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The amount of overlapping in partial coverings of space by equal spheres

Published online by Cambridge University Press:  26 February 2010

P. Erdős ,
L. Few  and
C. A. Rogers
P. Erdős
Affiliation:
University College, and London.
L. Few
Affiliation:
University College, and London.
C. A. Rogers
Affiliation:
University College, and London.
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Extract

We say that a system ∑ of equal spheres S1S2, … covers a proportion θ of n-dimensional space, if the limit, as the side of the cube C tends to infinity, of the ratio

of the volume of C covered by the spheres to the volume of C, exists and has the value θ. We say that such a system ∑ has density δ, if the corresponding ratio

has the limit δ as the side of the cube C tends to infinity. We confine our attention to systems ∑ for which both limits exist. It is clear that δ = θ, if no two spheres of the system overlap, i.e. if we have a. packing; and that, in general, the difference δ-θ is a measure of the amount of overlapping.

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Type
Research Article
Information
Mathematika , Volume 11 , Issue 2 , December 1964 , pp. 171 - 184
Copyright
Copyright © University College London 1964

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