Tiling with Three Polygons is Undecidable

TL;DR

This paper proves that determining whether three polygons can tile the plane through isometries is undecidable, improving previous results that required five polygons, thus advancing understanding of tiling problems' computational complexity.

Contribution

It establishes that tiling with just three polygons is undecidable, reducing the number of polygons needed from five in prior work.

Findings

Tiling with three polygons is co-RE-complete and undecidable.

The result applies to tilings allowing or forbidding reflections.

This advances the complexity understanding of polygon tiling problems.

Abstract

We prove that the following problem is co-RE-complete and thus undecidable: given three simple polygons, is there a tiling of the plane where every tile is an isometry of one of the three polygons (either allowing or forbidding reflections)? This result improves on the best previous construction which requires five polygons.