Tilings and quasiperiodicity
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 bar right arrow 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. (C) 1999 Elsevier Science B.V. All rights reserved.
tilings quasiperiodicity combinatorics 2d words