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186 geometry/kissing.number.p




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This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

186 geometry/kissing.number.p


How many n-dimensional unit spheres can be packed around one unit sphere?

geometry/kissing.number.s

From the Feb. 1992 issue of Scientific American:

Kissing Numbers

Dimension    Lower limit   Upper limit
   1*            2             2
   2*            6             6
   3*           12            12
   4            24            25
   5            40            46
   6            72            82
   7           126           140
   8*          240           240
   9           306           380
  10           500           595
  11           582           915
  12           840         1,416
  13         1,130         2,233
  14         1,582         3,492
  15         2,564         5,431
  16         4,320         8,313
  17         5,346        12,215
  18         7,398        17,877
  19        10,688        25,901
  20        17,400        37,974
  21        27,720        56,852
  22        49,896        86,537
  23        93,150       128,096
  24*      196,560       196,560

* = dimensions for which the answer is known.

REFERENCES (from the Sci. Am. article)

The Problem of the Thirteen Spheres. John Leech in Mathematical Gazette,
Vol. 40, No. 331, pages 22-23; February 1956
Sphere Packings, Lattics and Groups. John Horton Conway and Neil J. A.
Sloane. Springer-Verlag, 1988.
Sphere Packings and Spherical Geometry--Kepler's Conjecture and Beyond,
preprint. Wu-Yi Hsiang. Center for Pure and Applied Mathematics,
University of California, Berkeley, July 1991.
--
David Radcliffe
radcliff@csd4.csd.uwm.edu

 

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