2003/01/31 by Michael Baake, Uwe Grimm
Materials Science · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Graph theory and applications #Quasicrystal Structures and Properties #math-ph #math.CO #math.MG #math.MP #msc:05A15 #msc:11R99 #msc:52C20 #msc:52C23
paper · pdf · doi:10.1524/zkri.219.2.72.26322
published as Zeitschrift f. Kristallographie 219 (2004) 72-80 · 8 pages, references updated, see also math.MG/0511147 and math.MG/0511306 for related work on single and multiple coincidences
openalex publication_date 2004/02/01 · arxiv created 2006/06/26 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Abstract The first step in investigating colour symmetries for periodic and aperiodic systems is the determination of all colouring schemes that are compatible with the symmetry group of the underlying structure, or with a subgroup of it. For an important class of colourings of planar structures, this mainly combinatorial question can be addressed with methods of algebraic number theory. We present the corresponding results for all planar modules with N -fold symmetry that emerge as the rings of integers in cyclotomic fields with class number one. The counting functions are multiplicative and can be encapsulated in Dirichlet series generating functions, which turn out to be the Dedekind zeta functions of the corresponding cyclotomic fields.