Polytope Contractions within Weyl Group Symmetries

A general scheme for constructing polytopes is implemented here specifically for the classes of the most important 3D polytopes, namely those whose vertices are labeled by integers relative to a particular basis, here called the ω-basis. The actual number of non-isomorphic polytopes of the same group has no limit. To put practical bounds on the number of polytopes to consider for each group we limit our consideration to polytopes with dominant point (vertex) that contains only nonnegative integers in ω-basis. A natural place to start the consideration of polytopes from is the generic dominant weight which were all three coordinates are the lowest positive integer numbers. Contraction is a continuous change of one or several coordinates to zero.


Introduction
Contractions of reflection generated polytopes in 3D real Euclidean space R 3 were introduced in [2]. Although the method is quite general, in terms of the dimension of the space and the reflection group symmetry, it has been used specifically for polytopes with icosahedral symmetry only [2]. More precisely, a polytope is being deformed into another polytope by the contraction process, while the icosahedral symmetry is preserved at all stages of the process.
In this paper we consider contractions of the polytopes with symmetries that include the Weyl reflection groups associated with the simple Lie algebras of rank 3 [5,6,10]. Those are the Lie algebras of type A 3 , B 3 , and C 3 . The Weyl group of A 3 is isomorphic to the symmetry group of 4 elements, and is thus of order 24. The Weyl groups of B 3 and C 3 are isomorphic to each other. Therefore their polytopes have a certain resemblance, for example as a cube and regular octahedron. Nevertheless their contractions are sufficiently different.
A motivation for the polytope contractions can be found in [1,2,9]. The analogy with the definition of contractions of Lie algebras established some 60 years ago [7], is in itself a motivation for our present undertaking.
Sooner or later one may envisage the use of the following property of polytopes, which is a direct consequence of their contractions. A polytope whose vertices are situated on a sphere of a certain radius, say R, is diminished during the contraction. In order to keep the radius unchanged, coordinates of the remaining vertices have to be correspondingly increased. During that process the overall symmetry of the polytope does not change. If we let the coordinates of one vertex vibrate, then the coordinates of the rest of the vertices have to vibrate in opposite phase to preserve the radius of the polytope [3].

Preliminary
In this section we recall certain facts about the symmetry groups considered in this paper.

W (A 3 ) Symmetry Group
The symmetry group of W (A 3 ) is the Weyl group of the simple Lie algebra A 3 . Its order is 4! = 24. It is known to be isomorphic to the permutation group of four elements. It is generated by the reflections r 1 , r 2 , r 3 in the mirrors m 1 , m 2 , m 3 , intersecting at the origin of R 3 of the real Euclidean space of dimension 3 (see for example [6,10]). More details about 3D polytopes of this kind are found in Table 2 in [4].
The simple roots α 1 , α 2 , α 3 are the normal vectors to the three mirrors. It is convenient to show the simple roots as nodes of a Coxeter diagram (see Fig. 1). Relative Fig. 1 Coxeter diagram of simple roots for A 3 . Nodes describe simple roots and a link between them indicates an angle between vectors. An angle is equal 2π 3 . If there is no link between nodes it means that the angle is π 2 angles of the mirrors are also conveniently specified in the case of W (A 3 ) in terms of the matrix of scalar products of simple roots (the Cartan matrix). (2.1) The simple roots are normalized to √ 2. The normalization is a matter of convenience which makes the Cartan matrix contain the smallest integers as matrix elements.
In the paper we also make extensive use of the dual basis which is the reciprocal of the α-basis. It is defined by Thus we have [4] The vectors of the ω-basis are not of the same length, ω 1 , ω 1 = ω 2 , ω 2 = ω 3 , ω 3 . The reflections of W (A 3 ) r 1 , r 2 , r 3 in the mirrors m 1 , m 2 , m 3 respectively, transform a vector (x, y, z) ∈ R 3 given in the ω-basis, as follows,

W (B 3 ) Symmetry Group
The symmetry group W (B 3 ) is generated by the reflections r 1 , r 2 , r 3 in mirrors intersecting at the origin of R 3 defined by their normal vectors α 1 , α 2 , α 3 . The Coxeter diagram for B 3 is shown in Fig. 2. Fig. 2 Coxeter diagram of simple roots for B 3 . Nodes describe simple roots and a link between them indicates an angle between vectors. One of the angle is equal to 2π 3 . If there is no link between nodes it means that the angle is π 2 . A double line describes the angle 3π 4 . A black node means a shorter root vector The relative position of the mirrors is given by the relative position of their normals. The Cartan matrix of B 3 differs from that of A 3 (2.1).
The simple roots are of different length and are normalized to √ 2 and 1.
The reflections of W (B 3 ) r 1 , r 2 , r 3 in mirrors m 1 , m 2 , m 3 respectively, transform a vector (x, y, z) ∈ R 3 given in the ω-basis, as follows,

