Hypersphere Packing
In two dimensions, there are two periodic circle packings for identical circles: square lattice and hexagonal lattice. In 1940, Fejes Tóth proved that the hexagonal lattice is the densest of all possible plane packings (Conway and Sloane 1993, pp. 8-9).
The analog of face-centered cubic packing is the densest lattice packing in four and five dimensions. In eight dimensions, the densest lattice packing is made up
of two copies of face-centered cubic. In six and seven dimensions, the densest lattice
packings are cross sections of the eight-dimensional
case. In 24 dimensions, the densest packing appears to be the Leech
lattice. For high dimensions (
-D), the
densest known packings are nonlattice.
The densest lattice packings of hyperspheres in 
dimensions are
known rigorously for 
, 2, ..., 8,
and have packing densities 
summarized
in the following table, which also gives the corresponding Hermite
constants 
(Gruber and Lekkerkerker 1987,
p. 518; Hilbert and Cohn-Vossen 1999, p. 47; Finch) and relevant literature
citations.
 |  |  | reference |
| 2 |  |  | Kepler 1611, 1619; Lagrange 1773 |
| 3 |  | 2 | Kepler 1611, 1619; Gauss 1840 |
| 4 |  | 4 | Korkin and Zolotarev 1877 |
| 5 |  | 8 | Korkin and Zolotarev 1877 |
| 6 |  |  | Blichfeldt 1934, Barnes 1957, Vetčinkin 1980 |
| 7 |  | 64 | Blichfeldt 1934, Watson 1966, Vetčinkin 1980 |
| 8 |  | 256 | Blichfeldt 1934, Watson 1966, Vetčinkin 1980 |
The packing densities 
of the densest
known non-lattice packings of hyperspheres in dimensions up to 10 are given
by Conway and Sloane (1995), However, there are no proofs that any packing in dimensions
greater than 3 is optimal (Sloane 1998).
The largest number of unit circles which can touch a given unit circle is six. For spheres,
the maximum number is 12. Newton considered this question long before a proof was
published in 1874. The maximum number of hyperspheres that can touch another in 
dimensions is the so-called kissing
number.
The following example illustrates the sometimes counterintuitive properties of hypersphere packings. Draw unit 
-spheres in an 
-dimensional space centered at all 
coordinates.
Now place an additional hypersphere at the origin
tangent to the other hyperspheres. For values of
between 2 and 8, the central hypersphere
is contained inside the hypercube with polytope
vertices at the centers of the other spheres. However, for 
, the central
hypersphere just touches the hypercube
of centers, and for 
, the central
hypersphere is partially outside the hypercube.
This fact can be demonstrated by finding the distance from the origin to the center of one of the 
hyperspheres,
which is given by
 |
The radius of the central sphere is therefore 
. Now,
the distance from the origin to the center of a facet bounding
the hypercube is always 1 (one hypersphere radius),
so the center hypersphere is tangent to the hypercube
when 
, or 
, and partially
outside it for 
.
Barnes, E. S. "The Complete Enumeration of Extreme Senary Forms." Philos. Trans. Roy. Soc. London A 249, 461-506, 1957.
Blichfeldt, H. F. "The Minimum Value of Positive Quadratic Forms in Six, Seven, and Eight Variables." Math. Z. 39, 1-15, 1934.
Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, 1993.
Conway, J. H. and Sloane, N. J. A. "What Are All the Best Sphere Packings in Low Dimensions?" Disc. Comput. Geom. 13, 383-403, 1995.
Finch, S. R. "Hermite's Constants." §2.7 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 506-508, 2003.
Gardner, M. Martin Gardner's New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 89-90, 1966.
Gauss, C. F. "Recursion der 'Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber, Dr. der Philosophie, ordentl. Professor an der Universität in Freiburg, 1831, 248 S. in 4." J. reine angew. Math. 20, 312-320, 1840.
Gruber, P. M. and Lekkerkerker, C. G. Geometry of Numbers. Amsterdam, Netherlands: North-Holland, 1987.
Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 47, 1999.
Kepler, J. "Strena seu de nive sexangula." Frankfurt, Germany: Tampach, 1611. Reprinted in Gesammelte Werke, Vol. 4 (Ed. M. Caspar and F. Hammer). Oxford, England: Clarendon Press, 1966.
Kepler, J. Harmonice Mundi, Libri V. Linz, Austria: Kugellagerungen, 1619. Reprinted in Gesammelte Werke, Vol. 6 (Ed. M. Caspar). Munich, Germany: Weltharmonik, 1939.
Korkin, A. and Zolotarev, G. "Sur les formes quadratiques positives." Math. Ann. 11, 242-292, 1877.
Lagrange, J.-L. "Recherches d'arithmétique." Nouv. Mém. Acad. Roy. Soc. Belles Lettres (Berlin), pp. 265-312, 1773. Reprinted in Oeuvres, Vol. 3, pp. 693-758.
Schnell, U. and Wills, J. M. "Densest Packings of More than Three 
-Spheres are Nonplanar." Disc. Comput. Geom. 24,
539-549, 2000.
Sloane, N. J. A. "Kepler's Conjecture Confirmed." Nature 395, 435-436, 1998.
Vetčinkin, N. M. "Uniqueness of Classes of Positive Quadratic Forms on Which Values of the Hermite Constants are Attained for 
."
Trudy Mat. Inst. Steklov 148, 65-76, 1978. English translation in Proc.
Steklov Inst. Math. 148, 63-74, 1980.
Watson, G. L. "On the Minimum of a Positive Quadratic Form in 
(
) Variables.
Verification of Blichfeldt's Calculation." Proc. Cambridge Philos. Soc. 62,
719, 1966.
Weisstein, Eric W. "Hypersphere Packing." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HyperspherePacking.html
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Keller's conjecture
