Also called Soddy's Hexlet. Consider three mutually tangent Spheres , , and . Then construct a chain of Spheres tangent to each of , , and threading and interlocking with the ring. Surprisingly, every chain closes into a ``necklace'' after six Spheres regardless of where the first Sphere is placed. This is a special case of Kollros' Theorem. The centers of a Soddy hexlet always lie on an Ellipse (Ogilvy 1990, p. 63).
See also Coxeter's Loxodromic Sequence of Tangent Circles, Kollros' Theorem, Steiner Chain
References
Coxeter, H. S. M. ``Interlocking Rings of Spheres.'' Scripta Math. 18, 113-121, 1952.
Gosset, T. ``The Hexlet.'' Nature 139, 251-252, 1937.
Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 49-50, 1976.
Morley, F. ``The Hexlet.'' Nature 139, 72-73, 1937.
Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 60-72, 1990.
Soddy, F. ``The Bowl of Integers and the Hexlet.'' Nature 139, 77-79, 1937.
Soddy, F. ``The Hexlet.'' Nature 139, 154 and 252, 1937.