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A note on tilings and translation surfaces

Published online by Cambridge University Press:  13 December 2005

JEAN-MARC GAMBAUDO
Affiliation:
Centro de Modelamiento Matemático, U.M.I. CNRS 2807, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile (e-mail: gambaudo@dim.uchile.cl)
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Abstract

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Consider a tiling $\mathcal T$ of the two-dimensional Euclidean space made with copies up to translation of a finite number of polygons meeting each other full edge to full edge. In this paper, we prove that, associated with $\mathcal T$, there exists a tiling of a (compact) translation surface made with copies up to translation of some of the polygons used to construct $\mathcal T$. Furthermore, when $\mathcal T$ is repetitive, there exists a tiling of a translation surface, made with copies up to translation of arbitrarily large polygons chosen in a finite collection of patches of $\mathcal T$; each of these patches contain copies of all the polygons used to construct $\mathcal T$.

Information

Type
Research Article
Copyright
2005 Cambridge University Press