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.
Department of Physics, College of Science, Sultan Qaboos University, PO Box 36, Al-Khoud, 123 Muscat, Sultanate of Oman, Oman.
Mehmet
M
Koca
Department of Physics, College of Science, Sultan Qaboos University, PO Box 36, Al-Khoud, 123 Muscat, Sultanate of Oman, Oman.
Nazife
N
Ozdes Koca
Department of Physics, Gaziantep University, Gaziantep 27310, Turkey.
Ramazan
R
Koc
Coxeter–Weyl groups; aperiodic tilings; cut-and-project technique; lattices; quasicrystallography; strip projection