The lattice triple packing of spheres in Euclidean space
HTML articles powered by AMS MathViewer
- by G. B. Purdy PDF
- Trans. Amer. Math. Soc. 181 (1973), 457-470 Request permission
Abstract:
We say that a lattice $\Lambda$ in $n$-dimensional Euclidean space ${E_n}$ provides a $k$-fold packing for spheres of radius 1 if, when open spheres of radius 1 are centered at the points of $\Lambda$, no point of space lies in more than $k$ spheres. The multiple packing constant $\Delta _k^{(n)}$ is the smallest determinant of any lattice with this property. In the plane, the first three multiple packing constants $\Delta _2^{(2)},\Delta _3^{(2)}$, and $\Delta _4^{(2)}$ are known, due to the work of Blundon, Few, and Heppes. In ${E_3},\Delta _2^{(3)}$ is known, because of work by Few and Kanagasabapathy, but no other multiple packing constants are known. We show that $\Delta _3^{(3)} \leqslant 8\sqrt {38} /27$ and give evidence that $\Delta _3^{(3)} = 8\sqrt {38} /27$. We show, in fact, that a lattice with determinant $8\sqrt {38} /27$ gives a local minimum of the determinant among lattices providing a $3$-fold packing for the unit sphere in ${E_3}$.References
-
L. Few, Ph. D. Thesis, London, 1953.
- L. Few, Multiple packing of spheres: A survey, Proc. Colloquium on Convexity (Copenhagen, 1965) Kobenhavns Univ. Mat. Inst., Copenhagen, 1967, pp. 88–93. MR 0215193
- L. Few, The double packing of spheres, J. London Math. Soc. 28 (1953), 297–304. MR 55711, DOI 10.1112/jlms/s1-28.3.297
- L. Few and P. Kanagasabapathy, The double packing of spheres, J. London Math. Soc. 44 (1969), 141–146. MR 243430, DOI 10.1112/jlms/s1-44.1.141
- A. Heppes, Mehrfache gitterförmige Kreislagerungen in der Ebene, Acta Math. Acad. Sci. Hungar. 10 (1959), 141–148 (unbound insert) (German, with Russian summary). MR 105066, DOI 10.1007/BF02063295 L. E. Dickson, Studies in the theory of numbers, Univ. of Chicago Press, Chicago, Ill., 1939.
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 181 (1973), 457-470
- MSC: Primary 52A45
- DOI: https://doi.org/10.1090/S0002-9947-1973-0377706-4
- MathSciNet review: 0377706