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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
  • 126 Downloads

Abstract

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.

Keywords

Euclidean tilings Branched surfaces Translation surfaces 

Mathematics Subject Classification

52C20 57M12 

Notes

Acknowledgments

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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