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Barycentric subdivision is an example of a subdivision rule with one edge type ( that gets subdivided into two edges ) and one tile type ( a triangle that gets subdivided into 6 smaller triangles ) . Any triangulated surface is a barycentric subdivision complex .
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The Penrose tiling can be generated by a subdivision rule on a set of four tile types ( the curved lines in the table below only help to show how the tiles fit together ) :
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Certain rational maps give rise to finite subdivision rules . This includes most Lattès maps .
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Every prime , non @-@ split alternating knot or link complement has a subdivision rule , with some tiles that do not subdivide , corresponding to the boundary of the link complement . The subdivision rules show what the night sky would look like to someone living in a knot complement ; because the universe wraps aroun... |
The subdivision rule looks different for different geometries . This is a subdivision rule for the trefoil knot , which is not a hyperbolic knot :
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And this is the subdivision rule for the Borromean rings , which is hyperbolic :
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In each case , the subdivision rule would act on some tiling of a sphere ( i.e. the night sky ) , but it is easier to just draw a small part of the night sky , corresponding to a single tile being repeatedly subdivided . This is what happens for the trefoil knot :
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And for the Borromean rings :
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= = Subdivision Rules in Higher Dimensions = =
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Subdivision rules can easily be generalized to other dimensions . For instance , barycentric subdivision is used in all dimensions . Also , binary subdivision can be generalized to other dimensions ( where hypercubes get divided by every midplane ) , as in the proof of the Heine @-@ Borel theorem .
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= = Rigorous definition = =
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A finite subdivision rule <formula> consists of the following .
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1 . A finite 2 @-@ dimensional CW complex <formula> , called the subdivision complex , with a fixed cell structure such that <formula> is the union of its closed 2 @-@ cells . We assume that for each closed 2 @-@ cell <formula> of <formula> there is a CW structure <formula> on a closed 2 @-@ disk such that <formula> h... |
2 . A finite two dimensional CW complex <formula> , which is a subdivision of <formula> .
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3.A continuous cellular map <formula> called the subdivision map , whose restriction to every open cell is a homeomorphism .
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Each CW complex <formula> in the definition above ( with its given characteristic map <formula> ) is called a tile type .
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An <formula> -complex for a subdivision rule <formula> is a 2 @-@ dimensional CW complex <formula> which is the union of its closed 2 @-@ cells , together with a continuous cellular map <formula> whose restriction to each open cell is a homeomorphism . We can subdivide <formula> into a complex <formula> by requiring t... |
Binary subdivision is one example :
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The subdivision complex can be created by gluing together the opposite edges of the square , making the subdivision complex <formula> into a torus . The subdivision map <formula> is the doubling map on the torus , wrapping the meridian around itself twice and the longitude around itself twice . This is a four @-@ fold... |
= = Quasi @-@ isometry properties = =
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Subdivision rules can be used to study the quasi @-@ isometry properties of certain spaces . Given a subdivision rule <formula> and subdivision complex <formula> , we can construct a graph called the history graph that records the action of the subdivision rule . The graph consists of the dual graphs of every stage <f... |
The quasi @-@ isometry properties of the history graph can be studied using subdivision rules . For instance , the history graph is quasi @-@ isometric to hyperbolic space exactly when the subdivision rule is conformal , as described in the combinatorial Riemann mapping theorem .
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= = Applications = =
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Islamic Girih tiles in Islamic architecture are self @-@ similar tilings that can be modeled with finite subdivision rules . In 2007 , Peter J. Lu of Harvard University and Professor Paul J. Steinhardt of Princeton University published a paper in the journal Science suggesting that girih tilings possessed properties c... |
Subdivision surfaces in computer graphics use subdivision rules to refine a surface to any given level of precision . These subdivision surfaces ( such as the Catmull @-@ Clark subdivision surface ) take a polygon mesh ( the kind used in 3D animated movies ) and refines it to a mesh with more polygons by adding and sh... |
Subdivision rules were applied by Cannon , Floyd and Parry ( 2000 ) to the study of large @-@ scale growth patterns of biological organisms . Cannon , Floyd and Parry produced a mathematical growth model which demonstrated that some systems determined by simple finite subdivision rules can results in objects ( in thei... |
= = Cannon 's conjecture = =
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Cannon , Floyd , and Parry first studied finite subdivision rules in an attempt to prove the following conjecture :
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Cannon 's conjecture : Every Gromov hyperbolic group with a 2 @-@ sphere at infinity acts geometrically on hyperbolic 3 @-@ space .
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Here , a geometric action is a cocompact , properly discontinuous action by isometries . This conjecture was partially solved by Grigori Perelman in his proof of the Geometrization conjecture , which states ( in part ) than any Gromov hyperbolic group that is a 3 @-@ manifold group must act geometrically on hyperbolic... |
Cannon and Swenson showed that a hyperbolic group with a 2 @-@ sphere at infinity has an associated subdivision rule . If this subdivision rule is conformal in a certain sense , the group will be a 3 @-@ manifold group with the geometry of hyperbolic 3 @-@ space .
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= = Combinatorial Riemann Mapping Theorem = =
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Subdivision rules give a sequence of tilings of a surface , and tilings give an idea of distance , length , and area ( by letting each tile have length and area 1 ) . In the limit , the distances that come from these tilings may converge in some sense to an analytic structure on the surface . The Combinatorial Riemann... |
Its statement needs some background . A tiling <formula> of a ring <formula> ( i.e. , a closed annulus ) gives two invariants , <formula> and <formula> , called approximate moduli . These are similar to the classical modulus of a ring . They are defined by the use of weight functions . A weight function <formula> assi... |
<formula>
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<formula> .
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Note that they are invariant under scaling of the metric .
