Nazife Ozdes Koca

Catalan solids derived from three-dimensional-root systems and quaternions

Catalan solids are the duals of the Archimedean solids, the vertices of which can be obtained from the Coxeter–Dynkin diagrams A3, B3, and H3 whose simple roots can be represented by quaternions. The respective Weyl groups W(A3), W(B3), and W(H3) acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result from the orbits derived from fundamental weights. The Platonic solids are dual to each other; however, the duals of the Archimedean solids are the Catalan solids whose vertices can be written as the union of the orbits, up to some scale factors, obtained by applying the above Weyl groups on the fundamental highest weights (100), (010), and (011) for each diagram. The faces are represented by the orbits derived from the weights (010), (110), (101), (011), and (111), which correspond to the vertices of the Archimedean solids. Representations of the Weyl groups W(A3), W(B3), and W(H3) by the quaternions sim...

Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system

Abstract Vertices of the 4-dimensional semi-regular polytope, snub 24-cell and its symmetry group ( W ( D 4 ) / C 2 ) : S 3 of order 576 are represented in terms of quaternions with unit norm. It follows from the icosian representation of E 8 root system. A simple method is employed to construct the E 8 root system in terms of icosians which decomposes into two copies of the quaternionic root system of the Coxeter group W ( H 4 ) , while one set is the elements of the binary icosahedral group the other set is a scaled copy of the first. The quaternionic root system of H 4 splits as the vertices of 24-cell and the snub 24-cell under the symmetry group of the snub 24-cell which is one of the maximal subgroups of the group W ( H 4 ) as well as W ( F 4 ) . It is noted that the group is isomorphic to the semi-direct product of the proper rotation subgroup of the Weyl group of D 4 with symmetric group of order 3 denoted by ( W ( D 4 ) / C 2 ) : S 3 , the Coxeter notation for which is [ 3 , 4 , 3 + ] . We analyze the vertex structure of the snub 24-cell and decompose the orbits of W ( H 4 ) under the orbits of ( W ( D 4 ) / C 2 ) : S 3 . The cell structure of the snub 24-cell has been explicitly analyzed with quaternions by using the subgroups of the group ( W ( D 4 ) / C 2 ) : S 3 . In particular, it has been shown that the dual polytopes 600-cell with 120 vertices and 120-cell with 600 vertices decompose as 120 = 24 + 96 and 600 = 24 + 96 + 192 + 288 respectively under the group ( W ( D 4 ) / C 2 ) : S 3 . The dual polytope of the snub 24-cell is explicitly constructed. Decompositions of the Archimedean W ( H 4 ) polytopes under the symmetry of the group ( W ( D 4 ) / C 2 ) : S 3 are given in the appendix.

Affine coxeter group Wa(A4), quaternions, and decagonal quasicrystals

We introduce a technique of projection onto the Coxeter plane of an arbitrary higher-dimensional lattice described by the affine Coxeter group. The Coxeter plane is determined by the simple roots of the Coxeter graph I2(h) where h is the Coxeter number of the Coxeter group W(G) which embeds the dihedral group Dh of order 2h as a maximal subgroup. As a simple application, we demonstrate projections of the root and weight lattices of A4 onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine Wa(A4) can be decomposed into two orthogonal spaces whose point group is the dihedral group D5 which acts in both spaces faithfully. The strip projections of the root and weight lattices can be taken as models for the decagonal quasicrystals. The paper also revises the quaternionic descriptions of the root and weight lattices, described by the affine Coxeter group Wa(A3), which correspond to the face centered cubic (fcc) lattice and body centered cubic (bcc) lattice respectively. Extensions of these lattices to higher dimensions lead to the root and weight lattices of the group Wa(An), n ≥ 4. We also note that the projection of the Voronoi cell of the root lattice of Wa(A4) describes a framework of nested decagram growing with the power of the golden ratio recently discovered in the Islamic arts.

Twelvefold symmetric quasicrystallography from the lattices F4, B6 and E6

One possible way to obtain the quasicrystallographic structure is the projection of the higher-dimensional lattice into two- or three-dimensional subspaces. Here a general technique applicable to any higher-dimensional lattice is introduced. The Coxeter number and the integers of the Coxeter exponents of a Coxeter–Weyl group play a crucial role in determining the plane onto which the lattice is to be projected. The quasicrystal structures display the dihedral symmetry of order twice that of the Coxeter number. The eigenvectors and the corresponding eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors in n-dimensional Euclidean space which lead to suitable choices for the projection subspaces. The maximal dihedral subgroup of the Coxeter–Weyl group is identified to determine the symmetry of the quasicrystal structure. Examples are given for 12-fold symmetric quasicrystal structures obtained by projecting the higher-dimensional lattices determined by the affine Coxeter–Weyl groups Wa(F4), Wa(B6) and Wa(E6). These groups share the same Coxeter number h = 12 with different Coxeter exponents. The dihedral subgroup D12 of the Coxeter groups can be obtained by defining two generators R1 and R2 as the products of generators of the Coxeter–Weyl groups. The reflection generators R1 and R2 operate in the Coxeter planes where the Coxeter element R1R2 of the Coxeter–Weyl group represents the rotation of order 12. The canonical (strip, equivalently, cut-and-project technique) projections of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with fourfold and sixfold symmetry. It is noted that the quasicrystal structures obtained from the lattices Wa(F4) and Wa(B6) are compatible with some experimental results.One possible way to obtain the quasicrystallographic structures is the projections of the higher dimensional lattices into 2D or 3D subspaces. In this work we introduce a general technique applicable to any higher dimensional lattice. We point out that the Coxeter number and the Coxeter exponents of a Coxeter-Weyl group play a crucial role in determining the plane onto which the lattice to be projected as well as the dihedral symmetry of the quasicrystal structure. The eigenvectors and eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors in nD Euclidean space which lead suitable choices for the projection subspaces. The maximal dihedral subgroup of the Coxeter-Weyl group is identified to determine the symmetry of the quasicrystal structure. We give examples for 12-fold symmetric quasicrystal structures obtained by projecting the higher dimensional lattices determined by the affine Coxeter-Weyl groups Wa(F4), Wa(B6) and Wa(E6) . These groups share the same Coxeter number h=12 with different Coxeter exponents. The dihedral subgroup D12 of the Coxeter groups can be obtained by defining two generators R1 and R2 as the products of generators of the Coxeter-Weyl groups. The reflection generators R1 and R2 operate in the Coxeter planes where the Coxeter element R1R2 of the Coxeter group represents the rotation of order 12. The canonical projections (strip projections) of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with 4-fold and 6-fold symmetry. We note that the quasicrystal structures obtained from the lattices Wa(F4) and Wa(B6) and are compatible with the experimental results.

