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Tilings colourful - only even more so

Baake, Michael; Grimm, Uwe and Scheffer, Max (2004). Tilings colourful - only even more so. Journal of Non-Crystalline Solids, 334-33 pp. 83–85.

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DOI (Digital Object Identifier) Link: https://doi.org/10.1016/j.jnoncrysol.2003.11.017
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Abstract

We consider colour symmetries for planar tilings of certain n-fold rotational symmetry. The colourings are such that one colour occupies a submodule of n-fold symmetry, while the other colours encode the cosets. We determine the possible numbers of colours and count inequivalent colouring solutions with those numbers. The corresponding Dirichlet series generating functions are zeta functions of cyclotomic fields. The cases with phi(n)<=8, where phi is Euler's totient function, have been completely presented in previous publications. The same methods can be employed to extend the classification to all cases where the cyclotomic integers have class number one. Several examples for symmetries with phi(n)>8 are discussed here.

Item Type: Journal Article
Copyright Holders: 2004 Elsevier
ISSN: 0022-3093
Keywords: quasicrystals; aperiodic tilings; colour symmetry; cyclotomic fields; Dirichlet Series
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 6741
Depositing User: Uwe Grimm
Date Deposited: 13 Feb 2007
Last Modified: 29 Nov 2016 21:07
URI: http://oro.open.ac.uk/id/eprint/6741
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