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

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(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.