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, 334-335 pp. 83–85.

DOI: https://doi.org/10.1016/j.jnoncrysol.2003.11.017


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 φ(n)<=8, where φ 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 φ(n)>8 are discussed here.

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