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Six and Seven Dimensional Non-Lattice Sphere Packings

Published online by Cambridge University Press:  20 November 2018

John Leech
John Leech*
Affiliation:
University of Stirling, Stirling, Scotland
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The densest lattice packings of equal spheres in Euclidean spaces En of n dimensions are known for n ⩽ 8. However, it is not known for any n ⩾ 3 whether there can be any non-lattice sphere packing with density exceeding that of the densest lattice packing. W. Barlow described [1] a non-lattice packing in E3 with the same density as the densest lattice packing, and I described [6] three non-lattice packings in E5 which also have this property.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 2 , April 1969 , pp. 151 - 155
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Barlow, W., Probable nature of the internal symmetry of crystals. Nature 29 (1883) 186–188.Google Scholar
2. Coxeter, H.S.M., Extreme forms. Canad. J. Math. 3 (1951) 391–441.Google Scholar
3. Coxeter, H.S.M., Twelve geometric essays. Southern Illinois Univ. Press, 1968).Google Scholar
4. Leech, J., Some sphere packings in higher space. Canad. J. Math. 16 (1964) 657–682.Google Scholar
5. Leech, J., Notes on sphere packings. Canad. J. Math. 19 (1967) 251–267.Google Scholar
6. Leech, J., Five-dimensional non-lattice sphere packings. Canad. Math. Bull. 10 (1967) 387–393.Google Scholar