Kissing Number
The number of equivalent hyperspheres in n dimensions which can touch an equivalent hypersphere without any intersections, also sometimes called the Newton number, contact number, coordination number, or ligancy. Newton correctly believed that the kissing number in three dimensions was 12, but the first proofs were not produced until the 19th century by Bender, Hoppe, and Günther. More concise proofs were published by Schütte and van der Waerden and Leech. After packing 12 spheres around the central one (which can be done, for example, by arranging the spheres so that their points of tangency with the central sphere correspond to the vertices of an icosahedron), there is a significant amount of free space left (above figure), although not enough to fit a 13th sphere. Exact values for lattice packings are known for n = 1 to 9 and n = 24 (Conway and Sloane 1993, Sloane and Nebe). Odlyzko and Sloane found the exact value for 24-D. Exact values for general packings are known for n = 1, 2, 3, 4, 8, and 24. Musin developed a bounding method in 2003 to prove the 24-dimensional case, and his method also provides proofs for three and four dimensions. The arrangement of n points on the surface of a sphere, corresponding to the placement of n identical spheres around a central sphere (not necessarily of the same radius) is called a spherical code. The following table gives the largest known kissing numbers in dimension D for lattice (L) and nonlattice (NL) packings (if a nonlattice packing with higher number exists). In nonlattice packings, the kissing number may vary from sphere to sphere, so the largest value is given below. A more extensive and up-to-date tabulation is maintained by Sloane and Nebe. Here, the nonlattice bounds for D = 13 and 14 were proved by Zinov'ev and Ericson. D | L | NL | D | L | NL 1 | 2 | | 13 | >=918 | >=1154 2 | 6 | | 14 | >=1422 | >=1606 3 | 12 | | 15 | >=2340 | 4 | 24 | | 16 | >=4320 | 5 | 40 | | 17 | >=5346 | 6 | 72 | | 18 | >=7398 | 7 | 126 | | 19 | >=10668 | 8 | 240 | | 20 | >=17400 | 9 | 272 | >=306 | 21 | >=27720 | 10 | >=336 | >=500 | 22 | >=49896 | 11 | >=438 | >=582 | 23 | >=93150 | 12 | >=756 | >=840 | 24 | 196560 | The lattices having maximal packing numbers in 12 and 24 dimensions have special names: the Coxeter-Todd lattice and Leech lattice, respectively. The general form of the lower bound of n-dimensional lattice densities given by η>=(ζ(n))/2^(n - 1), where ζ(n) is the Riemann zeta function, is known as the Minkowski-Hlawka theorem.