W (C 3 ) Symmetry Group
As in the previous subsections the symmetry group W (C 3 ) is generated by the reflections r 1 , r 2 , r 3 in mirrors intersecting at the origin of R 3 defined by their normal vectors α 1 , α 2 , α 3 . The Coxeter diagram for C 3 is shown in Fig. 3. The Cartan matrix of C 3 has the following form: Coxeter diagram of simple roots for C 3 . Nodes describe simple roots and a link between them indicates an angle between vectors. One of the angle is equal to 2π 3 . If there is no link between nodes it means that the angle is π 2 . A double line describes the angle 3π 4 . Black nodes describes shorter root vectors The simple roots, as for the B 3 case, are of different length and are normalized to √ 2 and 1.
The difference between B 3 and C 3 is that the long roots of B 3 are the short roots of C 3 and short root of B 3 is a long root of C 3 . The reflections of the W (C 3 ) r 1 , r 2 , r 3 in the mirrors m 1 , m 2 , m 3 respectively, transform a vector (x, y, z) ∈ R 3 given in the ω-basis, as follows, For any of the three groups we can choose a point with coordinates (a, b, c) in the ω-basis. Applying to it reflections of its Weyl group in every way as long as distinct points are created we form an orbit of the Weyl group. The points of such an orbit are interpreted as vertices of a polytope in R 3 . Its vertices are equidistant from the origin. The number of points belong to the orbits of each Weyl group are summarized in Table 1.

The Family of W (A ), W (B ) and W (C ) Polytopes
Any orbit of the symmetry group is specified by its unique point, called the dominant point, which has non-negative coordinates in the ω-basis. The orbit of the point  The length of the edges meeting at the dominant point P = aω 1 + bω 2 + cω 3 is shown below, where the scalar products of the ω's are taken from (2.2), (2.4), (2.6), respectively for each type of polytope.
We present polytopes which preserve tetrahedral W (A 3 ) and octahedral W (B 3 ), W (C 3 ) symmetries. We choose the three values 1, 1 3 , 1 6 for the parameters a, b, c. The six resulting polytopes are shown in Fig. 4, 5 and 6.   Fig. 4 one notices that the polytopes with dominant weight, where the first and third coordinate are interchanged are the same. This is an effect of a symmetry of group A 3 visible in the Coxeter diagrams in Fig. 1.
The number of edges and 2-faces in each polytope can be read from Table 2.

Contraction of W (A 3 ), W (B 3 ), W (C 3 ) Polytopes
Most often the polytopes of interest have the seed point (a, b, c) with integer coordinates relative to the ω-basis. However here, during the contraction process [8], the coordinates of the point will be allowed to change continuously, taking positive non-integer values as well. Let us emphasize that at every stage of that continuous contraction process of the polytope, exact symmetry of the polytope is preserved. However the distances between the vertices and the origin vary through the contraction process.   (1, 1, 1). In the first step we change only one of the lengths of the three edges in the polytope. For example we reduce one of the edges to 1/10. Fig. 7, 8 and 9 present the first step of the contraction.  W (B 3 ). The first row shows the initial polytope before contraction. The second row shows the polytopes with one coordinate changed  (a, b, 0), (0, b, c) and (a, 0, c) In this section we consider polytopes after the first contraction where one of the coordinates in the ω-basis is zero. Transforming only one of the dominant coordinates to zero, one gets polytopes of several kinds. If the dominant point of the polytope has one coordinate equal to zero its properties are quite different (see Table 2). Fig. 10, 11 and 12 show the next step of contractions.

Contractions of Radii of Polytopes
Since the vertices of a polytope generated by the reflections of the Weyl group are equidistant from the origin, the radii of a polytope with dominant point (a, b, c) can be found by the formula Using the matrices ( ω i , ω j ) from (2.2), (2.4) and (2.6) we have the exact form of the square of the radius of a polytope with dominant point (a, b, c) for each group Table 3 we present some examples of radii for a few dominant points for each group considered in the paper. In Fig. 13 we show how the length of the radii for the polytopes with dominant weights (1,1,1), (1,1,0) and (1,0,0) is changed.    We can also combine the contractions (6.1) and (6.2), as for example,

Concluding Remarks and Open Problems
• There are seven cases of semisimple Lie groups of rank 3. The three which are simple among them have been described in this paper. A description of the four which are semisimple and not simple would also be interesting to consider. • The polytopes we consider here were summarized in Table 2 by [4]. In that paper corresponding results are found also for polytopes of dimension 4 in Table 3 and 4.
• During the reduction of the symmetry process the initial group may be reduced to its subgroup. When the two groups coincide we speak about contraction [8] if the second group is smaller we would speak of ,branching rules' [10]. • A chain of polytopes attached to each other by equal sides would be interesting to classify. • The polytopes with the same symmetry formed by more than one orbit in the outer shell can also be created and would certainly find applications in science.