More Optimal Packings of Equal Circles in a Square

The densest packings of n equal circles in a square have been determined earlier for n ≤ 20 and for n = 25, 36 . Several of these packings have been proved with the aid of a computer. The computer-aided approach is further developed here and the range is extended to n ≤ 27 . The optimal packings are depicted. Received February 11, 1998, and in revised form December 17, 1998.     

Densest packings of eleven congruent circles in a circle

In 1969 Pirl provided the densest packings ofn equal circles in a circle forn 10. We will prove the optimality for the packings that were conjectured forn=11. The proof is based on elementary combinatorial and analytical techniques.     

Dense Crystalline Dimer Packings of Regular Tetrahedra

We present the densest known packing of regular tetrahedra with density $\phi =\frac{4000}{4671}=0.856347\ldots\,We present the densest known packing of regular tetrahedra with density f = \frac40004671=0.856347? \phi =\frac{4000}{4671}=0.856347\ldots\,. Like the recently discovered packings of Kallus et al. and Torquato–Jiao, 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.     

Curved Hexagonal Packings of Equal Disks in a Circle

For each k ≥ 1 and corresponding hexagonal number h(k) = 3k(k+1)+1, we introduce packings of h(k) equal disks inside a circle which we call the curved hexagonal packings. The curved hexagonal packing of 7 disks (k = 1, m(1)=1) is well known and one of the 19 disks (k = 2, m(2)=1) has been previously conjectured to be optimal. New curved hexagonal packings of 37, 61, and 91 disks (k = 3, 4, and 5, m(3)=1, m(4)=3, and m(5)=12) were the densest we obtained on a computer using a so-called ``billiards' simulation algorithm. A curved hexagonal packing pattern is invariant under a rotation. For , the density (covering fraction) of curved hexagonal packings tends to . The limit is smaller than the density of the known optimum disk packing in the infinite plane. We found disk configurations that are denser than curved hexagonal packings for 127, 169, and 217 disks (k = 6, 7, and 8). In addition to new packings for h(k) disks, we present the new packings we found for h(k)+1 and h(k)-1 disks for k up to 5, i.e., for 36, 38, 60, 62, 90, and 92 disks. The additional packings show the ``tightness' of the curved hexagonal pattern for k ≤ 5: deleting a disk does not change the optimum packing and its quality significantly, but adding a disk causes a substantial rearrangement in the optimum packing and substantially decreases the quality. Received May 15, 1995, and in revised form March 5, 1996.     

Towards a proof of the 24-cell conjecture

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 lattice packing D₄ is the densest sphere packing in 4 dimensions.     

Lattices of Optimal Finite Lattice Packings

We consider finite lattice ball packings with respect to parametric density and show that densest packings are attained in critical lattices if the number of translates and the density parameter are sufficiently large. A corresponding result is not valid for general centrally symmetric convex bodies.     

Perfect, strongly eutactic lattices are periodic extreme

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?8 and d=24.     

Lattices of Optimal Finite Lattice Packings

We consider finite lattice ball packings with respect to parametric density and show that densest packings are attained in critical lattices if the number of translates and the density parameter are sufficiently large. A corresponding result is not valid for general centrally symmetric convex bodies.The second author was partially supported by a DAAD Postdoc fellowship and the hospitality of Peking University during his work.     

Finite and Uniform Stability of Sphere Packings

The main purpose of this paper is to discuss how firm or steady certain known ball packing are, thinking of them as structures. This is closely related to the property of being locally maximally dense. Among other things we show that many of the usual best-known candidates, for the most dense packings with congruent spherical balls, have the property of being uniformly stable, i.e., for a sufficiently small ε > 0 every finite rearrangement of the balls of this packing, where no ball is moved more than ε , is the identity rearrangement. For example, the lattice packings D _d and A _d for d ≥ 3 in E ^d are all uniformly stable. The methods developed here can work for many other packings as well. We also give a construction to show that the densest cubic lattice ball packing in E ^d for d ≥ 2 is not uniformly stable. A packing of balls is called finitely stable if any finite subfamily of the packing is fixed by its neighbors. If a packing is uniformly stable, then it is finitely stable. On the other hand, the cubic lattice packings mentioned above, which are not uniformly stable, are nevertheless finitely stable. Received April 22, 1996, and in revised form October 11, 1996.     

On Large Lattice Packings of Spheres

The packing density of large lattice packings of spheres in Euclidean E ^d measured by the parametric density depends on the parameter and on the shape of the convex hull P of the sphere centers; in particular on the isoperimetric coefficient of P and on the second term in the Ehrhart polynomial of the lattice polytope P. We show in E ^d , d 2, that flat or spherelike polytopes generate less dense packings, whereas polytopes with suitably chosen large facets generate dense packings. This indicates that large lattice packings in E ³ of high parametric density may be good models for real crystals.     

On the translative packing densities of tetrahedra and cuboctahedra

In 1900, as a part of his 18th problem, Hilbert asked the question to determine the density of the densest tetrahedron packings. However, up to now no mathematician knows the density δ^t(T)${\delta }^t(T)$ of the densest translative tetrahedron packings and the density δ^c(T)${\delta }^c(T)$ of the densest congruent tetrahedron packings. This paper presents a local method to estimate the density of the densest translative packings of a general convex solid. As an application, we obtain the upper bound in

0.3673469?≤δ^t(T)≤0.3840610?,$0.3673469?\le {\delta }^t(T)\le 0.3840610?,$

where the lower bound was established by Groemer in 1962, which corrected a mistake of Minkowski. For the density δ^t(C)${\delta }^t(C)$ of the densest translative cuboctahedron packings, we obtain the upper bound in

0.9183673?≤δ^t(C)≤0.9601527?.$0.9183673?\le {\delta }^t(C)\le 0.9601527?.$

In both cases we conjecture the lower bounds to be the correct answer.     

