Let $L \subset {\Bbb R}^3$ be the union of unit balls, whose centres lie on the $z$-axis, and are equidistant with distance $2d \in [2, 2\sqrt{2}]$. Then a packing of unit balls in ${\Bbb R}^3$ consisting of translates of $L$ has a density at most $π/(3d\sqrt{3-d^2})$, with equality for a certain lattice packing of unit balls. Let $L \subset {\Bbb R}^4$ be the union of unit balls, whose centres lie on the $x_3x_4$ coordinate plane, and form either a square lattice or a regular triangular lattice, of edge length $2$. Then a packing of unit balls in ${\Bbb R}^4$ consisting of translates of $L$ has a density at most $π^2/16$, with equality for the densest lattice packing of unit balls in ${\Bbb R}^4$. This is the first class of non-lattice packings of unit balls in ${\Bbb R}^4$, for which this conjectured upper bound for the packing density of balls is proved. Our main tool for the proof is a theorem on $(r,R)$-systems in ${\Bbb R}^2$. If $R/r \le 2 \sqrt{2}$, then the Delone triangulation associated to this $(r,R)$-system has the following property. The average area of a Delone triangle is at least $\min \{ V_0, 2r^2 \} $, where $V_0$ is the infimum of the areas of the non-obtuse Delone triangles. This general theorem has applications also in other problems about packings: namely for $2r^2 \ge V_0$ it is sufficient to deal only with the non-obtuse Delone triangles, which is in general a much easier task. Still we give a proof of an unpublished theorem of L. Fejes Tóth and E (=J.) Székely: for the $2$-dimensional analogue of our question about equidistant strings of unit balls, we determine the densest packing of translates of an equidistant string of unit circles with distance $2d$, for the first non-trivial interval $2d \in (2{\sqrt{3}},4)$.
Based on Minkowski's work on critical lattices of 3-dimensional convex bodies we present an efficient algorithm for computing the density of a densest lattice packing of an arbitrary 3-polytope. As an application we calculate densest lattice packings of all regular and Archimedean polytopes.
We consider packings of the plane using discs of radius 1 and r=0.545151... . The value of r admits compact packings in which each hole in the packing is formed by three discs which are tangent to each other. We prove that the largest density possible is that of the compact packing shown in figure 1.
We investigate, by "a la Marcinkiewicz" techniques applied to the (asymptotic) density function, how dense systems of equal spheres of $\rb^{n}, n \geq 1,$ can be partitioned at infinity in order to allow the computation of their density as a true limit and not a limsup. The density of a packing of equal balls is the norm 1 of the characteristic function of the systems of balls in the sense of Marcinkiewicz. Existence Theorems for densest sphere packings and completely saturated sphere packings of maximal density are given new direct proofs.
In this paper we study the locally optimal geodesic ball packings with equal balls to the $\mathbf{S}^2\!\times\!\mathbf{R}$ space groups having rotation point groups and their generators are screw motions. We determine and visualize the densest simply transitive geodesic ball arrangements for the above space groups, moreover we compute their optimal densities and radii. The densest packing is derived from the $\mathbf{S}^2\!\times\!\mathbf{R}$ space group $\mathbf{3qe.~I.~3}$ with packing density $\approx 0.7278$. E. Molnár has shown, that the Thurston geometries have an unified interpretatio…
In 1900, as a part of his 18th problem, Hilbert proposed the question to determine the densest congruent (or translative) packings of a given solid, such as the unit ball or the regular tetrahedron of unit edges. Up to now, our knowledge about this problem is still very limited, excepting the ball case. It is conjectured that, for some particular solids such as tetrahedra, cuboctahedra and octahedra, their maximal translative packing densities and their maximal lattice packing densities are identical. To attack this conjecture, this paper suggests a computer approach to determine the maximal l…
The ball (or sphere) packing problem with equal balls, without any symmetry assumption, in a $3$-dimensional space of constant curvature was settled by Böröczky and Florian for the hyperbolic space $\HYP$ in \cite{BF64} and by proving the famous Kepler conjecture by Hales \cite{H} for the Euclidean space $\EUC$. The goal of this paper is to extend the problem of finding the densest geodesic ball (or sphere) packing for the other $3$-dimensional homogeneous geometries (Thurston geometries) $$ \SXR,~\HXR,~\SLR,~\NIL,~\SOL, $$ where a transitive symmetry group of the ball packing is assumed, one …
We consider circle packings in the plane with circles of sizes $1$, $r\simeq 0.834$ and $s\simeq 0.651$. These sizes are algebraic numbers which allow a compact packing, that is, a packing in which each hole is formed by three mutually tangent circles. Compact packings are believed to maximize the density when there are possible. We prove that it is indeed the case for these sizes. The proof should be generalizable to other sizes which allow compact packings and is a first step towards a general result.
