Structural aspects of tilings

TL;DR

This paper explores the structural properties of tilings generated by tile-sets, analyzing their finite patterns through combinatorial and topological methods, and reveals key results about periodicity and countability.

Contribution

It introduces a pattern preorder and uses Cantor-Bendixson rank to analyze tiling structures, providing new insights into the countable and uncountable cases.

Findings

Tile-sets producing only periodic tilings have finitely many such tilings.

Existence of a tiling with exactly one vector of periodicity in the countable case.

Distinct structures emerge in uncountable tiling sets.

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.