LIPIcs - Leibniz International Proceedings in Informatics. 25th International Symposium on Theoretical Aspects of Computer Science
Doi
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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