Chiral Polyhedra Derived From Coxeter Diagrams and Quaternions

There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their duals the Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. In this paper we construct the chiral polyhedra and their dual solids in a systematic way. We use the proper rotational subgroups of the Coxeter groups $W(A_1 \oplus A_1 \oplus A_1)$, $W(A_3)$, $W(B_3)$ and $W(H_3)$ to derive the orbits representing the solids of interest. They lead to the polyhedra tetrahedron, icosahedron, snub cube, and snub dodecahedron respectively. We prove that the tetrahedron and icosahedron can be transformed to their mirror images by the proper rotational octahedral group $\frac{W(B_3)}{C_2}$ so they are not classified in the class of chiral polyhedra. It is noted that the snub cube and the snub dodecahedron can be derived from the vectors, which are non-linear combinations of the simple roots, by the actions of the proper rotation groups $\frac{W(B_3)}{C_2}$ and $\frac{W(H_3)}{C_2}$ respectively. Their duals are constructed as the unions of three orbits of the groups of concern. We also construct the polyhedra, quasiregular in general, by combining chiral polyhedra with their mirror images. As a by product we obtain the pyritohedral group as the subgroup the Coxeter group $W(H_3)$ and discuss the constructions of pyritohedrons. We employ a method which describes the Coxeter groups and their orbits in terms of quaternions.

W H C respectively. Their duals are constructed as the unions of three orbits of the groups of concern. We also construct the polyhedra, quasiregular in general, by combining chiral polyhedra with their mirror images. As a by product we obtain the pyritohedral group as the subgroup the Coxeter group and discuss the constructions of pyritohedrons. We employ a method which describes the Coxeter groups and their orbits in terms of quaternions. 3 ( W H ) a) electronic-mail: kocam@squ.edu.om b) electronic-mail: nazife@squ.edu.om c) electronic-mail: m054946@squ.edu.om

Introduction
It seems that the Coxeter groups and their orbits [1] derived from the Coxeter diagrams describe the molecular structures [2], viral symmetries [3], crystallographic and quasi crystallographic materials [4]. Chirality is a very interesting topic in molecular chemistry and physics. A number of molecules display one type of chirality; they are either leftoriented or right-oriented molecules. In fundamental physics chirality plays very important role. For example a massless Dirac particle has to be either in the left handed state or in the right handed state. No Lorentz transformation exist transforming one state to the other state. The weak interactions which is described by the standard model of high energy physics is invariant under one type of chiral transformations. In three dimensional Euclidean space, which will be the topic of this paper, the chirality is defined as follows: the object which can not be transformed to its mirror image by other than the proper rotations and translations are called chiral objects. For this reason the chiral objects lack the plane and/or central inversion symmetry. In two previous papers we have constructed the vertices of the Platonic-Archimedean solids [5] and the dual solids of the Archimedean solids, the Catalan solids [6], using the quaternionic representations of the rank-3 Coxeter groups. Two of the 13 Archimedean solids, the snub cube and snub dodecahedron are the chiral polyhedra whose symmetries are the proper rotational subgroups of the octahedral group and the icosahedral group respectively.
In this paper we use a similar technique of references [5][6] to construct the vertices of the chiral Archimedean solids, snub cube, snub dodecahedron and their duals. They have been constructed by employing several techniques [7][8] but it seems that the method in what follows has not been studied earlier in this context. We follow a systematic method for the construction of the chiral polyhedra. First we begin with the Coxeter diagrams which lead to the tetrahedron and icosahedron respectively and prove that they possess larger proper rotational symmetries which transform them to their mirror images so that they are not chiral solids. We organize the paper as follows. In Sec.2 we construct the Coxeter groupsW 1 1

