Hexlet

Also called Soddy's Hexlet. Consider three mutually tangent Spheres $A$, $B$, and $C$. Then construct a chain of Spheres tangent to each of $A$, $B$, and $C$ threading and interlocking with the $A-B-C$ 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.