Periodicity and decidability of tilings of $\mathbb{Z}^{2}$

TL;DR

This paper proves that any finite tile set that can tile the 2D integer grid does so periodically, making the tiling problem decidable for such sets.

Contribution

It establishes that all finite tilings of  are periodic and that the tiling problem for these sets is decidable, resolving longstanding questions.

Findings

All finite tiles that tile  admit periodic tilings

The tiling problem for finite sets in  is decidable

Periodic tilings can be constructed for any such tile set

Abstract

We prove that any finite set $F\subset {\mathbb{Z}^2}$ that tiles ${\mathbb{Z}^2}$ by translations also admits a periodic tiling. As a consequence, the problem whether a given finite set $F$ tiles ${\mathbb{Z}^2}$ is decidable.