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Tilings and quasiperiodicity. (English) Zbl 1062.05502
Summary: Quasiperiodic tilings are those tilings in which finite patterns appear regularly in the plane. This property is a generalization of periodicity; it was introduced for representing quasicrystals and it is also motivated by the study of quasiperiodic words. We prove that if a tile set can tile the plane, then it can tile the plane quasiperiodically – a surprising positive result that does not hold for periodicity. In order to compare the regularity of quasiperiodic tilings, we introduce and study a quasiperiodicity function and prove that it is bounded by \(x\mapsto x+c\) if and only if the considered tiling is periodic. Finally, we prove that if a tile set can be used to form a quasiperiodic tiling which is not periodic, then it can form an uncountable number of tilings.

MSC:
05B45 Combinatorial aspects of tessellation and tiling problems
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
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