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. This beautiful and amazing result due to Soddy is a special case of Kollros' theorem. It can be demonstrated using inversion of six identical spheres around an equal center sphere, all of which are sandwiched between two planes. This result was given in a Sangaku problem from Kanagawa Prefecture in 1822, more than a century before it was published by Soddy.
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