Xeones last been seen?_
_"His body dispatched with honor by the Persian captain Orontes to that temple of Athens called by the Hellenes Persephone of the Veil."_
_At this the Spartan Dekton elevated his hand in clemency. "This stranger speaks true." His comrade Xeones' ashes, he confirmed, had been restored to Sparta, delivered months prior to this day's battle by a priestess of that very temple._
_Hearing this, all strength fled my knees. I sank upon the earth, overcome by the apprehension of my own and our army's annihilation and by the irony of discovering myself now before the Spartans in that selfsame posture which the man Xeones had been compelled to assume before the warriors of Asia, that of the vanquished and the enslaved._
_The general Mardonius had perished in the battle at Plataea, and the captain Orontes as well._
_Yet now the Spartans believed me, my life was spared._
_I was held at Plataea in the custody of the Hellenic allies, treated with consideration and courtesy, for most of the following month, then assigned as a captive interpreter to the staff of the Allied Congress._
_This document, in the end, preserved my life._
**A** _n aside, as to the battle. His Majesty may recall the name Aristodemos, the Spartan officer mentioned on several occasions by the man Xeones as an envoy and, later, as among the Three Hundred at the Hot Gates. This man alone among the Peers survived, having been evacuated due to field blindness prior to the final morning._
_Upon this Aristodemos' return alive to Sparta, he was forced to endure at the hands of the citizenry such scorn as a coward or_ tresante, _"trembler," that, now at Plataea, discovering the opportunity to redeem himself, he displayed such spectacular heroism, excelling all upon the field, as to eradicate forever his former disgrace._
_The Spartans, however, spurned Aristodemos for their prize of valor, awarding this to three other warriors, Posidonius, Philokyon and Amompharetus. The commanders adjudged Aristodemos' heroics reckless and unsound, striving in blood madness alone in front of the line, clearly seeking death before his comrades' eyes to expiate the infamy of his survival at Thermopylae. The valor of Posidonius, Philokyon and Amompharetus they reckoned superior, being that of men who wish to live yet still fight magnificently._
**T** _o return to my own lot. I was detained at Athens for two summers, serving in such capacities as translator and scribe as permitted me to witness firsthand the extraordinary and unprecedented transformation there taking place._
_The ruined city rose again. With astonishing celerity the walls and port were rebuilt, the buildings of assembly and commerce, the courts and magistracies, the houses and shops and markets and factories. A second conflagration now consumed all Hellas, in particular the city of Athena, and this was the blaze of boldness and self-assurance. The hand of heaven, it seemed, had set itself in benediction upon each man's shoulder, banishing all timorousness and irresolution. Overnight the Greeks had seized the stage of destiny. They had defeated the mightiest army and navy in history. What lesser undertaking could now daunt them? What enterprise could they not dare?_
_The Athenian fleet drove His Majesty's warships back to Asia, clearing_ _the Aegean. Trade boomed. The treasure and commerce of the world flooded into Athens._
_Yet massive as was this economic recrudescence, it paled alongside the effects of victory upon the individuals, the commons of the populace themselves. A dynamism of optimism and enterprise fired each man with belief in himself and his gods. Each citizen-warrior who had endured trial of arms in the phalanx or pulled an oar under fire on the sea now deemed himself deserving of full inclusion in all affairs and discourse of the city._
_That peculiar Hellenic form of government called_ democratia, _rule of the people, had plunged its roots deep, nurtured by the blood of war; now with victory the shoot burst forth into full flower. In the Assembly and the courts, the marketplace and the magistracies, the commons thrust themselves forward with vigor and confidence._
_To the Greeks, victory was proof of the might and majesty of their gods. These deities, which to our more civilized understanding appear vain and passion-possessed, riddled with folly and so prey to humanlike faults and foibles as to be unworthy of being called divine, to the Greeks embodied and personified their belief in that which was, if grander than human in scale, yet human in spirit and essence. The Greeks' sculpture and athletics celebrated the human form, their literature and music human passion, their discourse and philosophy human reason._
_In the flush of triumph the arts exploded. No man's home, however humble, reascended from the ashes without some crowning mural, statue or memorial in thanksgiving to the gods and to the valor of their own arms. Theater and the chorus throve. The tragedies of Aeschylus and Phrynichus drew hordes to the precincts of the_ theatron, _where noble and common, citizen and foreigner, took their stations, attending in rapt and often transported awe to works whose stature, the Greeks professed, would endure forever._
**I** _n the fall of my second year of captivity I was repatriated upon receipt of His Majesty's ransom, along with a number of other officers of the Empire, and returned to Asia._
_Restored to His Majesty's service, I reassumed my responsibilities re_ _cording the affairs of the Empire. Chance, or perhaps the hand of God Ahura Mazda, found me toward the close of the following summer in the port city of Sidon, there assigned to assist in the interrogation of a ship's master of Aegina, a Greek whose galley had been driven by storm to Egypt and there been captured by Phoenician warships of His Majesty's fleet. Examining this officer's logs, I came upon an entry indicating a sea passage, the summer previous, from Epidaurus Limera, a port of Lakedaemon, to Thermopylae._
_At my urging, His Majesty's officers pressed their interrogation upon this point. The Aeginetan captain declared that his vessel had been among those employed to convey a party of Spartan officers and envoys to the dedication of a monument to the memory of the Three Hundred._
_Also on board, the captain stated, was a party of Spartan women, the wives and relations of a number of the fallen._
_No commerce was permitted, the captain reported, between himself and his officers and these gentlewomen. I questioned the man strenuously, but could determine neither by evidence nor by surmise if among these were included the ladies Arete and Paraleia, or the wives of any of the warriors mentioned in the papers of the man Xeones._
_His vessel beached at the mouth of the Spercheios, the captain stated, at the eastern terminus of the very plain where His Majesty's army had encamped during the assault upon the Hot Gates. The memorial party there disembarked and proceeded the final distance on foot._
_Three corpses of Greek warriors, the ship's master reported, had been recovered by the natives months earlier at the upper margins of the Trachinian plain, the very pastureland upon which His Majesty's pavilion had been sited. These remains had been preserved piously by the citizens of Trachis and were restored now with honor to the Lakedaemonians._
_Though certainty remains ever elusive in such matters, the bodies, common sense testifies, can have been none other than those of the Spartan Knight Doreion, the Skirite "Hound" and the outlaw known as Ball Player, who participated in the night raid upon His Majesty's pavilion._
_The ashes of one other body, that of a warrior of Lakedaemon returned from Athens, were borne by the Aeginetan vessel. The captain could provide no intelligence as to the identity of these remains. My heart, however,_ _leapt at the possibility that they might be those of our narrator. I pressed the sea captain for further intelligence._
_At the Hot Gates themselves, this officer declared, these final bodies and the urn of ashes were interred in the burial mound of the Lakedaemonian precinct, sited upon a knoll directly above the sea. Scrupulous interrogation of the captain as to the topography of the site permits me to conclude with near certainty that this hillock is the same whereupon the final defenders perished._
_No athletic games were celebrated in memoriam, but only a simple solemn service sung in thanksgiving to Zeus Savior, Apollo, Eros and the Muses. It was all over, the ship's master stated, in less than an hour._
_The captain's preoccupations upon the site were understandably more for the tide and the security of his vessel than with the memorial events transpiring. One instance, however, struck him, he said, as singular to the point of recollection. A woman among the Spartan party had held herself discrete from the others and chose to linger, solitary, upon the site after her sisters had reassembled in preparation to depart. In fact this lady tarried so late that the captain was compelled to dispatch one of his seamen to summon her away._
_I inquired earnestly after the name of this woman. The captain, not surprisingly, had neither inquired nor been informed. I pressed the question, seeking any peculiarities of dress or person which might assist in mounting a supposition as to her identity. The captain insisted that there was nothing._
_"What about her face?" I persisted. "Was she young or old? Of what age or appearance?"_
_"I cannot say," the man replied._
_"Why not?"_
_"Her face was hidden," the ship's master declared. "All but her eyes obscured by a veil."_
**I** _inquired further as to the monuments themselves, the stones and their inscriptions. The captain reported what he recalled, which was little. The stone over the Spartans' grave, he recollected, bore verses composed_ _by the poet Simonides, who himself stood present that day to assist in the dedication._
_"Can you recall the epitaph upon the stone?" I inquired. "Or were the verses too lengthy for memory to retain?"_
_"Not at all," the captain replied. "The lines were composed Spartan style. Short. Nothing wasted."_
_So spare were they, he testified, that even one of as poor a memory as himself encountered no difficulty in their recollection._
O xein angellein Lakedaimoniois hoti tede
keimetha tois keinon rhemasi peithomenoi
_These verses have I rendered thus, as best I can:_
Tell the Spartans, stranger passing by,
that here obedient to their laws we lie.
