4D Pyritohedral Symmetry

We describe an extension of the pyritohedral symmetry in 3D to 4-dimensional Euclidean space and construct the group elements of the 4D pyritohedral group of order 576 in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W (F4) and W (H4), implying that it is a group relevant to the crystallographic as well as quasicrystallographic structures in 4-dimensions. We derive the vertices of the 24 pseudoicosahedra, 24 tetrahedra and the 96 triangular pyramids forming the facets of the pseudo snub 24cell. It turns out that the relevant lattice is the root lattice of W (D4). The vertices of the dual polytope of the pseudo snub 24-cell consists of the union of three sets: 24-cell, another 24-cell and a new pseudo snub 24-cell. We also derive a new representation for the symmetry group of the pseudo snub 24-cell and the corresponding vertices of the polytopes.


Introduction
attices in higher dimensions described by the affine Coxeter groups, when projected into lower dimensions, may represent the quasicrystal structures [1][2][3][4][5].It is known that the 4 A lattice projects into the aperiodic lattice with 5- fold symmetry [1].There is no doubt that the projections of the higher dimensional lattices may have some implications in physics.The exceptional Coxeter-Weyl group ) ( 4 F W describes the symmetry of the unique self-dual polytope, the 24-cell, which is the Voronoi cell (Wigner-Seitz cell) of the 4 F lattice.The noncrystallographic Coxeter group ) ( 4 H W is the symmetry of the famous 600-cell and its dual 120-cell [6][7].
In this work we construct the 4D pyritohedral group from 4 D diagram.In technical terms the group   its dual polytope, which are related to the lattice ) ( 4 D W . All rank 4 Coxeter-Weyl groups can be represented, in compact forms, by quaternion pairs [9].This paper is organized as follows.In Section 2 we introduce the 4D -pyritohedral symmetry derived from D4 diagram.In Section 3 we construct the group   and apply it to a vector to generate the vertices of a polytope which we call "pseudo snub 24-cell".We find the facets of the pseudo snub 24-cell which consist of the pseudoicosahedra, tetrahedra and triangular pyramids.The vertices of the dual polytope of the pseudo snub 24-cell are constructed.Finally, in Section 4 we present a brief discussion on the physical implications of our technique.

4D crystals with the pyritohedral symmetry derived from D4 diagram
In the paper [10] [6], [8].The snub 24-cell is a convex uniform polytope in four dimensions consisting of 120 regular tetrahedral and 24 icosahedral cells.It has 96 vertices at each of which five tetrahedra and three icosahedra meet.Snub 24-cell can be constructed from the 24-cell by dividing the edges in the golden ratio and truncating it in a certain way.This truncation transforms the 24 octahedral cells of the 24-cell to the 24 icosahedral cells of the snub 24-cell; the truncated vertices become 24 tetrahedral cells and the gaps in between are filled in by another 96 tetrahedra.D is shown in Figure 1 with the quaternionic simple roots.

Construction of the symmetry group of snub 24-cell
The corresponding weights are determined as The set T is given by the group elements  = {±1, ± 1 , ± 2 , ± 3 , and is called the binary tetrahedral group of order 24 .Another set of 24 quaternions is defined by In terms of quaternionic simple roots, the group generators of ) ( 4 D W can be written as They generate the Coxeter-Weyl group ) ( 4 D W of order 192 [12].The subsets of the quaternions are defined as follows: These subsets are useful to denote the Coxeter-Weyl group Note that the subset of the Coxeter-Weyl group represents the proper subgroup and can be directly generated by the rotation generators  2  1 ,  2  3 ,  2  4 .Let us impose the Dynkin diagram symmetry which is the permutation group  3 of the simple roots  1 ,  3 and  4 as shown in Figure 2.
} , , , { , even number of (-) sign, is invariant under conjugation by the group  3 .We first note that the extension of the group of eq. ( 7) by the cyclic group of order 3 generated by the generator [, ] is a group of order 288 which can be denoted by The extension of the group by full permutation group 3 S is given as the semi-direct product of two groups as: As we will see in the next section this is the symmetry group of the snub 24-cell as well as that of any pseudo snub 24cell.

