Bravais colourings of planar modules with N-fold symmetry

Details Bravais colourings of planar modules with N-fold symmetry Bravais colourings of planar modules with N-fold symmetry

2003-01-03

arxiv.org/abs/math/0301021v3

Zeitschrift f. Kristallographie 219 (2004) 72-80

Mathematics

Combinatorics

8 pages, references updated, see also math.MG/0511147 and math.MG/0511306 for related work on single and multiple coincidences

Scientific paper

10.1524/zkri.219.2.72.26322

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.

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