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Expectation values of the spin operator [MATH] are obtained by inserting the matrix [MATH] as the [MATH] ’th matrix in the string of transfer matrices in the trace, with [MATH] given by |
[EQUATION] We are interested in calculating correlation functions of spins across a series of disordered bonds at sites [MATH] and [MATH] , where we presume [MATH] . Exploiting the cyclicity of the trace for bounded operators and the fact that [MATH] and [MATH] commute, we can write |
[EQUATION] where it is convenient to work in the basis where the [MATH] and [MATH] are simultaneously diagonalized, in which case |
[EQUATION] Straightforward calculations then show that [MATH] and with multiple disordered bonds between the sites [MATH] and [MATH] |
[EQUATION] where the [MATH] correspond to the intervening disordered bonds, and where the correlation length [MATH] one would have obtained in the absence of disorder, which is given by |
[EQUATION] Note that the fixed points of ( 18 ) correspond to the disorder parameter settling on [MATH] , so that the factors in the product return [MATH] , where [MATH] denotes the number of bonds corresponding to [MATH] . This has the transparent physical interpretation of preserving the usual correlation up to a sig... |
# Source: arxiv 1805.12579 # Title: Algorithms and Geometric Constructions # Sections: all # Downloaded: 2026-03-03T02:44:56.773033+00:00 |
Algorithms and Geometric Constructions Abstract It is well known that several classical geometry problems (e.g., angle trisection) are unsolvable by compass and straightedge constructions. But what kind of object is proven to be non-existing by usual arguments? These arguments refer to an intuitive idea of a geometric ... |
Introduction The notion of an algorithm as an intuitively clear notion that precedes any formalization, has a rather short history. The first examples of what we now call algorithms were given already by Euclid and al-Khwârizmî. But the general idea of an algorithm seems to appear only in 1912 when Borel considered “le... |
, p. 162] . The formal definition of a representative class of algorithms was given in 1930s (in the classical works of Gödel, Church, Kleene, Turing, Post and others); the famous Church–Turing thesis claims that the class of algorithms provided by these formal definitions is representative (for each algorithm in the i... |
In this paper we look at the history of another related notion: the notion of a geometric construction . One may consider geometric constructions as a special type of algorithms that deal with geometric objects. Euclid provided many examples of geometric constructions by compass and straightedge (ruler); later these co... |
However, historically this was not the case and the impossibility proofs appeared without an exact definition of a “geometric construction”. These proofs used the algebraic approach: For example, to show that the cube cannot be doubled, one proves that [MATH] cannot be obtained from rationals by arithmetic operations a... |
, “pour reconnaitre si la construction d’un problème de Géometrie peut s’effectuer avec la règle et le compas, if faut chercher s’il est possible de faire dépendre les racines de l’equation à laquelle il conduit de celles d’un système d’équations du second degré”. This is said in the first paragraph of the paper and th... |
Several other interesting results were obtained in 19th century. It was shown that all constructions by compass and straightedge can be performed using the compass only (the Mohr–Mascheroni theorem ) if we agree that a line is represented by a pair of points on this line. Another famous result from 19th century, the Po... |
Later geometric construction became a popular topic of recreational mathematics (see, e.g., ). In most of the expositions the general notion of a geometric construction is still taken as granted, without a formal definition, even in the nonexistence proofs (e.g., when explaining Hilbert’s proof that the center of a cir... |
; see below Section about problems with this argument). Sometimes a definition for some restricted class of geometric construction is given (see, e.g., |
). In an attempt to provide a formal definition is made, still it remains ambiguous with respect to the use of “arbitrary points” (see Section ). Baston and Bostock |
observe that the intuitive idea of a “geometric construction” has no adequate formal definition and discuss several examples but do not attempt to give a formal definition that is close to the intuitive notion. It seems that even today people still consider the intuitive notion of a “geometric construction algorithm” a... |
, especially the first arxiv version). In Section we consider a naïve approach that identifies constructible points with the so-called “derivable” points. Then in Sections and we explain why this approach contradicts our intuition. In Section we suggest a more suitable definition, and finally in Section we note that th... |
Derivable Points and Straight-Line Programs At first it seems that the definition of a geometric construction is straightforward. We have three classes of geometric objects: points, lines and circles. Then we consider some operations that can be performed on these objects. We need to obtain some object (the goal of our... |
