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Covering space by spheres

Published online by Cambridge University Press:  26 February 2010

L. Few
L. Few
Affiliation:
University College, London.
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Extract

Let Λ be a lattice in three-dimensional space with the property that the spheres of radius 1 centred at the points of Λ cover the whole of space. In other words, every point of space is at a distance not more than 1 from some point of Λ. It was proved by Bambah that then

equality occurring if and only if Λ is a body-centred cubic lattice with the side of the cube equal to 4/√5. Another way of stating the result is to say that the least density of covering of three-dimensional space by equal spheres, subject to the condition that the centres of the spheres form a lattice, is . Another proof of Bambah's result was given recently by Barnes. Both proofs depend on the theory of reduction of ternary quadratic forms.

Type
Research Article
Information
Mathematika , Volume 3 , Issue 2 , December 1956 , pp. 136 - 139
Copyright
Copyright © University College London 1956

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References

* Bambah, R. P., Proc. Nat. Inst. Sci. India, 20 (1954), 25–52.Google Scholar

† Barnes, E. S., Canadian J. of Math., 8 (1956), 293–304.CrossRefGoogle Scholar

‡ An appeal to this could easily be avoided.