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A sequence <formula> of tilings is conformal ( <formula> ) if mesh approaches 0 and :
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For each ring <formula> , the approximate moduli <formula> and <formula> , for all <formula> sufficiently large , lie in a single interval of the form <formula> ; and
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Given a point <formula> in the surface , a neighborhood <formula> of <formula> , and an integer <formula> , there is a ring <formula> in <formula> separating x from the complement of <formula> , such that for all large <formula> the approximate moduli of <formula> are all greater than <formula> .
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= = = Statement of theorem = = =
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If a sequence <formula> of tilings of a surface is conformal ( <formula> ) in the above sense , then there is a conformal structure on the surface and a constant <formula> depending only on <formula> in which the classical moduli and approximate moduli ( from <formula> for <formula> sufficiently large ) of any given a... |
= = = Consequences = = =
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The Combinatorial Riemann Mapping Theorem implies that a group <formula> acts geometrically on <formula> if and only if it is Gromov hyperbolic , it has a sphere at infinity , and the natural subdivision rule on the sphere gives rise to a sequence of tilings that is conformal in the sense above . Thus , Cannon 's conj... |
= Ash Crimson =
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Ash Crimson ( アッシュ ・ クリムゾン , Asshu Kurimuzon ) is a video game character in The King of Fighters fighting game series developed by SNK Playmore . His first appearance was in The King of Fighters 2003 as leader of its Hero Team . Ash , a teenager , participates in the series ' fighting tournaments . He employs a person... |
SNK Playmore staff created Ash as an evil main character , in contrast to previous main characters . His moves have been reworked in every game in which he has appeared , gaining new techniques as a result of his actions or modified to balance the character roster . Critical response to Ash has been largely negative d... |
= = Appearances = =
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Introduced in The King of Fighters 2003 , Ash leads the New Hero Team of Duo Lon and Shen Woo which participates in a new King of Fighters tournament . After the tournament , he ambushes host Chizuru Kagura and steals her powers , telling Iori Yagami he will be his next opponent . In The King of Fighters XI , the char... |
In The King of Fighters XIII Ash is the only character not part of a team , and it is learned that he works for Those From the Past ( seen in 2003 and XI ) . However , he has a conflict with organization leader Saiki at the end of the game . Ash is the final boss in The King of Fighters XIII as " Ash driven insane by ... |
Ash has made appearances in various other King of Fighters media as well . In the fourth chapter of the anime The King of Fighters : Another Day , he sets fire to a city to lure Kyo into a trap . He is also prominent in the manhua adaptations of The King of Fighters 2003 by Wing Yang and King Tung . In the 2003 tourna... |
= = Creation and development = =
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Ash was designed as an " attractive evil character " , in contrast to previous King of Fighters heroes . The supervising designer created the character as desired , with few changes since conception . When The King of Fighters 2003 was released and Ash introduced , the staff did not want to disclose information about ... |
In The King of Fighters XII Ash was the character the staff worked on the most , reworking his movements and speech to be more consistent with the rest of the cast . The King of Fighters XIII producer Kei Yamamoto jokingly called him a character players could use to " show off " . His EX version in the game makes more... |
= = Reception = =
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Ash Crimson 's character had a mixed reception in video @-@ game publications . When he was introduced , GameSpy 's Christian Nutt called him a " clone " of Guile from the Street Fighter series because of their similar movesets but Ash 's team was praised as new characters with new elements in the series . Lucas M. Th... |
In an interview with Ignition Entertainment director of business development Shane Bettehausen , Alex Lucard of Diehard GameFan said that North American SNK fans detested Ash and complained about his inclusion in The King of Fighters XII without a storyline while popular series characters were overlooked . Bettehausen... |
= Cunard Building =
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Not to be confused with the 1921 Cunard Building ( New York City )
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The Cunard Building is a Grade II * listed building in Liverpool , England . It is located at the Pier Head and along with the neighbouring Liver Building and Port of Liverpool Building is one of Liverpool 's Three Graces , which line the city 's waterfront . It is also part of Liverpool 's UNESCO designated World Her... |
It was designed by William Edward Willink and Philip Coldwell Thicknesse and was constructed between 1914 and 1917 . The building 's style is a mix of Italian Renaissance and Greek Revival , and its development has been particularly influenced by Italian palace design . The building is noted for the ornate sculptures ... |
The building was , from its construction until the 1960s , the headquarters of the Cunard Line , and the building still retains the name of its original tenants . It was also home to Cunard 's passenger facilities for trans @-@ Atlantic journeys that departed from Liverpool . Today , the building is owned by the Merse... |
= = History = =
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In 1914 the Cunard Steamship Company commissioned the construction of new headquarters for the company . Cunard 's expansion had meant that they had outgrown their previous offices , which were also in Liverpool , and the site chosen for construction was at the former George 's Dock , in between the Liver Building and... |
In 1934 the Cunard Steamship Company merged with the White Star Line to form Cunard White Star Line , which became the largest passenger steamship company in the world , helping to make Liverpool one of the most important centres of the British trans @-@ Atlantic ocean liner industry . The Cunard building subsequently... |
The building remained the headquarters of Cunard until the 1960s , when they decided to relocate their UK operations to Southampton on England 's south coast and their global headquarters to New York . Cunard subsequently sold the building to Prudential plc in 1969 . In 1965 the Cunard Building was awarded Grade II * ... |
In October 2013 Liverpool City Council approved the acquisition of the Cunard Building for use as offices and as a cruise liner terminal . The council projected that the building would accommodate 1000 staff relocated from Millennium House and leases in the Capital Building , saving an estimated £ 1 @.@ 3 million . Th... |
= = Architectural design = =
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