puts it, Formally, one can set up the problem as follows. Define a configuration to be a finite collection [MATH] of points, lines, and circles in the Euclidean plane. Define a construction step to be one of the following operations to enlarge the collection [MATH] |
(Straightedge) Given two distinct points [MATH] [MATH] in [MATH] , form the line [MATH] that connects [MATH] and [MATH] , and add it to [MATH] |
(Compass) Given two distinct points [MATH] [MATH] in [MATH] , and given a third point [MATH] in [MATH] (which may or may not equal [MATH] or [MATH] ), form the circle with centre [MATH] and radius equal to the length [MATH] of the line segment joining [MATH] and [MATH] , and add it to [MATH] |
(Intersection) Given two distinct curves [MATH] [MATH] in [MATH] (thus [MATH] is either a line or a circle in [MATH] , and similarly for [MATH] ), select a point [MATH] that is common to both [MATH] and [MATH] (there are at most two such points), and add it to [MATH] |
We say that a point, line, or circle is constructible by straightedge and compass from a configuration [MATH] if it can be obtained from [MATH] after applying a finite number of construction steps. |
We can even try to define the geometric construction algorithm as a straight-line program, a sequence of assignments whose left-hand side is a fresh variable and the right-hand side contains the name of the allowed operation and the names of objects (variables) to which this operation is applied. A definition of this t... |
where this type of programs is called [MATH] Baston and Bostock use the name “derivable” for objects that can be obtained in this way starting from given objects. In other words, starting with some set of given objects, they consider its closure, i.e., the minimal set of objects that contains the given ones and is clos... |
Baston and Bostock note that the intuitive notion of a “constructible” point (that they intentionally leave without any definition) may differ from the formal notion of a derivable point in both directions. We discuss the possible differences in the following sections. |
Uniformity and Tests There are some problems with the approach based on the derivability. First of all, this appoach is “non-uniform”. Asking a high school student to construct, say, a center of an inscribed circle of a triangle [MATH] , we expect the solution to be some specific construction that works for all triangl... |
includes two operations in his list of “operating capabilities”: (6) [MATH] , choose an intersection point of two intersecting circles; |
(7) [MATH] , given an intersection point of two circles, find the other intersection point. (p. 65; similar operations for lines and circles are also in the list.) In this way the straight-line program is no more deterministic. We may guarantee only that some run of the program produces the required object, or guarante... |
Bieberbach , p. 26] describes this problem as follows: Noch mag aber ausdrücklich hervorgehoben werden, daß es sich in diesem Paragraphen ebenso wie bei den Poncelet–Steinerschen Konstruktionen stets um ein Konstruieren in der orientierten Ebene handelt. Es soll bei jedem gegebenen und bei jedem konstruierten Punkt fes... |
Bieberbach then adds (on p. 151): H.Tietze hat (l.c.,§4) zu dem Ergebnis dieses Paragraph die folgende Bemerkung gemacht: Bei der Ausführung der Konstruktion erhält man im Schnitt eines Kreises mit einer Geraden oder einem anderen Kreis stets mehrere Punkte. Bei der Fortsetzung der Konstruktion bedarf es dann noch eine... |
Indeed, Tietze studied this question in several papers, starting from 1909 . He noted the necessity of ordering tests for standard constructions already in |
amd improves the exposition in . Then 24 , pp. 228–229] he considers also the question: which distances can be constructed without ordering tests? Tietze notes that one can construct a distance that is [MATH] times longer that a given one (just consider the distance between the intersection points of two circles with c... |
One could give up and consider the non-uniform setting only. As Manin 16 , p. 209] puts it, “we ignore how to choose the required point from the set of points obtained by the construction” ( <<остаётся в стороне вопрос о [MATH] выборе из построенной совокупности точек>> ). Another approach is to replace straight-line p... |
To save the Mohr–Mascheroni construction, one may consider programs that allow loops. This was suggested, e.g., by Engeler . Here we should specify what kind of data structures are allowed (e.g., whether we allow dynamic arrays of geometric objects or not). In this way we encounter another problem, at least if we consi... |
observed that having four different points [MATH] in a general position (no three points lie on a line, no two connecting lines are parallel), we can enumerate all (rational) points and therefore all rational lines. Then we can wait until a line parallel to [MATH] appears (we assume that we may test whether two given l... |
for details. Arbitrary Points Let us now consider the other (and probably more serious) reason why the notion of a derivable object differs from the intuitive notion of a constructible object. Recall the statement about angle trisection as stated by Tao |
