Structural aspects of tilings

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.