An algorithm for generating the sphere coordinates in a three-dimensional osculatory packing

Author:
David W. Boyd

Journal:
Math. Comp. **27** (1973), 369-377

MSC:
Primary 52A45; Secondary 52-04

MathSciNet review:
0338937

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Abstract: This paper develops an efficient algorithm which generates the pentaspherical coordinates of the spheres in an osculatory packing of the three-dimensional unit sphere. The algorithm has a tree-like structure and is easily modified so that, given a prescribed bound, it counts the number of spheres in the packing whose curvatures are less than this bound. The algorithm has been used to produce heuristic estimates of the exponent *M* of the packing, and these indicate that *M* is approximately 2.42.

**[1]**David W. Boyd,*On the exponent of an osculatory packing*, Canad. J. Math.**23**(1971), 355–363. MR**0271842****[2]**David W. Boyd,*Improved bounds for the disk-packing constant*, Aequationes Math.**9**(1973), 99–106. MR**0317180****[3]**David W. Boyd,*The osculatory packing of a three dimensional sphere*, Canad. J. Math.**25**(1973), 303–322. MR**0320897****[4]**D. G. Larman,*On the exponent of convergence of a packing of spheres*, Mathematika**13**(1966), 57–59. MR**0202054****[5]**Z. A. Melzak,*On the solid-packing constant for circles*, Math. comp.**23**(1969), 169–172. MR**0244866**, 10.1090/S0025-5718-1969-0244866-8**[6]**F. Soddy, "The bowl of integers and the hexlet,"*Nature*, v. 139, 1937, pp. 77-79.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1973-0338937-6

Keywords:
Sphere packing,
osculatory packing,
exponent of packing,
algorithm,
computer study

Article copyright:
© Copyright 1973
American Mathematical Society