Consider two mutually tangent (externally) spheres 
and 
together with a larger sphere 
inside which 
and 
are internally tangent. Then construct a chain of spheres each tangent externally
to 
, 
and internally to 
(so that 
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 (1937) 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 (Wells 1991, pp. 120 and 232). This result was given in a Sangaku problem from Kanagawa Prefecture in 1822, more than a century before it was published by Soddy (Rothman 1998).
Moreover, the centers of the six spheres in the necklace and their six points of contact all lie in a plane. Furthermore, there are two planes which touch each of
the six spheres, one on either side of the necklace. Finally, the radii 
of the spheres are related by
 |
(Rothman 1998).
Soddy's bowl of integers contains an infinite number of nested hexlets. The centers of a Soddy hexlet always lie on an ellipse (Ogilvy 1990, p. 63).
