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Group Theoretical Analysis of Quasicrystallography from Projections of Higher Dimensional Lattices Bn

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.