: for some triangle [MATH] the trisectors of angle [MATH] are not derivable from [MATH] . (Tao uses the word “constructible”, but we keep this name for the intuitive notion, following |
.) Tao interprets this statement as the impossibility of angle trisection with a compass and straightedge, and for a good reason. |
On the other hand, the center of a circle is not derivable from the circle itself, for the obvious reason that no operation can be applied to enlarge the collection that consists only of the circle. Should we then say that the center of a given circle cannot be constructed by straightedge and compass? Probably not, sin... |
Looking at the corresponding standard constructions, we notice that they involve another type of steps, “choosing an arbitrary point” (on the circle or elsewhere). But we cannot just add the operation “add an arbitrary point” to the list of allowed operations, since all points would become derivable. So what are the “a... |
Tietze does not consider the problem of arbitrary points; in he writes: 18. Was wir nicht in unsere Betrachtungen einbezogen haben, sind willkürlich gewählte Elemente, wie sie bei manchen Konstruktionen eine Rolle spielen. Diese Rolle ist aber nicht ganz so einfach, als es bisweilen dargestellt wird, weil beispielsweis... |
Baston and Bostock do not even attempt to give a formal definition of a constructible point. However, they acknowledge that the use of “arbitrary” points in necessary for some constructions, and make some (rather vague, to be honest) remarks about that (p. 1018): |
Result 1. Given three distinct collinear points [MATH] [MATH] , and [MATH] such that [MATH] is the midpoint of the segment [MATH] , then it is possible using a ruler alone to construct the midpoint [MATH] of the segment [MATH] |
Here we see that [MATH] [the left-hand side is the closure of points [MATH] [MATH] [MATH] ] so, although the midpoint [MATH] is meant to be constructible from the set [MATH] , it is not actually derivable. Of course the constructibility of [MATH] depends on the (not unreasonable) ability to choose arbitrary points not ... |
The footnote “ ” says “The interested reader may wish to consult [4,p. 79] where an elementary approach to the use of arbitrary points can be found.” The reference points to the book |
; however, the corresponding explanations (p. 46 of the Russian original version) are also far from being clear: One often uses arbitrary elements in the geometric constructions. The option of adding arbitrary elements to the current configuration needs to be restricted, and there restrictions are usually formulated as... |
Restriction A One may consider an arbitrary point of the plane outside the given line as constructed Restriction B One may consider an arbitrary point on a given line that differs from the already constructed points in this line, as constructed |
The items A and B are essential for many construction problems. [MATH] Of course, when proving the correctness of a construction that uses arbitrary elements we should not use any special properties of these elements and should consider them as essentially arbitrary points. |
Another popular exposition 13 , p. 18] explains the use of arbitrary elements (in the context of finding the center of a given circle by a straightedge only) as follows: |
What is a construction by straightedge alone? It is a finite succession of steps, each requiring that either a straight line be drawn, or a point of intersection of two lines or of a line and the given circle be found. A straight line can be drawn through two points chosen more or less arbitrarily. For example, a step ... |
Both quotes speak about proofs that the constructed point has the desired properties (though do not specify what kind of proofs they have in mind). This is an old theme that goes back to Hilbert. In his classical book on the foundations of geometry |
there is a chapter “Geometrical constructions based upon the axioms I–V” that speaks, e.g., about “those problems in geometrical constructions, which may be solved by the assistance of only the axioms I–V”, but there are no exact definitions of what does it mean. |
Probably the most detailed exposition of the role of arbitrary points is given by Manin in (an encyclopedia of elementary mathematics addressed to advanced high school students and teachers): |
It is convenient to describe the construction process inductively. We start with a finite set of points of the plane and want to obtain another finite set of points. The construction process adds some new points to the existing ones, and then selects the answer to our problem among the points constructed. The second (s... |
By a construction step we mean the process of adding one new point. To find this new point, we perform some operations. B y d e f i n i t i o n , a compass and straightedge construction may use only the following operations (note that operations 1 and 2 can be used several times while operation 3 is used once during th... |
1. Drawing a line through two points from a current set of points . (By the current set of points we mean the result of the preceding construction step or the initial set for the first step.) |
