Detecting combinatorial hierarchy in tilings using derived Voronoi tessellations

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 filings 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 filings via their dual graphs. Examples of combinatonally 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 Voronoi tiling"-a retiling in terms of reappearances of an initial patch of tiles. Using derived Voronoi filings, we obtain a sufficient condition for a tiling to be combinatorially substitutive.