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.
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).
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.
Barlow, W. "Probable Nature of the Internal Symmetry of Crystals." Nature29, 186-188, 1883.Conway, J. H.
and Sloane, N. J. A. Sphere
Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, 1993.Coxeter,
H. S. M. "Close-Packing and so Forth." Illinois J. Math.2,
746-758, 1958.Coxeter, H. S. M. "Close Packing of Equal
Spheres." Section 22.4 in Introduction
to Geometry, 2nd ed. New York: Wiley, pp. 405-411, 1961.Coxeter,
H. S. M. "The Problem of Packing a Number of Equal Nonoverlapping
Circles on a Sphere." Trans. New York Acad. Sci.24, 320-331,
1962.Critchlow, K. Order
in Space: A Design Source Book. New York: Viking Press, 1970.Cundy,
H. and Rollett, A. Mathematical
Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 195-197, 1989.Eppstein,
D. "Covering and Packing." http://www.ics.uci.edu/~eppstein/junkyard/cover.html.Fejes
Tóth, G. "Über einen geometrischen Satz." Math. Z.46,
78-83, 1940.Fejes Tóth, G. Lagerungen
in der Ebene, auf der Kugel und in Raum, 2nd ed. Berlin: Springer-Verlag,
1972.Friedman, E. "Spheres in Cubes." http://www.stetson.edu/~efriedma/sphincub/.Gardner,
M. "Packing Spheres." Ch. 7 in Martin
Gardner's New Mathematical Diversions from Scientific American. New York:
Simon and Schuster, pp. 82-90, 1966.Gardner, M. "Packing Spheres."
Ch. 10 in The
Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. New
York: W. W. Norton, pp. 128-136, 2001.Gauss, C. F.
"Besprechung des Buchs von L. A. Seeber: Intersuchungen über
die Eigenschaften der positiven ternären quadratischen Formen usw." Göttingsche
Gelehrte Anzeigen (1831, July 9)2, 188-196, 1876.Gensane,
T. "Dense Packings of Equal Spheres in a Cube." Electronic J. Combinatorics11,
No. 1, R33, 2004. http://www.combinatorics.org/Volume_11/PDF/v11i1r33.pdf.Goldberg,
M. "On the Densest Packing of Equal Spheres in a Cube." Math. Mag.44,
199-208, 1971.Hales, T. C. "The Sphere Packing Problem."
J. Comput. Appl. Math44, 41-76, 1992.Hilbert, D. and
Cohn-Vossen, S. Geometry
and the Imagination. New York: Chelsea, pp. 45-53, 1999.Jaeger,
H. M. and Nagel, S. R. "Physics of Granular States." Science255,
1524, 1992.Le Lionnais, F. Les
nombres remarquables. Paris: Hermann, p. 31, 1983.Lindsey,
J. H. II. "Sphere Packing in ." 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. Packing
and Covering. Cambridge, England: Cambridge University Press, 1964.Schaer,
J. "On the Densest Packing of Spheres in a Cube." Can. Math. Bul.9,
265-270, 1966.Sigrist, F. "Sphere Packing." Math. Intell.5,
34-38, 1983.Sloane, N. J. A. "The Packing of Spheres."
Sci. Amer.250, 116-125, 1984.Sloane, N. J. A.
"The Sphere Packing Problem." Proc. Internat. Congress Math., Vol. 3
(Berlin, 1998).Doc. Math. Extra Volume ICM 1998, 387-396, 1998. http://www.research.att.com/~njas/doc/icm.ps.Sloane,
N. J. A. Sequence A093825 in "The
On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, pp. 202-203, 1999.Stewart,
I. The
Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press,
pp. 69-82, 1987.Thompson, T. M. From
Error-Correcting Codes Through Sphere Packings to Simple Groups. Washington,
DC: Math. Assoc. Amer., 1984.Torquato, S.; Truskett, T. M.; and
Debenedetti, P. G. "Is Random Close Packing of Spheres Well Defined?"
Phys. Lev. Lett.84, 2064-2067, 2000.van Dam, E.; den
Hertog, D.; Husslage, B.; and Rennen, G. "Maximin Designs (Dimensions: 3)."
Mar. 31, 2006. http://www.spacefillingdesigns.nl/.Weisstein,
E. W. "Books about Sphere Packings." http://www.ericweisstein.com/encyclopedias/books/SpherePackings.html.Wells,
D. The
Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England:
Penguin Books, p. 29, 1986.Wells, D. The
Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
237-238, 1991.Zong, C. and Talbot, J. Sphere
Packings. New York: Springer-Verlag, 1999.
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
satisfies
.
-dimensional hyperspheres) which can touch an equivalent sphere
(hypersphere) without intersections is called the 
-dimensional kissing number.
(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.
. This is the lattice formed by carbon
atoms in a diamond (Conway and Sloane 1993, p. 113).
,
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.