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- W1968872938 abstract "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:" @default.
- W1968872938 created "2016-06-24" @default.
- W1968872938 creator A5081680960 @default.
- W1968872938 date "2011-02-09" @default.
- W1968872938 modified "2023-09-27" @default.
- W1968872938 title "Equivalence of Self-similar and Pseudo-self-similar Tiling Spaces in ℝ2" @default.
- W1968872938 cites W1516153062 @default.
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- W1968872938 doi "https://doi.org/10.1007/s00454-011-9327-y" @default.
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