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
(Sloane's A093825;
Steinhaus 1999, p. 202; Wells 1986, p. 29; Wells 1991, p. 237).
Gauss (1831) 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) | | cubic close packing |  | 0.7405 | Steinhaus (1999, p. 202), Wells (1986, p. 29; 1991, p. 237) | | 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.
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