On tilings of the plane
type: Item
instance of: scholarly article
title: On tilings of the plane
zbMATH Open document ID: 0634.52011
DOI: 10.1007/BF00181602
author: Andreas W. M. Dress
author: D. Huson
published in: Geometriae Dedicata
publication date: 1987-01-01T00:00:00Z
review text: The authors' abstract. ``The paper discusses homeomorphic types of (periodic) tilings of the plane in terms of their associated Delaney symbol. Such a symbol consists of a (finite) set \({\mathcal D}\) on which three involutions \(\sigma_ 0,\sigma_ 1\) and \(\sigma_ 2\) act from the right such that \(\sigma_ 0\sigma_ 2=\sigma_ 2\sigma_ 0\) and there are two maps \(m_{01},m_{12}: {\mathcal D}\to {\mathbb{N}}\) satisfying certain compatibility conditions. It is shown how the barycentric subdivision of a tiling can be used to define its Delaney symbol and that the symbol characterizes the tiling up to (equivariant) homeomorphisms. Furthermore, it is shown how properties of the tiling can be recognized from corresponding properties of the symbol and how this technique can be used to enumerate various types of tilings with specific properties. If necessary, this enumeration can be done by appropriate computer programs. Among other results, we have been able to vindicate the results by \textit{B. Grünbaum}, \textit{H.-D. Loeckenhoff}, \textit{G. C. Shephard} and \textit{A. H. Temesvari} [Geom. Dedicata 19, 109-174 (1985; Zbl 0572.52024)]. Finally, some recursive enumeration formulas, based on the Delaney symbol technique, are stated.''
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05B45
zbMATH DE Number: 4032507
zbMATH Keywords: Euclidean plane
zbMATH Keywords: tilings
zbMATH Keywords: Delaney symbol
zbMATH Keywords: enumeration
reviewed by: Pier Luigi Papini
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00181602
OpenAlex ID: W2076566697
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description: scientific article; zbMATH DE number 4032507