The Sphere-Packing Problem (original) (raw)

Convex Maximization Formulation of General Sphere Packing Problem

The Bulletin of Irkutsk State University. Series Mathematics

We consider a general sphere packing problem which is to pack nonoverlapping spheres (balls) with the maximum volume into a convex set. This problem has important applications in science and technology. We prove that this problem is equivalent to the convex maximization problem which belongs to a class of global optimization. We derive necessary and sufficient conditions for inscribing a finite number of balls into a convex compact set. In two dimensional case, the sphere packing problem is a classical circle packing problem. We show that 200 years old Malfatti's problem [11] is a particular case of the circle packing problem. We also survey existing algorithms for solving the circle packing problems as well as their industrial applications.

A Literature Review on Circle and Sphere Packing Problems: Models and Methodologies

Advances in Operations Research, 2009

This paper reviews the most relevant literature on efficient models and methods for packing circular objects/items into Euclidean plane regions where the objects/items and regions are either two-or three-dimensional. These packing problems are NP hard optimization problems with a wide variety of applications. They have been tackled using various approaches-based algorithms ranging from computer-aided optimality proofs, to branch-and-bound procedures, to constructive approaches, to multi-start nonconvex minimization, to billiard simulation, to multiphase heuristics, and metaheuristics.

A Short History of Packing Problems

2000

Packing problems have arisen throughout the history of science. In particular the problem of the densest close packing of spheres has become celebrated as the Kepler Problem. The history of the subject is briefly reviewed, from both mathematical and physical (or practical) standpoints.

Sphere Packing: Asymptotic Behavior and Existence of Solution

2000

Lattices in n-dimensional Euclidean spaces may be parameterized by the non-compact symmetric space SL(n, R)/SO(n, R). We consider sphere packings determined by lattices and study the density function in the symmetric space, showing that the density function ρ(A k ) decreases to 0 if A k is a sequence of matrices in SL(n, R) with lim k→∞ A k = ∞. As a consequence, we give a simple prove that the optimal solution for the sphere packing problem is attained.

A Lower Bound on the Density of Sphere Packings via Graph Theory

International Mathematics Research Notices

Using graph-theoretic methods we give a new proof that for all sufficiently large n, there exist sphere packings in R n of density at least cn2 −n exceeding the classical Minkowski bound by a factor linear in n. This matches up to a constant the best known lower bounds on the density of sphere packings due to Rogers [9], Davenport-Rogers [4], and Ball [2]. The suggested method makes it possible to describe the points of such a packing with complexity exp(n log n), which is significantly lower than in the other approaches.

Locally Optimal 2-Periodic Sphere Packings

Discrete & Computational Geometry, 2019

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 surpasses the density of the optimal lattices in these dimensions. A partial enumeration is performed in d = 6.

JAMS 1 Sphere Packing: asymptotic behavior and existence of solution

2003

Abstract. Lattices in n-dimensional Euclidean spaces may be parameterized by the non-compact symmetric space SL(n, R)/SO(n, R). We consider sphere packings determined by lattices and study the density function in the symmetric space, showing that the density function ρ(Ak) decreases to 0 if Ak is a sequence of matrices in SL(n, R) with limk→ ∞ ‖Ak ‖ = ∞. As a consequence, we give a simple prove that the optimal solution for the sphere packing problem is attained. The sphere packing problem is one of the famous open problems in mathematics. In short, it asks about the densest way a set of equal spheres can be packed in space n-dimensional Euclidean space R n, without overlapping one the other. In this context, the density means the proportion between the covered and the uncovered amount of space. It has many variations: one could replace spheres of equal radii by spheres of radii 0 < a ≤ r ≤ b bounded from above and below, replace spheres by a collection of identical (preferably c...