Document Open Access Logo

Tiling with Three Polygons Is Undecidable

Authors Erik D. Demaine , Stefan Langerman

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2025.39.pdf
  • Filesize: 0.82 MB
  • 16 pages

Document Identifiers

Related Versions

Subject Classification

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

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.

Supplementary Materials

References

  1. Robert Berger. The undecidability of the domino problem. Memoirs of the American Mathematical Society, 66:1-72, 1966. URL: https://doi.org/10.1090/memo/0066.
  2. Siddhartha Bhattacharya. Periodicity and decidability of tilings of ℤ². American Journal of Mathematics, 142(1):255-266, 2020. URL: https://doi.org/10.1353/ajm.2020.0006.
  3. Vasco Brattka, Peter Hertling, and Klaus Weihrauch. A tutorial on computable analysis. In S. Barry Cooper, Benedikt Löwe, and Andrea Sorbi, editors, New Computational Paradigms, pages 425-491. Springer, New York, NY, 2008. URL: https://doi.org/10.1007/978-0-387-68546-5_18.
  4. Erik D. Demaine and Stefan Langerman. edemaine/three-tiles, 2024. Software, (visited on 2025-06-05). URL: https://github.com/edemaine/three-tiles
    Software Heritage Logo archived version
    full metadata available at: https://doi.org/10.4230/artifacts.23339
  5. D. Girault-Beauquier and M. Nivat. Tiling the plane with one tile. In Topology and Category Theory in Computer Science. Oxford University Press, August 1991. URL: https://doi.org/10.1093/oso/9780198537601.003.0012.
  6. Solomon W. Golomb. Tiling with sets of polyominoes. Journal of Combinatorial Theory, 9(1):60-71, 1970. URL: https://doi.org/10.1016/S0021-9800(70)80055-2.
  7. Rachel Greenfeld and Terence Tao. Undecidable translational tilings with only two tiles, or one nonabelian tile. Discrete & Computational Geometry, 70(4):1652-1706, 2023. URL: https://doi.org/10.1007/s00454-022-00426-4.
  8. Rachel Greenfeld and Terence Tao. Undecidability of translational monotilings. Journal of the European Mathematical Society, 2024. To appear. URL: https://arxiv.org/abs/2309.09504.
  9. Branko Grünbaum and G. C. Shephard. Tilings and Patterns. W. H. Freeman, 1987. Google Scholar
  10. Emmanuel Jeandel and Michaël Rao. An aperiodic set of 11 Wang tiles. Advances in Combinatorics, January 2021. URL: https://doi.org/10.19086/aic.18614.
  11. Nicolas Ollinger. Tiling the plane with a fixed number of polyominoes. In Proceedings of the 3rd International Conference on Language and Automata Theory and Applications (LATA 2009), volume 5457 of Lecture Notes in Computer Science, pages 638-647, Tarragona, Spain, April 2009. Springer. URL: https://doi.org/10.1007/978-3-642-00982-2_54.
  12. H. G. Rice. Classes of recursively enumerable sets and their decision problems. Transactions of the American Mathematical Society, 74(2):358-366, 1953. URL: https://doi.org/10.2307/1990888.
  13. Raphael M. Robinson. Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae, 12(3):177-209, September 1971. URL: https://doi.org/10.1007/BF01418780.
  14. David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. An aperiodic monotile, 2023. URL: https://doi.org/10.48550/arXiv.2303.10798.
  15. Hao Wang. Proving theorems by pattern recognition - II. Bell Systems Technical Journal, 40(1):1-41, 1961. URL: https://doi.org/10.1002/j.1538-7305.1961.tb03975.x.
  16. Damien Woods and Turlough Neary. The complexity of small universal turing machines: A survey. Theoretical Computer Science, 410(4):443-450, 2009. URL: https://doi.org/10.1016/j.tcs.2008.09.051.
  17. Chao Yang. Tiling the plane with a set of ten polyominoes. International Journal of Computational Geometry & Applications, 33(03n04):55-64, 2023. URL: https://doi.org/10.1142/S0218195923500012.
  18. Chao Yang and Zhujun Zhang. A proof of Ollinger’s conjecture: undecidability of tiling the plane with a set of 8 polyominoes, 2024. URL: https://doi.org/10.48550/arXiv.2403.13472.
  19. Chao Yang and Zhujun Zhang. Undecidability of translational tiling of the 4-dimensional space with a set of 4 polyhypercubes, 2024. URL: https://doi.org/10.48550/arXiv.2409.00846.
  20. Jed Chang-Chun Yang. Computational Complexity and Decidability of Tileability. PhD thesis, University of California, Los Angeles, 2013. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact