Quasicrystalline combinatorics

Baake, Michael and Grimm, Uwe (2003). Quasicrystalline combinatorics. In: Gazeau, J.-P.; Kerner, R.; Antoine, J.-P.; Metens, S. and Thibon, J.-Y. eds. Group 24: Physical and Mathematical Aspects of Symmetries. Institute of Physics Conference Series (173). Bristol, UK: IOP Publishing Ltd, pp. 193–200.

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Several combinatorial problems of (quasi)crystallography are reviewed with special emphasis on a unified approach, valid for both crystals and quasicrystals. In particular, we consider planar sublattices, similarity sublattices, coincidence sublattices and their module counterparts. The corresponding counting functions are encapsulated in Dirichlet series generating functions, with worked out results for the square lattice and the Tuebingen triangle tiling. Finally, we discuss a novel approach to central and averaged shelling for these examples, also involving Dirichlet series.

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