Kissing number problem

Encyclopedia : K : KI : KIS : Kissing number problem

In geometry, the kissing number is the maximum number of spheres of radius 1 that can simultaneously touch the unit sphere in n-dimensional Euclidean space. The kissing number problem seeks the kissing number as a function of n.

Known kissing numbers

In one dimension, the kissing number is obviously 2:

It is easy to see (and to prove) that in two dimensions the kissing number is 6.

In three dimensions the answer is not so clear. It is easy to arrange 12 spheres so that each touches a central sphere, but there is a lot of space left over, and it is not obvious that there is no way to pack in a 13th sphere. (In fact, there is so much extra space that any two of the 12 outer spheres can exchange places without any of the outer spheres losing contact with the center one.) This was the subject of a famous disagreement between mathematicians Isaac Newton and David Gregory. Newton thought that the limit was 12, and Gregory that a 13th could fit. The question was not resolved until 1874; Newton was correct.

In four dimensions, it has been 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.Florian Pfender, Gunter M. Ziegler [Kissing numbers, sphere packings, and some unexpected proofs]

The kissing number in n dimensions is unknown for n > 4, except for n = 8 (240), and n = 24 (196,560).Levenshtein, V. I. Boundaries for packings in n-dimensional Euclidean space. (Russian) Dokl. Akad. Nauk SSSR 245 (1979), no. 6, 1299—1303 Odlyzko, A. M., Sloane, N. J. A. New bounds on the number of unit spheres that can touch a unit sphere in n dimensions. J. Combin. Theory Ser. A 26 (1979), no. 2, 210—214 The results in these dimensions stems from the existence of highly symmetrical lattices: the E₈ lattice and the Leech lattice. In fact, the only way to arrange spheres in these dimensions with the above kissing numbers is to center them at the minimal vectors in these lattices. There is no space whatsoever for any additional balls.

Rough volume estimates show that kissing number in n dimensions grows exponentially. The base of exponential growth is not known.

Some known bounds

The following table lists some known bounds on the kissing number is various dimensions. The dimensions in which the kissing number is known are listed in boldface.

Dimension	Lower bound	Upper bound
1	2
2	6
3	12
4	24
5	40	46
6	72	82
7	126	140
8	240
9	306	380
10	500	595
11	582	915
12	840	1,416
13	1,130	2,233
14	1,582	3,492
15	2,564	5,431
16	4,320	8,313
17	5,346	12,215
18	7,398	17,877
19	10,688	25,901
20	17,400	37,974
21	27,720	56,852
22	49,896	86,537
23	93,150	128,096
24	196,560

See also

• Sphere packing

References