No Weak Local Rules for the 4p-Fold Tilings

TL;DR

This paper proves that 4p-fold tilings do not admit weak local rules, extending previous results and clarifying the limitations of local rule characterizations for these aperiodic tilings.

Contribution

It establishes that all 4p-fold tilings lack weak local rules, resolving a long-standing open problem in the theory of aperiodic tilings.

Findings

4p-fold tilings do not admit weak local rules

Extends previous results for 8- and 12-fold tilings to all 4p-fold cases

Clarifies limitations of local rule characterizations in aperiodic tilings

Abstract

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showed in 1988 that the 8-fold tilings do not admit weak local rules, and Le showed the same for the 12-fold tilings (unpublished).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 case for all the 4p-fold tilings.