The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained for selected values of N up to N = 1054. In the predecessor to this paper [A.B. Hopkins, F.H. Stillinger and S. Torquato, Phys. Rev. E 81 041305 (2010)], we described our method for finding the putative densest packings of N spheres in d-dimensional Euclidean space Rd and presented those packings in R2 for values of N up to N = 348. We analyze the properties and characteristics of the densest local packings in R3 and employ knowledge…
In this paper, upper bounds for the densities of the densest translative tetrahedron packings and the densest translative cubooctahedron packings are obtained.
Using transversality and a dimension reduction argument, a result of A. Bezdek and W. Kuperberg is applied to polycylinders $\mathbb{D}^2\times \mathbb{R}^n$, showing that the optimal packing density is $π/\sqrt{12}$ in any dimension.
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…
Packings of regular convex polygons ($n$-gons) that are sufficiently dense have been studied extensively in the context of modeling physical and biological systems as well as discrete and computational geometry. Former results were mainly regarding densest lattice or double-lattice configurations. Here we consider all two-dimensional crystallographic symmetry groups (plane groups) by restricting the configuration space of the general packing problem of congruent copies of a compact subset of the two-dimensional Euclidean space to particular isomorphism classes of the discrete group of isometri…
The densest local packings of N identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained using a nonlinear programming method operating in conjunction with a stochastic search of configuration space. Knowledge of Rmin(N) in d-dimensional Euclidean space allows for the construction both of a realizability condition for pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings. In this paper, we focus on the two-dimensional circular disk problem. We find and present the putative dens…
The Sausage Catastrophe of J. Wills (1983) is the observation that in $d=3$ and $d=4$, the densest packing of $n$ spheres in $\mathbb{R}^{d}$ is a sausage for small values of $n$ and jumps to a full-dimensional packing for large $n$ without passing through any intermediate dimensions. Let $n_{d}^{*}$ be the smallest value of $n$ for which the densest packing of $n$ spheres in $\mathbb{R}^{d}$ is full-dimensional and $N_{d}^{*}$ be the smallest value of $N$ for which the densest packing of $N$ spheres in $\mathbb{R}^{d}$ is full-dimensional for all $N\geq N_{d}^{*}$. We extend the work of Gandi…
In this paper we study the regular prism tilings and construct ball packings by geodesic balls related to the above tilings in the projective model of $\mathbf{Nil}$ geometry. Packings are generated by action of the discrete prism groups $\mathbf{pq2_{1}}$. We prove that these groups are realized by prism tilings in $\mathbf{Nil}$ space if $(p,q)=(3,6), (4,4), (6,3)$ and determine packing density formulae for geodesic ball packings generated by the above prism groups. Moreover, studying these formulae we determine the conjectured maximal dense packing arrangements and their densities and visua…
In this research announcement we outline the methods used in our recent proof that the Leech lattice is the unique densest lattice in R^24. Complete details will appear elsewhere, but here we illustrate our techniques by applying them to the case of lattice packings in R^2, and we discuss the obstacles that arise in higher dimensions.
