Quasicrystallography from B_n lattices

We present a group theoretical analysis of the hypercubic lattice described by the affine Coxeter-Weyl group W_a (B_n). An h-fold symmetric quasicrystal structure follows from the hyperqubic lattice whose point group is described by the Coxeter-Weyl group W (B_n) with the Coxeter number h=2n. Higher dimensional cubic lattices are explicitly constructed for n 4,5,6 by identifying their rank-3 Coxeter subgroups and maximal dihedral subgroups. Decomposition of their Voronoi cells under the respective rank-3 subgroups W (A₃), W (H₂)×W (A₁) and W (H₃)lead to the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron respectively. Projection of the lattice B₄ describes a quasicrystal structure with 8-fold symmetry. The B₅ lattice leads to quasicrystals with both 5fold and 10 fold symmetries. The lattice B₆ projects on a 12-fold symmetric quasicrystal as well as a 3D icosahedral quasicrystal depending on the choice of subspace of projections. The projected sets of lattice points are compatible with the available experimental data.