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Sphere Packing

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A sphere packing is an arrangement of congruent closed balls of positive radius with pairwise disjoint interiors. Its packing density eta is the asymptotic fraction of space occupied by the balls. If the limiting fraction does not exist, an upper limiting density can be used. For periodic packings, the density can be computed in a unit cell. In three dimensions, examples include packings on simple and body-centered cubic lattices, cubic close packing, and hexagonal close packing. It was hypothesized by Kepler in 1611 that close packing (cubic or hexagonal, which have equivalent packing densities) is the densest possible, and this assertion is known as the Kepler conjecture. The problem of finding the densest packing of spheres (not necessarily periodic) is therefore known as the Kepler problem, where

eta_(Kepler)=eta_(FCC)=eta_(HCP)=pi/(3sqrt(2)) approx 74.048%

(OEIS A093825; Steinhaus 1999, p. 202; Wells 1986, p. 29; Wells 1991, p. 237).

In 1831, Gauss managed to prove that the face-centered cubic is the densest lattice packing in three dimensions (Conway and Sloane 1993, p. 9), but the general conjecture remained open for many decades.

While the Kepler conjecture is intuitively obvious, the proof remained surprisingly elusive. Rogers (1958), a well-known researcher on the problem, remarked that "many mathematicians believe, and all physicists know" that the actual answer is 74.048% (Conway and Sloane 1993, p. 3). For packings in three dimensions, C. A. Rogers (1958) showed that the maximum possible packing density eta_(max) satisfies

eta_(max)<sqrt(18)(cos^(-1)1/3-1/3pi) approx 77.96355700%

(Le Lionnais 1983), and this result was subsequently improved to 77.844% (Lindsey 1986), then 77.836% (Muder 1988). A proof of the full conjecture was published by Hales (2005, p. 1067).

Interestingly, the packing density in ellipsoid packing can exceed eta_(Kepler).

The maximum number of equivalent spheres (or n-dimensional hyperspheres) which can touch an equivalent sphere (hypersphere) without intersections is called the n-dimensional kissing number.

The packing densities for several types of sphere packings are summarized in the following table. In a 1972 personal communication to Martin Gardner, Ulam conjectured that in their densest packing, spheres allow more empty space than the densest packing of any other identical convex solids (Gardner 2001, p. 135).

packinganalytic etaetareference
reported rigid packing--0.0555Gardner (1966)
tetrahedral packing(pisqrt(3))/(16)0.3401Hilbert and Cohn-Vossen (1999, pp. 48-50)
cubic latticepi/60.5236
simple hexagonal latticepi/(3sqrt(3))0.6046
random (typical)--0.6400Jaeger and Nagel (1992)
face-centered cubic close packingpi/(3sqrt(2))0.7405Steinhaus (1999, p. 202), Wells (1986, p. 29; 1991, p. 237)
body-centered cubic lattice(pisqrt(3))/80.6802
hexagonal close packingpi/(3sqrt(2))0.7405Steinhaus (1999, p. 202), Wells (1986, p. 29; 1991, p. 237)

Gardner (1966) reported a rigid packing with packing density eta approx 0.0555, significantly lower than that reported by Hilbert and Cohn-Vossen (1999, p. 51). A necessary condition for each sphere to be unable to move while all other spheres are held fixed is that it have at least four contacts and that all its contact directions, taken together, are not contained in a closed hemisphere. This condition alone does not establish rigidity against collective motions.

Hilbert and Cohn-Vossen (1999, pp. 48-50) consider a tetrahedral packing in which each sphere touches four neighbors and the density is pisqrt(3)/16 approx 0.3401. Its centers form two translates of a face-centered cubic lattice, rather than a point lattice. This is the periodic arrangement formed by carbon atoms in a diamond (Conway and Sloane 1993, p. 113).

Densities obtained by protocols for random close packing depend on the procedure used. Torquato et al. (2000) discuss values around 0.60, 0.642-0.649, and 0.68 and argue against a universal random close packing density. Compressing a random packing gives polyhedra with an average of 13.3 faces (Coxeter 1958, 1961).

For sphere packing inside a cube, see Goldberg (1971), Schaer (1966), Gensane (2004), and Friedman. Gensane (2004, pp. 1 and 5) improved Goldberg's configurations for n=11, 12, and all n from n=15 to n=26 except for n=18. The optimality of these configurations was unproved.

See also

Cannonball Problem, Circle Packing, Cubic Close Packing, Cuboctahedron, Dodecahedral Conjecture, Ellipsoid Packing, Hemisphere, Hermite Constants, Hexagonal Close Packing, Hypersphere, Hypersphere Packing, Kepler Conjecture, Kepler Problem, Kissing Number, Local Density, Local Density Conjecture, Random Close Packing, Reuleaux Tetrahedron, Space-Filling Polyhedron, Sphere, Spherical Design, Sphericon, Stella Octangula, Tangent Spheres, Triangular Orthobicupola, Unit Cell

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References

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Sphere Packing

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Weisstein, Eric W. "Sphere Packing." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SpherePacking.html

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