SNUB 24-CELL DERIVED FROM THE COXETER–WEYL GROUP W(D4)

Snub 24-cell is the unique uniform chiral polytope in four dimensions consisting of 24 icosahedral and 120 tetrahedral cells. The vertices of the four-dimensional semiregular polytope snub 24-cell and its symmetry group (W(D4)/C2): S3 of order 576 are obtained from the quaternionic representation of the Coxeter–Weyl group W(D4). The symmetry group is an extension of the proper subgroup of the Coxeter–Weyl group W(D4) by the permutation symmetry of the Coxeter–Dynkin diagram D4. The 96 vertices of the snub 24-cell are obtained as the orbit of the group when it acts on the vector Λ = (τ, 1, τ, τ) or on the vector Λ = (σ, 1, σ, σ) in the Dynkin basis with and . The two different sets represent the mirror images of the snub 24-cell. When two mirror images are combined it leads to a quasiregular four-dimensional polytope invariant under the Coxeter–Weyl group W(F4). Each vertex of the new polytope is shared by one cube and three truncated octahedra. Dual of the snub 24-cell is also constructed. Relevance of these structures to the Coxeter groups W(H4) and W(E8) has been pointed out.

4D-POLYTOPES AND THEIR DUAL POLYTOPES OF THE COXETER GROUP W(A4) REPRESENTED BY QUATERNIONS

Four-dimensional A4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter–Weyl group W(A4) where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary W(A4) orbit into three dimensions is made using the subgroup W(A3). A generalization of the Catalan solids for 3D-polyhedra has been developed and dual polytopes of the uniform A4 polytopes have been constructed.

Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional lattices Bn.

A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group W(a)(B(n)) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup D(h) of W(B(n)) with h = 2n representing the Coxeter number describes the h-fold symmetric aperiodic tilings. Higher-dimensional cubic lattices are explicitly constructed for n = 4, 5, 6. Their rank-3 Coxeter subgroups and maximal dihedral subgroups are identified. It is explicitly shown that when their Voronoi cells are decomposed under the respective rank-3 subgroups W(A(3)), W(H(2)) × W(A(1)) and W(H(3)) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the lattice B(4) onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. The B(5) lattice is used to describe both fivefold and tenfold symmetries. The lattice B(6) can describe aperiodic tilings with 12-fold symmetry as well as a three-dimensional icosahedral symmetry depending on the choice of subspace of projections. The novel structures from the projected sets of lattice points are compatible with the available experimental data.

SU(5) grand unified theory, its polytopes and 5-fold symmetric aperiodic tiling

We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell (0001)A4 and the rectified 5-cell (0100)A4 derived from the SU(5) Coxeter–Dynkin diagram. The off-diagonal gauge bosons are associated with the root polytope (1001)A4 whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the SU(5) charge conservation. The Dynkin diagram symmetry of the SU(5) diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root lattice consists of the union of the polytopes (1000)A4 + (0100)A4 + (0010)A4 + (0001)A4 whose facets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the root lattice as the alternating 5-cell and the rectified 5-cell, a kind of dual to the Voronoi cell. The vertices of the Delone cells closest to the origin consist of the root vectors representing the gauge bosons. The faces of the rhombohedra project onto the Coxeter plane as thick and thin rhombs leading to Penrose-like ti...

QUATERNIONIC CONSTRUCTION OF THE W(F4) POLYTOPES WITH THEIR DUAL POLYTOPES AND BRANCHING UNDER THE SUBGROUPS W(B4) AND W(B3) × W(A1)

Four-dimensional F4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter–Weyl group W(F4 ) where the group elements and the vertices of the polytopes are represented by quaternions. Branchings of an arbitrary W(F4 ) orbit under the Coxeter groups W(B4) and W(B3) × W(A1) have been presented. The role of group theoretical technique and the use of quaternions have been emphasized.

Branching of the W(H 4) polytopes and their dual polytopes under the coxeter groups W(A 4) and W(H 3) represented by quaternions

dimensional H4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter- Weyl group W (H4) , where the group elements and the vertices of the polytopes are represented by quater- nions. Projection of an arbitrary W (H4) orbit into three dimensions is made preserving the icosahedral subgroup W (H3) and the tetrahedral subgroup W (A3) . The latter follows a branching under the Cox- eter group W (A4) . The dual polytopes of the semi-regular and quasi-regular H4 polytopes have been constructed.