In geometry, a kissing number is defined as the number of unit spheres that touch another given unit sphere. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another. Other names for kissing number that have been used are Newton number (after the originator of the problem), and contact number.
The kissing number problem seeks the maximum possible kissing number for n-dimensional Euclidean space as a function of n.
In one dimension, the kissing number is 2:
It is easy to see (and to prove) that in two dimensions the kissing number is 6.
Kissing number problem in three dimensions (*)
In four dimensions, it was known for some time that the answer is either 24 or 25. It is easy to produce a packing of 24 spheres around a central sphere (one can place the spheres at the vertices of a suitably scaled 24-cell centered at the origin). As in the three-dimensional case, there is a lot of space left over—even more, in fact, than for n = 3—so the situation was even less clear. Finally, in 2003, Oleg Musin proved the kissing number for n = 4 to be 24, using a subtle trick.
The kissing number in n dimensions is unknown for n > 4, except for n = 8 (240), and n = 24 (196,560). The results in these dimensions stem from the existence of highly symmetrical lattices: the E8 lattice and the Leech lattice.
If arrangements are restricted to regular arrangements, in which the centres of the spheres all lie on points in a lattice, then the kissing number is known for n = 1 to 9 and n = 24 dimensions. For 5, 6 and 7 dimensions the arrangement with the highest known kissing number is the optimal lattice arrangement, but the existence of a non-lattice arrangement with a higher kissing number has not been excluded.
Some known bounds
The following table lists some known bounds on the kissing number in various dimensions. The dimensions in which the kissing number is known are listed in boldface.
Rough volume estimates show that kissing number in n dimensions grows exponentially in n. The base of exponential growth is not known. The grey area in the above plot represents the possible values between known upper and lower bounds. Circles represent values that are known exactly.
* Sphere packing
1. ^ Conway, John H.; Neil J.A. Sloane (1999). Sphere Packings, Lattices and Groups (3rd ed.). New York: Springer-Verlag. p. 21. ISBN 0-387-98585-9.
* T. Aste and D. Weaire "The Pursuit of Perfect Packing" (Institute Of Physics Publishing London 2000) ISBN 0-7503-0648-3