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Equivalence of Self-similar and Pseudo-self-similar Tiling Spaces in ℝ

Equivalence of Self-similar and Pseudo-self-similar Tiling Spaces in ℝ,10.1007/s00454-011-9327-y,Discrete & Computational Geometry,Betseygail Rand

Equivalence of Self-similar and Pseudo-self-similar Tiling Spaces in ℝ  
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Given a tiling T, one may form a related tiling, called the derived Voronoi tiling of T, based on a patch of tiles in T. Similarly, for a tiling space X, one can identify a patch which appears regularly in all tilings in X, and form a derived Voronoi space of tilings, based on that patch. Each tiling or tiling space is defined with an associated group of rigid motions, G. In the case where G is a group of translations, a series of results have been proved demonstrating the equivalence of a tiling to its derived Voronoi tilings. Here we generalize these results: first, to groups which include both translations and rotations, and second, to tiling spaces as well. We say that two tilings, or two tiling spaces, are mutually locally derivable, or MLD, if they are related by local operations. A tiling or tiling space with a high degree of hierarchical structure is called a self-similar tiling or self-similar tiling space, respectively. A tiling or tiling space is pseudo-self-similar if it has a weaker degree of hierarchical structure. Finally, given a tiling, we are interested in the set of all derived Voronoi tilings, and whether it contains two elements related by a similar expansion. If so, the set contains a scaled pair (and the set of derived Voronoi spaces contains a scaled pair of spaces, as well). In this paper, we show: 1.  A tiling and its derived Voronoi tiling are MLD, as are tiling space and its derived Voronoi space. 2.  If T or X is self-similar, then its derived Voronoi tiling or derived Voronoi space has a scaled pair. 3.  If a derived Voronoi tiling, or derived Voronoi tiling space, contains a scaled pair, then the original tiling or tiling space is pseudo-self-similar. 4.  If a tiling or tiling space is pseudo-self-similar, then it is MLD to a self-similar tiling or tiling space.
Journal: Discrete & Computational Geometry - DCG , vol. 46, no. 1, pp. 1-28, 2011
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