**ACKNOWLEDGMENTS**
It goes without saying that a work which attempts to imagine vanished worlds and cultures owes everything to the original literary sources, in this case Homer, Herodotus, Plutarch, Pausanias, Diodorus, Plato, Thucydides, Xenophon, and on and on. They're the real stuff, without which nothing.
Almost as indispensable, however, have been the extraordinary scholars and historians of our own time, whose published wisdom I have looted shamelessly. I hope they will forgive the author of this work of far less rigorous scholarship than their own if he acknowledges with gratitude and by name a number of these distinguished classicists—Paul Cartledge, G. L. Cawkwell, Victor Davis Hansen, Donald Kagan, John Keegan, H. D. F. Kitto, J. F. Lazenby, E. V. Pritchett, W. K. Pritchett and, especially, Mary Renault.
In addition, I would like to thank two colleagues whose personal counsel and direction have been indispensable:
First, Hunter B. Armstrong, Director of the International Hoplology Society, for graciously sharing his expertise in hoplite weapons, tactics and practice and for his invaluable insights into, and imaginative reconstructions of, ancient battle. Himself an acclaimed weapons athlete, Mr. Armstrong's combatant's-eye-view assisted immeasurably in reimagining the experience of Greek heavy-infantry warfare.
Finally, my profound gratitude to Dr. Ippokratis Kantzios, Assistant Professor of Greek Language and Literature at the Richard Stockton College of New Jersey, for his generous and encyclopedic assistance through all aspects of this undertaking, acting not only as guide and mentor for historical and linguistic authenticity and as translator (free as well as exact) of the epigraph and of passages and terms throughout this book, but for numerous other sage and inspired contributions. There's not a page in the book that doesn't owe something to you, Hip. Thanks for your innumerable creative contributions, your unfailing encouragement and your ever-Olympian counsel.
ALSO BY STEVEN PRESSFIELD
FICTION
_The Virtues of War_
_Last of the Amazons_
_Tides of War_
_The Legend of Bagger Vance_
NONFICTION
_The War of Art_
WARS CHANGE. WARRIORS DON'T.
STEVEN PRESSFIELD,
the "master storyteller" _(Publishers Weekly)_ and bestselling author, returns with a stunning, plausible near-future thriller about the rise of a privately financed and global military industrial complex.
THE PROFESSION • A Thriller • $25.00 (Canada: $28.95) • 978-0-385-52873-3
_"The Profession_ is chilling because it rhymes just enough with today to make us wonder whether this future will be, or only might be... A ripping read."
—NATHANIEL FICK, author of the
_New York Times_ bestseller _One Bullet Way_
"Pressfield dominates the military thriller genre."
— _LIBRARY JOURNAL_
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_The Profession_
**1**
**A BROTHER**
**MY MOST ANCIENT MEMORY** is of a battlefield. I don't know where. Asia maybe. North Africa. A plain between the hills and the sea.
The hour was dusk; the fight, which had gone on all day, was over. I was alive. I was looking for my brother. Already I knew he was dead. If he were among the living, he would have found me. I would not have had to look for him.
Across the field, which stretched for thousands of yards in every direction, you could see the elevations of ground where clashes had concentrated. Men stood and lay upon these. The dying and the dead sprawled across the lower ground, the depressions and the sunken traces. Carrion birds were coming down with the night—crows and ravens from the hills, gulls from the sea.
I found my brother's body, broken beneath the wheels of a battle wagon. Three stone columns stood above it on an eminence—a shrine or gate of some kind. The vehicle's frame had been hacked through by axes and beaten apart by the blows of clubs; the traces were still on fire. All that remained aboveground of my brother was his left arm and hand, which still clutched the battle-axe by which I recognized him. Two village women approached, seeking plunder. "Touch this man," I told them, "and I will cut your hearts out."
I stripped my cloak and wrapped my brother's body in it. The dames helped me settle him in the earth. As I scraped black dirt over my brother's bones, the eldest caught my arm. "Pray first," she said.
We did. I stood at the foot of my brother's open grave. I don't know what I expected to feel: grief maybe, despair. Instead what ascended from that aperture to hell were such waves of love as I have never known in this life or any other. Do not tell me death is real. It is not. I have sustained my heart for ages with the love my brother passed on to me, dead as he was.
While I prayed, a commander passed on horseback. "Soldier," he asked, "whom do you bury?" I told him. He reined in, he and his lieutenants, and bared his head. Who was he? Did I know him? When the last spadeful of earth had been mounded atop my brother's grave, the general's eyes met mine. He said nothing, yet I knew he had felt what I had, and it had moved him.
I am a warrior. What I narrate in these pages is between me and other warriors. I will say things that only they will credit and only they understand.
A warrior, once he reckons his calling and endures its initiation, seeks three things.
First, a field of conflict. This sphere must be worthy. It must own honor. It must merit the blood he will donate to it.
Second, a warrior seeks comrades. Brothers-in-arms, with whom he willingly undergoes the trial of death. Such men he recognizes at once and infallibly, by signs others cannot know.
Last, a warrior seeks a leader. A leader defines the cause for which the warrior offers sacrifice. Nor is this dumb obedience, as of a beast or a slave, but the knowing heart's pursuit of vision and significance. The greatest commanders never issue orders. Rather, they compel by their own acts and virtue the emulation of those they command. The great champions throw leadership back on you. They make you answer: Who am I? What do I seek? What is the meaning of my existence in this life?
I fight for money. Why? Because gold purges vanity and self-importance from the fight. Shall we lay down our lives, you and I, for a flag, a tribe, a notion of the Almighty? I did, once. No more. My gods now are Ares and Eris. Strife. I fight for the fight itself. Pay me. Pay my brother.
