Bravais colourings of planar modules with N-fold symmetry

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.

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

References in corpus (4)

• Planar coincidences for N-fold symmetry
• Combinatorial problems of (quasi-)crystallography
• Colourings of planar quasicrystals
• Lattice Color Groups of Quasicrystals

Cited by in corpus (13)

• Solution of the coincidence problem in dimensions $d\le 4$
• Multiple planar coincidences with N-fold symmetry
• Coincidences in 4 dimensions
• Similarity and Coincidence Isometries for Modules
• Similar submodules and coincidence site modules
• Perfect colourings of cyclotomic integers
• On Bravais Colorings Associated with Periodic and Non-Periodic Crystals
• Similarity Isometries of Point Packings
• Counting perfect colourings of plane regular tilings
• On Bravais colourings of cyclotomic integers with class number one
• Geometric enumeration problems for lattices and embedded $\mathbb{Z}$-modules
• Ten colours in quasiperiodic and regular hyperbolic tilings
• Symmetries and color symmetries of a family of tilings with a singular point