and
in terms of quaternions. In Sec.3 we obtain the proper rotation subgroup of the Coxeter group and determine the vertices of the tetrahedron by imposing some conditions on the general vector expressed in terms of simple roots of the diagram . We prove that the tetrahedron can be transformed to its mirror image by the proper octahedral rotation group . In Sec.4 we discuss similar problem for the Coxeter-Dynkin diagram leading to an icosahedron and again prove that it can be transformed by the group C to its mirror image which indicates that neither tetrahedron nor icosahedron are chiral solids. Here we also discuss the properties of the pyritohedral group and the constructions of the pyritohedrons. The Sec.5 deals with the construction of the snub Cube and its dual pentagonal icositetrahedron from the proper rotational octahedral symmetry 3 2 using the same technique employed in Sec.3 and Sec.4. In Sec.6 we repeat a similar work for the constructions of the snub dodecahedron and its dual pentagonal hexacontahedron from the proper icosahedral group 3 2 5 which is isomorphic to the group of even permutations of five letters. In the concluding Sec.7 we point out that our technique can be extended to determine the chiral polyhedra in higher dimensions.
where ij  and ijk  are the Kronecker and Levi-Civita symbols and summation over the repeated indices is implicit. The unit quaternions form a group isomorphic to the unitary group . With the definition of the scalar product ) quaternions generate the four-dimensional Euclidean space. The Coxeter diagram can be represented by its quaternionic roots in Fig.1 with the The Cartan matrix and its inverse are given as follows , 2 0 0 For any Coxeter diagram, the simple roots i  and their dual vectors i  satisfy the scalar We note also that they can be expressed in terms of each other: Let be an arbitrary quaternionic simple root. Then the reflection of an arbitrary vector with respect to the plane orthogonal to the simple root   is given by [10] [ , Our notations for the rotary reflections and the proper rotations will be [ , ] p q  and [ , ] p q respectively where and p q are arbitrary quaternions. The Coxeter group is generated by three commutative group elements The next Coxeter group which will be used is the tetrahedral groupW A . Its diagram with its quaternionic roots is shown in Fig.2. The Cartan matrix of the Coxeter diagram and its inverse matrix are given respectively by the matrices 3 A , 2 1 0 The group elements of the Coxeter group which is isomorphic to the tetrahedral group of order 24 can be written compactly by the set [11] 3 Here represent respectively the binary tetrahedral group of order 24 and the coset representative where O is the binary octahedral group of quaternions of order 48 [11]. The Coxeter diagram 3 B leading to the octahedral group is shown in Fig. 3. The Cartan matrix of the Coxeter diagram 3 B and its inverse matrix are given by The generators, [ , ] r e e    (13) generate the octahedral group which can be written as 3 3 A shorthand notation could be . Note that we have three maximal subgroups of the octahedral group , namely, the tetrahedral group , the chiral octahedral group consisting of the , and the pyritohedral group consisting of the The pyritohedral symmetry represents the symmetry of the pyritohedrons, an irregular dodecahedron, with irregular pentagonal faces which occurs in iron pyrites. The Coxeter diagram leading to the icosahedral group is shown in Fig. 4. The Cartan matrix of the diagram and its inverse are given as follows: The generators, and I is the set of 120 quaternionic elements of the binary icosahedral group [10]. The chiral icosahedral group is represented by the set which is isomorphic to the even permutations of five letters. Note also that the pyritohedral group is a maximal subgroup of the Coxeter groupW H . All finite subgroups of the groups O in terms of quaternions can be found in reference [12].
A general vector in the dual space is represented by the vector . We will use the notation O W for the orbit of the Coxeter groupW G generated from the vector ( )  where the letter G represents the Coxeter diagram. We follow the Dynkin notation to represent an arbitrary vector in the dual space and drop the basis vectors In the Lie algebraic representation theory the components of the vector  are called the Dynkin indices [13] which are non-negative integers if it represents the highest weight vector. Here we are not restricted to the integer values of the Dynkin indices. They can be any real number. When the components of the vector in the dual space are non integers values we will separate them by commas otherwise no commas will be used. For an arbitrary Coxeter diagram of rank 3 we define the fundamental orbits as (18) Any linear combination of the basis vectors i  over the real numbers will, in general, lead to quasi regular polyhedra under the action of the Coxeter group. In the next four sections we discuss a systematic construction of chiral polyhedra and their dual solids. In our construction tetrahedron and icosahedron will also occur but we prove that they are not chiral polyhedra.