I served once beneath a great commander who asked in council one night, of me and my comrades, if we believed our calling to be a species of penance—a hell or purgatory through which we must pass, again and again, in expurgation of some crime committed eons gone.
"I do," he said. He offered us as recompense for this passage "an unmarked grave on a hill with no name, for a cause we cannot understand, in the service of those who hate us."
Not one of us hesitated to embrace this.
**BOOK
ONE**
**EUPHRATES**
**2**
**ESPRESSO STREET**
**NINETY MILES SOUTH OF** Nazirabad, we sight a convoy of six vehicles speeding west and flying the black-and-yellow death's-head pennant of CounterArmor. The date is 15 August 2032. In that country, when you run into other Americans, you don't ask who they're working for, where they're from, or what they're up to. You help them.
We brake beside the CounterArmor vehicles in the lee of a thirty-foot sand berm. The team is pipeline security Their chief is a black dude, about forty, with a Chicago accent. "The whole goddam city's gone over!"
"Over to who?" I ask. A gale is shrieking, the last shreds of a sandstorm that has knocked out satellite and VHF comms for the past two and a half hours.
"Whoever the hell wants it!"
The CounterArmor commander's vehicle is a desert-tan Chevy Simoom with a reinforced-steel X-frame and a .50-caliber mounted topside. My own team is six men in three vehicles—two Lada Neva up-armors and one RT-7, an Iraq-era 7-ton truck configured for air defense. The outfit is part of Force Insertion, the largest private military force in the world and the one to whom all of western Iran has been contracted. I'm in command of the group, which is a standard MRT, Mobile Response Team. The overall contract is with ExxonMobil and BP.
The CounterArmor trucks are fleeing west for the Iraq border. The Turks have invaded, the chief is telling us. Or maybe it's the Russians. Tactical nukes have been used, near Qom and Kashan in the No-Go Zone; or maybe that's false too. "Get in behind us," he shouts. "We're gonna need every gun we can get."
I tell him our team has orders to enter the city. Five American engineers, civilian contractors, are trapped there, along with the TCN—Third Country Nationals—security detail assigned to protect them. Our instructions are to get them out, along with a technical brief they have prepared for the commanding general's eyes only.
"You can't go back there," the chief says.
"Watch us."
Nazirabad is a Shiite city of about three hundred thousand. They're all Shiite cities in Iran. You can tell a Shiite city by the billboards and the vehicles, which are plastered with pix of their saints, Ali and Hussein. A Shiite truck or bus is festooned with religious amulets and geegaws. Reflectorized pinwheels dangle from the rearview and outboard mirrors; framed portraits adorn the dash; every square inch is crazy quilted with talismans and mandalas, good luck charms and magic gimcracks.
Anyway, that's what we're seeing now—forty minutes after leaving the CounterArmor convoy—as Iranian civilian cars, trucks, and buses flood past on the highway, fleeing. Comms are still out, whether from the nukes, the storm, or man-made jamtech, we can't tell. Our orders are to rescue the engineers. Beyond that, we know nothing. We don't know what we're riding into or what our chances are of getting out. This is the bitch of modern warfare. Every technological breakthrough spawns its dedicated countermeasure, with each generation getting cheaper and more accessible. X knocks out Y; before you know it, you're back to deadfalls and punji stakes.
So we're relieved, forty miles south of the city, when two Little Bird choppers—the kind used by the Legion, one of Force Insertion's subcontractors—show up topside and communicate to us by line-of-sight that other friendlies are up ahead. Twenty minutes later we pick up radio traffic from Legion vehicles heading our way and, half an hour after that, two black bulletproofs—GMC Kodiaks with cork tires and gun-slit windows—roll up and brake, coated with gray dust. An operator springs down, wearing a tuxedo jacket and white linen shirt over cargo pants and boots. We can see, in the distance, the three-level overpass south of the city. The merc comes up, grinning in his black tie. My #2, Chutes Savarese, hails him.
"Where's the party?"
"We brought it, babies."
The merc introduces himself as Chris Candelaria and shakes my hand and the others'. His ring says SEAL Team Six. He wears another that I don't see, under the Nomex glove on his left hand: the Wharton School. The team he's leading is from DSF, Dienstleister Schwarze Flagge, the crack German-South African outfit that evolved in the twenties out of the Zimbabwean Selous Scouts. He just got out of Isfahan five hours ago, he says. Dried blood paints both his hands and arms; the shoulder of his jacket has been charred through; he's got a dust-caked battle dressing on his neck, above an ear whose bottom third is scorched black and slathered with green combat antiseptic. But he's grinning. Like me, he wears a beard. His hair is long and falls in a cascade of black ringlets.
"You guys going in there?" he asks. From our rise south of the highway, we can see Dragonfly drones in swarms over the city Every punk-ass gang and militia is flying these little fuckers, some the size of kites, others no bigger than pie plates. The streaks of their rockets—high-explosive and flechette—blow away in the wind. "Want some help?"
The merc and I do a quick map orientation, marking the in-city locations and the routes, order, and sequence we'll use to approach them. What about supporting fires, I ask. Our team has zero; has he got Close Air Support, drones, anything?
The cupboard is bare, the merc says. "It's just you and me, partner. We are officially OOO"—On Our Own—"and SOL." Shit Outa Luck.
The contractor has a case of Jack Daniel's in the lead Kodiak. Standing at the rear doors, he passes us two bottles for each vehicle. He's got cups but no ice. He introduces the rest of his team, who are more comm guys than trigger pullers. I note two DSFers packing Heckler & Koch 416s, German superguns, with 40 mm grenade launchers underslung. On the truck's roof squats a donut satlink receiver in a fiberglass cover; inside the vehicle I note a bank of tech gear, including a Xenor encryption box.
"What kind of team are you leading?" I ask.
"We're a financial unit. I'm specking oil and gas contracts. Haven't had a rifle in my hands for seven years!"
I'm laughing now. So is Chutes. "Thanks for the help, bro."
"I'm coming from an embassy ball," says Chris, indicating his tux. He nods toward the trucks and guns. "We grabbed this shit and ran."
He tells us Isfahan is burning. Tehran too. Mobs are storming the U.S. embassy—and the embassies of the Russians and the Chinese. He doesn't know who's attacking whom. He has caught snatches on al-Alam, the Iranian satellite channel, about a rising in Saudi Arabia; the fear in the West, says the report, is of a Shiite sweep across southern Iraq and into the Eastern Province of Saudi Arabia. Or maybe it's all bullshit. The one thing the merc can tell us for sure is the nearest safety is across two hundred miles of hell. "Salter's at Kirkuk with two armatures, moving toward the Iranian border. If we can get to him, we're home free."
