The kissing problem asks how many spheres can be arranged tangent to a given sphere, if they all have the same size and their interiors cannot overlap. The maximum such number in n dimensions is called the n-dimensional kissing number. Equivalently, we can ask how many points can be arranged on the surface of a sphere such that no two distinct points form an angle of less than 60 degrees with the center of the sphere. See [4] for more information about the kissing problem.
This table shows the best upper and lower bounds known for the kissing numbers in dimensions up to 48 as well as 72. It was originally based on a table of lower bounds that was part of the Nebe-Sloane Catalogue of Lattices, and before that Tables I.2(a) and I.2(b) in [4]. Please let me know of any improvements that should be incorporated.
A plot of the data is available here, and the data is archived here, along with more information about many of the configurations.
Dimension | Lower bound | Upper bound | Ratio | References |
---|---|---|---|---|
1 | 2 | 1 | ||
2 | 6 | 1 | ||
3 | 12 | 1 | [20] | |
4 | 24 | 1 | [19, 16] | |
5 | 40 | 44 | 1.100 | [8, 15] |
6 | 72 | 77 | 1.070 | [8, 10] |
7 | 126 | 134 | 1.064 | [8, 15] |
8 | 240 | 1 | [8, 13, 18] | |
9 | 306 | 363 | 1.187 | [12, 14] |
10 | 510 | 553 | 1.085 | [6, 14] |
11 | 592 | 868 | 1.467 | [6, 9] |
12 | 840 | 1355 | 1.614 | [12, 9] |
13 | 1154 | 2064 | 1.789 | [21, 9] |
14 | 1932 | 3174 | 1.643 | [6, 9] |
15 | 2564 | 4853 | 1.893 | [12, 9] |
16 | 4320 | 7320 | 1.695 | [1, 9] |
17 | 5730 | 10978 | 1.916 | [3, 9] |
18 | 7654 | 16406 | 2.144 | [3, 9] |
19 | 11692 | 24417 | 2.089 | [3, 9] |
20 | 19448 | 36195 | 1.862 | [3, 9] |
21 | 29768 | 53524 | 1.799 | [3, 9] |
22 | 49896 | 80810 | 1.620 | [11, 9] |
23 | 93150 | 122351 | 1.314 | [11, 9] |
24 | 196560 | 1 | [11, 13, 18] | |
25 | 197048 | 265006 | 1.345 | [7, 9] |
26 | 198512 | 367775 | 1.853 | [7, 9] |
27 | 199976 | 522212 | 2.612 | [7, 9] |
28 | 204368 | 752292 | 3.682 | [7, 9] |
29 | 208272 | 1075991 | 5.167 | [7, 9] |
30 | 219984 | 1537707 | 6.991 | [7, 9] |
31 | 232874 | 2213487 | 9.506 | [7, 9] |
32 | 345408 | 3162316 | 9.156 | [2, 9] |
33 | 360640 | 4494570 | 12.47 | [2, 9] |
34 | 380868 | 6422593 | 16.87 | [2, 9] |
35 | 409548 | 9162403 | 22.38 | [2, 9] |
36 | 484568 | 13017098 | 26.87 | [2, 9] |
37 | 494312 | 18498316 | 37.43 | [2, 9] |
38 | 566652 | 26496684 | 46.77 | [2, 9] |
39 | 755988 | 37826766 | 50.04 | [2, 9] |
40 | 1064368 | 53589200 | 50.35 | [2, 9] |
41 | 1170384 | 76287040 | 65.19 | [2, 9] |
42 | 1250676 | 108404055 | 86.68 | [2, 9] |
43 | 1745692 | 153813582 | 88.12 | [2, 9] |
44 | 2948552 | 220788272 | 74.89 | [5, 9] |
45 | 3047160 | 316735249 | 104.0 | [2, 9] |
46 | 5318060 | 441900184 | 83.10 | [2, 9] |
47 | 9741412 | 621658419 | 63.82 | [2, 9] |
48 | 52416000 | 867897072 | 16.56 | [12, 9] |
72 | 6218175600 | 2545617287927 | 409.4 | [17, 9] |
References
- E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of Abelian groups, J. Austral. Math. Soc. 1 (1959), 47–63, doi:10.1017/S1446788700025064.
- A. E. Brouwer, Bounds for binary constant weight codes, https://www.win.tue.nl/~aeb/codes/Andw.html.
- H. Cohn and A. Li, Improved kissing numbers in seventeen through twenty-one dimensions, preprint, 2024, arXiv:2411.04916.
- J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, third edition, Grundlehren der Mathematischen Wissenschaften 290, Springer-Verlag, New York, 1999, doi:10.1007/978-1-4757-6568-7.
- Y. Edel, E. M. Rains, and N. J. A. Sloane, On kissing numbers in dimensions 32 to 128, Electron. J. Combin. 5 (1998), Research Paper 22, 5 pp., doi:10.37236/1360, arXiv:math/0207291.
- M. Ganzhinov, Highly symmetric lines, preprint, 2022, arXiv:2207.08266.
- K. Kallal, T. Kan, and E. Wang, Improved lower bounds for kissing numbers in dimensions 25 through 31, SIAM J. Discrete Math. 31 (2017), 1895–1908, doi:10.1137/16M1095810, arXiv:1608.07270.
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- H. D. Mittelmann and F. Vallentin, High-accuracy semidefinite programming bounds for kissing numbers, Experiment. Math. 19 (2010), 175–179, doi:10.1080/10586458.2010.10129070, arXiv:0902.1105.
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- L. Schläfli, Theorie der vielfachen Kontinuität [1852], Denkschriften der Schweizerischen naturforschenden Gesellschaft, vol. 38, J. H. Graf, Bern, 1901.
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