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Some Sphere Packings in Higher Space

Published online by Cambridge University Press:  20 November 2018

John Leech
John Leech*
Affiliation:
Glasgow University
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 657 - 682
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Blichfeldt, H. F., The minimum values of quadratic forms and the closest packing of spheres, Math. Ann., 101 (1929), 605–608.Google Scholar
2. Blichfeldt, H. F., The minimum values of positive quadratic forms in six, seven and eight variables, Math. Z., 39 (1935), 1–15.Google Scholar
3. Coxeter, H. S. M., Twelve points in PG(5, 3) with 95040 self-transformations, Proc. Roy. Soc. (London), A, 247 (1958), 279–293. 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.Google Scholar
5. Coxeter, H. S. M. and Todd, J. A., An extreme duodenary form, Can. J. Math., 5 (1953), 384–392.Google Scholar
6. Golay, M. J. E., Notes on digital coding, Proc. Inst. Radio Engrs., 37 (1949), 657.Google Scholar
7. Paige, L. J., A note on the Mathieu groups, Can. J. Math. 9 (1957), 15–18.Google Scholar
8. Paley, R. E. A. C., On orthogonal matrices, J. Math. & Phys., 12 (1933), 311–320.Google Scholar
9. Perron, O., Bemerkungen über die Verteilung der quadratischen Reste, Math. Z., 56 (1952), 122–130.Google Scholar
10. Rogers, C. A., The packing of equal spheres, Proc. London Math. Soc. (3), 8 (1958), 609– 620.Google Scholar
11. Rogers, C. A., An asymptotic expansion for certain Schläfli functions, J. London Math. Soc, 36 (1961), 78–80.Google Scholar
12. Sylvester, J. J., Thoughts on inverse orthogonal matrices … , Phil. Mag. (4), 34 (1867), 461–475.Google Scholar
13. Todd, J. A., On representations of the Mathieu groups as collineation groups, J. London Math. Soc, 34 (1959), 406–416.Google Scholar