A self-similar aperiodic set of 19 Wang tiles

  • Sébastien Labbé
Original Paper


We define a Wang tile set \(\mathcal {U}\) of cardinality 19 and show that the set \(\Omega _\mathcal {U}\) of all valid Wang tilings \(\mathbb {Z}^2\rightarrow \mathcal {U}\) is self-similar, aperiodic and is a minimal subshift of \(\mathcal {U}^{\mathbb {Z}^2}\). Thus \(\mathcal {U}\) is the second smallest self-similar aperiodic Wang tile set known after Ammann’s set of 16 Wang tiles. The proof is based on the unique composition property. We prove the existence of an expansive, primitive and recognizable 2-dimensional morphism \(\omega :\Omega _\mathcal {U}\rightarrow \Omega _\mathcal {U}\) that is onto up to a shift. The proof of recognizability is done in two steps using at each step the same criteria (the existence of marker tiles) for proving the existence of a recognizable one-dimensional substitution that sends each tile either on a single tile or on a domino of two tiles.


Wang tiles Tilings Aperiodic Self-similar Recognizability 

Mathematics Subject Classification (2010)

Primary 52C23 Secondary 37B50 



I want to thank Michaël Rao for the talk he made (Combinatorics on words and tilings, CRM, Montréal, April 2017) from which this work is originated. I want to thank Vincent Delecroix for many helpful discussions at LaBRI in Bordeaux during the preparation of this article. I am also thankful to Jörg Thuswaldner, Henk Bruin, for inviting me to present this work (Substitutions and tiling spaces, University of Vienna, September 2017) and to Pierre Arnoux and Shigeki Akiyama for the same reason (Tiling and Recurrence, CIRM, Marseille, December 2017). I want to thank Michael Baake for very helpful discussions on inflation rules and model sets and Shigeki Akiyama for its enthusiasm toward this project. The author is grateful to the comments of the referee which leaded to a great improvement in the presentation while reducing its size and simplifying many technical proofs into simpler ones. I also wish to thank David Renault for his careful reading of a preliminary version and his valuable comments. I acknowledge financial support from the Laboratoire International Franco-Québécois de Recherche en Combinatoire (LIRCO), the Agence Nationale de la Recherche “Dynamique des algorithmes du pgcd : une approche Algorithmique, Analytique, Arithmétique et Symbolique (Dyna3S)” (ANR-13-BS02-0003) and the Horizon 2020 European Research Infrastructure project OpenDreamKit (676541).


  1. 1.
    Ammann, R., Grünbaum, B., Shephard, G.C.: Aperiodic tiles. Discrete Comput. Geom. 8(1), 1–25 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Akiyama, S.: A note on aperiodic Ammann tiles. Discrete Comput. Geom. 48(3), 702–710 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Akiyama, S., Tan, B., Yuasa, H.: On B. Mossé’s unilateral recognizability theorem (2017). arXiv:1801.03536
  4. 4.
    Berthé, V., Delecroix, V.: Beyond substitutive dynamical systems: \(S\)-adic expansions. In: Numeration and Substitution 2012, RIMS Kôkyûroku Bessatsu, B46, pp. 81–123. Research Institute for Mathematical Sciences (RIMS), Kyoto, (2014)Google Scholar
  5. 5.
    Berger, R.: The Undecidability of the Domino Problem. ProQuest LLC, Ann Arbor, MI, Ph.D. Thesis, Harvard University (1965)Google Scholar
  6. 6.
    Baake, M., Grimm, U.: Aperiodic Order. Vol. 1, Volume 149 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2013)Google Scholar
  7. 7.
    Berthé, V., Rigo, M. (eds.): Combinatorics, Automata and Number Theory, Volume 135 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2010)Google Scholar
  8. 8.
    Berthé, V., Steiner, W., Thuswaldner, J., Yassawi, R.: Recognizability for sequences of morphisms (2017). arXiv:1705.00167
  9. 9.
    Charlier, E.: Abstract Numeration Systems: Recognizability, Decidability, Multidimensional S-automatic Words, and Real Numbers. Ph.D. Thesis, Université de Liège, Liège, Belgique (2009)Google Scholar
  10. 10.
    Charlier, E., Kärki, T., Rigo, M.: Multidimensional generalized automatic sequences and shape-symmetric morphic words. Discrete Math. 310(6–7), 1238–1252 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Culik II, K.: An aperiodic set of \(13\) Wang tiles. Discrete Math. 160(1–3), 245–251 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Durand, F.: A characterization of substitutive sequences using return words. Discrete Math. 179(1–3), 89–101 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Frank, N.P.: Detecting combinatorial hierarchy in tilings using derived Voronoï tesselations. Discrete Comput. Geom. 29(3), 459–476 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Frank, N.P.: Introduction to hierarchical tiling dynamical systems. In: Tiling and Recurrence, December 4–8 2017, CIRM, Marseille Luminy, France. (2017)
  15. 15.
    Frank, N.P., Sadun, L.: Fusion: a general framework for hierarchical tilings of \(\mathbb{R}^d\). Geom. Dedicata 171, 149–186 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gurobi Optimization, LLC: Gurobi Optimizer Reference Manual (Version 8.0.0). (2018)
  17. 17.
    Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W. H. Freeman and Company, New York (1987)zbMATHGoogle Scholar
  18. 18.
    Jeandel, E., Rao, M.: An aperiodic set of 11 Wang tiles (2015). arXiv:1506.06492
  19. 19.
    Kari, J.: A small aperiodic set of Wang tiles. Discrete Math. 160(1–3), 259–264 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Knuth, D.E.: The Art of Computer Programming. Vol. 1: Fundamental Algorithms. Second Printing. Addison-Wesley Publishing Co. (1969)Google Scholar
  21. 21.
    Labbé, S.: S. Labbé’s Research Code (Version 0.4.2). (2018). Accessed 25 July 2018
  22. 22.
    Labbé, S.: Substitutive structure of Jeandel–Raoaperiodic tilings (2018) (In preparation)Google Scholar
  23. 23.
    Mossé, B.: Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theor. Comput. Sci. 99(2), 327–334 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mozes, S.: Tilings, substitution systems and dynamical systems generated by them. J. Anal. Math. 53, 139–186 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ollinger, N.: Two-by-two substitution systems and the undecidability of the domino problem. In: Logic and Theory of Algorithms, Volume 5028 of Lecture Notes in Computer Science, pp. 476–485. Springer, Berlin (2008)Google Scholar
  26. 26.
    Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Invent. Math. 12, 177–209 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sage Developers: SageMath, the Sage Mathematics Software System (Version 8.2). (2018). Accessed 25 July 2018
  28. 28.
    Schmidt, K.: Multi-dimensional symbolic dynamical systems. In: Codes, Systems, and Graphical Models (Minneapolis, MN, 1999), Volume 123 of IMA Volumes in Mathematics and its Applications, pp. 67–82. Springer, New York (2001)Google Scholar
  29. 29.
    Solomyak, B.: Dynamics of self-similar tilings. Ergodic Theory Dyn. Syst. 17(3), 695–738 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Solomyak, B.: Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20(2), 265–279 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wang, H.: Proving theorems by pattern recognition—II. Bell Syst. Tech. J. 40(1), 1–41 (1961)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.CNRS, LaBRI, UMR 5800TalenceFrance

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