Twelvefold symmetric quasicrystallography from the latticesF4,B6andE6

Abstract

One possible way to obtain the quasicrystallographic structure is the projection of the higher-dimensional lattice into two- or three-dimensional subspaces. Here a general technique applicable to any higher-dimensional lattice is introduced. The Coxeter number and the integers of the Coxeter exponents of a Coxeter–Weyl group play a crucial role in determining the plane onto which the lattice is to be projected. The quasicrystal structures display the dihedral symmetry of order twice that of the Coxeter number. The eigenvectors and the corresponding eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors inn-dimensional Euclidean space which lead to 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. Examples are given for 12-fold symmetric quasicrystal structures obtained by projecting the higher-dimensional lattices determined by the affine Coxeter–Weyl groupsWa(F4),Wa(B6) andWa(E6). These groups share the same Coxeter numberh= 12 with different Coxeter exponents. The dihedral subgroupD12of the Coxeter groups can be obtained by defining two generatorsR1andR2as the products of generators of the Coxeter–Weyl groups. The reflection generatorsR1andR2operate in the Coxeter planes where the Coxeter elementR1R2of the Coxeter–Weyl group represents the rotation of order 12. The canonical (strip, equivalently, cut-and-project technique) projections of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with fourfold and sixfold symmetry. It is noted that the quasicrystal structures obtained from the latticesWa(F4) andWa(B6) are compatible with some experimental results.