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).
Bender, C. "Bestimmung der grössten Anzahl gleich Kugeln, welche sich auf eine Kugel von demselben Radius, wie die übrigen, auflegen
lassen." Archiv Math. Physik (Grunert)56, 302-306, 1874.Cohn,
H. "Kissing Numbers." https://cohn.mit.edu/kissing-numbers/.Conway,
J. H. and Sloane, N. J. A. "The Kissing Number Problem"
and "Bounds on Kissing Numbers." §1.2 and Ch. 13 in Sphere
Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 21-24
and 337-339, 1993.Edel, Y.; Rains, E. M.; Sloane, N. J. A.
"On Kissing Numbers in Dimensions 32 to 128." Elec. J. Combin.5,
No. 1, R22, 1-5, 1998. https://doi.org/10.37236/1360.Günther,
S. "Ein stereometrisches Problem." Archiv Math. Physik57,
209-215, 1875.Hoppe, R. "Bemerkung der Redaction." Archiv
Math. Physik. (Grunert)56, 307-312, 1874.Kabatyanskii, G. A.
and Levenshtein, V. I. "Bounds for Packing on a Sphere and in Space."
Problems Inform. Transm.14, 1-17, 1978.Kuperberg, G.
"Average Kissing Numbers for Sphere Packings." Preprint.Kuperberg,
G. and Schramm, O. "Average Kissing Numbers for Non-Congruent Sphere Packings."
Math. Res. Let.1, 339-344, 1994.Leech, J. "The Problem
of Thirteen Spheres." Math. Gaz.40, 22-23, 1956.Musin,
O. R. "The Kissing Number in Four Dimensions." Ann. Math.168,
1-32, 2008. https://doi.org/10.4007/annals.2008.168.1.Odlyzko,
A. M. and Sloane, N. J. A. "New Bounds on the Number of Unit
Spheres That Can Touch a Unit Sphere in Dimensions." J. Combin. Th. A26, 210-214,
1979.OpenAI. "Improved Bounds for Binary and Spherical Codes."
Ch. 2 in Ten Advances in Mathematics and Theoretical Computer Science.
Aug. 1, 2026. https://cdn.openai.com/pdf/ten-proofs-oai.pdf.Pfender,
F. and Ziegler, G. "Kissing Numbers, Sphere Packings, and Some Unexpected Proofs."
Not. Amer. Math. Soc.51, 873-883, 2004.Schütte,
K. and van der Waerden, B. L. "Das Problem der dreizehn Kugeln." Math.
Ann.125, 325-334, 1953.Sloane, N. J. A. Sequence
A001116/M1585 in "The On-Line Encyclopedia
of Integer Sequences."Stewart, I. The
Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press,
pp. 82-84, 1987.Wells, D. The
Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England:
Penguin Books, p. 84, 1986.Zinov'ev, V. A. and Ericson, T.
"New Lower Bounds for Contact Numbers in Small Dimensions." Prob. Inform.
Transm.35, 287-294, 1999.Zong, C. and Talbot, J. Sphere
Packings. New York: Springer-Verlag, 1999.
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,
was published by Schütte and van der Waerden (1953),
followed by a shorter proof by Leech (1956).
. This
smaller sphere touches the central sphere
and four surrounding spheres. Since 
, it does not provide a 13th unit
sphere.
, 
, 
, 
, 
, 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.
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.
,
an AI-generated proof given by OpenAI (2026) established
is little-o notation. This improves the optimized
classical exponent 
of Kabatyanskii and Levenshtein (1978).
Dimensions." J. Combin. Th. A 26, 210-214,
1979.OpenAI. "Improved Bounds for Binary and Spherical Codes."
Ch. 2 in Ten Advances in Mathematics and Theoretical Computer Science.
Aug. 1, 2026.