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- Title
Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional lattices B<sub>n</sub>.
- Authors
Koca, Mehmet; Ozdes Koca, Nazife; Koc, Ramazan
- Abstract
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 = 2 n 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 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.
- Subjects
APERIODIC tilings; LATTICE theory; COXETER groups; SUBSPACES (Mathematics); MATHEMATICAL symmetry
- Publication
Acta Crystallographica. Section A, Foundations & Advances, 2015, Vol 71, Issue 2, p175
- ISSN
2053-2733
- Publication type
Article
- DOI
10.1107/S2053273314025492