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

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

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]**D. W. Boyd, "On the exponent of an osculatory packing,"*Canad. J. Math.*, v. 23, 1971, pp. 355-363. MR**42**#6723. MR**0271842 (42:6723)****[2]**D. W. Boyd, "Improved bounds for the disk-packing constant,"*Aequationes Math.*(To appear.) MR**0317180 (47:5728)****[3]**D. W. Boyd, "The osculatory packing of a three dimensional sphere,"*Canad. J. Math.*(To appear.) MR**0320897 (47:9430)****[4]**D. G. Larman, "On the exponent of convergence of a packing of spheres,"*Mathematika*, v. 13, 1966, pp. 57-59. MR**34**#1928. MR**0202054 (34:1928)****[5]**Z. A. Melzak, "On the solid-packing constant for circles,"*Math. Comp.*, v. 23, 1969, pp. 169-172. MR**39**#6179. MR**0244866 (39:6179)****[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