Construction of the vertices of the pseudo snub 24-cell
The affine Coxeter group   ( 4 ) =<  0 ,  1 ,  2 ,  3 ,  4 > can be generated by five generators by introducing  0 as shown in Figure 3.  0 represents the reflection with respect to the hyperplane bisecting the line from the origin to the highest root The action of  0 on a general vector  is given as Applying the group Wa (D4) on a simple root we can generate the root lattice.We now derive the vertices of pseudo snub 24-cell in terms of the root lattice vectors of D4.The lattice of D4 is self-dual so that we can express the dual lattice vectors in terms of the weight vectors [14].Table 1 shows the lattice vectors in terms of the root & weight vectors and quaternions.
Table 1.The construction of the lattice from Wa (D4).

Lattice
Root lattice (Real lattice) Weight lattice (Reciprocal lattice) Vectors in terms of quaternions We impose the Dynkin-diagram symmetry  3 on the vector  , that is, 3 S    as shown in Figure 2. The group  3 permutes the weight vectors  1 ,  3   4 but leaves  2 invariant.Similarly, the generators  1 ,  3   4 are permuted and the generator  2 is left invariant under the group conjugation of  3 .This implies that the vector  takes the form Factorizing by  2 and defining the rational number  =  1  2 the vector  reads in terms of quaternions Note that the sum of the coefficients of the quaternionic units is an even integer as we mentioned earlier.For 0 and 2 which is known to be the 24-cell.For these particular values of x, the group   generates 24 vertices of ′.The 24-cell has 24 vertices and 24 octahedral cells where 6 octahedra meet at one vertex.The polytope 24-cell constitutes the unit cell of the  4 lattice.In terms of the quaternionic sets when T represents the unit cell of the root lattice, then the set represents its Voronoi cell.Applying the group elements represented by eq. ( 9) on the vector in (11) we obtain 96 vertices of pseudo snub 24-cell as the orbit of the group   The mirror image of the pseudo snub 24-cell can be obtained by applying any reflection generator of  4 .For example, applying  2 on () in ( 12) interchanges 1 2 e e  and leaves the other quaternionic units unchanged.Then under the action of the mirror operator  2 one can obtain the mirror image of the pseudo snub 24-cell in (12).Both the pseudo snub 24-cell and its mirror image lie in the  4 lattice.It is clear that for 0 and 2 the vertices of (12) reduces to the set ′.Excluding these values of x, the set of vertices represent a pseudo snub 24-cell.Only in the limit ) the vertices given in (13) represent the snub 24-cell [8].Since in this case x is not a rational number, the vertices do not belong to the lattice  4 .
Table 2 summarizes the action of the group (11) for certain x values.

Determination of facets of pseudo 24-cell
It is known that every vertex of the snub 24-cell is surrounded by three icosahedra and five tetrahedra.We shall prove that the facets of the pseudo snub 24-cell consist of pseudo icosahedra, tetrahedra and triangular pyramids.In the pseudo snub 24-cell three pseudoicosahedra, one tetrahedron and four triangular pyramids meet at the same vertex.Now we discuss the details of this structure.It is evident from the  4 diagram that each of the following sets of rotation generators ( 1  2 ,  2  3 ), ( 3  2 ,  2  4 ) and ( 4  2 ,  2  1 ) generate a proper subgroup of the tetrahedral group of order 12 as shown in Figure 4. Let us discuss one of these groups acts on the vertex  .Let us take the group ] , [ , (case 1 in Figure 4) which implies that the group consists of 12 elements leaving the weight vector T  4  invariant.The group generators transform quaternionic units as follows: are determined as: ( (1 ) , ( 1) (1 ) , (1 2 ) , (1 ) (1 2 ) , (   is left invariant by the generators in ( 14) the center of the polyhedron of (15) can be taken as 4  up to a scale factor.One may check that the set in ( 15 (case 3 in Figure 4) respectively are applied to the vector  , one generates two more pseudo icosahedra with the centers represented by 1  and 3  respectively.The groups generating the vertices of the second and the third pseudo icosahedra can be written respectively as .These groups are isomorphic to the pyritohedral group . Now we continue to discuss the polyhedral facets having  as a vertex.Let us consider the following five sets of rotational generators obtained from the generators of the Coxeter-Weyl group ( 4 ): The first two sets of generators are invariant under the conjugation of the permutation group  3 but the next three sets of generators are permuted among each other.Let us determine the vertices of the polyhedra under the action of five sets.
1.The set of vertices 2 ( represents a tetrahedron of edge length 2x as shown in Figure 5(b).Its center can be represented by (1) =  2 = 1 +  1 up to some scale factor.Since the Dynkin diagram symmetry  3 also leaves  2 = 1 +  1 invariant, the group  2 × 2 generated by the generators ( 1  3 ,  3  4 ,  4  1 ) can be extended by the  3 symmetry to a group  of order 24 isomorphic to the tetrahedral group [10].Since , the group leaves the vertices of the tetrahedron in (17) invariant (Figure 5b).Note that this not a pyritohedral symmetry.This tells us that the number of tetrahedra generated by the conjugate tetrahedral groups is also 24.
Extension of the group by the generator [1, −1] leads to the group that is isomorphic to the octahedral group [10].
2. The set of vertices .) ( determines a triangular pyramid with a base of equilateral triangle with sides 2x and the other edges of length √2( 2 +  + 1).The triangular pyramid is depicted in Figure 5 (c).The hyperplane determined by these four vertices in (18) is orthogonal to the vector defines another triangular pyramid with the same edge lengths as above.The vector orthogonal to the hyperplane determined by the vertices in (19) is also determines a triangular pyramid as above.The vector which is orthogonal to the hyperplane determined by the vectors of (20) can be computed as Now their symmetries are more transparent under the permutation group:  3 permutes  1 ,  3   4 but leaves  2 invariant; ) (1 p and ) (2 p are invariant under the group  3 but the others are permuted to each other.Pseudo snub 24-cell consists of  0 = 96 vertices,  3 = 24 + 24 + 96 = 144 cells consisting of pseudo icosahedra, tetrahedra and triangular pyramids respectively.It has  1 = 432 edges and  2 = 480 faces.These numbers satisfy the Euler characteristic formula  0 −  1 + 2 −  3 = 0. Table 3 summarizes the facets of pseudo snub 24-cell.