He means our Force Insertion commander, Gen. James Salter. An armature is the equivalent of the old conventional-army airmobile division. The word comes from Latin, meaning equipment or armor. Force Insertion has, along the Iraq-Iran border, four armatures with all supporting arms including artillery (105- and 177 mm howitzers), drone and truckborne antiarmor, and air defense in the form of mobile Chinese I-SAM rocket trucks. Salter's air assault complement, we know, is at near full strength, meaning each armature has three battalions of extended-range Black Hawk and up-gunned War Hawk choppers, a battalion of heavy Chinooks, plus seventy-two owner-operated AH-64 Apache attack helicopters, all outfitted with the latest aftermarket Chinese, Czech, and Israeli missile technology, American and Indian avionics and satcomms, and flown by American, Russian, South African, Australian, Polish, and British mercs, most of whom have in the old days been majors, lieutenant colonels, and colonels in their respective conventional air forces. Our new friend eyes our ragged-ass gear, which looks like it came from Operation Iraqi Freedom, and the faces of our guys—Chutes Savarese, Junk Olsen, Adrian "Q" Quinones, Marcus Aurelius "Mac" Jones, and Tony Singh, our six-foot-four Hindu from Sri Lanka. He indicates the city.
"Gentlemen, as Sarpedon said to Glaucus, 'Let us go forth and win glory—or cede it to others.' "
Chutes is grinning. "What's your name again, man?"
"Chris Candelaria."
"Chris, you're my kinda dude."
They bump elbows. In we go.
Nazirabad is situated at the juncture of two highways—8, which runs north-south, and 41, east-west. The three-level interchange and its security station, Checkpoint 290, is the funnel through which all motorized entry and egress is channeled. There's an industrial slum to the north called Ali City, from which most of the bad actors come—tribal militias, criminal gangs, Mahdi revivalists, cabals of displaced army officers, as well as Jaish al-Sha'b, "Army of the People," which has replaced AQP—al-Qaeda in Persia—plus every imaginable hue of nationalist, separatist, and irredentist forces, including foreign fighters—Turks, Chechens, Syrians, Saudis, Uzbeks, Tajiks, Uighurs, as well as Shiite Kurds, Afghan Hazaras, and Lebanese.
As recently as ten years ago, Nazirabad was a secure, attractive tourist destination. Brochures called it the "city of artists." The Old Town had four souks, one entirely for tiles, another for decorative ironwork—gates, lamps, chandeliers. Nazirabad had two synagogues, believe it or not, and a Christian bookstore. A woman could walk alone and bareheaded, even after dark. Eighteen months ago, when our team deployed, a foreigner could still get a private villa, with cook, driver, and laundress. No more. In the space of ten weeks, since the start of the third Iran-Iraq war, the place has degenerated to a level of violence equal to Baghdad or Ramadi twenty-five years earlier—and the last half year has been even worse.
We take side streets into the city, bypassing Checkpoint 290. The sun is dropping fast. Mac has made radio contact with our engineers; they have abandoned the company compound and made their way to a safe house (actually the home of their supervisor's father) on Espresso Street, a well-to-do boulevard so named because it has the only Starbucks within five hundred miles. The only problem is that Espresso Street has become the epicenter for whatever conflagration is currently consuming the city.
We approach from the west, so the sun is behind us. We can see Iranian-badged Hind gunships overhead, putting out rocket and machine-gun fire—probably at sniper teams on rooftops—and see the propellant trails of heat seekers and SFRs, shoulder-fired rockets, corkscrewing up in response. I'm navigating by the electrical power lines, which run along central thoroughfares and are the only objects taller than three stories in the city. In an urban firefight, you can't simply race to the action like a fire truck toward a burning building. You have to patrol up to it, employing "movement to contact," which basically means keep advancing until somebody starts shooting at you. Our engineers are talking us in over line-of-sight squad radios, which work for two seconds and then break up as buildings and vehicles intervene. "How close are you to the fight?" I speak into my mike.
"We _are_ the fight!" comes the answer.
Espresso Street, when we enter it, is as broad as a boulevard and sizzling with spent shell casings, smoking bricks, rubble, and blocks of concrete-and-rebar and is pocked by craters from which ruptured water-main fluid floods, mingling with raw sewage, garbage, and gasoline to form an inch-deep burning lake across the welcome-to-hell cityscape. We pass one Russian-built Iranian T-79 tank coming out, protecting two gun trucks with wounded regulars inside and on top. Local civilians are running up to my window and Chutes's, shouting that there are snipers on such and such a rooftop or drone swarms above such and such a block. Unarmed boys race on foot toward the action, just for the excitement. We see a press pickup, with "TV" on the windshield in masking tape. Chutes is my driver. The boom box blares Bloodstone's "Death or Dismemberment":
_Eat me, beat me_
_Wolf me down and excrete me_
_I'm here for your ass, motherfucker_
Cars are burning in the middle of the street; we're jinking around downed phone poles. Adrenaline is flooding through me; I can tell because the pulse hemometer on my wrist reads out at 180/15. But my subjective experience is the opposite. I'm cool. The hotter it gets outside, the cooler I become. This is nothing I can take credit for or claim to have achieved by virtue of training or application of will. I was born this way. When I was nine years old and my old man would wallop the tar out of me for some infraction of his demented code of honor, I would stare up at him icy eyed and not feel a twinge of rage, even though I could have and would have killed him on the spot if I had taken a notion to. I was remote. I was detached. I felt like another person was inside me. This other person was me, only stronger and crueler, more cunning and more deadly.
I never told anyone about this secret me. I was afraid they might think I was crazy, or try to take this other me away, or convince me that I should be ashamed of him. I wasn't. I loved him. In sports or fistfights, in moments of crisis or decision, I cut loose my conventional self and let this inner me take over. He never hesitated. He never second-guessed. Later, in combat, when I began to experience fragments of recall that were clearly not from this lifetime, I knew at once that these memories were connected to my secret self. They were his memories. I was only the temporary vessel in which they were housed.
This secret self is whom I surrender to now, entering Nazirabad. I become him. I feel fear. At times it threatens to overwhelm me. But my secret self pays no attention. I hear the gunships overhead and see the vapor trails of their rockets. A man must be crazy, I think, to head of his own free will _toward_ that. But at the same time, no force beneath heaven can keep me away. This is what I was born for. I'm geeked out of my skull—and I'm curious. I want to see what's up ahead. How bad is it? What kind of fucked-up shit will we run into this time?
In action there is no such thing as thought, only instinct. We blow past two sharp exchanges of fire and suddenly there's the house. Our engineers have spray-painted on the compound wall.
**THIS IS IT!!!**
in block letters two feet high. I wave to the other trucks: Keep moving. We can't burst in on our own engineers or their bodyguards will shoot us, not to mention the wicked stream of fire that is pouring onto the compound from rooftops and circling drones and is now zeroing on us.
"I know the place!" Chutes is shouting across at me. We accelerate past the rusty front gates—seven feet tall and perforated like cheese graters by .50-cal fire—and swerve hard left into an alley. Chutes is bawling across the seat at me. I can't hear a word but I understand from his gestures: there's a way in, from the rear, which we can access from one of the myriad cross-channels if we can find it. I'm raising the engineers and telling them to hold their fire when they see the rear gate blow in. "I will throw a flash-bang," I enunciate with exaggerated clarity into the Motorola mike duct-taped to the right shoulder of my Kev-lite vest. "When you see the flash, run straight out to us."
