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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
DOI: https://doi.org/10.1090/S0002-9947-1973-0377706-4
MathSciNet review: 0377706
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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}$.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0377706-4
Keywords: Spheres, lattice packing, multiple packing
Article copyright: © Copyright 1973 American Mathematical Society