  Affine subgroups having the same Coxeter number with the affine Coxeter
groups W(An), W(Dn), and W(En) are constructed by graph folding technique. The
affine groups W(Cn) and W(Bn) are obtained from the Coxeter groups W(A2n-1) and
W(D2n-1) respectively. The affine groups W(E6), W(D6) and W(E8) lead to the
affine groups W(F4), W(H3), and W(H4) respectively by graph folding. The latter
two are the non-crystallographic groups where W(H3) plays a special role in the
quasicrystallographic structures with icosahedral symmetry. A general
construction of the affine dihedral subgroups is introduced, some of which,
describe the existing planar quasicrystallography. In the construction of the
root systems, sets of orthonormal vectors are used but a special non-orthogonal
set of vectors in the formulation of the root system of W(An) is also
introduced which has practical applications in the construction of the lattices
An and An* and their Delone and Voronoi cells.
