Tilings and quasiperiodicity

Abstract

Quasiperiodic tilings are those tilings in which finite patterns appear regularly in the plane. This property is a generalization of the 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 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↦x+c if and only if the considered tiling is periodic. At last, 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.KeywordsLeft BorderPeriodicity VectorInfinite PathTiling ProblemInfinite WordThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.