6 research outputs found
Local Rules for Computable Planar Tilings
Aperiodic tilings are non-periodic tilings characterized by local
constraints. They play a key role in the proof of the undecidability of the
domino problem (1964) and naturally model quasicrystals (discovered in 1982). A
central question is to characterize, among a class of non-periodic tilings, the
aperiodic ones. In this paper, we answer this question for the well-studied
class of non-periodic tilings obtained by digitizing irrational vector spaces.
Namely, we prove that such tilings are aperiodic if and only if the digitized
vector spaces are computable.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
Weak local rules for planar octagonal tilings
We provide an effective characterization of the planar octagonal tilings
which admit weak local rules. As a corollary, we show that they are all based
on quadratic irrationalities, as conjectured by Thang Le in the 90s.Comment: 23 pages, 6 figure
No Weak Local Rules for the 4p-Fold Tilings
International audiencePlanar tilings with n-fold rotational symmetry are commonly used to model the long range order of quasicrystals. In this context, it is important to know which tilings are characterized only by local rules. Local rules are constraints on the way neighboor tiles can fit together. They aim to model finite-range energetic interactions which stabilize quasicrystals. They are said to be weak if they moreover allow the tilings to have small variations which do not affect the long range order. On the one hand, Socolar showed in 1990 that the n-fold planar tilings do admit weak local rules when n is not divisible by 4 (the n = 5 case corresponds to the Penrose tilings and is known since 1974). On the other hand, Burkov showed in 1988 that the 8-fold tilings do not admit weak local rules, and Le showed the same for the 12-fold tilings (unpublished). We here finally close the matter of weak local rules for the n-fold tilings by showing that they do not admit weak local rules as soon as n is divisible by 4