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

Abstract

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 (•).