The determination of the densest packings of regular tetrahedra (one of the five Platonic solids) is attracting great attention as evidenced by the rapid pace at which packing records are being broken and the fascinating packing structures that have emerged. Here we provide the most general analytical formulation to date to construct dense periodic packings of tetrahedra with four particles per fundamental cell. This analysis results in six-parameter family of dense tetrahedron packings that includes as special cases recently discovered "dimer" packings of tetrahedra, including the densest kno…
The determination of the densest packings of regular tetrahedra (one of the five Platonic solids) is attracting great attention as evidenced by the rapid pace at which packing records are being broken and the fascinating packing structures that have emerged. We have discovered the densest known packings of regular tetrahedra with a density $ϕ= {12250/14319} = 0.855506...$. These packings are special cases of an analytical two-parameter family of dense periodic packings with four particles per fundamental cell that we have constructed here. From this family, we can recover a set of recent packi…
Particle packing problems have fascinated people since the dawn of civilization, and continue to intrigue mathematicians and scientists. Resurgent interest has been spurred by the recent proof of Kepler's conjecture: the face-centered cubic lattice provides the densest packing of equal spheres with a packing fraction $ϕ\approx0.7405$ \cite{Kepler_Hales}. Here we report on the densest known packings of congruent ellipsoids. The family of new packings are crystal (periodic) arrangements of nearly spherically-shaped ellipsoids, and always surpass the densest lattice packing. A remarkable maximum …
In this paper we construct a new family of lattice packings for superballs in three dimensions (unit balls for the $l^p_3$ norm) with $p \in (1, 1.58]$. We conjecture that the family also exists for $p \in (1.58, \log_2 3 = 1.5849625\ldots]$. Like in the densest lattice packing of regular octahedra, each superball in our family of lattice packings has $14$ neighbors.
Polyhedra and their arrangements have intrigued humankind since the ancient Greeks and are today important motifs in condensed matter, with application to many classes of liquids and solids. Yet, little is known about the thermodynamically stable phases of polyhedrally-shaped building blocks, such as faceted nanoparticles and colloids. Although hard particles are known to organize due to entropy alone, and some unusual phases are reported in the literature, the role of entropic forces in connection with polyhedral shape is not well understood. Here, we study thermodynamic self-assembly of a fa…
This review paper is devoted to the problems of sphere packings in 4 dimensions. The main goal is to find reasonable approaches for solutions to problems related to densest sphere packings in 4-dimensional Euclidean space. We consider two long-standing open problems: the uniqueness of maximum kissing arrangements in 4 dimensions and the 24-cell conjecture. Note that a proof of the 24-cell conjecture also proves that the checkerboard lattice packing D4 is the densest sphere packing in 4 dimensions.
In $n$-dimensional hyperbolic space $\mathbf{H}^n$ $(n\ge2)$ there are $3$-types of spheres (balls): the sphere, horosphere and hypersphere. If $n=2,3$ we know an universal upper bound of the ball packing densities, where each ball volume is related to the volume of the corresponding Dirichlet-Voronoi (D-V) cell. E.g. in $\mathbf{H}^3$ the densest horoball packing is derived from the $\{3,3,6\}$ Coxeter tiling consisting of ideal regular simplices $T_{reg}^\infty$ with dihedral angles $\fracπ{3}$. The density of this packing is $δ_3^\infty\approx 0.85328$ and this provides a very rough upper b…
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.
In this paper, we describe and visualize the densest ball and horoball packing configurations belonging to the simply truncated $3$-dimensional hyperbolic Coxeter orthoschemes with parallel faces. These beautiful packing arrangements describe and show the very interesting structure of the mentioned orthoschemes and the corresponding Coxeter groups. We use for the visualization the Beltrami-Cayley-Klein ball model of $3$-dimensional hyperbolic space $\boldsymbol{H}^3$ and the pictures were made by the Python software.
