2012/09/30 by Mehmet Koca, Nazife Ozdes Koca, Nazife O. Koca +1
Chemistry · Materials Science · Mathematics · Physics and Astronomy · #Affine transformation #Artin group #Coxeter complex #Coxeter element #Coxeter group #Crystal Structures and Properties #Lattice (music) #Longest element of a Coxeter group #Point group #Quasicrystal Structures and Properties #Synthesis and Properties of Aromatic Compounds #cond-mat.other #math-ph #math.MP
paper · pdf · doi:10.1142/s0219887814500315
published as International Journal of Geometric Methods in Modern Physics (IJGMMP) 11 (2014), World Scientific Publishing Company · 26 pages, 17 figures
arxiv created 2013/03/26 · openalex publication_date 2013/12/23 · arxiv updated 2014/03/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 D h of order 2h as a maximal subgroup. As a simple application, we demonstrate projections of the root and weight lattices of A 4 onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine W a (A 4 ) can be decomposed into two orthogonal spaces whose point group is the dihedral group D 5 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 (A 3 ), 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 W a (A n ), n ≥ 4. We also note that the projection of the Voronoi cell of the root lattice of W a (A 4 ) describes a framework of nested decagram growing with the power of the golden ratio recently discovered in the Islamic arts.