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Equivalence of Selfsimilar and Pseudoselfsimilar Tiling Spaces in ℝ
Equivalence of Selfsimilar and Pseudoselfsimilar Tiling Spaces in ℝ,10.1007/s004540119327y,Discrete & Computational Geometry,Betseygail Rand
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Equivalence of Selfsimilar and Pseudoselfsimilar Tiling Spaces in ℝ
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Betseygail Rand
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 selfsimilar tiling or selfsimilar tiling space, respectively. A tiling or tiling space is pseudoselfsimilar 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 selfsimilar, 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 pseudoselfsimilar. 4. If a tiling or tiling space is pseudoselfsimilar, then it is MLD to a selfsimilar tiling or tiling space.
Journal:
Discrete & Computational Geometry  DCG
, vol. 46, no. 1, pp. 128, 2011
DOI:
10.1007/s004540119327y
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