Nothing works in combat the way you think it will. It takes us almost twenty minutes to cover the two hundred feet to the rear compound gate. By then our besieged engineers are too petrified to stick a toe out. Snipers are on every rooftop, with rocketmen racing up, more every minute. We break in the gate with our Lada Neva's tailboard and are just starting in reverse across the court to the rear entry of the house, when one of the security men—a Fijian, no taller than five-foot-two, probably making forty bucks a day—appears in the rear egress, shouting, "IED! IED!" and pointing to the dirt drive over which our truck is about to roll. Three 177 mm artillery shells sit unburied, big as life, with their wires exposed, lining the south side of the lane. "Brake!" I shout to Chutes. The Fijian dives back into the house, one nanosecond ahead of a fusillade of 7.62 fire that blows the jamb and the security door to powder. Chutes powers the Lada Neva back out through the gate, to safety behind the four-foot-thick main wall. "What the fuck do we do now?"
There's nothing for us but to go in on foot, blasting every hajji triggerman on every rooftop as we go.
Junk, Q, and I bolt back through the gate. The rear door of the compound is about sixty yards away. We dash past a line of parked Toyota trucks and Tata/LUK compgas cars; I can hear the sheet metal shredding as gunfire pursues us. We can hear Iranian voices everywhere, not just on the overlooming buildings, but on the flat roof of the compound building itself. They know exactly what's happening. They could trigger the IEDs right now and take us out, but they're greedy—they want our vehicles, too. We dive into an alley and take cover beneath two gigantic air-conditioning units supported by pipe stanchions. I can hear one gunner on an overhead taunting us in English. Another in a baseball cap pops up behind the roof wall and starts firing. I jack myself left-handed into the clear and blow his head off. "Man!" shouts Junk, whooping.
I'm shouting to the Fijian, telling him we're coming. "All!" I cry, pantomiming to the group. "Run for it!" Does he understand English? I can't see him. Where are the engineers? The Fijian pops out again, flashing five fingers and a thumbs-up. Again gunfire shreds his nest as he plunges back inside. Junk and I spring out. Hajjis are pouring gunfire from rooftops and upper-story windows. Out of the corner of my eye, I see Junk take a hit and drop. "Go!" he's shouting. "I'm okay!"
I reach the rear door. The engineers and two other Fijians crouch back in the dimness, which is dense with brick dust and smoke from the pulverized walls of the building. The engineers wear body armor and blue civilian Kevlars. The security men have on nothing thicker than khaki; one dude is in flip-flops. "There!" I point to the alley where Junk and Q have taken cover. "Now!"
The mob gets ten steps and the first 177 blows. It's the far one, the one closest to the rear gate. The dumb-bastard triggerman has pushed the wrong button. The blast still knocks every one of us flat and annihilates our hearing. I'm in the rear, driving the engineers forward. Everybody's still alive. My head is ringing like a Chinese gong, but I can still hear the Iranians on the roofline cursing their numb-nuts triggerman. Our gang plunges to cover under the air-conditioning units. Quinones is kneeling between the A/Cs, firing at the V of two tenement rooftops above us. The enemy keeps popping up between clotheslines and satellite dishes. Every time Q pings one and the pink spray blows out of their heads and they drop away out of sight, the engineers yelp with terror and relief. Q and Junk are cross-decking now, firing over each other's shoulders. We're halfway to the compound wall, halfway to safety "Move now!" I shout.
The group bolts to the gate and wall, to Chris, Chutes, and the others. Q and I are dragging one engineer, who has lost sight and hearing from the concussion of the 177, with Junk hopping on one leg and hanging on to one of the security men. We plunge back to safety just as full darkness falls.
Chris's two Kodiaks, which we had held in reserve two blocks back, have come forward now, ready to take us out. They're revving in the alley, thirty feet north, in the safe zone shielded by an adjacent building. Enemy 5.56 and 7.62 fire is ripping into the wall above us. Now the rocket rounds start flying; our Lada Nevas and 7-ton truck have to pull back. Someone is helping Junk into the first vehicle. I hear one of Chris's DSF men shouting in a German accent, "Who? Who's missing?" For a moment I think they mean Junk. I look over. Junk is okay. Then I realize they're talking about someone else. I turn back toward the compound. On the ground beside the air-conditioning units crawls one of our Fijians.
Sonofabitch! The man is in the dirt, clawing his way toward cover. Furious fire rakes the ground around him. I see him scramble face-first into a cooking ditch, just as a full burst from an AK takes him square between the shoulder blades. Both elbows fly rearward, then flop; his neck snaps; he crashes face-first into the dirt. He stops moving. "Brake!" I'm shouting to Chutes. "Q! Chris!"
"Go! Get out!" Chris Candelaria is calling, waving the vehicles to pull back. He has packed Junk's wound and stripped his own tourniquet, worn lanyard-style around his neck; he's cinching it around Junk's thigh as he and the German DSF man help him toward the first Lada Neva.
"We're going back!" I shout.
"What?"
"The Fijian. We're not leaving him!"
A shoulder-fired rocket whistles overhead and blows the hell out of a house across Espresso Street. What little hearing I have left is now gone.
"We're not leaving without him."
It's my secret me who's talking. He has made the decision.
"What the fuck are you talking about?" This is Chutes, my tightest mate and most trusted brother. He sticks his jaw six inches from mine.
"We're going back," I tell him.
Junk curses. "He's not our guy, chief! We don't even know who the fuck he is!"
"He's dead!" says Chutes. "There's two more 177s in there, waiting to blow!"
Other faces stare at me.
"Leave the body," cries the Fijian team leader. "The man would say so himself if he could!"
I tell the Fijians he's not theirs, he's ours.
Chris Candelaria's two Kodiaks are hauling ass now; they know they're targets. Iranian rocket gunners are trying to blow down the building that protects our flank. As their rounds scream in, blocks of concrete the size of bowling balls sail a hundred feet into the air and fall back, crashing all around us. We scramble into the slit trench of sewage. Guys are trying to crawl up inside their helmets. I'm peering around the corner, back into the compound.
Chutes clutches my sleeve. "Bro, listen to me. We got the engineers, we got the report... that's what we came here for." He points past the gate to the compound, to the fresh enemy streaming in along the rooflines. "We go back in there, somebody's gonna die."
There's no fear in Chutes's voice. He's just stating the truth.
I meet his eyes.
"Fuck you," he says, jamming fresh magazines into his belly rig. "You hear me, bro? Fuck you!"
Back we go. Chris Candelaria comes with us. We can hear the enemy hooting with anticipation. In the interval they have brought up a Russian PKM, which fires Eastern Bloc 7.62 rounds with a nutsack-shriveling _rat-a-tat_ sound, and these are tearing the hell out of the open space we have to cross. The foe has got his second wind now. He is going after our Lada Nevas and the 7-ton truck, which have stayed behind to cover us. Rockets are zinging across the compound like Roman candles.
We grab the dead Fijian and haul him facedown from the dirt behind the blown-down cookhouse. The IEDs never blow. Chutes curses me all the way back to the gate, curses me when I bag the security man's effects and lash them around my waist with 550 cord. And he curses me all the way out of town.
Two hours later, our team has reached safety in Husseinabad, in the fortified compound of an Iranian police chief whose real name is Gholamhossein Mattaki, but whom everyone calls Col. Achmed. Col. Achmed has his own doctor, his cousin Rajeef. Rajeef has a pharmacy and a little surgical suite in a side building of Col. Achmed's compound. Rajeef is our team doctor. He supplies all our pills and powders. We call him "Medicare."
