Undecidability of Translational Tiling of the Plane with Orthogonally Convex Polyominoes

TL;DR

This paper proves that it is undecidable whether a set of seven orthogonally convex polyominoes can tile the plane through translation, extending undecidability results to convex shapes.

Contribution

It establishes the first undecidability result for translational tiling of the plane using convex tiles, specifically with a set of seven orthogonally convex polyominoes.

Findings

Undecidability of tiling with 7 convex polyominoes

Extension of undecidability results to convex shapes

Addresses open question in tiling theory

Abstract

The first undecidability result on the tiling is the undecidability of translational tiling of the plane with Wang tiles, where there is an additional color matching requirement. Later, researchers obtained several undecidability results on translational tiling problems where the tilings are subject to the geometric shapes of the tiles only. However, all these results are proved by constructing tiles with extremely concave shapes. It is natural to ask: can we obtain undecidability results of translational tiling with convex tiles? Towards answering this question, we prove the undecidability of translational tiling of the plane with a set of 7 orthogonally convex polyominoes.