H-fold Symmetric quasicrystallography from affine Coxeter groups

One can obtain the quasicrystallographic structures from the projection of the higher dimensional lattices into 2D or 3D subspaces. Here we introduce a general technique applicable to any higher dimensional lattice described by the affine Coxeter groups. It is pointed out that the Coxeter number h and the Coxeter exponents play an important role in determining the principal planes onto which the lattice to be projected. The quasicrystal structures obtained by projection display the dihedral symmetry of order 2h . The projection subspaces are determined by using the eigenvectors and the corresponding eigenvalues of the Cartan matrix. Examples are given for 12-fold symmetric quasicrystal structures obtained by projections of the lattices determined by the affine Coxeter-Weyl groups W_a (F₄), W_a (B₆), and W_a (E₆). The reflection generators R₁ and R₂ of the dihedral group D₁₂ can be obtained as the products of the generators of the Coxeter-Weyl groups. It is noted that the quasicrystal structures obtained from the lattices W_a (F₄) and W_a (B₆) are compatible with the experimental data.