Kissing number problem
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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.
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[edit] 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.[1]
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.[2]
The kissing number in n dimensions is unknown for n > 4, except for n = 8 (240), and n = 24 (196,560).[3][4] The results in these dimensions stem 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.
[edit] 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.
| 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 | |
[edit] See also
[edit] References
- ^ Conway, John H.; Neil J.A. Sloane (1999). Sphere Packings, Lattices and Groups, 3rd ed., New York: Springer-Verlag. ISBN 0-387-98585-9.
- ^ Florian Pfender, Gunter M. Ziegler Kissing numbers, sphere packings, and some unexpected proofs
- ^ 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
- T. Aste and D. Weaire "The Pursuit of Perfect Packing" (Institute Of Physics Publishing London 2000) ISBN 0-7503-0648-3
- Eric W. Weisstein, Kissing Number at MathWorld.
- Table of the Highest Kissing Numbers Presently Known maintained by Gabriele Nebe and Neil Sloane
