Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T11:55:23.342Z Has data issue: false hasContentIssue false
  • English
  • Français

Sets of Equally Inclined Spheres

Published online by Cambridge University Press:  20 November 2018

J. G. Mauldon
J. G. Mauldon*
Affiliation:
University of California at Berkeley Corpus Christi College, Oxford
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In n-dimensional Euclidean space En, where we shall throughout assume that n ≥ 2, the maximum number of (n — 1)-dimensional spheres which can be mutually orthogonal is n + 2, and it is well known that the sum of the squares of the reciprocals of their radii is zero, so that the spheres cannot be all real. The maximum number of such spheres which can be mutually tangent is also n + 2, and in 1936 Soddy (7) indicated the beautiful relation connecting their radii. These two formulae are the particular cases γ = 0, γ - 1 of Theorem 1 below, which gives the relation connecting the radii of a set of n + 2 such spheres when every pair is inclined at a given non-zero angle 0, where γ is written for cos θ.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 509 - 516
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Coxeter, H. S. M., Interlocked rings of spheres, Scripta Mathematical, 18 (1952), 113–121.Google Scholar
2. Coxeter, H. S. M., An introduction to geometry (New York, 1961).Google Scholar
3. Coxeter, H. S. M., Regular polytopes (London, 1948).Google Scholar
4. Coxeter, H. S. M., Non-Euclidean geometry (Toronto, 1957).Google Scholar
5. Gosset, T., The kiss precise, Nature, 139 (1937), 62.Google Scholar
6. Mauldon, J. G., Equally inclined spheres, Proc. Camb. Phil. Soc. (to appear).Google Scholar
7. Soddy, F., The kiss precise, Nature, 187 (1936), 1021.Google Scholar