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Kissing numbers

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

  1. 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.
  2. A. E. Brouwer, Bounds for binary constant weight codes, https://www.win.tue.nl/~aeb/codes/Andw.html.
  3. H. Cohn and A. Li, Improved kissing numbers in seventeen through twenty-one dimensions, preprint, 2024, arXiv:2411.04916.
  4. 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.
  5. 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.
  6. M. Ganzhinov, Highly symmetric lines, preprint, 2022, arXiv:2207.08266.
  7. 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.
  8. A. Korkine and G. Zolotareff, Sur les formes quadratiques, Math. Ann. 6 (1873), 366–389, doi:10.1007/BF01442795.
  9. D. de Laat and N. Leijenhorst, Solving clustered low-rank semidefinite programs arising from polynomial optimization, preprint, 2022, arXiv:2202.12077.
  10. D. de Laat, N. Leijenhorst, and W. H. H. de Muinck Keizer, Optimality and uniqueness of the D4 root system, preprint, 2024, arXiv:2404.18794.
  11. J. Leech, Notes on sphere packings, Canadian J. Math. 19 (1967), 251–267, doi:10.4153/CJM-1967-017-0.
  12. J. Leech and N. J. A. Sloane, Sphere packings and error-correcting codes, Canadian J. Math. 23 (1971), 718–745, doi:10.4153/CJM-1971-081-3.
  13. V. I. Levenšteĭn, On bounds for packings in n-dimensional Euclidean space (Russian), Dokl. Akad. Nauk SSSR 245 (1979), 1299–1303; English translation in Soviet Math. Dokl. 20 (1979), 417–421.
  14. F. C. Machado and F. M. de Oliveira Filho, Improving the semidefinite programming bound for the kissing number by exploiting polynomial symmetry, Exp. Math. 27 (2018), 362–369, doi:10.1080/10586458.2017.1286273, arXiv:1609.05167.
  15. 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.
  16. O. R. Musin, The kissing number in four dimensions, Ann. of Math. (2) 168 (2008), 1–32, doi:10.4007/annals.2008.168.1, arXiv:math/0309430.
  17. G. Nebe, An even unimodular 72-dimensional lattice of minimum 8, J. Reine Angew. Math. 673 (2012), 237–247, doi:10.1515/crelle.2011.175, arXiv:1008.2862.
  18. A. M. Odlyzko and N. J. A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere in n dimensions, J. Combin. Theory Ser. A 26 (1979), 210–214, doi:10.1016/0097-3165(79)90074-8.
  19. L. Schläfli, Theorie der vielfachen Kontinuität [1852], Denkschriften der Schweizerischen naturforschenden Gesellschaft, vol. 38, J. H. Graf, Bern, 1901.
  20. K. Schütte and B. L. van der Waerden, Das Problem der dreizehn Kugeln, Math. Ann. 125 (1953), 325–334, doi:10.1007/BF01343127.
  21. V. A. Zinoviev and T. Ericson, New lower bounds for contact numbers in small dimensions (Russian), Problemy Peredachi Informatsii 35 (1999), 3–11; English translation in Problems Inform. Transmission 35 (1999), 287–294.