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 through a continuous movement 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. [cite book |first=John H. |last=Conway |authorlink=John Horton Conway |coauthors=Neil J.A. Sloane |year=1999 |title=Sphere Packings, Lattices and Groups |edition=3rd ed. |publisher=Springer-Verlag |location=New York |id=ISBN 0-387-98585-9] 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. [citation|last1=Pfender|first1=Florian|last2=Ziegler|first2=Günter M.|authorlink2=Günter M. Ziegler|title=Kissing numbers, sphere packings, and some unexpected proofs|journal=Notices of the American Mathematical Society|date=September 2004|pages=873–883|url=http://www.ams.org/notices/200408/fea-pfender.pdf.]

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 stem from the existence of highly symmetrical lattices: the "E"₈ lattice and the Leech lattice.

ome 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.

ee also

*Sphere packing

Notes

References

* T. Aste and D. Weaire "The Pursuit of Perfect Packing" (Institute Of Physics Publishing London 2000) ISBN 0-7503-0648-3
*MathWorld | urlname=KissingNumber |title=Kissing Number
* [http://www.research.att.com/~njas/lattices/kiss.html Table of the Highest Kissing Numbers Presently Known] maintained by Gabriele Nebe and Neil Sloane (lower bounds)
* Christine Bachoc and Frank Vallentin. " [http://arxiv.org/abs/math.MG/0608426 New upper bounds for kissing numbers from semidefinite programming] ".