Local Rules for Computable Planar Tilings

TL;DR

This paper characterizes aperiodic tilings derived from digitized irrational vector spaces, showing they are aperiodic precisely when these vector spaces are computable, linking tiling properties to computability theory.

Contribution

It establishes a precise criterion for aperiodicity in digitized irrational vector space tilings based on their computability, advancing understanding of aperiodic tilings.

Findings

Aperiodic tilings from digitized irrational vector spaces are characterized by computability.

The paper proves the if and only if condition for aperiodicity based on computability.

Provides a link between tiling theory and computability theory.

Abstract

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.