Soddy's hexlet

Consider two mutually tangent (externally) spheres A and B together with a larger sphere C inside which A and B are internally tangent. Then construct a chain of spheres each tangent externally to A, B and internally to C (so that C encloses the chain as well as the two original spheres). Surprisingly, every such chain closes into a "necklace" after six spheres, regardless of where the first sphere is placed.

bowl of integers | Coxeter's loxodromic sequence of tangent circles | daisy | Kollros' theorem | seven circles theorem | Steiner chain | tangent spheres