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.
Microsoft Research New England（微软研究院新英格兰）；Department of Mathematics, Massachusetts Institute of Technology（麻省理工学院数学系）；Massachusetts Institute of Technology（麻省理工学院）
We study the relationship between local and global density for sphere packings, and in particular the convergence of packing densities in large, compact regions to the Euclidean limit. We axiomatize key properties of sphere packing bounds by the concept of a packing bound function, and we study the special case of sandwich functions, which give a framework for inequalities given by the Lovász sandwich theorem. We show that every packing bound function tends to a Euclidean limit on rectifiable sets, generalizing the work of Borodachov, Hardin, and Saff. Linear and semidefinite programming bound…
Compact packings are specific packings of spheres which can be seen as tilings and are good candidates to maximize the density. We show that the compact packings of the Euclidean space with two sizes of spheres are exactly those obtained by filling with smaller spheres the octahedral holes of a close-packing of equal spheres.
In "Dense Sphere Packings: A Blueprint for Formal Proofs" Hales proves that for every packing of unit spheres, the density in a ball of radius $r$ is at most $π/\sqrt{18}+c/r$ for some constant $c$. When $r$ tends to infinity, this gives a proof to the famous Kepler conjecture. As formulated by Hales, $c$ depends on the packing. We follow the proofs of Hales to calculate a constant $c'$ independent of the sphere packing that exists as mentioned in "A Formal Proof of the Kepler Conjecture" by Hales et al..
A sphere packing of the three-dimensional Euclidean space is compact if it has only tetrahedral holes, that is, any local maximum of the distance to the spheres is at equal distance to exactly four spheres. This papers describes all the compact packings with spheres of three different sizes. They are close-compact packings of unit spheres with holes filled in four different ways by smaller spheres.
We define three-point bounds for sphere packing that refine the linear programming bound, and we compute these bounds numerically using semidefinite programming by choosing a truncation radius for the three-point function. As a result, we obtain new upper bounds on the sphere packing density in dimension 4 through 7 and 9 through 16. We also give a different three-point bound for lattice packing and conjecture that this second bound is sharp in dimension 4.
We algorithmically characterize the maximal contact number problem for finite congruent lattice sphere packings in $\mathbb{R}^d$ and show that in $\mathbb{R}^3$ this problem is equivalent to determining the maximal coordination of a monatomic crystal.
This paper is an exposition, written for the Nieuw Archief voor Wiskunde, about the two recent breakthrough results in the theory of sphere packings. It includes an interview with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Maryna Viazovska.
We introduce an analog of Bianchi groups for rational quaternion algebras and use it to construct sphere packings that are analogs of the Apollonian circle packing known as integral crystallographic packings.
The contact number of a packing of finitely many balls in Euclidean $d$-space is the number of touching pairs of balls in the packing. A prominent subfamily of sphere packings is formed by the so-called totally separable sphere packings: here, a packing of balls in Euclidean $d$-space is called totally separable if any two balls can be separated by a hyperplane such that it is disjoint from the interior of each ball in the packing. Bezdek, Szalkai and Szalkai (Discrete Math. 339(2): 668-676, 2016) upper bounded the contact numbers of totally separable packings of $n$ unit balls in Euclidean $d…
This expository paper describes Viazovska's breakthrough solution of the sphere packing problem in eight dimensions, as well as its extension to twenty-four dimensions by Cohn, Kumar, Miller, Radchenko, and Viazovska.
We study some sequences of functions of one real variable and conjecture that they converge uniformly to functions with certain positivity and growth properties. Our conjectures imply a conjecture of Cohn and Elkies, which in turn implies the complete solution to the sphere packing problem in dimensions 8 and 24. We give numerical evidence for these conjectures as well as some arithmetic properties of the hypothetical limiting functions. The conjectures are of greatest interest in dimension 24, in light of Viazovska's recent solution to the Cohn-Elkies conjecture (and consequently the sphere p…
Inversive geometry can be used to generate exactly self-similar space-filling sphere packings. We present a construction method in two dimensions and generalize it to search for packings in higher dimensions. We newly discover 29 three-dimensional and 13 four-dimensional topologies of which 10 and 5, respectively, are bearings. To distinguish and characterize the packing topologies, we numerically estimate their fractal dimensions and we analyze their contact networks.
