Projection of Polyhedra onto Coxeter Planes Described with Quaternions

3-dimensional convex uniform polyhedra have been projected onto their corresponding Coxeter planes defined by the simple roots of the Coxeter diagram . The projected vertices of a polyhedron onto its corresponding Coxeter plane constitute an orbit of the dihedral group . In this paper, we use the quaternionic representations of the Coxeter group elements and the vertices of the polyhedra.


Introduction
ymmetries of objects play a vital role in the analysis of structure, bonding and spectroscopy of molecules in several scientific disciplines.Group theory provides a great tool for understanding symmetry phenomena in physics.Using group theory to study the symmetry of polyhedra is helpful for physicists to develop an understanding of crystallography and quasi-crystallography.In this paper we use the rank-3 Coxeter-Dynkin diagrams, a special technique in Lie algebra, to construct the symmetries of polyhedra.
A symmetry group may contain transformations such as reflections, rotations, rotary reflections and inversions which leave the object unchanged.Coxeter groups are abstractions of the symmetry groups only generated by reflections group elements.They are useful in describing the symmetry of regular and semi-regular polytopes in arbitrary dimensions [1].
The list of convex uniform polyhedra contains Platonic solid, Archimedean solids and their duals, the Catalan solids.These special polyhedra have been successfully used to describe the crystals in physics, molecular symmetry in chemistry and some viral structures in biology [2].To describe the symmetry of all convex uniform polyhedra we use three main finite Coxeter groups 3 , 3 , and 3 .
Invoking quaternions is an efficient tool for representing elements of the Coxeter groups of rank-3 and rank-4.Quaternions have been used to represent rotations and reflections since their discovery in 1843 [3] and have now become used to represent the Coxeter groups [4].They are also used to symbolize the simple roots and weights of Coxeter-Dynkin diagrams.Furthermore the vertices of polyhedra are written in terms of pure imaginary quaternions.Quaternions are used because of their compactness, stability and simplicity, which make this method easy to use when compared to other methods that are based on orthogonal matrices or Euler angles.
The objective of this paper is to project the vertices of several 3D polyhedra onto their corresponding 2D Coxeter planes.We define a Coxeter plane by the simple roots of the Coxeter diagram where is the Coxeter number.When projected onto the Coxeter plane, the vertices of any polyhedron form orbits of the dihedral group .

S
Orthogonal projection of higher dimensional lattices onto 2 or 3 dimensional subspaces can be related to quasi-crystal structures.
We organize the paper as follows.In section 2, definitions of finite reflection groups, quaternions, and Coxeter groups are given.We then describe the Platonic solids, the Archimedean solids and their duals with respectively their symmetry groups in section 3.In section 4, we study the projections of the convex uniform polyhedra.Finally, section 5 contains our conclusive remarks.

Finite Reflection Groups
The dihedral group is the group which involves the rotation through an angle , which is expressible as the product of two reflections and whose mirrors have an angle of between them.Here, plane symmetries preserve a regular -sided polygon where .The reflections and generate a dihedral group , which is a subgroup of the symmetric group of order .
A hypercomplex number is defined as quaternion, which can be written in general as with , being the set of real numbers, and the quaternionic units which satisfy where ij  and ijk  are the Kronecker and Levi-Civita symbols, and where summation over the repeated indices is implicit.
Any quaternion can be split into two parts, a scalar part: , and a vector part: .The following definitions will be useful in our discussions.
The conjugate of a quaternion is defined by

