A group theoretical discussion on the hypercubic lattice described by theaffine Coxeter-Weyl group Wa(Bn) has been presented. When the lattice isprojected onto the Coxeter plane it is noted that the maximal dihedral subgroupDh of W(Bn) with h = 2n representing the Coxeter number describes the h-foldsymmetric quasicrystallography. Higher dimensional cubic lattices areexplicitly constructed for n = 4, 5, 6. Their rank 3 Coxeter subgroups andmaximal dihedral subgroups are identified. It has been explicitly shown thatwhen their Voronoi cells are decomposed under the respective rank 3 subgroupsW(A3),W(H2) x W(A1) and W(H3) one obtains the rhombic dodecahedron, rhombicicosahedron and rhombic triacontahedron respectively. Projection of the latticeB4 onto the Coxeter plane represents quasicrystal structures with 8-foldsymmetry. The B5 lattice is used to describe the quasicrystals with both 5-foldand 10-fold symmetries. The lattice B6 can describe a 12-fold symmetricquasicrystal as well as a 3D icosahedral quasicrystal depending on the choiceof subspace of projections. The novel structures from the projected sets oflattice points are compatible with the available experimental data.
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