Symmetry of the Pyritohedron and Lattices

The pyritohedron consisting of twelve identical but non regular pentagonal faces and its dual pseudoicosahedron that possess the pyritohedral (Th) symmetry play an essential role in understanding the crystallographic structures with the pyritohedral symmetry. The pyritohedral symmetry takes a simpler form in terms of quaternionic representation. We discuss the 3D crystals with the pyritohedral symmetry which can be derived from the Coxeter-Dynkin diagram of D3.


Introduction
ymmetry describes the periodic repetition of structural features.Any system exhibits symmetry if the action of the symmetry operations leaves the system apparently unchanged.Crystals possess a regular, repetitive internal structure, therefore they have symmetry.Coxeter groups are the symmetry groups generated by reflections [1].They describe the symmetry of regular and semi-regular polytopes in arbitrary dimensions [1].The Coxeter-Weyl groups acting as discrete groups in 3D Euclidean space generate orbits representing vertices of certain polyhedra [2][3].Rank-3 Coxeter-Weyl groups ( 3 )and ( 3 ) ≈ ( 3 ) define the point tetrahedral and octahedral symmetries of the cubic lattices [4].In this work we use Coxeter-Dynkin diagram  3 and construct the pyritohedral group and the related polyhedra in terms of quaternions by finding the vertices of the pseudoicosahedron with pyritohedral symmetry.
We organize the paper as follows.In Section 2 we introduce quaternions and their relevance to O (3) and O (4) transformations.The sets of quaternions defining the binary tetrahedral and binary octahedral groups are given.Section 3 explains Coxeter-Dynkin diagram of  3 in which the simple roots (lattice generating vectors) and the group generators are expressed in terms of imaginary quaternionic units and the reflection planes of the diagram  3 are identified as certain planes of the unit cube.Section 4 deals with the construction of the icosahedron, dodecahedron, pseudoicosahedron and pyritohedron from the pyritohedral group derived from the  3 diagram as its automorphism group.It is noted that the truncated octahedron, the Wigner-Seitz cell of the BCC lattice, splits into two pseudoicosahedra which are mirror images of each other.In Section 5 we discuss the pseudoicosahedron and its dual pyritohedron to give a description of those crystals possessing pyritohedral symmetry.

S 2. Quaternions and their Relevance to the Isometries of the O (3) and O (4) Transformations
Quaternions are vectors in four dimensions provided with a rule for multiplication that is associative but not commutative, distributive through addition, contains an identity, and for each nonzero vector in four dimensions has a unique inverse [5] The quaternionic imaginary units   , (, , ) = (1, 2, 3) satisfy the relation: where ijk  is the Levi-Civita symbol that is completely antisymmetric in the indices.
A real quaternion q can be written in general as with  0 ,  1 ,  2 ,  3 ∈ ℝ, ℝ being the set of real numbers;  0 is called the scalar part and ( 1  1 +  2  2 +  3  3 ) is the vector part of a quaternion.The conjugate of a quaternion is defined as The norm of unit quaternion q is || = √ ̅ =1 and  −1 =  ̅.
The scalar product of two arbitrary quaternions  and  is defined as The transformations of an arbitrary quaternion  →  and  →  ̅  define orthogonal transformation of the group (4).It is clear that the above transformations preserve the norm  ̅ =  0 2 +  1 2 +  2 2 +  3 2 .We define the above transformations as abstract group operations by the notations Dropping also t, the pair of quaternions define a set closed under multiplication.
The inverse elements take the forms [6]: With the choice of qp  , the orthogonal transformations define a three parameter subgroup pp  where the sandwiching operator [ , ] pprepresents the rotations around the vector () = ( 1 ,  2 ,  3 ) and [ , ] pp  is a rotary inversion [1].
Reflection of a vector Λ represented as a quaternion with respect to a plane orthogonal to the unit pure quaternion  can be written as If we apply another reflection to the vector Λ it will lead to a rotation.The product of two reflections is a rotation.We will display some of the finite subgroups of quaternions related to the tetrahedral and octahedral groups.
The set  is given by the group elements and is called the binary tetrahedral group of order 24.Another set of 24 quaternions is defined by The set  =  ∪  ′ forms binary octahedral group of order 48.These sets play an important role in the definition of the Pyritohedral subgroup.

Finite Coxeter Groups, Cartan Matrix and Root Systems
In 1934 Coxeter classified all finite Euclidean reflection groups [7][8].A Coxeter group is a group () which has a presentation with a very special form where  1 ,  2 , … ,   are the reflection generators of the group.
All information of a root system can be encoded by Coxeter-Dynkin diagrams and the Cartan matrix C. The simple roots  1 ,  2 , … ,   of the Coxeter-Dynkin diagram are the vectors orthogonal to certain hyperplanes with respect to which the simple reflection acts as an arbitrary vector  as [1]: The Cartan matrix is a square integer matrix that links the simple roots of a given group through the following relation: The inverse of the Cartan matrix is related to the metric of the dual (reciprocal) space as: The basis vectors in the dual space are called the weight vectors, denoted by   , satisfying the scalar product:   = (  ,   ).The simple roots and weight vectors are related to each other by ) =   and   =     .

Coxeter-Dynkin diagram of 𝑾(𝑫 𝟑 )
The Coxeter-Dynkin diagram of  3 with the quaternionic simple roots is shown in Figure 1.The angle between two connected roots is 120°, otherwise they are orthogonal.An arbitrary quaternion  when reflected by the operator   with respect to the hyperplane orthogonal to the quaternion   , is given in terms of quaternion multiplication [6] as The generators of the Coxeter group ( 3 ) are then given in the notation of ( 14) by .
They generate the Coxeter-Weyl group ( 3 ) of order 24, isomorphic to the tetrahedral group, the elements of which can be written compactly by the notation Here  and ′ are the sets of quaternions given in (8)(9).When the simple roots are chosen as in Figure 1, then the weight vectors are determined as Using the orbit definition ( 3 )( 1  2  3 ) ≔ ( 1  2  3 )  3 all the orbits can be determined.

