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Some Sphere Packings in Higher Space

  • John Leech (a1)
Extract

This paper is concerned with the packing of equal spheres in Euclidean spaces [n] of n > 8 dimensions. To be precise, a packing is a distribution of spheres any two of which have at most a point of contact in common. If the centres of the spheres form a lattice, the packing is said to be a lattice packing. The densest lattice packings are known for spaces of up to eight dimensions (1, 2), but not for any space of more than eight dimensions. Further, although non-lattice packings are known in [3] and [5] which have the same density as the densest lattice packings, none is known which has greater density than the densest lattice packings in any space of up to eight dimensions, neither, for any space of more than two dimensions, has it been shown that they do not exist.

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References
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1. Blichfeldt, H. F., The minimum values of quadratic forms and the closest packing of spheres, Math. Ann., 101 (1929), 605608.
2. Blichfeldt, H. F., The minimum values of positive quadratic forms in six, seven and eight variables, Math. Z., 39 (1935), 115.
3. Coxeter, H. S. M., Twelve points in PG(5, 3) with 95040 self-transformations, Proc. Roy. Soc. (London), A, 247 (1958), 279293. 4# An upper bound for the number of equal non-overlapping spheres that can touch another of the same size, Proc. Symposia Pure Math., vol. VII (Providence, 1963), 53-71.
5. Coxeter, H. S. M. and Todd, J. A., An extreme duodenary form, Can. J. Math., 5 (1953), 384392.
6. Golay, M. J. E., Notes on digital coding, Proc. Inst. Radio Engrs., 37 (1949), 657.
7. Paige, L. J., A note on the Mathieu groups, Can. J. Math. 9 (1957), 1518.
8. Paley, R. E. A. C., On orthogonal matrices, J. Math. & Phys., 12 (1933), 311320.
9. Perron, O., Bemerkungen über die Verteilung der quadratischen Reste, Math. Z., 56 (1952), 122130.
10. Rogers, C. A., The packing of equal spheres, Proc. London Math. Soc. (3), 8 (1958), 609 620.
11. Rogers, C. A., An asymptotic expansion for certain Schläfli functions, J. London Math. Soc, 36 (1961), 7880.
12. Sylvester, J. J., Thoughts on inverse orthogonal matrices … , Phil. Mag. (4), 34 (1867), 461475.
13. Todd, J. A., On representations of the Mathieu groups as collineation groups, J. London Math. Soc, 34 (1959), 406416.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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