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Chains, froths and a ten-bead necklace: systems of circles and spheres

Published online by Cambridge University Press:  01 August 2016

Michael Fox
Michael Fox*
Affiliation:
2 Leam Road, Leamington Spa, Warwickshire CV31 3PA, e-mail: mdfox@foxleam.freeserve.co.uk
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Extract

Figures 1–4 set the scene, with configurations of tangent circles that may perhaps be new. The labels are curvatures: the curvature or bend being the reciprocal of the radius. By convention, a circle that surrounds another has negative bend and radius. In the first part of this paper I give some easy formulae for configurations like these (see Note a). I show how we can develop fractal structures—I call them froths—having an infinity of tangent circles in each ‘triangular’ region; and give methods for finding integral froths. The second part extends these ideas to systems of tangent spheres, which may also be integral. Throughout, an integral object is one whose bend is an integer, circles and spheres are generally referred to by their bends, and examples are marked with bullets (•).

Type
Research Article
Information
The Mathematical Gazette , Volume 84 , Issue 500 , July 2000 , pp. 242 - 259
Copyright
Copyright © The Mathematical Association 2000

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