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A self-similar aperiodic set of 19 Wang tiles

  • Sébastien Labbé
Original Paper
  • 38 Downloads

Abstract

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.

Keywords

Wang tiles Tilings Aperiodic Self-similar Recognizability 

Mathematics Subject Classification (2010)

Primary 52C23 Secondary 37B50 

Notes

Acknowledgements

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).

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.CNRS, LaBRI, UMR 5800TalenceFrance

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