Apollonius spheres

Any four mutually tangent spheres determine six points of tangency. A pair of tangencies (t_i, t_j) is said to be opposite if the two spheres determining t_i are distinct from the two spheres determining t_j. The six tangencies are therefore grouped into three opposite pairs corresponding to the three ways of partitioning four spheres into two pairs. These three pairs of opposite tangencies are coincident. A special case of tangent spheres is given by Soddy's hexlet, which consists of a chain of six spheres externally tangent to two mutually tangent spheres and internally tangent to a circumsphere.

Apollonian gasket | bowl of integers | hexlet | Soddy circles | sphere | tangent circles | tetrahedron | Wada basin