The number of equivalent hyperspheres in 
dimensions which can touch an equivalent hypersphere
without any intersections, also sometimes called the Newton number, contact number,
coordination number, or ligancy. Newton correctly believed that the kissing number
in three dimensions was 12, but the first proofs were not produced until the 19th
century (Conway and Sloane 1993, p. 21) by Bender (1874), Hoppe (1874), and
Günther (1875). More concise proofs were published by Schütte and van der
Waerden (1953) and Leech (1956). After packing 12 spheres around the central one
(which can be done, for example, by arranging the spheres so that their points of
tangency with the central sphere correspond to the vertices of an icosahedron),
there is a significant amount of free space left (above figure), although not enough
to fit a 13th sphere.
Exact values for lattice packings are known for 
to 9 and 
(Conway and Sloane 1993, Sloane and Nebe). Odlyzko and
Sloane (1979) found the exact value for 24-D.
Exact values for general packings are known for 
, 2, 3, 4, 8, and 24. Musin developed a bounding method in
2003 to prove the 24-dimensional case, and his method also provides proofs for three
and four dimensions (Pfender and Ziegler 2004).
The arrangement of 
points on the surface of a sphere, corresponding to the placement of 
identical spheres around a central sphere (not necessarily
of the same radius) is called a spherical code.
The following table gives the largest known kissing numbers in dimension 
for lattice (
) and nonlattice (
) packings (if a nonlattice packing with higher number exists).
In nonlattice packings, the kissing number may vary from sphere to sphere, so the
largest value is given below (Conway and Sloane 1993, p. 15). A more extensive
and up-to-date tabulation is maintained by Sloane and Nebe. Here, the nonlattice
bounds for 
and 14 were proved by Zinov'ev and Ericson (1999).
 |  |  |  |  |  |
| 1 | 2 | 13 |  |  | |
| 2 | 6 | 14 |  |  | |
| 3 | 12 | 15 |  | ||
| 4 | 24 | 16 |  | ||
| 5 | 40 | 17 |  | ||
| 6 | 72 | 18 |  | ||
| 7 | 126 | 19 |  | ||
| 8 | 240 | 20 |  | ||
| 9 | 272 |  | 21 |  | |
| 10 |  |  | 22 |  | |
| 11 |  |  | 23 |  | |
| 12 |  |  | 24 |  |
The lattices having maximal packing numbers in 12 and 24 dimensions have special names: the Coxeter-Todd lattice and Leech
lattice, respectively. The general form of the lower bound of 
-dimensional lattice densities given by
 |
where 
is the Riemann zeta function, is known as
the Minkowski-Hlawka theorem.
Dimensions." J. Combin. Th. A 26, 210-214,
1979.Pfender, F. and Ziegler, G. "Kissing Numbers, Sphere Packings,
and Some Unexpected Proofs." Not. Amer. Math. Soc. 51, 873-883,
2004.Schütte, K. and van der Waerden, B. L. "Das Problem
der dreizehn Kugeln." Math. Ann. 125, 325-334, 1953.Sloane,
N. J. A. Sequence