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CORR
2008
Springer

Structural aspects of tilings

8 years 10 months ago
Structural aspects of tilings
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.
Alexis Ballier, Bruno Durand, Emmanuel Jeandel
Added 09 Dec 2010
Updated 09 Dec 2010
Type Journal
Year 2008
Where CORR
Authors Alexis Ballier, Bruno Durand, Emmanuel Jeandel
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