Affine Dihedral Subgroups of Higher Dimensional Cubic Lattices Zn\mathbb{Z}^nZn and Quasicrystallography (original) (raw)

Affine coxeter group W a (A 4 ), quaternions, and decagonal quasicrystals

International Journal of Geometric Methods in Modern Physics, 2014

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

Twelvefold symmetric quasicrystallography from the lattices F 4 , B 6 and E 6

Acta Crystallographica Section A Foundations and Advances, 2014

Quasicrystallography may take a century to settle down as commented by Prof Freeman J. Dyson in an address to the American Mathematical Society. One possible way to obtain the quasicrystallographic structures is the projections of the higher dimensional lattices in 2D or 3D. This paper introduces a general technique for the projections of the lattices described by the affine Coxeter groups and, as examples, apply it to the projections of the lattices described by the affine Coxeter groups

From affine A ₄ to affine H ₂: group-theoretical analysis of fivefold symmetric tilings

Acta Crystallographica, 2022

The projections of the lattices, may be used as models of quasicrystals, and the particular affine extension of the 2 symmetry as a subgroup of 4 , discussed in the work, presents a different perspective to 5-fold symmetric quasicrystallography. Affine 2 is obtained as the subgroup of the affine 4. The infinite discrete group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of 4 whose Voronoi cells tessellate the 4D Euclidean space possessing the affine 4 symmetry. Facets of the Voronoi cells of the root and weight lattices are identified. Four adjacent rhombohedral facets of the Voronoi cell (0) of 4 project into the decagonal orbit of 2 as thick and thin rhombuses where long diagonals of the rhombohedra serve as reflection line segments of the reflection operators of 2. It is shown that the thick and thin rhombuses constitute the finite-fragments of the tiles of the Coxeter plane with the action of the affine 2 symmetry. Projection of the Voronoi cell of the weight lattice onto the Coxeter plane tessellates the plane with four different tiles: thick and thin rhombuses with different edge lengths obtained from the projection of the square faces and two types of hexagons obtained from the projection of the hexagonal faces of the Voronoi cell. Structure of the local dihedral symmetry 2 fixing a particular point on the Coxeter plane is determined

12-fold Quasicrystallography from affine F4, B6, and E6

One possible way to obtain the quasicrystallographic structure is the projection of the higher dimensional lattice 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 integers of the Coxeter exponents of a Coxeter-Weyl group play a crucial role in determining the plane onto which the lattice to be projected. The quasicrystal structures display the dihedral symmetry of order twice the Coxeter number. The eigenvectors and the corresponding 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 4 6 6

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

Acta crystallographica. Section A, Foundations and advances, 2015

A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group Wa(Bn) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup Dh of W(Bn) 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(A3), W(H2) × W(A1) and W(H3) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the lattice B4 onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. The B5 lattice is used to describe both fivefold and tenfold symmetries. The lattice B6 can describe aperiodic tilings with 12-fold symmetry as well as a...

From affine A 4 to affine H 2: group-theoretical analysis of fivefold symmetric tilings

Acta Crystallographica Section A Foundations and Advances

The projections of lattices may be used as models of quasicrystals, and the particular affine extension of the H 2 symmetry as a subgroup of A 4, discussed in this work, presents a different perspective on fivefold symmetric quasicrystallography. Affine H 2 is obtained as the subgroup of affine A 4. The infinite discrete group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of A 4 whose Voronoi cells tessellate the 4D Euclidean space possessing the affine A 4 symmetry. Facets of the Voronoi cells of the root and weight lattices are identified. Four adjacent rhombohedral facets of the Voronoi cell V(0) of A 4 project into the decagonal orbit of H 2 as thick and thin rhombuses where long diagonals of the rhombohedra serve as reflection line segments of the reflection operators of H 2. It is shown that the thick and thin rhombuses constitute the finite fragments of the tiles of the Coxeter plane with the action of the affine H 2 sy...

The space groups of axial crystals and quasicrystals

Reviews of Modern Physics, 1991

We compute all the three-dimensional quasicrystallographic space groups with n-fold axial point groups and standard lattices by a method that treats crystals and quasicrystals on an equal footing. We do not rely on projecting higher-dimensional crystallographic space groups, our results are valid for arbitrary n, and our analysis is elementary. We regard space groups as a scheme for classifying diffraction patterns to be carried out in three-dimensional reciprocal space. The familiar three-dimensional crystallographic space groups with axial point groups emerge simply and directly as special cases of the general n-fold three-dimensional quasicrystallographic treatment with n=3, 4, and 6. We give a general discussion of extinctions in quasicrystals and give a simple (three-dimensional) geometrical specification of the extinctions for each axial space group. The paper is intended both for people trying to systematize quasicrystal diffraction patterns and for people interested in a simple alternative approach to the computation of crystallographic or quasicrystallographic space groups.

From Affine A_4A_4A4 to Affine H2H_2H_2: Group Theoretical Analysis of Penrose-like Tilings

arXiv (Cornell University), 2021

Remarks on symmetries of 2D-quasicrystals

International Journal of Computer Mathematics, 2008

The paper presents a strict mathematical proof of the classifications of finite groups of symmetries of 2-dimensional quasicrystals given in .

Affine Dihedral Subgroups of Higher Dimensional Cubic Lattices Zn\mathbb{Z}^nZn and Quasicrystallography (original) (raw)

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