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

Boyd, David W.

doi:10.1090/S0025-5718-1973-0338937-6

An algorithm for generating the sphere coordinates in a three-dimensional osculatory packing
HTML articles powered by AMS MathViewer

by David W. Boyd PDF

Math. Comp. 27 (1973), 369-377 Request permission

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

David W. Boyd, On the exponent of an osculatory packing, Canadian J. Math. 23 (1971), 355–363. MR 271842, DOI 10.4153/CJM-1971-036-8
David W. Boyd, Improved bounds for the disk-packing constant, Aequationes Math. 9 (1973), 99–106. MR 317180, DOI 10.1007/BF01838194
David W. Boyd, The osculatory packing of a three dimensional sphere, Canadian J. Math. 25 (1973), 303–322. MR 320897, DOI 10.4153/CJM-1973-030-5
D. G. Larman, On the exponent of convergence of a packing of spheres, Mathematika 13 (1966), 57–59. MR 202054, DOI 10.1112/S0025579300004216
Z. A. Melzak, On the solid-packing constant for circles, Math. comp. 23 (1969), 169–172. MR 0244866, DOI 10.1090/S0025-5718-1969-0244866-8

Nature