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oai:drops-oai.dagstuhl.de:1334 2024-03-06T10:32:56Z ddc:004 open_access

Structural aspects of tilings Ballier, Alexis Durand, Bruno Jeandal, Emmanuel Tiling domino patterns tiling preorder tiling structure 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Alexis Ballier and Bruno Durand and Emmanuel Jeandal 2008 Is Part Of LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2008.1334 urn:nbn:de:0030-drops-13343 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1334 eng https://creativecommons.org/licenses/by-nd/3.0/legalcode