Detecting Combinatorial Hierarchy in Tilings Using Derived Voronoi Tessellations

   Abstract. Tilings of R ² can display hierarchy similar to that seen in the limit sequences of substitutions. Self-similarity for tilings has been used as the standard generalization, but this viewpoint is limited because such tilings are analogous to limit points of constant-length substitutions. To generalize limit points of non-constant-length substitutions, we define hierarchy for infinite, labelled graphs, then extend this definition to tilings via their dual graphs. Examples of combinatorially substitutive tilings that are not self-similar are given. We then find a sufficient condition for detecting combinatorial hierarchy that is motivated by the characterization by Durand of substitutive sequences. That characterization relies upon the construction of the ``derived sequence'—a recoding in terms of reappearances of an initial block. Following this, we define the ``derived Vorono? tiling'—a retiling in terms of reappearances of an initial patch of tiles. Using derived Vorono? tilings, we obtain a sufficient condition for a tiling to be combinatorially substitutive.