Abstract
Abstract It has been known for a long time, see Pedoe (1970), that the curvatures of four mutually touching circles are related by a single straightforward equation. The result was FB01rst obtained by Descartes (1901). In this chapter we show that inFB01nite chains of touching circles, all with integer curvatures, can be formed. This means that the circles involved all have rational radii. Furthermore, we show how to obtain all such chains. In the FB01rst section we deal with the singular case when one of the circles degenerates into a line. In the next section we deal with the case of sequences of mutually touching circles in two dimensions. We then go on to show how to generalise to sequences of mutually touching spheres in three dimensions, and FB01nally to mutually touching hyperspheres in four dimensions. InFB01nite chains exist in all of these cases, and recurrence relations are established to show how they are formed.
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