Disk-like tiles derived from complex bases
type: Item
instance of: scholarly article
title: Disk-like tiles derived from complex bases
zbMATH Open document ID: 1063.11006
author: Jun Luo
author: Zuoling Zhou
publication date: 2005-01-25T00:00:00Z
review text: For each positive integer \(k\), let \(T_{k}\) be the set of complex numbers representable in the form \(\sum_{j\geq 1}d_{j}(-k+{\mathbf i})^{-j}\) with \( d_{j}\in \{0,1,2,\ldots ,k^{2}\}\). The translates of \(T_{k}\) by Gaussian integers tile the plane, \(T_{k}\) is affinely equivalent to \(\bigcup_{j=1}^{k^{2}}(T_{k}+j)\), and the Hausdorff dimension of the boundary of \(T_{k}\) is a number strictly between \(1\) and \(2\), which has been determined by \textit{W. J. Gilbert} [Can. Math. Bull. 29, 495--500 (1986; Zbl 0564.10007)]. Here the authors show that \(T_{k}\) is homeomorphic to the closed unit disk if and only if \(k\neq 2\). It is noted that this may also be inferred from a result of \textit{C. Bandt} and \textit{Y. Wang} [Discrete Comput. Geom. 26, No. 4, 591--601 (2001; Zbl 1020.52018)], but in a quite different way.
reviewed by: Johann Linhart
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 52C20
Mathematics Subject Classification ID: 54F65
Mathematics Subject Classification ID: 57M15
zbMATH DE Number: 2130978
zbMATH Keywords: self-affine tile
zbMATH Keywords: radix representation
zbMATH Keywords: fractals
MaRDI profile type: Publication
full work available at URL: s10114-004-0354-y
OpenAlex ID: W2062243990
cites work: Q4050689
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cites work: Q3247607
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cites work: TOPOLOGICAL STRUCTURE OF SELF-SIMILAR SETS
cites work: Classification of Self-Affine Lattice Tilings
cites work: Disk-like self-affine tiles in \(\mathbb{R}^2\)
DOI: 10.1007/S10114-004-0354-Y
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 2130978