Maximum density space packing with parallel strings of spheres

A string of spheres is a sequence of nonoverlapping unit spheres inR ³ whose centers are collinear and such that each sphere is tangent to exactly two other spheres. We prove that if a packing with spheres inR ³ consists of parallel translates of a string of spheres, then the density of the packing is smaller than or equal to . This density is attained in the well-known densest lattice sphere packing. A long-standing conjecture is that this density is maximum among all sphere packings in space, to which our proof can be considered a partial result. The work of A. Bezdek and E. Makai was partially supported by the Hungarian National Foundation for Scientific Research under Grant Number 1238.     

Densest Packing of Equal Spheres in Hyperbolic Space

Abstract. We propose a method to analyze the density of packings of spheres of fixed radius in the hyperbolic space of any dimension m≥ 2 , and prove that for all but countably many radii, optimally dense packings must have low symmetry.     

Densest Packing of Equal Spheres in Hyperbolic Space

   Abstract. We propose a method to analyze the density of packings of spheres of fixed radius in the hyperbolic space of any dimension m≥ 2 , and prove that for all but countably many radii, optimally dense packings must have low symmetry.     

Dense lattice packings of spheres in Euclidean spaces of dimension n⩽16 *

We present a number of lattice packings of equal spheres in ⁿ for n16 For n15, these packings have the same density as the densest known lattice packings. For n=16, the packing described here is denser than the known ones.It should be pointed out that the 16-dimensional lattice described here is equivalent to one found by E. S. Barnes and G. E. Wall, J. Aust. Math. Soc.,1, 47–63 (1959); see also J. Leech and N. J. A. Sloane, Can. J. Math.,23, 718–745 (1971) — Translator.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, 144–146, 1979.     

Horoball packings and their densities by generalized simplicial density function in the hyperbolic space

The aim of this paper is to determine the locally densest horoball packing arrangements and their densities with respect to fully asymptotic tetrahedra with at least one plane of symmetry in hyperbolic 3-space $\overline{\mathbf{H}}^{3}$ extended with its absolute figure, where the ideal centers of horoballs give rise to vertices of a fully asymptotic tetrahedron. We allow horoballs of different types at the various vertices. Moreover, we generalize the notion of the simplicial density function in the extended hyperbolic space $\overline{\mathbf{H}}^{n}$ (n≧2), and prove that, in this sense, the well known B?r?czky–Florian density upper bound for “congruent horoball” packings of ? $\overline{\mathbf{H}}^{3}$ does not remain valid to the fully asymptotic tetrahedra. The density of this locally densest packing is ≈0.874994, may be surprisingly larger than the B?r?czky–Florian density upper bound ≈0.853276 but our local ball arrangement seems not to have extension to the whole hyperbolic space.     

Highly saturated packings and reduced coverings

We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packingP with congruent replicas of a bodyK isn-saturated if non–1 members of it can be replaced withn replicas ofK, and it is completely saturated if it isn-saturated for eachn1. Similarly, a coveringC with congruent replicas of a bodyK isn-reduced if non members of it can be replaced byn–1 replicas ofK without uncovering a portion of the space, and its is completely reduced if it isn-reduced for eachn1. We prove that every bodyK ind-dimensional Euclidean or hyperbolic space admits both ann-saturated packing and ann-reduced covering with replicas ofK. Under some assumptions onKE ^d (somewhat weaker than convexity), we prove the existence of completely saturated packings and completely reduced coverings, but in general, the problem of existence of completely saturated packings, and completely reduced coverings remains unsolved. Also, we investigate some problems related to the the densities ofn-saturated packings andn-reduced coverings. Among other things, we prove that there exists an upper bound for the density of ad+2-reduced covering ofE ^d with congruent balls, and we produce some density bounds for then-saturated packings andn-reduced coverings of the plane with congruent circles.     

Note on an Inequality of Wegner

Wegner gave a geometric characterization of all so-called Groemer packing of n ≥ 2 unit discs in E ² that are densest packings of n unit discs with respect to the convex hull of the discs. In this paper we provide a number theoretic characterization of all n satisfying that such a "Wegner packing" of n unit discs exists, and show that the proportion of these n is 23/24 among all natural numbers.     

VORONOI POLYHEDRA OF UNIT BALL PACKINGS WITH SMALL SURFACE AREA

In this note, first, we give a very short new proof of the theorem which yields a lower bound for the surface area of Voronoi cells of unit ball packings in E ^d and implies Rogers' upper bound for the density of unit ball packings in E ^d for all d ≥ 2. Second we sharpen locally a classical result of Gauss by finding the locally smallest surface area Voronoi cells of lattice unit ball packings in E ³. This revised version was published online in August 2006 with corrections to the Cover Date.     

Densest Geodesic Ball Packings to <Emphasis Type="Bold">S</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Bold">R</Emphasis> space groups generated by screw motions

In this paper we study the locally optimal geodesic ball packings with equal balls to the S ² × 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 S ² × R space group 3qe. I. 3 with packing density ≈0.7278. E. Molnár has shown in [9] that the Thurston geometries have an unified interpretation in the real projective 3-sphere \({\mathcal{PS}^3}\). In our work we shall use this projective model of S ² × R geometry.