12-Fold Symmetric Quasicrystallography from the lattices F₄, B₆, and E₆

One possible way to obtain the quasicrystallographic structure is the projection of the higher dimensional lattice 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 integers of the Coxeter exponents of a Coxeter-Weyl group play a crucial role in determining the plane onto which the lattice to be projected. The quasicrystal structures display the dihedral symmetry of order twice the Coxeter number. The eigenvectors and the corresponding 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 W_a(F₄),W_a (B₆), and W_a (E₆) . These groups share the same Coxeter number h =12with 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 R₁ R₂ of the Coxeter-Weyl group represents the rotation of order 12. The canonical projections (strip projection, equivalently, cut and project technique) 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 W_a (F₄) and W_a (B₆) are compatible with some experimental results.