I have driven flat out to Col. Achmed's, to get Junk (and two engineers whom we discover have been wounded in the dash across the compound) under serious medical care. Our medic Tony is a superb under-fire practitioner, and the DSF tech is good too. But neither one is a surgeon—and neither one has Dr. R's goody-box of Vicodin and Percoset, Ultram, Fentanyl, OxyContin, and plain old central Asian smack. I also want to bury our Fijian in a site that won't be desecrated. While our guys rehydrate and wolf down a meal of lamb and lentils, I grab Chutes and Chris Candelaria and organize a powwow with Col. Achmed. We still don't know who's invading whom, how dangerous the situation is, or what the hell is happening east in Isfahan and Tehran.
Achmed is not just a police chief and commander in the paramilitary Masij. He is a hereditary tribal leader and a grandson and great-grandson of Harul and Arishi sheikhs; he is responsible for the safety and welfare of several thousand men, women, and children of allied families, clans, and subtribes. He's a serious man.
"Get out now," he tells us, in the tone that your favorite uncle would take if he were looking out for you. We are in the third of his four storage buildings. The place looks like Costco. On racks and pallets sit unopened packing boxes of air conditioners and computer printers; cartons of Pennzoil, Pampers, and paper towels. Achmed has cases of Evian water, V8, Gatorade; crates of Nike running shoes, T-shirts, and tracksuits. The IRGC, the Iranian Revolutionary Guards Corps, smuggles in half the goods sold in Iran; the stuff comes in, billions of dollars' worth, by launch and lighter from Kish and Qeshm, "free trade" islands across the gulf, via the port of Bandar Ganayeh—not to mention whatever the colonel's minions have looted from ExxonMobil and BP. Guns and ammo are everywhere, in and out of crates—M4-40s, mortars, boxes of 5.56 and 7.62 NATO cartridges. Col. Achmed's family, he tells us, is packing up. The women and children will flee to southern Iraq and then to Syria. Achmed's sons and goons fill the room, armed to the teeth. "Don't wait even for morning," the colonel tells me, Chutes, and Chris. "If you do, you and your men will be massacred."
I ask him who will come after us.
"Me," he says with a smile. "Everyone."
Col. Achmed explains.
"They will not be able to help themselves. First they will come for your weapons and everything of yours that they can steal, then for honor, to avenge the humiliation you and your countrymen have inflicted upon our national manhood simply by your presence and your blue eyes. Next they will come to get you before others do, for the greatest honor goes to him who strikes first, while those who hesitate will be accounted cowards."
I ask him what will happen in the next week or ten days. He gives it to me in Revolutionspeak, but the gist is this: Shiite Iran—meaning those Revolutionary Guardsmen, army colonels, patriots, tribesmen, and true believers who have been biding their time throughout this long, phony war will unite now with their Iraqi Shiite coreligionists and, casting off the yoke of the West and its hirelings, strike east along the arc of the Shiite Crescent that runs from the Dasht-i-Margo—the Afghan Desert of Death—across Iran to the Eastern Province of Saudi Arabia (not coincidentally the swath of real estate that contains the richest petroleum reserves on earth) and purge this land of those who do not belong or believe.
"Do my men and I have time," I ask, "to finish dinner?"
Achmed's tribal code mandates hospitality. He helps us bury our Fijian, though he insists, first, on declaring the man a convert to Islam (which Col. Achmed can do, being a mullah as well as a tribal chief, and to which none of our Fijian's mates objects under the circumstances), then tops off our fuel tanks and loads us up with Neda spring water and sixteen-ounce cans of Beefaroni. One of Achmed's sons brings a tray of delicious homemade _sohan_ —pistachio candy.
Col. Achmed maps out the safest route to the frontier (the one he'll be chasing us on) and helps key into our GPSes the sequence of junctures—all unmarked desert and mountain tracks that we could never find without his help. Dr. Rajeef rigs mobile hospital beds for Junk and our two wounded engineers; he stocks us up with two dozen vials of morphine, plus sample packs of Demerol and Dilau-did with sterile syringes and a hundred ampules of methylephidrine, which we all need in our exhausted, postadrenalinized state.
We take our leave over cups of black Persian coffee. It's midnight. Our engines idle beyond the walls in the night. Achmed and his men leave us alone, for our final prep and words for each other.
"Chief." It's Chutes, stepping forward before the others. "I'm sorry for what I said back there in Nazirabad..."
"Forget it."
He apologizes for refusing, at first, to go back after the Fijian, whose name, we have learned, is Manasa Singh. Chris Candelaria seconds this. I thank them both. It takes guts to speak up in front of the others. The act is not without cost to proud men. I appreciate it and I tell them.
The men surround me in the headlight-lit court. Safety lies two hundred miles east, in the dark, across country none of us knows—back valleys and passes peopled by warriors who will know where we are, how many we are, and where we are heading. Every one of us knows this, and every one feels the fear in his bones.
"Because we went back when we didn't have to," I say, "we know something about ourselves that we didn't know before. You know now, Chris, that if you fall, I won't abandon you. I'll come back, if it costs me my life—and so will Q and so will Junk and so will Chutes. And we know the same about you."
A bottle makes the rounds.
"The contract we signed says nothing about honor. The company doesn't give a shit. But I do. I fight for money, yeah—but that's not why I'm here, and it's not why any of you are here either."
From inside the compound, Col. Achmed and his sons listen. Two hours from now they'll be hunting us as if we were animals. But for this moment they know us as men, and we know them.
"What we did today in Nazirabad," I tell my brothers, "would earn decorations for valor in any army in the world. You know what I'll give you for it?"
I grab my crotch.
Chris Candelaria laughs.
Chutes follows. The whole crew shakes their heads and rocks back and forth.
You have to lead men sometimes. As unit commander, you have to put words to the bonds of love they feel but may be too embarrassed to speak of—and to the secret aspirations of their hearts, which are invariably selfless and noble. More important, you have to take those actions yourself, first and alone, that they themselves know they should take, but they just haven't figured it out yet.
GATES OF FIRE
A Bantam Book
PUBLISHING HISTORY
Doubleday hardcover edition published October 1998
Bantam mass market edition published September 1999
Bantam trade paperback edition / October 2005
Published by
Bantam Dell
A Division of Random House, Inc.
New York, New York
This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.
All rights reserved
Copyright © 1998 by Steven Pressfield
Maps copyright © 1998 by David Cain
Title page illustration copyright © 1998 by David Cain
Excerpt from The Profession copyright © 2011 by Steven Pressfield.
Library of Congress Catalog Card Number: 98-19902
* * *
Bantam Books and the rooster colophon are registered trademarks of Random House, Inc.
* * *
www.bantamdell.com
This book contains an excerpt from the forthcoming book The Profession by Steven Pressfield. This excerpt has been set for this edition only and may not reflect the final content of the forthcoming edition.
eISBN: 978-0-553-90405-5
v3.0_r1
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 1,691
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The Mini-Beadbeater-16 is a high-energy cell disrupter.
