Towards a Characterization of Self-similar Tilings in Terms of Derived Voronoï Tessellations

In this paper, a technique for analyzing levels of hierarchy in a tiling T of Euclidean space is presented. Fixing a central configuration P of tiles in T , a “derived Voronöı” tessellation TP is constructed based on the locations of copies of P in T . A family of derived Voronöı tilings F(T ) is formed by allowing the central configurations to vary through an infinite number of possibilities. The family F(T ) will normally be an infinite one, but we show that for a self-similar tiling T it is finite up to similarity. In addition, we show that if the family F(T ) is finite up to similarity, then T is pseudo-self-similar. The relationship between self-similarity and pseudo-selfsimilarity is not well understood, and this is the obstruction to a complete characterization of self-similarity via our method. A discussion and conjecture on the connection between the two forms of hierarchy for tilings is provided.