Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The lattice triple packing of spheres in Euclidean space

Author: G. B. Purdy
Journal: Trans. Amer. Math. Soc. 181 (1973), 457-470
MSC: Primary 52A45
MathSciNet review: 0377706
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] L. Few, Ph. D. Thesis, London, 1953.
  • [2] 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
  • [3] L. Few, The double packing of spheres, J. London Math. Soc. 28 (1953), 297–304. MR 0055711
  • [4] L. Few and P. Kanagasabapathy, The double packing of spheres, J. London Math. Soc. 44 (1969), 141–146. MR 0243430
  • [5] A. Heppes, Mehrfache gitterförmige Kreislagerungen in der Ebene, Acta Math. Acad. Sci. Hungar. 10 (1959), 141–148 (unbound insert) (German, with Russian summary). MR 0105066
  • [6] L. E. Dickson, Studies in the theory of numbers, Univ. of Chicago Press, Chicago, Ill., 1939.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 52A45

Retrieve articles in all journals with MSC: 52A45

Additional Information

Keywords: Spheres, lattice packing, multiple packing
Article copyright: © Copyright 1973 American Mathematical Society