The Mini-Beadbeater-16 is a high-energy cell disrupter designed to handle four to sixteen samples (in 2ml tubes) at a time. An optional adaptor allows the use of 7ml vials (up to 8) as well.
disrupts microbial cells and plant and animal tissue by violently agitating four to sixteen 2 ml screw-cap tubes containing small ceramic or steel beads and disruption buffer.
completely homogenises resistant samples like yeast, spores or fibrous tissue are completely homogenized in around 3 minutes in 0.1 to 1 ml of extraction medium.
non-foaming, aerosol-free method preserves enzymes and organelles.
In the presence of nucleic acid extraction media such as phenol, Gu-SCN or a commercial kit solution, DNA or RNA is recovered in the highest possible yield. The method is ideal for PAGE, PCR applications and diagnostics.
Power: 230 volts, 50Hz, 3.7 amps.
Shaking pattern: Uses proven, more efficient near horizontal vial orientation with shaking motion top-to-bottom (not side-to-side or vortex-like) shaking patterns.
Capacity: four to sixteen screw-cap microvials (0.5, 1.5 and 2.0 ml) each handling 1 to 400 mg (wet weight) bio-sample.
Timer: 0-5 minute digital, with auto reset.
Recycle delay time: No imposed motor cool-down-time between each sample run.
The Mini-BeadBeater-16 uses standard screw-cap plastic microvials (snap top tubes are not recommended due the potential for aerosol contamination).
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,375
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Q: C# Cookie - Expires property won't set I have a page where a user logs in to a back-end application via a web service. The web service returns a session ID which I want to store in a cookie for 40 minutes, as after 40 minutes the back-end application automatically closes the session.
My code to write the cookie:
private void SetCookie()
{
Response.Cookies.Add(new HttpCookie("Cookie_SessionID"));
Response.Cookies["Cookie_SessionID"].Value = ni.NicheSessionID;
Response.Cookies["Cookie_SessionID"].Expires = DateTime.Now.AddMinutes(40);
//.... after a few more things
Response.Redirect(returnUrl);
}
Then on the receiving page I have this:
private HttpCookie GetCookie()
{
HttpCookie cookie = Request.Cookies["Cookie_SessionID"];
if (cookie != null && cookie.Value != null)
{
return cookie;
}
return null;
}
For some reason the cookie returned by GetCookie() always has an Expires value of 0001-01-01 00:00:00, even though when I view cookies in the browser it has the correct expiry time.
Having read this which states expired cookies are simply not sent to the server, I assume what could be happening is that the cookie is being written correctly but the browser is not sending the expiry date because it's actually unnecessary?...
My problem is that I want to capture precisely that - the cookie has 'expired' and so they have to log in again - but I need to display a message along the lines of "I know you have already logged in but you'll need to do it again" type thing.
Thanks
A: The HTTP protocol does not send cookie expiration dates to the server.
A: The browser will not send anything to the server except the cookie name and value. All of the other properties (expires, domain, path, httponly, ...) cannot be retrieved on requests after the cookie has been set.
If you want to display such a message then you will need some other mechanism of detecting that the user was logged in. You might set a presence cookie for a year or so and check to see if it exists.
The more accepted way to deal with this is to redirect the user to a login page when they try to access a protected resource and display some message along the lines of "You need to log in to view this page. If you were previously logged in, your session may have expired."
(Also note that you should be re-setting the cookie on every request, so that the user will not be logged out if they continue to use the site. It's not clear from your code whether you are doing this or not.)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,835
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Q: Crystal Reports - Change DataSource name We are using a 3rd party HR Tool (NuView) and are creating custom reports in Crystal. The source data is a csv file. In order for NuView to process the Crystal Report it REQUIRES that the:
*
*csv file name
*datasource name in Crystal
*the report name
be the same.
For example, if it was a report on ADDRESSES - Address.csv, Address#csv.fieldname, Address.rpt
Here's our dilemma - we are creating some complex reports and want to REUSE them. So, we want to take an existing CR report and copy it and then * CHANGE THE DATASOURCE * name - and we can not figure out how to to this?!?
We can change the source file (e.g., Address.csv) and the report name - but for some reason can not figure out how (or if it is possible) to change the Address#csv (which I'm calling the datasource name.
Just trying to save our dev staff some time from having to re-create very similar reports from scratch. We are using MS ACCESS/Jet to read the csv file (per NuView)
Any help is greatly appreciated.
A: From the Database Expert you can simply right click on it in the right hand pane selected tables and choose rename. Is that what you're looking for?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,113
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using System;
using System.Collections.Generic;
using System.Text;
namespace KLib
{
public class KRPCBatchRequest
{
public int requestId;
public KRPCRequest[] list_request;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,562
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67194成l人在线观看线路
A.M. TURING CENTENARY CELEBRATION WEBCAST
MORE ACM AWARDS
A.M. TURING AWARD LAUREATES BY...
YEAR OF THE AWARD
RESEARCH SUBJECT
October 11, 1943 in Newburyport, Mass.
Bachelor's degree in Electrical Engineering (Princeton University, 1965); M.Sc. in Electrical Engineering (University of Michigan, Ann Arbor, 1967); Ph.D. in Computer Science & Engineering (University of Michigan, Ann Arbor, 1971).
Assistant Professor of Computer Science (University of California at Berkeley, 1971--1976), Associate Professor (1976-1982), Professor (1982-1993), Professor of the Graduate School (1994-1999); Senior Lecturer (Massachusetts Institute of Technology, 2001-2) Adjunct Professor (2002-Present). Concurrently co-founded and held executive or advisory roles with companies including Relational Technology, Inc. (founded 1980, later Ingres Corporation), Illustra Corporation (founded 1992, later acquired by Informix where Stonebraker was Chief Technology Officer 1996-2000), Cohera Corporation (1997, acquired by PeopleSoft), StreamBase Systems (2003, acquired by Tibco in 2013), Vertica Systems (2005, acquired by HP), Goby (2008, acquired by Telenauv in 2012), SciDB (2008), VoltDB (2009), and Tamr (2013).
ACM System Software Award (1992); ACM SIGMOD Innovation Award (1994); National Academy of Engineering (elected 1998); IEEE John von Neumann Medal (2005); Alan M. Turing Award (2014).
Michael Stonebraker
United States – 2014
For fundamental contributions to the concepts and practices underlying modern database systems.
Short Annotated
ACM Turing Award
Lecture Video
Michael Stonebraker's contributions to the refinement and spread of database management technology are hard to overstate. He began work in this area as a young assistant professor at the University of California—Berkeley. After reading Edgar F. Codd's seminal papers on the relational model, Stonebraker started work with a colleague, Eugene Wong, to develop an efficient and practical implementation.
Stonebraker explains how he shifted his focus to database research after discovering the work of Ted Codd.
The result was INGRES, a name that reflected the project's original intention to produce a geographically-oriented system with graphical capabilities. This officially stood for "Interactive Graphic and Retrieval System" but echoed the name of a celebrated French painter.
A prototype of INGRES was working by 1974, but the project did not stop there. Over the next decade INGRES, and systems inspired by it, built a new commercial market of relational database systems. Today the relational database management system is one of computing's most important and widely used technologies, having replaced filing cabinets as the standard way of storing and retrieving information.
