  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.
