Weak colored local rules for planar tilings

Weak colored local rules for planar tilings

A linear subspace E of R has colored local rules if there exists a finite set of decorated tiles whose tilings are digitizations of E. The local rules are weak if the digitizations can slightly wander around E. We prove that a linear subspace has weak colored local rules if and only if it is computable. This goes beyond the previous results, all based on algebraic subspaces. We prove an analogo...

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-period...

No Weak Local Rules for the 4p-Fold Tilings

On the one hand, Socolar showed in 1990 that the n-fold planar tilings admit weak local rules when n is not divisible by 4 (the n = 10 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showed in 1988 that the 8-fold tilings do not admit weak local rules, and Le showed the same for the 12-fold tilings (unpublished).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 show that this is actually the...

Planar Dimer Tilings

Rhombic Penrose Tilings Can be 3-Colored