HAL : hal-00145800, version 2

arXiv : 0802.2828

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STACS 2008, Bordeaux : France (2008)

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v1 (11-05-2007)

v2 (20-02-2008)

Structural aspects of tilings

Alexis Ballier 1, Bruno Durand 1, Emmanuel Jeandel 1

(02/2008)

In this paper, we study the structure of the set of tilings produced by any given tile-set. For better understanding this structure, we address the set of finite patterns that each tiling contains. This set of patterns can be analyzed in two different contexts: the first one is combinatorial and the other topological. These two approaches have independent merits and, once combined, provide somehow surprising results. The particular case where the set of produced tilings is countable is deeply investigated while we prove that the uncountable case may have a completely different structure. We introduce a pattern preorder and also make use of Cantor-Bendixson rank. Our first main result is that a tile-set that produces only periodic tilings produces only a finite number of them. Our second main result exhibits a tiling with exactly one vector of periodicity in the countable case.

1 :	Laboratoire d'informatique Fondamentale de Marseille (LIF)
	CNRS : UMR6166 – Université de la Méditerranée - Aix-Marseille II – Université de Provence - Aix-Marseille I

Domaine

:

Informatique/Autre

Mots Clés : cantor-bendixson. orders. patterns. tilings. topology. domino – tiling preorder – tiling structure

hal-00145800, version 2

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Contributeur : Pascal Weil <>

Soumis le : Jeudi 14 Février 2008, 10:09:10

Dernière modification le : Mercredi 20 Février 2008, 15:07:04