Mehmet Koca, Nazife O. Koca, Ramazan Koc
The root and weight lattices of A4 are constructed with the quaternionic representation of the affine Coxeter-Weyl group W(A4). We develop a technique for the orthogonal projection of the lattice vectors on the Coxeter plane which is defined by the simple roots of the Coxeter graph I2(5)(also denoted by H2). The projected point set of the root lattice displays a generalized Penrose tiling with a point dihedral symmetry D5 of order 10 which can be used for the description of the decagonal quasicrystals. The paper also revises the quaternionic descriptions of the root and weight lattices of A3 which correspond to the face centered cubic (fcc) lattice and body centered cubic (bcc) lattice respectively. Extensions of these lattices to the 4D lead to the root and weight lattices of A4. We also note that the projection of the Voronoi cell of the root lattice of A4 describes a framework of nested decagrams growing with the power of the golden ratio recently discovered in the Islamic arts.
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http://arxiv.org/abs/1209.1878
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