Kissing number
In geometry, a kissing number is defined as the number of non-overlapping 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.
Known greatest kissing numbers
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.
In three dimensions the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two. 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 correctly thought that the limit was 12; Gregory thought that a 13th could fit. Some incomplete proofs that Newton was correct were offered in the nineteenth century, but the first correct proof did not appear until 1953.[1][2]
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.[3][4]
The kissing number in n dimensions is unknown for n > 4, except for n = 8 (240), and n = 24 (196,560).[5][6] 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 this restricted kissing number is known for n = 1 to 9 and n = 24 dimensions.[7] 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.[8] 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[3] | |
| 5 | 40 | 44 |
| 6 | 72 | 78 |
| 7 | 126 | 134 |
| 8 | 240 | |
| 9 | 306 | 364 |
| 10 | 500 | 554 |
| 11 | 582 | 870 |
| 12 | 840 | 1,357 |
| 13 | 1,154[9] | 2,069 |
| 14 | 1,606[9] | 3,183 |
| 15 | 2,564 | 4,866 |
| 16 | 4,320 | 7,355 |
| 17 | 5,346 | 11,072 |
| 18 | 7,398 | 16,572 |
| 19 | 10,688 | 24,812 |
| 20 | 17,400 | 36,764 |
| 21 | 27,720 | 54,584 |
| 22 | 49,896 | 82,340 |
| 23 | 93,150 | 124,416 |
| 24 | 196,560 | |
See also
Notes
- ^ Conway, John H. (1999). Sphere Packings, Lattices and Groups (3rd ed.). New York: Springer-Verlag. p. 21. ISBN 0-387-98585-9.
{{cite book}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Brass, Peter; Moser, W. O. J.; Pach, János (2005). Research problems in discrete geometry. Springer. p. 93. ISBN 978-0-387-23815-9.
- ^ a b O. R. Musin (2003). "The problem of the twenty-five spheres". Russ. Math. Surv. 58 (4): 794–795. doi:10.1070/RM2003v058n04ABEH000651.
- ^ Pfender, Florian; Ziegler, Günter M. (September 2004). "Kissing numbers, sphere packings, and some unexpected proofs" (PDF). Notices of the American Mathematical Society: 873–883..
- ^ Levenshtein, Vladimir I. (1979). "О границах для упаковок в n-мерном евклидовом пространстве". Doklady Akademii Nauk SSSR (in Russian). 245 (6): 1299–1303.
{{cite journal}}: Unknown parameter|trans_title=ignored (|trans-title=suggested) (help) - ^ 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
- ^ Weisstein, Eric W. "Kissing Number". MathWorld.
- ^ Mittelmann, Hans D.; Vallentin, Frank (2009). "High accuracy semidefinite programming bounds for kissing numbers". Experimental Mathematics. 19: 174–178. arXiv:0902.1105.
- ^ a b В. А. Зиновьев, Т. Эриксон (1999). "Новые нижние оценки на контактное число для небольших размерностей". Пробл. Передачи Информ. (in Russian). 35 (4): 3–11. English translation: V. A. Zinov'ev, T. Ericson (1999). "New Lower Bounds for Contact Numbers in Small Dimensions". Problems of Information Transmission. 35 (4): 287–294. MR 1737742.
References
- T. Aste and D. Weaire "The Pursuit of Perfect Packing" (Institute Of Physics Publishing London 2000) ISBN 0-7503-0648-3
- Table of the Highest Kissing Numbers Presently Known maintained by Gabriele Nebe and Neil Sloane (lower bounds)
- Christine Bachoc and Frank Vallentin. "New upper bounds for kissing numbers from semidefinite programming".