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Structural aspects of tilings. (English) Zbl 1258.05023
Albers, Susanne (ed.) et al., STACS 2008. 25th international symposium on theoretical aspects of computer science, Bordeaux, France, February 21–23, 2008. Wadern: Schloss Dagstuhl – Leibniz Zentrum für Informatik (ISBN 978-3-939897-06-4). LIPIcs – Leibniz International Proceedings in Informatics 1, 61-72, electronic only (2008).
Summary: In this paper, we study the structure of the set of tilings produced by any given tile-set. For a 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.
For the entire collection see [Zbl 1213.68020].

MSC:
05B45 Combinatorial aspects of tessellation and tiling problems
03E05 Other combinatorial set theory
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