We consider hard-disc mixtures with disc sizes within ratio $\sqrt{2}-1$, that is, the small disc exactly fits in the hole between four large discs. For each prescribed stoichiometry of large and small discs, the densest packings are rigorously determined via a computer-assisted proof. The density is maximal for the 1:1 stoichiometry: the large discs then form a square grid in each interstitial site of which a small disc nests. When there is an excess of large discs, the densest packings are made of a single phase which mixes the two types of discs in a chaotic way (it can be described by squa…
Molecular crystal structure prediction (CSP) seeks the most stable periodic structure given the chemical composition of a molecule and pressure-temperature conditions. Modern CSP solvers use global optimization methods to search for structures with minimal free energy within a complex energy landscape induced by intermolecular potentials. A major caveat of these methods is that initial configurations are random, making thus the search susceptible to convergence at local minima. Providing initial configurations that are densely packed with respect to the geometric representation of a molecule c…
The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by analytic constructions, the importance of an efficient numerical method for conducting \textit{de novo} (from-scratch) searches for dense packings becomes crucial. In this paper, we use the \textit{divide and concur} framework to develop a general search method for the solution of periodic constraint problems, and we apply it to the discovery of dense perio…
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 $S^2 \times R$ geometry can be derived by the direct product of the spherical plane $\bS^2$ and the real line $\bR$. J. Z. Farkas has classified and given the complete list of the space groups of $S^2 \times R$. The $S^2 \times R$ manifolds were classified by E. Molnár and J. Z. Farkas by similarity and diffeomorphism. In Szirmai we have studied the geodesic balls and their volumes in $S^2 \times R$ space, moreover we have introduced the notion of geodesic ball packing and its density and have determined the densest geodesic ball packing for generalized Coxeter space groups of $S^2 \times …
In this paper we consider the ball and horoball packings belonging to $3$-dimensional Coxeter tilings that are derived by simply truncated orthoschemes with parallel faces. The goal of this paper to determine the optimal ball and horoball packing arrangements and their densities for all above Coxeter tilings in hyperbolic 3-space $\mathbb{H}^3$. The centers of horoballs are required to lie at ideal vertices of the polyhedral cells constituting the tiling, and we allow horoballs of different types at the various vertices. We prove that the densest packing of the above cases is realized by horob…
The contact graph of a packing of translates of a convex body in Euclidean $d$-space $\mathbb E^d$ is the simple graph whose vertices are the members of the packing, and whose two vertices are connected by an edge if the two members touch each other. A packing of translates of a convex body is called totally separable, if any two members can be separated by a hyperplane in $\mathbb E^d$ disjoint from the interior of every packing element. We give upper bounds on the maximum vertex degree (called separable Hadwiger number) and the maximum number of edges (called maximum separable contact number…
We prove that the Leech lattice is the unique densest lattice in R^24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R^24 can exceed the Leech lattice's density by a factor of more than 1+1.65*10^(-30), and we give a new proof that E_8 is the unique densest lattice in R^8.
A packing of translates of a convex body in the $d$-dimensional Euclidean space $\mathbb{E}^d$ is said to be totally separable if any two packing elements can be separated by a hyperplane of $\mathbb{E}^{d}$ disjoint from the interior of every packing element. We call the packing $\mathcal P$ of translates of a centrally symmetric convex body $\mathbf{C}$ in $\mathbb{E}^d$ a $ρ$-separable packing for given $ρ\geq 1$ if in every ball concentric to a packing element of $\mathcal P$ having radius $ρ$ (measured in the norm generated by $\mathbf{C}$) the corresponding sub-packing of $\mathcal P$ is…
The sphere packing problem asks for the densest packing of unit balls in d-dimensional Euclidean space. This problem has its roots in geometry, number theory and it is part of Hilbert's 18th problem. In 1958 C. A. Rogers proved a non-trivial upper bound for the density of unit ball packings in d-dimensional Euclidean space for all d>0. In 1978 Kabatjanskii and Levenstein improved this bound for large d. In fact, Rogers' bound is the presently known best bound for 43>d>3, and above that the Kabatjanskii-Levenstein bound takes over. In this paper we improve Rogers' upper bound for the density of…
One of the basic problems in discrete geometry is to determine the most efficient packing of congruent replicas of a given convex set $K$ in the plane or in space. The most commonly used measure of efficiency is density. Several types of the problem arise depending on the type of isometries allowed for the packing: packing by translates, lattice packing, translates and point reflections, or all isometries. Due to its connections with number theory, crystallography, etc., lattice packing has been studied most extensively. In two dimensions the theory is fairly well developed, and there are seve…
Viazovska proved that the $E_8$ lattice sphere packing is the densest sphere packing in 8 dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations.