2. Drawing a circle whose center is a point from the current set that contains some other point from the current set. [Manin does not allow to draw a circle that has center [MATH] and radius [MATH] where [MATH] are already constructed, but this does not matter much.] [MATH] |
3. Choosing one intersection points of the lines and circles constructed and adding this point to the current set. Instead of operation 3 where some s p e c i f i c point is added, we sometimes need to use the following operation |
3a. Choosing an “arbitrary” point and adding it to the current set. One should specify what do we mean by “arbitrary” point. By definition, it means that the point can be “arbitrarily” chosen either on some line segment, or arc of a circle, or in some part of the plane bounded by segments or arcs (this part can also be... |
A compass and straightedge construction is a finite sequence of steps described above [MATH] Summarizing, let us assume that a finite set of points of the plane is given. We say that a point [MATH] |
is constructible (by compass and straightedge), if there exists a construction such that the resulting set of points contains [MATH] for all possible intermediate “arbitrary” choices. A construction problem is solvable if it requires to find a set of points where each points is constructible by compass and straightedge... |
Note that, being literally understood, this definition makes no sense. Indeed, a “construction” that should exist is a finite sequence of steps as described, so it (if understood in a natural sense) determines the “arbitrary” points that were added during operations of type 3a. So we cannot add the quantifier “for all ... |
How can we modify the definitions to make them rigorous? One of the possibilities is to consider the construction as a strategy in some game with explicitly defined rules. We discuss this approach in the next section. |
Game Definition The natural interpretation of the “arbitrary choice” is that the choice is made by an adversary. In other words, we consider a game with two players, Alice and Bob. We start with the non-uniform version of this game. |
Let [MATH] be some finite set of geometric objects (points, lines, and circles). To define which objects [MATH] are constructible starting from [MATH] , consider the following full information game. The position of the game is a finite set of geometric objects. The initial position is [MATH] . During the game, Alice an... |
Here are possible request types. Alice may ask Bob to add to the current position some straight line that goes through two different points from the current position. |
Alice may ask Bob to add to the current position a circle with center [MATH] and radius [MATH] , if [MATH] are points from the current position. |
Alice may ask Bob to add to the current position one or two points that form the intersection of two different objects (lines or circles) that already belong to the current position. |
If we stop here, we get exactly the notion of derivable points, though in a strange form of a “game” where Bob has no choice. To take the “arbitrary” points into account, we add one more operation: |
Alice specifies an open subset of the plane (say, an open circle), and Bob adds some point of this subset to the current position. |
The game (depending on [MATH] and [MATH] ) is now described, and the point [MATH] is constructible from [MATH] if Alice has a winning strategy in this game. |
Let us comment on the last operation. (1) Note that Alice cannot (directly) force Bob to choose some point on a line or on a circle, and this is often needed in the standard geometric constructions. But this is inessential since Alice can achieve this goal in several steps. First she asks to add points on both sides of... |
(2) On the other hand, according to our rules, Alice can specify with arbitrarily high precision where the new point should be (by choosing a small open set). A weaker (for Alice) option would be to allow her to choose a connected component of the complement of the union of all objects in the current position. Then Bob... |
Proposition 1 This restriction does not change the notion of a constructible point. Proof. Idea: Using the weaker option, Alice may force Bob to put enough points to make the set of derivable points dense, and then use the first three options to get a point in an arbitrary open set. |
Let us explain the details. First, she asks for an arbitrary point [MATH] , then for a point [MATH] that differs from [MATH] , then for line [MATH] , then for a point [MATH] outside line [MATH] (thus getting the triangle [MATH] ), then for the sides of this triangle, and then for a point [MATH] inside the triangle. (Al... |
Now the points [MATH] and [MATH] obtained as shown in the picture are derivable (after the projective transformation that moves [MATH] and [MATH] to infinity, the points [MATH] and [MATH] become the midpoints of [MATH] and [MATH] ). Repeating this construction, we get a dense set of derivable points on intervals [MATH]... |
Now, instead of asking Bob for a point in some open set [MATH] , Alice may force him to include one of the derivable points (from the dense set discussed above) that is in [MATH] |