̅
(3)  We define the inverse quaternion as  The scalar product of two quaternions and is given by , or as  The product of and ̅ gives a real non-negative number called the norm of .
The norm of does not equal zero and is the length of the quaternion.The unit quaternion is a normalized quaternion when | | .
-The binary tetrahedral (BT) group is represented by the following set of 24 quaternions [5].
This set represents the vertices of the 24-cell.
-The binary octahedral ( ) group consists of the union of set and [3] where The order of the group (10) is 48.
-The binary icosahedral ( ) group is of order 60×2=120.The union of two sets and represents the set of quaternions of the binary icosahedral group as The set in the quaterionic form is [6]: where the golden ratio and its algebraic conjugate respectively.
The vertices of the 600-cell [7] consists of the set of quaternions .However, the set represents the vertices of the snub 24-cell [8].
A Coxeter group is defined as an abstract group that is generated by elements subject to the relations Where and for all , so that .The pair is called a Coxeter system; it consists of a group and a set of generators .
The Cartan matrix is defined by The matrix elements are scalar products of two simple roots and satisfy the following criteria: 1. Every Cartan matrix has an inverse defined by where is the fundamental weight defined as the basis vectors of the dual space.The real and the dual space are in the same space.
2. The relation between and satisfies: The summation over repeated indices is implicit.
Finite Coxeter groups are generated by simple reflections as finite Euclidean reflection groups.A reflection of a vector in a finite dimensional Euclidean space is a linear transformation by a reflection generator r of the finite Coxeter group with respect to a hyperplane perpendicular to the root vector .This can be defined by the formula [5] (19) Here, is a vector which is usually called the root vector or simply root.This root vector is called a simple root in Lie algebra.The norm of the simple root is fixed as √ .
If is an arbitrary simple root written in terms of quaternions, then the reflection of an arbitrary vector with respect to the plane orthogonal to the simple root can be written as the product of quaternions √ ̅ √ (20) We then define this reflection operation by a different notation as * √ √ + (21) Here, the asterisk signifies that we take the conjugate of Λ.However, a rotation can be written in our notation as ̅ (22)

Polyhedra Platonic and Archimedean Solids
Platonic solids are made of regular polygons (triangle, square or pentagon) meeting in identical vertices.On the other hand, the Archimedean solids are convex polyhedra whose faces are regular polygons.One can obtain 5 of them by truncation of the Platonic solids and the others by expansion and snubification.The Archimedean solids are vertextransitive (vertices move to each other in any symmetry operation).The method explained in [4] gives only the vertices of 11 Archimedean solids.The other two chiral solid vertices are taken from [9].The first Platonic solid is the tetrahedron and its polyhedral group is the tetrahedral group where its group elements can be written as To obtain the orbit of the group, corresponding to the vertices of the tetrahedron, we use the method described in [6].
We apply the group elements on the highest weight written in a general form: which can be written in terms of quaternionic imaginary units as .
In [6], the values of and are found from eq (18).Replacing these values in Λ, we obtain: The seven special cases of are indicated by ones or zeros and called the Dynkin indices.The cube and the octahedron are dual of each other, and they also have the same polyhedral symmetry group, which is the octahedral group with 48 elements given by The Coxeter group of this polyhedral symmetry group is represented by the Coxeter-Dynkin diagram shown in Figure 2.  The five different Platonic solids are displayed in Table 1.

Catalan Solids
To construct the dual of a solid we connect vertices which go through the centers of the faces of each solid.The set of new polyhedra is called the Catalan solids.They are not semi-regular since they have more than one set of vertices and their faces are not regular polygons in general.They are made of scalene triangles, isosceles triangles, rhombuses, kites or irregular pentagons.Obviously they are still convex.
The symmetry groups of Catalan solids are the same as their Archimedean duals.Some of the Archimedean and Catalan solids do not have the full symmetry group of a Platonic solid but only the rotational symmetry group of the octahedron or the icosahedron.
The Catalan solids are face-transitive (faces move onto each other in any symmetry operation).In this paper, we take vertices of the Catalan solid from [7].The Archimedean and their dual Catalan solids are displayed in Tables 1-4.

Coxeter plane
The product of all reflections in a Coxeter group is called the Coxeter element: and its order is called Coxeter number .There is a unique plane on which the Coxeter element acts as a rotation by an angle .This plane is called the Coxeter plane [1].Every Coxeter group has a maximal subgroup isomorphic to the dihedral group of order 2h.We use the following technique to determine the Coxeter planes of the three main polyhedaral symmetry groups.
The Coxeter group elements are given in eq. ( 8) and the group is represented by the Coxeter-Dynkin diagram shown in Figure 1  Using eq. ( 14) the Cartan matrix of is ( √ √ ).
Using eq. ( 19), the generators and of correspond to a reflection at a hyperplane perpendicular to the root vectors given as (29) (30) In eqs (31 & 32) we use eq.( 19), replacing α with γ.In eq. ( 29) we define as a linear combination of and : (33 where equals zero, and in eq (30), is defined as where and equal zero.Combining the equations above we can get the values of coefficients , and :  are the simple roots and .
Using eq. ( 14) the Cartan matrix of D 6 is ( √ √ ) Using the method explained above for the I 2 (4), and for the I 2 (6) can be written in terms of quaternions as . The two components of and in the x-axis and y-axis are orthogonal to each other: where ̂ ̂ ̂ ̂ .
Using the argument introduced above, we define the Coxeter plane as 〈 〉 of with generators and respectively.The Cartan matrix of is The coefficients and are √ √ and the coefficient Then and can be written in terms of quaternions as From [4], , , and .The two orthogonal components of and are ̂ in x -axis and ̂ in y -axis

