Two Tiling is Undecidable

TL;DR

This paper proves that determining whether two polygonal tiles can tile the plane is undecidable, extending previous results and including new findings on single-tile tiling with local matching rules.

Contribution

It establishes the undecidability of two-tile tiling problems and introduces the first undecidability result for monotiling with local matching constraints.

Findings

Undecidability of tiling with two polygonal prototiles.

Undecidability of monotiling with local edge-to-edge matching rules.

Extension of previous three-tile undecidability results.

Abstract

We show that the following problem is undecidable: given two polygonal prototiles, determine whether the plane can be tiled with rotated and translated copies of them. This improves a result of Demaine and Langerman [SoCG 2025], who showed undecidability for three tiles. Along the way, we show that tiling with one prototile is undecidable if there can be edge-to-edge matching rules. This is the first result to show undecidability for monotiling with only local matching constraints.