No Weak Local Rules for the 4p-Fold Tilings

Nicolas Bedaride (1) , Thomas Fernique (2)

1 I2M - Institut de Mathématiques de Marseille
2 LIPN - Laboratoire d'Informatique de Paris-Nord

Planar 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showed in 1988 that the 8-fold tilings do not admit weak local rules, and Le showed the same for the 12-fold tilings (unpublished).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.

Domaines: Mathématiques [math]; Systèmes dynamiques [math.DS]; Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Nicolas Bedaride, Thomas Fernique. No Weak Local Rules for the 4p-Fold Tilings. Discrete and Computational Geometry, 2015, ⟨10.1007/s00454-015-9740-8⟩. ⟨hal-01219081⟩