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DOI: 10.4230/LIPIcs.STACS.2008.1334

Structural aspects of tilings

Authors: Alexis Ballier, Bruno Durand, and Emmanuel Jeandal

Published in: LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

Abstract

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.

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Alexis Ballier, Bruno Durand, and Emmanuel Jeandal. Structural aspects of tilings. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 61-72, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{ballier_et_al:LIPIcs.STACS.2008.1334,
  author =	{Ballier, Alexis and Durand, Bruno and Jeandal, Emmanuel},
  title =	{{Structural aspects of tilings}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{61--72},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Albers, Susanne and Weil, Pascal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1334},
  URN =		{urn:nbn:de:0030-drops-13343},
  doi =		{10.4230/LIPIcs.STACS.2008.1334},
  annote =	{Keywords: Tiling, domino, patterns, tiling preorder, tiling structure}
}

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Structural aspects of tilings

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