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

Koca, Mehmet; Koca, Nazife Ozdes; Koc, Ramazan

doi:10.1063/1.3356985

Cited by 21 publications

(26 citation statements)

References 12 publications

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“…These groups are discussed extensively in the references [14][15]. Qua representations of these groups can be classified as follows.…”

Section: Rank-3 Coxeter-weyl Groups and Polyhedramentioning

confidence: 99%

Coxeter Groups, Quaternions, Symmetries of Polyhedra and 4d Polytopes

Koca¹,

Koca²

2012

Mathematical Physics

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Emergence of the experimental evidence of E 8 in the analysis of one dimensional Ising-model invokes further studies of the Coxeter-Weyl groups generated by reflections regarding their applications to polytopes. The Coxeter group W (H 4 ) which describes the Platonic Polytopes 600-cell and 120-cell in 4D singles out in the mass relations of the bound states of the Ising model for it is a maximal subgroup of the Coxeter-Weyl group W (E 8 ). There exists a one-to-one correspondence between the finite subgroups of quaternions and the Coxeter-Weyl groups of rank 4 which facilitates the study of the rank-4 Coxeter-Weyl groups. In this paper we study the systematic classifications of the 3D-polyhedra and 4D-polytopes through their symmetries described by the rank-3 and rank-4 Coxeter-Weyl groups represented by finite groups of quaternions. We also develop a technique on the constructions of the duals of the polyhedra and the polytopes and give a number of examples. Applications of the rank-2 Coxeter groups have been briefly mentioned.

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“…These groups are discussed extensively in the references [14][15]. Qua representations of these groups can be classified as follows.…”

Section: Rank-3 Coxeter-weyl Groups and Polyhedramentioning

confidence: 99%

Coxeter Groups, Quaternions, Symmetries of Polyhedra and 4d Polytopes

Koca¹,

Koca²

2012

Mathematical Physics

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“…When expressed in terms of the unit vectors t 0 ; t 1 ; t 2 and t 3 and the component of an arbitrary vector along t 0 is deleted they will decompose under the octahedral group as two orbits, one with the vertices ðAEt 1 ; AEt 2 ; AEt 3 Þ representing the vertices of an octahedron and the other with vertices 1 2 ðAEt 1 AE t 2 AE t 3 Þ representing a cube. Union of these two orbits represents a rhombic dodecahedron (Koca et al, 2010) with 14 vertices as shown in Fig. 3. 3.2.…”

Section: The Coxeter-weyl Group W(b 4 ) and Projection Of Its Fundamementioning

confidence: 99%

Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional latticesB_n

Koca¹,

Koca²,

Koc³

2015

Acta Cryst Sect A

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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.

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“…represent the tetrahedron, octahedron, cuboctahedron (vertices represented by the non-zero roots of 3 A ), and the truncated octahedron respectively. One can find the vertices of the duals of these polyhedra represented by quaternions in a simple manner [29,30]. The dual of the tetrahedron 3 (1, 0, 0) A , for example, is the tetrahedron represented by 3 (0, 0,1) A .…”

Section: Fig2 Coxeter Diagram Of 3 B With Quaternionic Simple Rootsmentioning

confidence: 99%

“…can be blockdiagonalized in terms of 22  matrices corresponding to two different representations of the graph 2 H . To complete this let us define 3   and convert the coefficients in(29)(30) into a matrix then the block diagonalized Cartan matrix turns out to be…”

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

Koca

Koc

2014

Int. J. Geom. Methods Mod. Phys.

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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 2 () Ih where h is the Coxeter number of the Coxeter group () WGwhich embeds the dihedral group h D of order 2h as a maximal subgroup. As a simple application we demonstrate projections of the root and weight lattices of A 4 onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine 4 () a WAcan be decomposed into two orthogonal spaces whose point groups is the dihedral group 5 D 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 3 () a WA, 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 () an WA, 4 n  . We also note that the projection of the Voronoi cell of the root lattice of

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

Cited by 21 publications

References 12 publications

Coxeter Groups, Quaternions, Symmetries of Polyhedra and 4d Polytopes

Coxeter Groups, Quaternions, Symmetries of Polyhedra and 4d Polytopes

Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional latticesB_n

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

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

Cited by 21 publications

References 12 publications

Coxeter Groups, Quaternions, Symmetries of Polyhedra and 4d Polytopes

Coxeter Groups, Quaternions, Symmetries of Polyhedra and 4d Polytopes

Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional latticesBn

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

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Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional latticesB_n

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