Coxeter-Dynkin diagram 𝑾(𝑫 𝟑 ) with Dynkin diagram symmetry
The symmetry of the union of the orbits (010  consists of 8 rotations by 120 0 around the 4 diagonals of the cube, 3 rotations by 180 0 around the x, y and z axes, and the unit element [9].Pyrite crystals often occur in the forms of cubes with striated faces as in Figure 4, octahedra and pyritohedra (a solid similar to dodecahedron but with non-regular pentagonal faces) or some combinations of these forms.

Construction of the vertices of the pseudoicosahedron
We denote a general vector of D3 by Applying some of the rotation elements of the pyritohedral symmetry on a general vector of , one can generate five triangles sharing the vertex  as shown in Figure 5(a), which consist of two equilateral and three isosceles triangles.The vertices ( Suppose the Dynkin-diagram symmetry as shown diagrammatically in Figure 2 leaves the general vector  invariant    .This symmetry exchanges  2 and  3 , so that From (19) one gets , 3 2 a a  and a general vector can be written in terms of quaternions as   is a parameter which could be computed from (19).If we take a general value for x in the vector given in (21), the action of the pyritohedral group on  will generate the set of 12 vectors (apart from the scale factor a1): Factoring ( 19) by a1 one obtains the equation . ( The vectors here have the same norm as in (23a).The icosahedron represented by (23a) is shown in Figure 6(a).Table 1 summarizes the action of the pyritohedral group on  for various values of x:  Table 1.The action of the pyritohedral group for various values of x.
, ( 1) , ( 1) The vertices of the dual of the icosahedron, say the set of vectors of (23a), can be determined as [9]: The vectors in (25b) represent the vertices of a cube which is invariant under the pyritohedral symmetry.Similarly, the other 12 vertices of (25a) form another orbit under the pyritohedral symmetry.They are obtained for

Construction of the vertices of the pyritohedron from the pseudoicosahedron
We can compute the dual of the pseudoicosahedron in equation ( 22) for an arbitrary value of x.This can be achieved by determining the vectors normal to the faces of the pseudoicosahedron in (22).The normal vectors of the five triangles in Figure 5 (b) can be determined as follows: The normal vector of the triangle with the vertices Similarly, the normal vector of the triangle with the vertices . The vertices generated by the pyritohedral group from either 2  or 3  would lead to the vertices of a cube given in (25b).The vectors normal to the isosceles triangles shown in Figure 5 (b) can be computed as follows: (1 ) ,

  
The vectors 2  and 3  are in the same orbit under the pyritohedral group.Moreover,

  
. On the other hand, one can prove that  determine a pentagon, non-regular in general (four edges of the same length and one edge different), as shown in Figure 7.
The pyritohedron has a geometric degree of freedom with two limiting cases as shown in Figure 8.

Vertices of the Pseudoicosahedron and the Pyritohedron in a Lattice
The simple cubic lattice consists of the vector  1  1 +  2  2 +  3  3 where   ∈ ,  = 1,2,3.Now, we can discuss the pseudoicosahedron and its dual pyritohedron relevant to crystallography.Many candidates of pseudoicosahedra and its dual pyritohedra can be obtained as a structure in the simple cubic lattice.The vertices of pseudoicosahedron and its dual pyritohedron for various x (and corresponding h) values are given in Table 2 and are plotted in Figure 9.As mentioned in Section 4.2, when  → 0 (ℎ → 0 ) the pseudoicosahedron is converted to an octahedron and the pyritohedron becomes a cube.The set of vertices of the pseudoicosahedron and the pyritohedron belong to the simple cubic lattice.
The unit cubic cell can be stacked in the pseudoicosahedron or in the pyritohedron as long as the vertices are chosen from the simple cubic lattice.

Conclusion
The polyhedra possessing the pyritohedral symmetry have been constructed in terms of quaternions.Crystals with pyritohedral symmetry exist in the form of a stratified cube, an octahedron and a pyritohedron.It is expected that crystal structures in the form of a pseudoicosahedron as well as a pseudoicosidodecahedron may exist.
The relevance of the pseudoicosahedron to pyritohedral crystals could stimulate research in application to material science.Representation of the pyritohedral symmetry and crystal vectors in term of quaternions is more demanding, but may lead to a new understanding of crystal structures and their symmetries.We are anticipating that this finding will form a link between quaternions and crystallography.

Figure 4 .
Figure 4.The cube with stripes on its faces possesses the pyritohedral symmetry.

3 
of x will lead to five equilateral triangles sharing the vertex  .Substituting and  respectively for x in (22) we obtain two sets of 12 vertices as: vertices represent two mirror images of an icosahedron with a scale factor difference.Multiplying the vertices in (23b) by one obtains the following set of quaternions:

Figure 5 .
Figure 5. (a) Five triangles meeting at one vertex, (b) The normal vectors of the triangles surrounding .

.
also in one orbit under the pyritohedral group.They form a plane orthogonal to As these five vertices should determine the same plane then one can show that condition to determine the scale factor  .From this relation we obtain  = 1+2 2(1+) 2 where the scale factor implies that  ≠ 1 and  ≠

Figure 7 .
Figure 7.The plane determined by the vectors normal to the faces of the pseudoicosahedron in equation (22).