Packing, covering and tiling in two-dimensional spaces
type: Item
instance of: scholarly article
title: Packing, covering and tiling in two-dimensional spaces
zbMATH Open document ID: 1307.52012
published in: Expositiones Mathematicae
publication date: 2014-11-20T00:00:00Z
review text: Packing, covering and tiling is a classic subject in pure mathematics with a long and rich history. This subject has many applications in number theory, coding theory, crystallography, etc. The paper gives a comprehensive review for packing, covering and tiling by convex domains in the two-dimensional spaces. Most of proofs are omitted. There are 265 references in the paper. The paper has the following sections. {\parindent=8mm \begin{itemize}\item[0] Introduction. In this section history of the problem is discussed. Also this section contains some basic definitions. \item[1] Translative packings. The definitions of translative and lattice packings are discussed. Some estimations and exact formulaes for packing densities of two-dimensional convex domains are given. Also some classic open problems are formulated. \item[2] Translative coverings. This section contains definitions of translative and lattice coverings and their densities. Also this section contains some classical results on lattice coverings of two-dimensional convex domains. \item[3] Tilings. This section describes known results on classifications of some types of tilings such as parallelogon tilings, monohedral tilings, isohedral and isogonal tilings, etc. Also quasiperiodic tiligs and quasicrystals are discussed. \item[4] Congruent packings. \item[5] Congruent coverings. These sections contain basic definitions and basic properties of general packings and coverings by congruent copies of two-dimensional convex domains. \item[6] Simultaneous packing and coverings. This section is devoted to relatively new problems about translative simultaineous lattice packings and coverings. For any convex body \(K\) consider the number \(\varphi(K)=\min_{X} \varphi(K,X)\) where the minimum is over all discrete sets \(X\) such that \(K+X\) is a translative packing and \(\varphi(K,X)\) is the smallest number \(r\) such that \(rK+X\) is a covering. Also \(\varphi^*(K)\) is a similar value for simultaneous lattice packi…
Mathematics Subject Classification ID: 52C15
Mathematics Subject Classification ID: 52C20
Mathematics Subject Classification ID: 52C23
Mathematics Subject Classification ID: 52C26
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 11H31
zbMATH DE Number: 6371546
zbMATH Keywords: lattice
zbMATH Keywords: metric space
zbMATH Keywords: Archimedean tiling
zbMATH Keywords: Penrose tiling
zbMATH Keywords: quasicrystal
zbMATH Keywords: Steiner ratio
zbMATH Keywords: Tammes problem
zbMATH Keywords: spherical code
zbMATH Keywords: bin packing
zbMATH Keywords: Apollonian circle packing
author: Chuan Ming Zong
reviewed by: Anton Shutov
describes a project that uses: CirclePack
MaRDI profile type: Publication
full work available at URL: j.exmath.2013.12.002
OpenAlex ID: W2058738901
Wikidata QID: Q57256993
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Packing, covering and tiling in two-dimensional spaces (DOI)
DOI: 10.1016/J.EXMATH.2013.12.002