Abstract
In this paper, a technique for analyzing levels of hierarchy in a tiling \(\mathcal{T}\) of Euclidean space is presented. Fixing a central configuration P of tiles in \(\mathcal{T}\), a `derived Voronoï' tessellation \(\mathcal{T}\) P is constructed based on the locations of copies of P in \(\mathcal{T}\). A family of derived Voronoï tilings \(\mathcal{F}(\mathcal{T}{\text{)}}\) is formed by allowing the central configurations to vary through an infinite number of possibilities. The family \(\mathcal{F}(\mathcal{T}{\text{)}}\) will normally be an infinite one, but we show that for a self-similar tiling \(\mathcal{T}\) it is finite up to similarity. In addition, we show that if the family \(\mathcal{F}(\mathcal{T}{\text{)}}\) is finite up to similarity, then \(\mathcal{T}\) is pseudo-self-similar. The relationship between self-similarity and pseudo-self-similarity 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.
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Priebe, N.M. Towards a Characterization of Self-Similar Tilings in Terms of Derived Voronoï Tessellations. Geometriae Dedicata 79, 239–265 (2000). https://doi.org/10.1023/A:1005191014127
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DOI: https://doi.org/10.1023/A:1005191014127