A257479 Maximal kissing number in n dimensions: maximal number of unit spheres that can touch another unit sphere.

2, 6, 12, 24

Offset: 1

Author

Peter Woodward, Apr 25 2015

Keywords

• nonn
• bref

Comments

Two additional terms are known: a(8) = 240, a(24) = 196560 [Odlyzko and Sloane; Levenshtein].

Lower bounds for a(5) onwards are 40, 72, 126, 240 (exact), 306, 500, ... - N. J. A. Sloane, May 15 2015

It seems that, while n is even, a lower bound for a(n) can be written as f(n) = (2n + 2^n) - k(n)^2, where k(2) = 2, for n > 2, k(n) = 2^(n/2) - q, q = {2^t, 3*2^t}, t is an integer, t > 0, 2 <= q <= n, and f(n) <= a(n) (Note: for n <= 24, q = n at n = {4, 6, 8, 24}, q = 0 at n = 2). - Sergey Pavlov, Mar 17 2017

It also seems that, while n is even, an upper bound for a(n) can be written as f(n) = (2n + 2^n) - k(n)^2, where k(2) = 0, for n > 2, k(n) = 2^(n/2) - q, q = {2^t, 3*2^t}, t is an integer, t > 0, n <= q <= 2n, and f(n) >= a(n) (Note: for n <= 24, q = n at n = {2, 4, 12, 16}). - Sergey Pavlov, Mar 19 2017

Examples

			For a(2), the maximal number of pennies that can touch one penny is 6.
For a(3), the most spheres that can simultaneously touch a central sphere of the same radius is 12.
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., Chap. 3, esp. pp. xxi, 23, etc.
• Musin, Oleg Rustumovich. "The problem of the twenty-five spheres." Russian Mathematical Surveys 58.4 (2003): 794-795.

Links

• Eiichi Bannai and N. J. A. Sloane, Uniqueness of certain spherical codes, Canad. J. Math. 33 (1981), no. 2, 437-449.
• J. Leech, The problem of the thirteen spheres, Math. Gaz., 40 (1956), 22-23.
• V. I. Levenshtein, On bounds for packings in n-dimensional Euclidean space, Dokl. Akad. Nauk., 245 (1979), 1299-1303; English translation in Soviet Math. Doklady, 20 (1979), 417-421.
• Hans D. Mittelmann and Frank Vallentin, High accuracy semidefinite programming bounds for kissing numbers, arXiv:0902.1105 [math.OC], 2009; Exp. Math. (2009), no. 19, 174-178.
• G. Nebe and N. J. A. Sloane, Table of highest kissing numbers known
• 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), no. 2, 210-214.
• N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.
• Wikipedia, Kissing number problem

Crossrefs

Cf. A001116 (n-dimensional lattice), A002336 (n-dimensional laminated lattice), A028923 (n-dimensional lattice Kappa_n).

Cf. A008408.

Extensions

Entry revised by N. J. A. Sloane, May 08 2015