Substitutive Structure of Jeandel–Rao Aperiodic Tilings

  • Sébastien LabbéEmail author


Jeandel and Rao proved that 11 is the size of the smallest set of Wang tiles, i.e., unit squares with colored edges, that admit valid tilings (contiguous edges of adjacent tiles have the same color) of the plane, none of them being invariant under a nontrivial translation. We study herein the Wang shift \(\Omega _0\) made of all valid tilings using the set \(\mathcal {T}_0\) of 11 aperiodic Wang tiles discovered by Jeandel and Rao. We show that there exists a minimal subshift \(X_0\) of \(\Omega _0\) such that every tiling in \(X_0\) can be decomposed uniquely into 19 distinct patches of size ranging from 45 to 112 that are equivalent to a set of 19 self-similar aperiodic Wang tiles. We suggest that this provides an almost complete description of the substitutive structure of Jeandel–Rao tilings, as we believe that \(\Omega _0{\setminus } X_0\) is a null set for any shift-invariant probability measure on \(\Omega _0\). The proof is based on 12 elementary steps, 10 of which involve the same procedure allowing one to desubstitute Wang tilings from the existence of a subset of marker tiles, while the other 2 involve addition decorations to deal with fault lines and changing the base of the \(\mathbb {Z}^2\)-action through a shear conjugacy. Algorithms are provided to find markers, recognizable substitutions, and shear conjugacy from a set of Wang tiles.


Wang tiles Tilings Aperiodic Substitutions Markers 

Mathematics Subject Classification

Primary 52C23 Secondary 37B50 



I want to thank Vincent Delecroix for many helpful discussions at LaBRI in Bordeaux during the preparation of this article, including some hints on how to prove Proposition 11.5. I am grateful to Michaël Rao for providing me with the first proof of Proposition 5.2. This contribution would not have been possible without the Gurobi linear program solver [17], which was very helpful in solving many instances of tiling problems in seconds (instead of minutes or hours), but it turns out that Knuth’s dancing links algorithm [22] performs well to find markers. I acknowledge financial support from the Laboratoire International Franco-Québécois de Recherche en Combinatoire (LIRCO), the Agence Nationale de la Recherche Agence Nationale de la Recherche through the project CODYS (ANR-18-CE40-0007), and the Horizon 2020 European Research Infrastructure project OpenDreamKit (676541). I am very grateful to the anonymous referees for their in-depth reading and valuable comments, from which I learned and that led to a great improvement of the presentation. The revision of the article benefited from the support of the Erwin Schrödinger International Institute for Mathematics and Physics during my stay at the conference Numeration 2019 (Vienna).


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Authors and Affiliations

  1. 1.Université Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800TalenceFrance

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