Development of INGRES
Stonebraker led development of INGRES at Berkeley until 1985, supported by grant money and the labor of graduate and undergraduate students. Berkeley was particularly notable during this era as a place where theoretical research and system building came together with spectacular results. Further examples included the work on timesharing systems by Butler Lampson (winner 1992) and others and the Berkeley Software Distribution (BSD) of the Unix operating system, which gave rise to a commonly used form of open source licensing. These cultures and practices anticipated much of what we now associate with the open source software movement. Stonebraker remembers that "we would recruit the smartest freshmen and sophomores we could find, give them wonderful equipment, and they would basically die writing code for us."
Stonebraker partnered with Eugene Wong to develop INGRES, one of the first relational database management systems.
Stonebraker's work built on, and complemented, that of three other Turing award winners. Academic research into database management technology has had an unusually direct connection to the widely used industrial-strength systems underlying the websites, business applications, scientific breakthroughs, social media systems, and "big data" projects of the modern world. Charles W. Bachman (winner 1973) designed what is often called the first database management system in the early 1960s, and helped to define and popularize the concept of a database management system through his later work with the industry group CODASYL. Edgar F. Codd (winner 1981) developed an elegant and flexible way of storing and retrieving data, the relational model, which gradually eclipsed the network data model over the course of the 1980s. James Nicholas Gray (winner 1988) contributed to IBM's System R, an influential experimental implementation of the relational model, and later pioneered robust, high performance methods for record locking and transaction processing.
Legacy of INGRES
INGRES and System R together helped to turn relational systems from a laboratory curiosity into the default choice for even the most demanding data processing applications. While the IBM prototype targeted the company's multi-million dollar mainframes, INGRES was a Unix application suitable for relatively affordable minicomputers and was widely distributed to other universities where people used it, experimented with it, and extensively modified it.
INGRES brought a new kind of database technology to a new kind of computer. Database management systems were widely adopted by businesses from the early 1970s onwards as central hubs which managed the data used by many different application programs. These early commercial systems ran on mainframes and followed either Bachman's network model or a more restrictive hierarchical approach favored by IBM. In the mainframe world these approaches remained dominant throughout the 1980s so that, for example, IBM first commercialized its work in the area as a niche product for "decision support" analytical applications rather than workaday operational systems.
During the 1970s, minicomputers became a cost-effective alternative to mainframes for an ever widening range of applications. Thanks to INGRES and its derivatives, relational technology became the default choice for minicomputer databases, as the new technology was widely applied to transaction processing applications (keeping routine records of things like address changes or account updates) as well as analytical work. The commercial database systems of the 1970s required their users to navigate through data structures at a relatively low level, making explicit decisions about how to index and link records when the database was created and navigating record by record through these structures when retrieving information. Relational database systems shifted to a more abstract and flexible view of data. Only when querying the database did users specify how data from different tables should be combined. This shifted much of the responsibility for efficiently organizing and retrieving data from the user to the database management software, pushing hard against the limits of affordable hardware.
INGRES was a feat of virtuoso software engineering, prioritizing performance and reliability so that new features were added only once a way of implementing them efficiently had been discovered. By 1976 INGRES was rapidly executing queries written in its QUEL query language (roughly equivalent to the SEQUEL, later SQL, language introduced by IBM). It could be embedded in C programs or used interactively. Under the hood, INGRES implemented a variety of indexing and compression methods, automatically optimizing queries. The team had already begun to add support for transactions, so that related updates would occur together--or not at all--to enforce integrity constraints between related records in different tables, and to deal with the potential problems caused by simultaneous updates from different users. Additional features, such as crash recovery and efficient backup and restore capabilities, turned INGRES from a research project to an industrial-strength technology. This took a huge amount of additional work. As Stonebraker recalled, "We built an initial prototype, putting in the first 90% of the effort required to create a real system, and it more or less worked. I think that the thing that distinguished INGRES from the typical academic project, and in retrospect one of the smartest things we ever did, was to then put in the next 90% of the effort to make INGRES really work."
In the late 1970s, Stonebraker set up a company to support the commercial development of INGRES.
Students trained on the INGRES project, and in many cases using the INGRES code itself as a starting point, produced most of the leading minicomputer database packages. These included Britton-Lee (an early supplier of specialized parallel processing database management systems), the NonStop SQL product offered by Tandem Computers, and Sybase (whose SQL Server was later licensed by Microsoft). In 1980 Stonebraker himself co-founded Relational Technology, Inc. to produce its own commercial version of INGRES. His involvement with the firm was primarily as a consultant, though he worked there full time for around six months. It was a significant player in the database software market over the next decade, making an initial public offering in 1988 before being acquired in 1990.
By this point Stonebraker was already immersed in the development and commercialization of a successor system. Postgres added many features missing from existing relational systems, including support for rules to maintain consistent relationships between tables, support for complex "object-relational" data types, the replication of data across servers, and procedural languages to embed code fragments within the database management system to be triggered when specified conditions occured.
Stonebraker explains the goals of Postgres, the follow-up to INGRES.
Postgres was also used to experiment with other features of interest to database researchers. Techniques pioneered in Postgres were widely implemented, and in 1992 Stonebraker cofounded Illustra Information Technologies to market a commercial version. It was acquired in 1997 by Informix, which rebuilt its product line around the code.
Entrepreneurial Career
Stonebraker retired from Berkeley in 1994, retaining a connection as a "Professor of the Graduate School." In 1999 he moving to New Hampshire, soon taking up an adjunct appointment at MIT where he could focus on developing and commercializing new technologies without the obligation of regular faculty responsibilities. Since then he has cofounded a company every few years, focusing on the development of database management technologies specialized for particular areas such as data warehousing (Vertica), managing data streams captured by sensors (StreamBase Systems), and high-throughput transaction processing (VoltDB). However one of his latest ventures, SciDB, which focuses on handling massive arrays of scientific data, departs from the relational model as well as from traditional general purpose implementation techniques.
As an eloquent and authoritative commentator on trends in database technology, Stonebraker has defended the enduring power of the relational model against efforts by the "NoSQL" movement to promote the superiority of "post-relational" approaches. At the same time, he has been critical of the assumption that "one size fits all" when implementing relational database management systems and that dominant general purpose systems, such as Oracle, can serve the needs of all users.
Stonebraker is the only Turing award winner to have engaged in serial entrepreneurship on anything like this scale, giving him a distinctive perspective on the academic world. The connection of theory to practice has often been controversial in database research, despite the foundational contribution of mathematical logic to modern database management systems. Stonebraker has been critical of the insularity of some researchers, noting the attention given to such ideas as recursive querying or object-oriented databases suggests that "they are more interested in working on problems that are solvable, rather than problems that are important." His "advice to theoreticians" was "go spend some time in the real world and work on problems that people want solved." In contrast, "Knowing what I know now, I would never have started building INGRES, because it's too hard…. So I think my advice to my younger self would be to suspend your disbelief and just do it anyway. The way you climb Mt. Everest is one step at a time…"
(Quotations from Stonebraker are taken from his interview with Marianne Winslett, published in ACM SIGMOD Record, Vol.32, No. 2, June 2003 as "Michael Stonebraker Speaks Out.")
Author: Thomas Haigh
A.M. TURING
OF THE AWARD
2019 LAUREATES:
EDWIN E. CATMULL
PATRICK M. HANRAHAN
THE A.M. TURING AWARD
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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