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Tiling with Three Polygons Is Undecidable

Authors Erik D. Demaine , Stefan Langerman

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Problems, reductions and completeness
Keywords
  • plane tilings
  • polygons
  • undecidability
  • co-RE

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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.

Cite As Get BibTex

Erik D. Demaine and Stefan Langerman. Tiling with Three Polygons Is Undecidable. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.39

Author Details

Erik D. Demaine
  • Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA
Stefan Langerman
  • Computer Science Department, Université libre de Bruxelles, Belgium

Acknowledgements

This research was initiated at the Second Adriatic Workshop on Graphs and Probability, and continued at the Eleventh Annual Workshop on Geometry and Graphs. The authors thank all participants of both workshops for providing a stimulating research environment. They also thank Joshua Blömer for spotting a bug in the implementation and Figure 6.

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References

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