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Kissing Number


KissingNumber12

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 (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). 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 (1979) 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 (Pfender and Ziegler 2004).

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 (Conway and Sloane 1993, p. 15). 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 (1999).

DLNLDLNL
1213>=918>=1154
2614>=1422>=1606
31215>=2340
42416>=4320
54017>=5346
67218>=7398
712619>=10668
824020>=17400
9272>=30621>=27720
10>=336>=50022>=49896
11>=438>=58223>=93150
12>=756>=84024196560

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

 eta>=(zeta(n))/(2^(n-1)),

where zeta(n) is the Riemann zeta function, is known as the Minkowski-Hlawka theorem.


See also

Coxeter-Todd Lattice, Dodecahedral Conjecture, Hermite Constants, Hypersphere Packing, Kepler Conjecture, Leech Lattice, Minkowski-Hlawka Theorem, Sphere Packing

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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/Abstracts/v5i1r22.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.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."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.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.

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Kissing Number

Cite this as:

Weisstein, Eric W. "Kissing Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KissingNumber.html

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