A packing of translates of a convex domain in the Euclidean plane is said to be totally separable if any two packing elements can be separated by a line disjoint from the interior of every packing element. This notion was introduced by G. Fejes Tóth and L. Fejes Tóth (1973) and has attracted significant attention. In this paper we prove an analogue of Oler's inequality for totally separable translative packings of convex domains and then we derive from it some new results. This includes finding the largest density of totally separable translative packings of an arbitrary convex domain and find…
We study the contact structure of totally separable} packings of translates of a convex body $K$ in $\mathbb{R}^d$, that is, packings where any two touching bodies have a separating hyperplane that does not intersect the interior of any translate in the packing. The separable Hadwiger number $H_{\text{sep}}(K)$ of $K$ is defined to be the maximum number of translates touched by a single translate, with the maximum taken over all totally separable packings of translates of $K$. We show that for each $d\geq 8$, there exists a smooth and strictly convex $K$ in $\mathbb{R}^d$ with $H_{\text{sep}}(…
The famous Kepler conjecture has a less spectacular, two-dimensional equivalent: The theorem of Thue states that the densest circle packing in the Euclidean plane has a hexagonal structure. A common proof uses Voronoi cells and analyzes their area applying Jensen's inequality on convex functions to receive a local estimate which is globally valid. Based on the concept of Voronoi cells, we will introduce a new tessellation into so-called L-triangles which can be related to fundamental parallelograms of lattice circle packings. Therefore a globally disordered circle packing can be reduced to loc…
We present the densest known packing of regular tetrahedra with density phi = 4000/4671 = 0.856347... Like the recently discovered packings of Kallus et al. [arXiv:0910.5226] and Torquato-Jiao [arXiv:0912.4210], our packing is crystalline with a unit cell of four tetrahedra forming two triangular dipyramids (dimer clusters). We show that our packing has maximal density within a three-parameter family of dimer packings. Numerical compressions starting from random configurations suggest that the packing may be optimal at least for small cells with up to 16 tetrahedra and periodic boundaries.
Each packing of R^d by translates of the unit cube [0,1)^d admits a decomposition into at most two parts such that if a translate of the unit cube is covered by one of them, then it also belongs to such a part.
After having investigated the densest packings by congruent hyperballs to the regular prism tilings in the $n$-dimensional hyperbolic space $\mathbb{H}^n$ ($n \in \mathbb{N}, n \ge 3)$ we consider the dual covering problems and determine the least dense hyperball arrangements and their densities.
We introduce a parameter space for periodic point sets, given as unions of $m$ translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a periodic sphere packing with greater density. This applies in particular to the densest known lattice sphere packings in dimension $d\leq 8$ and $d=24$.
We present an efficient Monte Carlo method for the lattice sphere packing problem in d dimensions. We use this method to numerically discover de novo the densest lattice sphere packing in dimensions 9 through 20. Our method goes beyond previous methods not only in exploring higher dimensions but also in shedding light on the statistical mechanics underlying the problem in question. We observe evidence of a phase transition in the thermodynamic limit $d\to\infty$. In the dimensions explored in the present work, the results are consistent with a first-order crystallization transition, but leave …
Optimal Extensions of Cross-Sections: Sphere Packings in Dimensions 38 to 43
State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences（处理器重点实验室，中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）
For every convex disk $K$ (a convex compact subset of the plane, with non-void interior), the packing density $δ(K)$ and covering density $\vartheta(K)$ form an ordered pair of real numbers, {\em i.e.}, a point in ${\mathbb R}^2$. The set $Ω$ consisting of points assigned this way to all convex disks is the subject of this article. A few known inequalities on $δ(K)$ and $\vartheta(K)$ jointly outline a relatively small convex polygon $P$ that contains $Ω$, while the exact shape of $Ω$ remains a mystery. Here we describe explicitly a leaf-shaped convex region $Λ$ contained in $Ω$ and occupying …
The densest local packing (DLP) problem in d-dimensional Euclidean space Rd involves the placement of N nonoverlapping spheres of unit diameter near an additional fixed unit-diameter sphere such that the greatest distance from the center of the fixed sphere to the centers of any of the N surrounding spheres is minimized. Solutions to the DLP problem are relevant to the realizability of pair correlation functions for packings of nonoverlapping spheres and might prove useful in improving upon the best known upper bounds on the maximum packing fraction of sphere packings in dimensions greater tha…
After having investigated the regular prisms and prism tilings in the $\SLR$ space in the previous work \cite{Sz13-1} of the second author, we consider the problem of geodesic ball packings related to those tilings and their symmetry groups $\mathbf{pq2_1}$. $\SLR$ is one of the eight Thurston geometries that can be derived from the 3-dimensional Lie group of all $2\times 2$ real matrices with determinant one. In this paper we consider geodesic spheres and balls in $\SLR$ (even in $\mathbf{SL_{\mathrm{2}}R})$, if their radii $ρ\in [0, \fracπ{2})$, and determine their volumes. Moreover, we cons…
