Tilings and quasiperiodicity

✍ Scribed by Bruno Durand

• Book ID: 104326710
• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 788 KB
• Volume: 221
• Category: Article
• ISSN: 0304-3975
• DOI: 10.1016/s0304-3975(99)00027-4

✦ Synopsis

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 ~ 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.