Quasicrystallography from Bn 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(A(3)), W(H-2)xW(A(1)) and W(H-3) lead to the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron respectively. Projection of the lattice B-4 describes a quasicrystal structure with 8-fold symmetry. The B-5 lattice leads to quasicrystals with both 5-fold and 10 fold symmetries. The lattice B-6 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.