Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-29T02:36:35.320Z Has data issue: false hasContentIssue false
  • English
  • Français

Covering space by spheres

Published online by Cambridge University Press:  26 February 2010

L. Few
L. Few
Affiliation:
University College, London.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.