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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.

References [Enhancements On Off] (What's this?)

  • [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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Keywords: Sphere packing, osculatory packing, exponent of packing, algorithm, computer study
Article copyright: © Copyright 1973 American Mathematical Society

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