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From affine A4 to affine H2: group‐theoretical analysis of fivefold symmetric tilings

From affine A4 to affine H2: group‐theoretical analysis of fivefold symmetric tilings The projections of lattices may be used as models of quasicrystals, and the particular affine extension of the H2 symmetry as a subgroup of A4, discussed in this work, presents a different perspective on fivefold symmetric quasicrystallography. Affine H2 is obtained as the subgroup of affine A4. The infinite discrete group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of A4 whose Voronoi cells tessellate the 4D Euclidean space possessing the affine A4 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 A4 project into the decagonal orbit of H2 as thick and thin rhombuses where long diagonals of the rhombohedra serve as reflection line segments of the reflection operators of H2. 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 H2 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. The structure of the local dihedral symmetry H2 fixing a particular point on the Coxeter plane is determined. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Crystallographica Section A: Foundations and Advances Wiley

From affine A4 to affine H2: group‐theoretical analysis of fivefold symmetric tilings

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References (7)

Publisher
Wiley
Copyright
Copyright © 2022 Wiley Subscription Services, Inc., A Wiley Company
eISSN
2053-2733
DOI
10.1107/s2053273322002595
Publisher site
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Abstract

The projections of lattices may be used as models of quasicrystals, and the particular affine extension of the H2 symmetry as a subgroup of A4, discussed in this work, presents a different perspective on fivefold symmetric quasicrystallography. Affine H2 is obtained as the subgroup of affine A4. The infinite discrete group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of A4 whose Voronoi cells tessellate the 4D Euclidean space possessing the affine A4 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 A4 project into the decagonal orbit of H2 as thick and thin rhombuses where long diagonals of the rhombohedra serve as reflection line segments of the reflection operators of H2. 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 H2 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. The structure of the local dihedral symmetry H2 fixing a particular point on the Coxeter plane is determined.

Journal

Acta Crystallographica Section A: Foundations and Advances – Wiley

Published: May 1, 2022

Keywords: affine groups; quasicrystals; aperiodic tilings; lattices; projections of polytopes

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