Geometriae Dedicata

, Volume 173, Issue 1, pp 129–142 | Cite as

Tilings of the plane and Thurston semi-norm

  • Jean-René Chazottes
  • Jean-Marc Gambaudo
  • François Gautero
Original Paper


We show that the problem of tiling the Euclidean plane with a finite set of polygons (up to translation) boils down to prove the existence of zeros of a non-negative convex function defined on a finite-dimensional simplex. This function is a generalisation, in the framework of branched surfaces, of the Thurston semi-norm originally defined for compact \(3\)-manifolds.


Euclidean tilings Branched surfaces Translation surfaces 

Mathematics Subject Classification

52C20 57M12 



The authors acknowledge L. Sadun and Robert F. Williams for allowing the reproduction of some figures from [16]. They also thank an anonymous referee whose fruitful remarks led to a great improvement of the article.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Jean-René Chazottes
    • 1
  • Jean-Marc Gambaudo
    • 2
  • François Gautero
    • 3
  1. 1.Centre de Physique ThéoriqueCNRS-École PolytechniquePalaiseau CedexFrance
  2. 2.INLNUniversité Nice Sophia Antipolis-CNRSValbonneFrance
  3. 3.Laboratoire J.A. DieudonnéUniversité Nice Sophia Antipolis-CNRSNice Cedex 02France

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