12-fold Quasicrystallography from affine F4, B6, and E6

Abstract

One possible way to obtain the quasicrystallographic structures is the projections of the higher dimensional lattices into 2D or 3D subspaces. In this work we introduce a general technique applicable to any higher dimensional lattice. We point out that the Coxeter number and the Coxeter exponents of a Coxeter-Weyl group play a crucial role in determining the plane onto which the lattice to be projected as well as the dihedral symmetry of the quasicrystal structure. The eigenvectors and eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors in nD Euclidean space which lead suitable choices for the projection subspaces. The maximal dihedral subgroup of the Coxeter-Weyl group is identified to determine the symmetry of the quasicrystal structure. We give examples for 12-fold symmetric quasicrystal structures obtained by projecting the higher dimensional lattices determined by the affine Coxeter-Weyl groups Wa(F4), Wa(B6) and Wa(E6) . These groups share the same Coxeter number h=12 with different Coxeter exponents. The dihedral subgroup D12 of the Coxeter groups can be obtained by defining two generators R1 and R2 as the products of generators of the Coxeter-Weyl groups. The reflection generators R1 and R2 operate in the Coxeter planes where the Coxeter element R1R2 of the Coxeter group represents the rotation of order 12. The canonical projections (strip projections) of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with 4-fold and 6-fold symmetry. We note that the quasicrystal structures obtained from the lattices Wa(F4) and Wa(B6) and are compatible with the experimental results.