Tom Kennedy
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.
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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 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 provide a counterexample to a conjecture by B. Connelly about density of circle packings
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…
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 $L$ h…
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.
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…
The smallest three hyperbolic compact arithmetic 5-orbifolds can be derived from two compact Coxeter polytops which are combinatorially simplicial prisms (or complete orthoschemes of degree $d=1$) in the five dimensional hyperbolic space $\mathbf{H}^5$ (see \cite{BE} and \cite{EK}). The corresponding hyperbolic tilings are generated by reflections through their delimiting hyperplanes those involve the study of the relating densest hyperball (hypersphere) packings with congruent hyperballs. The analogous problem was discussed in \cite{Sz06-1} and \cite{Sz06-2} in the hyperbolic spaces $\mathbf{…
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 …
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.
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…
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 …
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。
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.
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.
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.
Motivated by the relation between particle shape and packing, we measure the volume fraction $ϕ$ occupied by the Platonic solids which are a class of polyhedron with congruent sides, vertices and dihedral angles. Tetrahedron, cube, octahedron, dodecahedron, and icosahedron shaped plastic dice were fluidized or mechanically vibrated to find stable random loose packing $ϕ_{rlp} = 0.51, 0.54, 0.52, 0.51, 0.50$ and densest packing $ϕ_{rcp} = 0.64, 0.67, 0.64, 0.63, 0.59$, respectively with standard deviation $\simeq \pm 0.01$. We find that $ϕ$ obtained by all protocols peak at the cube, which is t…
We consider packings of the plane using discs of radius 1 and r. A packing is compact if every disc D is tangent to a sequence of discs D_1, D_2, ..., D_n such that D_i is tangent to D_{i+1}. We prove that there are only nine values of r with r<1 for which such packings are possible. For each of the nine values we describe the possible compact packings.
In an old paper of the author the thinnest five-neighbour packing of translates of a convex disc (different from a parallelogram) was determined. The minimal density was $3/7$, and was attained for a certain packing of triangles. In that paper it was announced that for centrally symmetric convex plates (different from a parallelogram) the analogous minimal density was $9/14$, and was attained for a certain packing of affine regular hexagons, and a very sketchy idea of the proof was given. In this paper we give details of this proof.
A disc packing in the plane is compact if its contact graph is a triangulation. There are $9$ values of $r$ such that a compact packing by discs of radii $1$ and $r$ exists. We prove, for each of these $9$ values, that the maximal density over all the packings by discs of radii $1$ and $r$ is reached for a compact packing (we give it as well as its density).
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…
We provide, for any $r\in (0,1)$, lower and upper bounds on the maximal density of a packing in the Euclidean plane of discs of radius $1$ and $r$. The lower bounds are mostly folk, but the upper bounds improve the best previously known ones for any $r\in[0.11,0.74]$. For many values of $r$, this gives a fairly good idea of the exact maximum density. In particular, we get new intervals for $r$ which does not allow any packing more dense that the hexagonal packing of equal discs.
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