Define the packing density 
of a packing of spheres to be the fraction of a volume
filled by the spheres. In three dimensions, there are three periodic packings
for identical spheres: cubic lattice, face-centered cubic lattice, and hexagonal
lattice. 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
 |
(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 
satisfies
 |
(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 finally accomplished in a series of papers by Hales culminating in 1998.
Interestingly, the packing density in ellipsoid packing can exceed 
.
The maximum number of equivalent spheres (or 
-dimensional hyperspheres) which can touch an equivalent sphere
(hypersphere) without intersections is called the 
-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).
| packing | analytic  |  | reference |
| loosest possible | -- | 0.0555 | Gardner (1966) |
| tetrahedral lattice |  | 0.3401 | Hilbert and Cohn-Vossen (1999, pp. 48-50) |
| cubic lattice |  | 0.5236 | |
| hexagonal lattice |  | 0.6046 | |
| random | -- | 0.6400 | Jaeger and Nagel (1992) |
| face-centered cubic close packing |  | 0.7405 | Steinhaus (1999, p. 202), Wells (1986, p. 29; 1991, p. 237) |
| body-centered cubic close packing |  | 0.6801 | |
| hexagonal close packing |  | 0.7405 | Steinhaus (1999, p. 202), Wells (1986, p. 29; 1991, p. 237) |
The rigid packing with lowest density known has 
(Gardner 1966), significantly lower than that
reported by Hilbert and Cohn-Vossen (1999, p. 51). To be rigid, each sphere
must touch at least four others, and the four contact points cannot be in a single
hemisphere or all on one equator.
Hilbert and Cohn-Vossen (1999, pp. 48-50) consider a tetrahedral lattice packing in which each sphere touches four neighbors and the density is 
. This is the lattice formed by carbon
atoms in a diamond (Conway and Sloane 1993, p. 113).
Random close packing of spheres in three dimensions gives packing densities in the range 0.06 to 0.65 (Jaeger and Nagel 1992, Torquato et al. 2000). 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. The results of Gensane (2004) improve those
of Goldberg for 
,
12, and all 
from 
to 
except for 
and are almost certainly optimal.
." Math. 33, 137-147, 1986.Muder,
D. J. "Putting the Best Face of a Voronoi Polyhedron." Proc. London
Math. Soc. 56, 329-348, 1988.Rogers, C. A. "The
Packing of Equal Spheres." Proc. London Math. Soc. 8, 609-620,
1958.Rogers, C. A.