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. 
Mehmet Koca, Nazife Ozdes Koca and Ramazan Koc
math-ph cond-mat.mtrl-sci cond-mat.other math.MP