Two Tiling is Undecidable
Computational Geometry
2025-06-16 v1 Combinatorics
Metric Geometry
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.
Keywords
Cite
@article{arxiv.2506.11628,
title = {Two Tiling is Undecidable},
author = {Jack Stade},
journal= {arXiv preprint arXiv:2506.11628},
year = {2025}
}
Comments
37 pages, 48 figures