Kissing Number

The kissing number is the largest number of congruent nonoverlapping hyperspheres in -dimensional Euclidean space that can touch a congruent central hypersphere. It is also called the Newton number, contact number, coordination number, or ligancy. Equivalently, the points of tangency form a spherical code on with pairwise angles of at least . In the standard notation for spherical codes,

The three-dimensional problem was discussed by Newton and David Gregory in 1694. Newton asserted that 12 spheres could touch the central sphere, while Gregory considered whether 13 might be possible. A configuration of 12 is obtained by placing the points of tangency at the polyhedron vertices of a regular icosahedron, but enough space remains to make the impossibility of a 13th sphere far from visually obvious. The first generally accepted proof that was published by Schütte and van der Waerden (1953), followed by a shorter proof by Leech (1956).

The asymmetric kissing configuration shown above creates a larger hole containing a smaller sphere of radius . This smaller sphere touches the central sphere and four surrounding spheres. Since , it does not provide a 13th unit sphere.

The exact values currently known are , , , , , and . Musin (2008) proved using a modification of Delsarte's method. The values for and 24 are realized by the root lattice and the Leech lattice, respectively (Odlyzko and Sloane 1979). For every other , the exact value remains unknown.

The following table gives the best general lower bounds and upper bounds known for as of August 2026; equal entries are exact (Cohn). A continuously maintained table, including larger values of and references for each bound, is provided by Cohn.

1	2	2	13
2	6	6	14
3	12	12	15
4	24	24	16
5	40	44	17
6	72	77	18
7	126	134	19
8	240	240	20
9	306	363	21
10	510	553	22
11	593	868	23
12	840		24

As , an AI-generated proof given by OpenAI (2026) established

where is little-o notation. This improves the optimized classical exponent of Kabatyanskii and Levenshtein (1978).

Ho (2026) constructed 11948 kissing spheres in dimension 19 using a 1280-word subcode of a punctured binary Golay code. Kolosov (2026) proved that the 1280-word added-vector code in Ho's 19-dimensional construction is optimal among subsets of the specified 4096-word binary code with minimum Hamming distance at least 5. The upper bound partitions the code into 256 classes whose forbidden-distance graphs are Clebsch graphs, each with independence number 5. This establishes optimality within that code and does not give an upper bound of 11948 for the unrestricted 19-dimensional kissing number. The proof was co-developed with AI and its exact certificate was replayed by VibeMathed (2026), but independent specialist review had not been reported as of Sep. 7, 2026.