Packing, covering and tiling in two-dimensional spaces
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 packing and coverings. The section contains some estimations of \(\varphi(K)\) and \(\varphi^*(K)\) for various convex domains. \item[7] Multiple packings and coverings. This section discusses asymptotics of \(j\)-fold packing and covering densities as \(j\to\infty\). Also exact formulaes for \(j\)-fold packing and covering densities of two-dimensional unit discs for \(j\leq 8\) are given. \item[8] Finite packings. This section contains known results on four special problems: optimal packings of circles in square, triangle and circle, and the Tammes problem. Also some general results and approaches is discussed. \item[9] Finite coverings. This section is a brief account on covering analogues of results of the previous section. \item[10] Mixed packings. \item[11] Mixed coverings. These sections discuss packings and coverings of different convex domains in a given region or in the whole plane. Also, the bin packing problem from operation research is discussed. \item[12] Two topics on circle packings. This section is devoted to Apollonian circle packings and Koebe's theorem on connections of planar graphs and finite circle packings. \end{itemize}}
scientific article; zbMATH DE number 6371546
scientific article