  A group theoretical discussion on the hypercubic lattice described by the
affine Coxeter-Weyl group Wa(Bn) has been presented. When the lattice is
projected onto the Coxeter plane it is noted that the maximal dihedral subgroup
Dh of W(Bn) with h = 2n representing the Coxeter number describes the h-fold
symmetric quasicrystallography. Higher dimensional cubic lattices are
explicitly constructed for n = 4, 5, 6. Their rank 3 Coxeter subgroups and
maximal dihedral subgroups are identified. It has been explicitly shown that
when their Voronoi cells are decomposed under the respective rank 3 subgroups
W(A3),W(H2) x 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 quasicrystal structures with 8-fold
symmetry. The B5 lattice is used to describe the quasicrystals with both 5-fold
and 10-fold symmetries. The lattice B6 can describe a 12-fold symmetric
quasicrystal as well as a 3D icosahedral quasicrystal 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.
