TY - JOUR
T1 - Affine coxeter group Wa(A4), quaternions, and decagonal quasicrystals
AU - Koca, Mehmet
AU - Koca, Nazife Ozdes
AU - Koc, Ramazan
PY - 2014/4
Y1 - 2014/4
N2 - 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 I 2(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(A 4) can be decomposed into two orthogonal spaces whose point group 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 W a(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.
AB - 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 I 2(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(A 4) can be decomposed into two orthogonal spaces whose point group 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 W a(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.
KW - Quaternions
KW - affine Coxeter groups
KW - quasicrystal
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U2 - 10.1142/S0219887814500315
DO - 10.1142/S0219887814500315
M3 - Article
AN - SCOPUS:84899116454
SN - 0219-8878
VL - 11
JO - International Journal of Geometric Methods in Modern Physics
JF - International Journal of Geometric Methods in Modern Physics
IS - 4
M1 - 1450031
ER -