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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: [email protected]
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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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References

1. Beecroft, P., Properties of circles in mutual contact, Lady’s and Gentleman’s Diary (1842) pp. 91–96.Google Scholar
2. Courant, R. and Robbins, H., What is mathematics?, Oxford University Press (1941).Google Scholar
3. Coxeter, H. S. M., Interlocked rings of spheres, Scripta Mathematica 18 (1952) pp. 113–121.Google Scholar
4. Coxeter, H. S. M. and Greitzer, S. L., Geometry revisited, Math. Assoc. of America (1967).Google Scholar
5. Descartes, R., Œuvres, vol 1, Adam, C. and Tannery, P., Paris (1901).Google Scholar
6. Durell, C. V., Modern geometry, Macmillan (1920).Google Scholar
7. Fox, M. D., Formulae for the curvatures of circles in chains, Amer. Math. Monthly 87 (1980) pp. 708–715.Google Scholar
8. Johnson, R. A., Advanced Euclidean geometry, Dover Publications (1960).Google Scholar
9. Morley, F., The hexlet, Nature 139 (1937) pp. 72–73.Google Scholar
10. Russell, J. W., A sequel to elementary geometry, Oxford University Press (1907).Google Scholar
11. Soddy, F., The kiss precise, Nature 137 (1936) p. 1021.Google Scholar
12. Soddy, F., The hexlet, Nature 138 (1936) p. 958.Google Scholar
13. Wells, D., The Penguin dictionary of curious and interesting geometry, Penguin (1991).Google Scholar
14. Wilker, J. B., Four proofs of a generalization of the Descartes circle theorem, Amer. Math. Monthly 76 (1969) pp. 278–282.Google Scholar