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Randomly Packed and Solidly Packed Spheres

Published online by Cambridge University Press:  20 November 2018

E. N. Gilbert
E. N. Gilbert*
Affiliation:
Bell Telephone Laboratories, Murray Hill, New Jersey
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In the classical packing problem unit spheres are placed without overlapping in D-dimensional space. When D = 2, the densest packing is a familiar regular arrangement of circles, each circle touching six others. In this packing, circles cover a fraction π/√12 = 0.9069 . . . of the area of the plane. The densest packing is not known for D ≥ 3.

Most of the packings to be considered here use spheres of many different sizes. In this way greater densities are obtainable; small spheres can fill up some of the space left over after large spheres have been packed.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 286 - 298
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Bernai, J. D., Geometry of the structure of monatomic liquids, Nature, 185 (1960), 68–70.Google Scholar
2. Blichfeldt, H. F., The minimum value of quadratic forms and the closest packing of spheres, Math. Ann., 101 (1929), 605–608.Google Scholar
3. Coxeter, H. S. M., Introduction to geometry (New York, 1961).Google Scholar
4. Fejes Tôth, L., Lagerungen in der Ebene, auf der Kugel und im Raum (Berlin, 1953).Google Scholar
5. Fejes, L. Tôth and J. Molnâr, Unterdeckung und Uberdeckung der Ebene durch Kreise, Math. Nachr., 18 (1958), 235–243.Google Scholar
6. Florian, August, Ausfüllung der Ebene durch Kreise, Rend. Circ. Mat. Palermo, II, 9 (1960), 300–312.Google Scholar
7. Hilbert, D. and Cohn-Vossen, S., Geometry and the imagination (New York, 1952).Google Scholar
8. Horsfield, H. T., The strength of asphalt mixtures, J. Soc. Chem. Ind., 53 (1934), 107T-115T.Google Scholar
9. Hudson, Douglas Rennie, Density and packing in an aggregate of mixed spheres, J. Appl. Phys., 20 (1949), 154–162.Google Scholar
10. Ney, P. E., A random interval filling problem, Ann. Math. Stat., 33 (1962), 702–718.Google Scholar
11. Rankin, R. A., On the closest packing of spheres in n dimensions, Ann. Math., 48 (1947), 1062–1081.Google Scholar
12. Rényi, Alfred, A one-dimensional problem of random space filling, Magyar Tud. Akad. Mat. Kutató Int. Közl, 3 (1958), 109–127.Google Scholar
13. Scott, G. D., The packing of spheres, Nature, 188 (1960), 908–909.Google Scholar
14. Westman, A. E. R. and Hugill, H. R., The packing of particles, J. Am. Ceram. Soc, 13 (1930), 767–779.Google Scholar
15. White, H. E. and Walton, S. F., Particle packing and particle shape, J. Am. Ceram. Soc, 20 (1937), 115–166.Google Scholar