Kollros' Theorem

For every ring containing $p$ Spheres, there exists a ring of $q$ Spheres, each touching each of the $p$ Spheres, where

$${1\over p}+{1\over q}={1\over 3}.$$

The Hexlet is a special case with $p=3$.

See also Hexlet, Sphere

References

Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., p. 50, 1976.