Kissing Number

The number of equivalent Hyperspheres in $n$-D 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 3-D was 12, but the first proofs were not produced until the 19th century (Conway and Sloane 1993, p. 21) by Bender (1874), Hoppe (1874), and Günther (1875). More concise proofs were published by Schütte and van der Waerden (1953) and Leech (1956). Exact values for lattice packings are known for $n=1$ to 9 and $n=24$ (Conway and Sloane 1992, Sloane and Nebe). Odlyzko and Sloane (1979) found the exact value for 24-D.

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 (Conway and Sloane 1993, p. 15). A more extensive and up-to-date tabulation is maintained by Sloane and Nebe.

$D$	$L$	NL	$D$	$L$	NL
1	2		13	$\geq 918$	$\geq 1,130$
2	6		14	$\geq 1,422$	$\geq 1,582$
3	12		15	$\geq 2,340$
4	24		16	$\geq 4,320$
5	40		17	$\geq 5,346$
6	72		18	$\geq 7,398$
7	126		19	$\geq 10,668$
8	240		20	$\geq 17,400$
9	272	$\geq 306$	21	$\geq 27,720$
10	$\geq 336$	$\geq 500$	22	$\geq 49,896$
11	$\geq 438$	$\geq 582$	23	$\geq 93,150$
12	$\geq 756$	$\geq 840$	24	196,560

The lattices having maximal packing numbers in 12- and 24-D have special names: the Coxeter-Todd Lattice and Leech Lattice, respectively. The general form of the lower bound of $n$-D lattice densities given by

$$\eta\geq{\zeta(n)\over 2^{n-1}},$$

where $\zeta(n)$ is the Riemann Zeta Function, is known as the Minkowski-Hlawka Theorem.

See also Coxeter-Todd Lattice, Hermite Constants, Hypersphere Packing, Leech Lattice, Minkowski-Hlawka Theorem

References

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.

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.'' Electronic J. Combinatorics 5, No. 1, R22, 1-5, 1998. http://www.combinatorics.org/Volume_5/v5i1toc.html.

Günther, S. ``Ein stereometrisches Problem.'' Archiv Math. Physik 57, 209-215, 1875.

Hoppe, R. ``Bemerkung der Redaction.'' Archiv Math. Physik. (Grunert) 56, 307-312, 1874.

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.

Odlyzko, A. M. and Sloane, N. J. A. ``New Bounds on the Number of Unit Spheres that Can Touch a Unit Sphere in $n$ Dimensions.'' J. Combin. Th. A 26, 210-214, 1979.

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 ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Sloane, N. J. A. and Nebe, G. ``Table of Highest Kissing Numbers Presently Known.'' http://www.research.att.com/~njas/lattices/kiss.html.

Stewart, I. The Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press, pp. 82-84, 1987.