Construction of the vertices of the dual polytope of the pseudo snub-24 cell
To construct the dual of the pseudo snub 24-cell we need to determine the centers (orthogonal vectors to the hyperplane) of the pseudoicosahedra, the tetrahedron and the four pyramids up to some scale factors.The centers of the first three pseudo icosahedra can be taken as the weight vectors  1 ,  3   4 as shown in Table 3.The other vectors in (22) can be taken as the center (orthogonal vector to the hyperplane) of the tetrahedron and the centers of the four pyramids up to some scale vectors.Let us denote by (),  = 1,2, … ,5, the centers of the respective tetrahedron and the pyramids and define (23) So we have eight vertices including  1 ,  3   4 .To determine the actual centers of these polyhedra the hyperplane defined by the eight vectors must be orthogonal to the vector  .This will determine the scale factors and the centers of the cells which can be written as: These eight vectors now determine the vertices of one facet of the dual polytope of the pseudo snub 24-cell.The center of this facet is the vector  .This is a convex solid with 8 vertices 15 edges and 9 faces possessing  3 symmetry.
One can generate the vertices of the dual polytope by applying the group  
list 96 vertices of the pseudo snub 24-cell omitting the overall factor  2 as follows:

1 2 3 Figure 4 .
Figure 4. Three proper subgroups of the tetrahedral group from the   diagram.
) is also left invariant under the group element set (15) is invariant under the larger group isomorphic to the pyritohedral group.If we define a new set of unit quaternions in (15) in terms of the new set of quaternions, then the set of vertices in (15) represent a pseudoicosahedron as shown in Figure5(a).When the set of generators  by these generators extended by the group  3 is the full group of symmetry
the facets of the dual polytope.One can display them as the union of three sets in (2-3) and () can be written as follows

‫األربعة‬ ‫األبعاد‬ ‫ذو‬ ‫االقليدي‬ ‫الفضاء‬ ‫في‬ ‫البايريت‬ ‫لمعدن‬ ‫المشابه‬ ‫التركيب‬ ‫ذو‬ ‫األوجه‬ ‫متعدد‬ ‫تماثل‬ ‫و‬ ‫القنوبي‬ ‫أمل‬ ‫كوجا،‬ ‫نظيفة‬ ‫كوجا‬ ‫محمد‬
we discussed the pyritohedral group D by the rotation generators and the Dynkin diagram symmetry.The straightforward generalization of this group to 4D is to start with the rotation generators of 4 D and impose the Dynkin diagram symmetry.We will see that the generated group from the diagram 4 D is nothing other than the group

Table 2 .
Action of the group   .