Equivalence of self-similar and pseudo-self-similar tiling spaces in \(\mathbb R^{2}\)
type: Item
instance of: scholarly article
title: Equivalence of self-similar and pseudo-self-similar tiling spaces in \(\mathbb R^{2}\)
zbMATH Open document ID: 1225.52008
author: Betseygail Rand
published in: Discrete \& Computational Geometry
publication date: 2011-06-10T00:00:00Z
review text: A derived Voronoi tiling of a given tiling \(T\) is a related tiling based on a patch of tiles in \(T\). Analogously, one may identify for a tiling space \(X\) a patch which appears regularly in all tilings in a tiling space \(X\) and forms a derived Voronoi space of tilings based on such patch. Two tilings or tiling spaces are called mutually locally derivable (MLD) if they are related by local operations. The author's main results are the following: {\parindent4mm \begin{itemize}\item[--] A tiling and its derived Voronoi tiling are MLD, as are the tiling space and its derived Voronoi space. \item[--]If \(T\) or \(X\) has high degree of hierarchical structure, then its derived Voronoi tiling or derived Voronoi space has a scaled pair. \item[--]If a derived Voronoi tiling, or derived Voronoi tiling space, contains a scaled pair, then the original tiling or tiling space has a weaker degree of hierarchical structure. \item[--]If a tiling or tiling space has a weaker degree of hierarchical structure, then it is MLD to a tiling or tiling space having high degree of hierarchical structure. \end{itemize}}
Mathematics Subject Classification ID: 52C20
zbMATH DE Number: 5906699
zbMATH Keywords: tiling space
zbMATH Keywords: equivalent tiling
zbMATH Keywords: mutually locally derivable tilings
zbMATH Keywords: scaled pair
zbMATH Keywords: self-similar tiling
zbMATH Keywords: pseudo-self-similar tiling
reviewed by: Josef Mikeš
MaRDI profile type: Publication
full work available at URL: s00454-011-9327-y
OpenAlex ID: W1968872938
cites work: Conjugacies for tiling dynamical systems
cites work: Towards a characterization of self-similar tilings in terms of derived Voronoï tessellations
cites work: Characterization of planar pseudo-self-similar tilings
DOI: 10.1007/S00454-011-9327-Y
description: scientific article; zbMATH DE number 5906699