  Quasicrystals described as the projections of higher dimensional cubic
lattices, and the particular affine extensions of the dihedral group $I_2(h)$
of order $2h$, $h=2n$ being the Coxeter number, as a subgroup of affine $B_n$
offers a different perspective to $h$-fold symmetric quasicrystallography.
Affine $I_2(h)$ is constructed as the subgroup of the affine $B_n$, the
symmetry of the cubic lattice $\mathbb{Z}^n$. The infinite discrete group with
local dihedral symmetry of order $2h$ operates on the concentric h-gons
obtained by projecting the Voronoi cell of the cubic lattice with $2^n$
vertices onto the Coxeter plane. Voronoi cells tile the space facet to facet,
consequently, leading to the tilings of the Coxeter plane with some overlaps of
the rhombic tiles. It is noted that the projected Voronoi cell is the overlap
of $h$ copy of the $h$-gons tiled with some rhombi and rotated by the angle
$2\pi/h$. After a general discussion on the lattice $\mathbb{Z}^n$ with the
affine symmetry $\tilde B_n$ and its affine dihedral subgroup $\tilde I_2(h)$
its projection onto the Coxeter plane has been worked out with some examples.
The cubic lattices with affine symmetry $\tilde B_n$ $(n=1,2,3,4,5)$ have been
presented and shown that the projection of the lattice $B_3$ leads to the
hexagonal lattice, the projection of the lattice $B_4$ describes the
Ammann-Beenker quasicrystal lattice with 8-fold local symmetry and the
projection of the lattice $B_5$ describes a quasicrystal structure with local
10-fold symmetry with thick and thin rhombi. It is then straight forward to
show that the projections of the cubic lattices with even higher dimensions
onto the Coxeter plane may lead to the quasicrystal structures with 12-fold,
18-fold symmetries and so on.
