Nonperiodicity implies unique composition for self-similar translationally finite tilings
type: Item
instance of: scholarly article
title: Nonperiodicity implies unique composition for self-similar translationally finite tilings
zbMATH Open document ID: 0919.52017
DOI: 10.1007/PL00009386
published in: Discrete \& Computational Geometry
publication date: 1999-09-12T00:00:00Z
review text: Let \({\mathcal T}\) be a translationally finite tiling of \(\mathbb{R}\) (a tiling which, up to translation, has finitely many patches of diameter less than a given number), which is self-similar (the tiles can be grouped into patches to form a new tiling that is isomorphic to the original one). The tiling \({\mathcal T}\) is said to have the unique composition property, if there is only one way to group its tiles to form a self-similar tiling. The author proves that if such a tiling \({\mathcal T}\) is nonperiodic (i.e. there is no nonzero vector \(v\) such that \({\mathcal T}+v={\mathcal T}\)), then it has a unique composition property. More generally he proves that \({\mathcal T}\) has the unique composition property modulo its translation symmetries. Thus the tilings obtained as the result of grouping the tiles of \({\mathcal T}\) into a self-similar tiling in two different ways, differ by a translation.
Mathematics Subject Classification ID: 52C22
zbMATH DE Number: 1239611
zbMATH Keywords: nonperiodic crystallographic
zbMATH Keywords: translationally finite tiling
zbMATH Keywords: self-similar
zbMATH Keywords: unique composition
author: B. Solomyak
reviewed by: Charles Leytem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1239611