Lee-Ad Gottlieb, Aryeh Kontorovich
We generalize the classical notion of packing a set by balls with identical radii to the case where the radii may be different. The largest number of such balls that fit inside the set without overlapping is called its {\em non-uniform packing number}. We show that the non-uniform packing number can be upper-bounded in terms of the {\em average} radius of the balls, resulting in bounds of the familiar classical form.
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The topic of totally separable sphere packings is surveyed with a focus on regular constructions, uniform tilings, and contact number problems. An enumeration of all regular totally separable sphere packings in $\mathbb{R}^2$, $\mathbb{R}^3$, and $\mathbb{R}^4$ which are based on convex uniform tessellations, honeycombs, and tetracombs, respectively, is presented, as well as a construction of a family of regular totally separable sphere packings in $\mathbb{R}^d$ that is not based on a convex uniform $d$-honeycomb for $d \geq 3$.
Packing unequal disks in the Euclidean plane
本文综述了欧几里得平面中不等圆盘装填的若干未解决问题，并按紧致性、共轭性、密度、均匀性和可计算性五个主题进行组织。
We prove upper bounds on the average kissing number $k(\mathcal{P})$ and contact number $C(\mathcal{P})$ of an arbitrary finite non-congruent sphere packing $\mathcal{P}$, and prove an upper bound on the packing density $δ(\mathcal{P})$ of an arbitrary infinite non-congruent sphere packing $\mathcal{P}$.
In \cite{Sz17-2} we considered hyperball packings in $3$-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing provides a decomposition of $\HYP$ into truncated tetrahedra. In order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyperball packings in truncated simplices. Therefore, in this paper we examine the doubly truncated Coxeter orthoscheme tilings and the corresponding congruent and non-congruent hyperball packings. We proved that related to the mentioned Coxeter tilin…
The Koebe circle packing theorem states that every finite planar graph can be realized as the nerve of a packing of (non-congruent) circles in R^3. We investigate the average kissing number of finite packings of non-congruent spheres in R^3 as a first restriction on the possible nerves of such packings. We show that the supremum k of the average kissing number for all packings satisfies 12.566 ~ 666/53 <= k < 8 + 4*sqrt(3) ~ 14.928 We obtain the upper bound by a resource exhaustion argument and the upper bound by a construction involving packings of spherical caps in S^3. Our result contradict…
In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular tetrahedron tilings. These are derived from the Coxeter simplex tilings $\{p,3,3\}$ $(7\le p \in \mathbb{N})$ and $\{5,3,3,3,3\}$ in $3$ and $5$-dimensional hyperbolic space. We determine the densest hyperball packing arrangements related to the above tilings. We find packing densities using congruent hyperballs and determine the smallest density upper bound of non-congruent hyperball packings generated by the above tilings.
On non-parallel cylinder packings
本文研究三维空间中非平行圆柱体堆积的密度问题，证明了Kuperberg关于上密度的猜想，并构造了任意接近下界$π/6$的下密度堆积。
Any self-similar directed graph iterated function system with probabilities, defined on m-dimensional Euclidean space, determines a unique list of self-similar Borel probability measures whose supports are the components of the attractor. Using an application of the Renewal Theorem we obtain an explicit calculable value for the power law behaviour of the qth packing moments of the self-similar measures at scale r as r tends to 0 in the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any real q if the strong separation condition (SSC) holds, (ii) for non-ne…
New lower bound for the optimal congruent geodesic ball packing density of screw motion groups in $\mathbf{H}^2\!\times\!\mathbf{R}$ space
该研究利用$\boldsymbol{\rm H}^2\boldsymbol{\times}\boldsymbol{\rm R}$的射影模型，得到了螺旋运动群生成的全等测地球堆积的新密度纪录约0.80529，还开发了确定对应最优堆积半径的程序。
Let a planar residual set be a set obtained by removing countably many disjoint topological disks from an open set in the plane. We prove that the residual set of a planar packing by curves that satisfy a certain lower curvature bound has Hausdorff dimension bounded away from 1, quantitatively, depending only on the curvature bound. As a corollary, the residual set of any circle packing has Hausdorff dimension uniformly bounded away from 1. This result generalizes the result of Larman, who obtained the same conclusion for circle packings inside a square. We also show that our theorem is optima…
In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular octahedron and cube tilings. These are derived from the Coxeter simplex tilings $\{p,3,4\}$ $(7\le p \in \mathbb{N})$ and $\{p,4,3\}$ $(5\le p \in \mathbb{N})$ in $3$-dimensional hyperbolic space $\mathbb{H}^3$. We determine the densest hyperball packing arrangement and its density with congruent and non-congruent hyperballs related to the above tilings in $\mathbb{H}^3$. We prove that the locally densest congruent or non-congruent hyperball configuration belongs to the regular truncate…
Supergroups of some hyperbolic space groups are classified as a continuation of our former works. Fundamental domains will be integer parts of truncated tetrahedra belonging to families F1 - F4, for a while, by the notation of E. Molnár et al. in $2006$. As an application, optimal congruent hyperball packings and coverings to the truncation base planes with their very good densities are computed. This covering density is better than the conjecture of L.~Fejes~Tóth for balls and horoballs in $1964$.
Sphere Packings in Higher Dimension (after Boaz Klartag)
本文解释Boaz Klartag如何通过概率方法证明高维球体堆积密度下界δ_n^L ≥ c n^2 2^{-n}，其中c>0为通用常数，显著改进了非格堆积的下界。
An extremal $k$-packing is a collection of $k$ mutually disjoint metric discs, embedded in a surface, whose radius is maximal for the given topology. We study compact non-orientable surfaces of genus $g\ge 3$ containing extremal $k$-packings.
It is shown that there is no quasi-sphere packing of the lattice grid Z^{d+1} or a co-compact hyperbolic lattice of H^{d+1} or the 3-regular tree \times Z, in R^d, for all d. A similar result is proved for some other graphs too. Rather than using a direct geometrical approach, the main tools we are using are from non-linear potential theory.
We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each ''hole'' is bounded by three pairwise tangent discs are called triangulated. There are 164 pairs $(r,s)$, $1{>}r{>}s$, allowing triangulated packings by discs of radii 1, $r$ and $s$. In this paper, we enhance existing methods of dealing with maximal-density packings in order to find ternary triangulated packings which maximize the density among all the packings with the same disc radii. We showed for 16 pairs that the density is maximized by a triangulated ternary pac…
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