Bravais colourings of planar modules with N-fold symmetry

Baake, Michael and Grimm, Uwe (2004). Bravais colourings of planar modules with N-fold symmetry. Zeitschrift für Kristallographie, 219(2) pp. 72–80.

DOI: https://doi.org/10.1524/zkri.219.2.72.26322

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.

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About

• Item ORO ID
• 2196
• Item Type
• Journal Item
• ISSN
• 0044-2968
• Extra Information
• Preprint version math.CO/0301021 available at http://uk.arxiv.org/abs/math.CO/0301021
• Keywords
• Colourings; Planar modules; Cyclotomic fields; Dirichlet series
• Academic Unit or School
• Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
  Faculty of Science, Technology, Engineering and Mathematics (STEM)
• Depositing User
• Uwe Grimm