3 The orbit as tetrahedron The tetrahedron with these vertices is shown in Fig.5. These are the vertices of a tetrahedron invariant under the rotation group given in (8). Of course the full symmetry of the tetrahedron is a group of order 24 isomorphic to the permutation group generated by reflections of the Coxeter-Dynkin diagram [6]. Now the mirror image of the tetrahedron of (20) can be determined applying the same group of elements in (8) Of course we know that the union of two orbits in (20) and (21) determines the vertices of a cube. The point here is that if we were restricted to the group of (8) then the tetrahedron in (20) would be a chiral solid. However this is not true because there exist additional rotational symmetries which exchange these two orbits of (20) and (21) proving that the tetrahedron is not a chiral solid. Now we discuss these additional symmetries. It is obvious that the Coxeter diagram in Fig.1 has an additional symmetry which permutes three diagrams. Indeed this symmetry extends the group to the proper octahedral rotation group as will be explained now. One of the generators of this symmetry is a 2-fold rotation leading to the transformation . It is straightforward to see that .This proves that by a proper rotation tetrahedron can be transformed to its mirror image therefore it is not a chiral solid. The generator d and those elements in (8) enlarge the symmetry to a group of order 8 which can be concisely written as the set of elements A cyclic subgroup of the symmetric group permutes three sets like those in (22) extending the group of order 4 to a group of order 24. Actually the larger group obtained by this extension is the chiral octahedral group of order 24 which can be symbolically written as (23) This is the proper rotational symmetry of the octahedron whose vertices are represented by the set of quaternions and the cube whose vertices are the union of the orbits .  Then one can obtain three more triangles by joining 1 3 r r  to the vertices and by drawing a line between 1   .This leads to two vectors . Here is an overall scale factor which can be adjusted accordingly. The  solution represents an octahedron which is not a chiral solid anyway. Let us study the orbit which is obtained from the vector . When expressed in terms of quaternions it will read which constitute the vertices of the icosahedron shown in Fig. 7. This is another icosahedron which is the mirror image of the icosahedron of (28 : A . This proves that two mirror images of the icosahedron are transformed to each other by rotations therefore the icosahedron is not a chiral solid rather it is achiral. When two orbits of (28) and (29) are combined one obtains a quasi regular polyhedron which can be obtained as the orbit of the group W B 3 ( )(1, ,0)  [14]. The quasi regular polyhedron represented by the combined vertices of (28-29) is shown in Fig. 8.It consists of two types of faces, squares of side  and isogonal hexagons of sides 1 and .  Figure 8. The quasi regular polyhedron represented by the vertices of (28-29).
Although we know that the dual of an icosahedron is a dodecahedron [6] here  . We note that the line joining these vectors is orthogonal to the vector , namely, 1 The centers of the faces #2, #4 and #5 can be determined by averaging the vertices representing these faces: Note that the last two orbits represent the vertices of two dual tetrahedra, when combined, represent a cube. These 20 vertices which decompose as three orbits under the tetrahedral group represent the vertices of a dodecahedron as shown in the Fig.9 which is also achiral solid. So far we have shown that, although, tetrahedron and icosahedron can be obtained as chiral solids there exists additional proper rotational group elements that convert them to their mirror images. Therefore they are not chiral solids. Although our main topic is to study the chiral objects systematically using the Coxeter diagrams, here with a brief digression, we construct the pyritohedron, a non regular dodecahedron, made by 12 irregular pentagons. If we plot the solid represented by the orbit 4 2 ( A b  in the first line of (31) we obtain an irregular icosahedron as shown in   .
There we see that two of the triangles are equilateral and the rest three are isosceles triangles. We determine the centers of the faces of this irregular icosahedron.
which leads to the pyritohedron as shown in Fig.12.   , ae be ae be   a are in the same plane determines that b a . Therefore the set of vertices of a pyritohedron has an arbitrary parameter and includes also dodecahedron and the rhombic dodecahedron [6], a Catalan solid, as members of the family for 2 2a     2 a and  respectively. The pyritohedron is facetransitive since the normal vectors of the faces form an orbit of size 12 under the pyritohedral group. It is an achiral solid.

The snub cube derived from the orbit
The snub cube is an Archimedean chiral solid. Its vertices and its dual solid can be determined employing the same method described in Sec. 3   Similar to the arguments discussed in Sec.4 we obtain four equilateral triangles and one square sharing the vertex (see Fig. 13) provided the following equations are satisfied 1  The snub cubes represented by these sets of vertices are depicted in Fig.14. Note that no proper rotational symmetry exists which transforms these two mirror images to each other so that they are truly chiral solids. One can combine the vertices of these two chiral solids in one solid which is achiral and it is depicted in Fig.15. This quasi regular solid can be obtained from the vector when  represents the normal of the plane containing these five points.
Then 38 vertices of the dual solid of the snub cube, the pentagonal icositetrahedron, are given in three orbits as follows The only difference of this from the one in Fig.13 is that in the present case the face #1 is an equilateral triangle whose center is represented by the vector 1  and the face #3 is a regular pentagon whose center is represented by the vector 3  . Assuming that the face #1 face #2, face #4 and face #5 are equilateral triangles which lie in the same orbit of size 60 one obtains the following equations: . Factoring by and defining