This definition of constructibility turns out to be equivalent to the negative definition suggested by Akopyan and Fedorov . They define non-constructibility as follows: an object [MATH] is non-constructible from a finite set [MATH] of objects if there exists a set [MATH] that is closed under the operations of adding p... |
Proposition 2 (Akopyan–Fedorov) This negative definition is equivalent to the game-theoretic definition given above. Proof. The equivalence is essentially proven as , Proposition 15, p. 9] , but Akopyan and Fedorov avoided stating explicitly the game-theoretic definition and spoke about “algorithms” instead (without an... |
Assume that [MATH] is non-constructible from [MATH] according to the negative definition. Then Bob can prevent Alice from winning by always choosing points from [MATH] when Alice asks for a point in an open set. Since [MATH] is dense, these points are enough. If Bob follows this strategy, then the current position will... |
On the other hand, assume that [MATH] in not constructible from [MATH] in the sense of the positive definition. Consider the following strategy for Alice. She takes some triangle [MATH] and point [MATH] inside it and asks Bob to add points [MATH] that belong to some small neighborhoods of [MATH] respectively. The size ... |
for every choice of Bob Alice has a winning strategy in the remaining game; there are some points [MATH] such that Alice does not have a winning strategy in the remaining game. |
In the first case Alice has a winning strategy in the entire game and [MATH] is constructible in the sense of the positive definition. In the second case we consider the set [MATH] of all objects derivable from [MATH] . As we have seen in the proof of the previous proposition, this set is dense. Therefore, [MATH] is no... |
The advantage of the game definition is that it can be reasonably extended to the uniform case. For the uniform case the game is no more a full-information game. Alice sees only the names (and types) of geometric objects in [MATH] , and assigns names to new objects produced by Bob. One should agree also how Alice can g... |
Formal Definitions Are Important In fact, the absence of formal definitions and exact statements is more dangerous than one could think. It turned out that some classical and well known arguments contain a serious gap that cannot be filled without changing the argument. This happened with a proof (attributed to Hilbert... |
) that one cannot find the center of a given circle using only a straightedge. It is reproduced in many popular books (see, e.g., |
) and all the arguments (at least in the four sources mentioned above) have the same gap. They all go as follows 13 , p. 18] Let the construction be performed in a plane [MATH] and imaging a transformation or mapping [MATH] of the plane [MATH] into another plane [MATH] such that: |
(a) straight lines in [MATH] transform into straight lines in [MATH] [MATH] (b) The circumference [MATH] of our circle is transformed into a circumference [MATH] for some circle in [MATH] |
As the steps called for in the construction are being performed in [MATH] , they are being faithfully copied in [MATH] . Thus when the construction in [MATH] terminates in the centre [MATH] of [MATH] , the “image” construction must terminate in the centre [MATH] of the circle [MATH] |
Therefore if one can exhibit a transformation [MATH] satisfying (a) and (b), but such that [MATH] is not the centre of [MATH] , then the impossibility of constructing the centre of a circle by ruler alone will be demonstrated. |
Such a transformation indeed exists, but the argument in the last paragraph has a gap. If we understand the notion of construction in a non-uniform way and require that the point was among the points constructed, the argument does not work since the center of [MATH] could be the image of some other constructed point. I... |
It is easy to correct the argument and make it work for the definition of constructibility given above (using the fact that there are many projective mappings that preserve the circle), but still one can say without much exaggeration that the first correct proof of this impossibility result appeared only in |
. One can add also that the stronger result about two circles that was claimed by Cauer and reproduced with a similar proof in , turned out to be plainly false as shown in |
, and the problems in the proof were noted already by Gram . It is not clear why Gram did not question the validity of the classical proof for one circle, since the argument is the same. Gram did not try to give a rigorous definition of the notion of a geometric construction, speaking instead about constructions in the... |
that also has no formal definitions. The weak version of Cauer’s result saying that for some pairs of circles one cannot construct their centers, can be saved and proven for the definition of constructibility discussed above (see |
and the popular exposition in ). It would be interesting to reconsider the other results claimed about geometric constructions (for example, in |
) to see whether the proofs work for some clearly defined notion of a geometric construction. Note that in some cases (e.g., for Tietze’s results) some definition of the geometric construction for the uniform case is needed (and the negative definition is not enough). |
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