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Scheffer: Colourings of planar quasicrystals

By Michael Baake, Uwe Grimm and Max Scheffer


The investigation of colour symmetries for periodic and aperiodic systems consists of two steps. The first concerns the computation of the possible numbers of colours and is mainly combinatorial in nature. The second is algebraic and determines the actual colour symmetry groups. Continuing previous work, we present the results of the combinatorial part for planar patterns with n-fold symmetry, where n = 7,9,15,16,20,24. This completes the cases with φ(n) ≤ 8, where φ is Euler’s totient function. Colour symmetries of crystals and, more recently, of quasicrystals continue to attract a lot of attention, simply because so little is known about their classification, see [4] for a recent review. A first step in this analysis consists in answering the question how many different colourings of an infinite point set exist which are compatible with its underlying symmetry. More precisely, one has to determine the possible numbers of colours, and to count the corresponding possibilities to distribute the colours over the point set (up to permutations). In this generality, the problem has not even been solved for simple lattices. One common restriction is to demand that one colour occupies a subset which is of the same Bravais type as the original set, while the other colours encode the cosets. In this situation, to which we will also restrict ourselves, several results are known and can be given in closed form [1, 2, 3, 4, 5]. Of particular interest are planar cases because, on the one hand, they show up in quasicrystalline T-phases, and, on the other hand, they are linked to the rather interesting classification of planar Bravais classes with n-fold symmetry [6]. They are unique for the following 29 choices of n

Year: 2013
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