  We introduce a technique of projection onto the Coxeter plane of an arbitrary
higher dimensional lattice described by the affine Coxeter group. The Coxeter
plane is determined by the simple roots of the Coxeter graph I2 (h) where h is
the Coxeter number of the Coxeter group W(G) which embeds the dihedral group Dh
of order 2h as a maximal subgroup. As a simple application we demonstrate
projections of the root and weight lattices of A4 onto the Coxeter plane using
the strip (canonical) projection method. We show that the crystal spaces of the
affine Wa(A4) can be decomposed into two orthogonal spaces whose point groups
is the dihedral group D5 which acts in both spaces faithfully. The strip
projections of the root and weight lattices can be taken as models for the
decagonal quasicrystals. The paper also revises the quaternionic descriptions
of the root and weight lattices, described by the affine Coxeter group Wa(A3),
which correspond to the face centered cubic (fcc) lattice and body centered
cubic (bcc) lattice respectively. Extensions of these lattices to higher
dimensions lead to the root and weight lattices of the group Wa(An), n>=4 . We
also note that the projection of the Voronoi cell of the root lattice of Wa(A4)
describes a framework of nested decagram growing with the power of the golden
ratio recently discovered in the Islamic arts.
