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An ellipsoid packing in E3 of unexpected high density

Part of: Discrete geometry

Published online by Cambridge University Press:  26 February 2010

J. M. Wills
J. M. Wills
Affiliation:
Math. Inst. Univ. Siegen, Hölderlinstr. 3, 5900 Siegen, Germany.
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Abstract.

A. Bezdek and W. Kuperberg constructed a nonlattice packing of congruent ellipsoids in Euclidean 3-space E3 with density 0·7459 …, which exceeds the density σL2 = 0·74048… of the densest lattice packing of spheres and hence of ellipsoids in E3. G. Kuperberg improved this to 0·7533… We improve this slightly to 0·7549…. In our case the quotient of the largest and the smallest halfaxis of the ellipsoids is <42, so the ellipsoids are not too degenerate. If one combines G. Kuperberg's refinement and ours, one obtains a packing density of 0·7585…

MSC classification

Secondary: 52C17: Packing and covering in $n$ dimensions
Type
Research Article
Information
Mathematika , Volume 38 , Issue 2 , December 1991 , pp. 318 - 320
Copyright
Copyright © University College London 1991

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References

1.Bezdek, A. and Kuperberg, W.. Packing Euclidean space with congruent cylinders and with congruent ellipsoids. In the “Victor Klee Festschrift”, edited by P. Gritzmann and B. Sturmkls, AMS and ACM, DIMACS, 4 (1991), 71–80.Google Scholar
2.Conway, J. H. and Sloane, N. J. A.. Sphere Packings, Lattices and Groups (Springer, New York, 1988).CrossRefGoogle Scholar
3.Gruber, P. M. and Lekkerkerker, C. G.. Geometry of Numbers (North Holland, Amsterdam, 1987).Google Scholar