̂ (44) Projection
In our work we use the orthogonal projection of polyhedra in three-dimensions onto their corresponding Coxeter planes.By taking the scalar product of an arbitrary vector of a polyhedron with ̂and ̂ of the Coxeter plane for , and we can determine the horizontal and vertical components of the projected vectors and of the orbit onto the Coxeter plane.

Projection of polyhedra having tetrahedral symmetry.
We can take the scalar product of the vertices of a polyhedron with ̂ ̂ in eqs (31  32) to get and of vertices of the polyhedra having tetrahedral symmetry.Let us now consider the tetrahedron which has tetrahedral symmetry.Here, .When the vertex is multiplied (as a scalar product) with ̂ ̂ the horizontal and vertical projections become respectively ./ . / When this projection is applied to all vertices of , we obtain the full projection of the tetrahedron in Figure 6.The solid line shows the projection of the front faces and the dashed line represents the projection of the back faces, and this applies to all following figures.The projection shows us 18 edges of the truncated tetrahedron with 8 faces (4 triangles and 4 hexagons) and 12 vertices.This projection an "edge first" projection with 8-fold symmetry.We can apply a similar projection to the dual of the truncated tetrahedron which has a symmetry group.The projected vertices are shown Figure 8. Applying the same projection to the other vertices of tetrahedral solids we can get their projections.Table 1 lists the results of projecting the Platonic solids onto their Coxeter planes.Table 2 lists the results of projecting solids having symmetry onto their Coxeter planes.We take the scalar product of the vertices of polyhedra that have the octahedral symmetry with ̂ ̂ of as defined in eqs.(39  40) to get their and components in the Coxeter plane.Table 3 lists the results of the projections of the Archimedean solids onto their Coxeter plane.
Taking the scalar product of the vertices of each polyhedron having an icosahedral symmetry with ̂ ̂ in eqs.(43  44) we get their and onto the (x,y)-plane of .The set of invariant points of the polyhedra having icosahedral symmetry projected onto the Coxeter plane of is listed in Table 3.

Conclusion
In this paper we projected the Platonic, regular Archimedean and Catalan polyhedra in 3 onto the Coxeter plane.We defined the Coxeter plane by the root system of the dihedral group generated by Coxeter-Dynkin diagram .Our projection preserves the h-fold symmetry of the root system which is a subgroup of its complete symmetry.Obviously, after projection, the regular polygons become irregular.However, it can be seen how the polygons still meet at the same vertices.The Dynkin indices chosen were 0's and 1's for a general vector.This work can be extended for those polyhedra whose vertices can be constructed by choosing arbitrary integer values of Dynkin indices.
Table 1.List of the Platonic solids and their projection onto Coxeter plane.

Figure 2 .
Figure 2. Coxeter-Dynkin Diagram of W(B 3 ).The other Platonic solids that are dual of each other are the icosahedron and the dodecahedron.Their polyhedral symmetry group is the icosahedral group of order 120 given by

Figure 4 .
Figure 4. Coxeter-Dynkin diagram of where are the simple roots and The norm and the scalar products are given as ‖ ‖ ‖ ‖ √ , ( ) √ (28) plane.However and are not orthogonal to each other.So, we define an orthogonal set of vertices given by ̂in x-axis , ̂ in y-axis and ̂ ̂ ̂ ̂ .Using the same argument as above, we define the Coxeter plane as 〈 〉 of with generators and where the Coxeter number of equals 6.The dihedral group represented by the Coxeter-Dynkin diagram is shown in Figure5.

Figure 8 .
Figure 8. "Edge first" projection of the dual of the truncated tetrahedron.

Table 4 .
List of the polyhedra (Archimedean & Catalan) having W(H 3 ) symmetry and their projection onto Coxeter plane.