Bounds for Local Density of Sphere Packings and the Kepler Conjecture
利用局部密度不等式方法得到了R^n中球堆积密度的上界.这种方法首先由L. Fejes-Toth在1956年证明了开普勒猜想，即R^3中的密积球填充是“炮弹填充”。L. Fejes-Toth，W. Y. Hsiang和T.海尔斯到开普勒猜想都是基于（不同的）局部密度不等式。最近T. Hales和S. P. Ferguson提出了一个修改过的Otrignal Hales方法来证明开普勒猜想的广泛细节。我们描述了特殊的局部密度不等式undelying黑尔斯和弗格森的方法，并勾勒出他们的证明的一些特点。
We prove a lower bound on the entropy of sphere packings of $\mathbb R^d$ of density $Θ(d \cdot 2^{-d})$. The entropy measures how plentiful such packings are, and our result is significantly stronger than the trivial lower bound that can be obtained from the mere existence of a dense packing. Our method also provides a new, statistical-physics-based proof of the $Ω(d \cdot 2^{-d})$ lower bound on the maximum sphere packing density by showing that the expected packing density of a random configuration from the hard sphere model is at least $(1+o_d(1)) \log(2/\sqrt{3}) d \cdot 2^{-d}$ when the …
The sphere packing problem is an old puzzle. We consider packings with m spheres in the unit cell (m-periodic packings). For the case m = 1 (lattice packings), Voronoi proved there are finitely many inequivalent local optima and presented an algorithm to enumerate them, and this computation has been implemented in up to d = 8 dimensions. We generalize Voronoi's method to m > 1 and present a procedure to enumerate all locally optimal 2-periodic sphere packings in any dimension, provided there are finitely many. We implement this computation in d = 3, 4, and 5 and show that no 2-periodic packing…
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 obtain new restrictions on the linear programming bound for sphere packing, by optimizing over spaces of modular forms to produce feasible points in the dual linear program. In contrast to the situation in dimensions 8 and 24, where the linear programming bound is sharp, we show that it comes nowhere near the best packing densities known in dimensions 12, 16, 20, 28, and 32. More generally, we provide a systematic technique for proving separations of this sort.
The Apollonian circle packing, generated from three mutually-tangent circles in the plane, has inspired over the past half-century the study of other classes of space-filling packings, both in two and in higher dimensions. Recently, Kontorovich and Nakamura introduced the notion of crystallographic sphere packings, $n$-dimensional packings of spheres with symmetry groups that are isometries of $\mathbb{H}^{n+1}$. There exist at least three sources which give rise to crystallographic packings, namely polyhedra, reflective extended Bianchi groups, and various higher dimensional quadratic forms. …
We say that a tiling separates discs of a packing in the Euclidean plane, if each tile contains exactly one member of the packing. It is a known elementary geometric problem to show that for each locally finite packing of circular discs, there exists a separating tiling with convex polygons. In this paper we show that this separating property remains true for circle packings on the sphere and in the hyperbolic plane. Moreover, we show that in the Euclidean plane circles are the only convex discs, whose packings with similar copies can be always separated by polygonal tilings. The analogous sta…
We establish a precise relation between the modular bootstrap, used to constrain the spectrum of 2D CFTs, and the sphere packing problem in Euclidean geometry. The modular bootstrap bound for chiral algebra $U(1)^c$ maps exactly to the Cohn-Elkies linear programming bound on the sphere packing density in $d=2c$ dimensions. We also show that the analytic functionals developed earlier for the correlator conformal bootstrap can be adapted to this context. For $c=4$ and $c=12$, these functionals exactly reproduce the "magic functions" used recently by Viazovska [1] and Cohn et al. [2] to solve the…
We consider the sets of dimensions for which there is an optimal sphere packing with special regularity properties (respectively, a lattice, or a periodic set with a given bound on the number of translations, or an arbitrary periodic set). We show that all these sets are oracle-computable, given an oracle that orders an associated set of spherical codes by increasing Kolmogorov complexity.
We introduce the notion of a "crystallographic sphere packing," defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit for the first time an infinite family of conformally-inequivalent such with all radii being reciprocals of integers. We then prove a result in the opposite direction: the "superintegral" ones exist only in finitely many "commensurability classes," all in dimensions below 30.
