Circle packing

Summary

A circle packing is an arrangement of circles in the plane, or on a surface, whose discs do not overlap. One studies dense packings of equal circles, where the hexagonal packing is optimal by Thue's theorem. One also studies packings with prescribed tangencies, where the circle packing theorem realizes every planar graph. Circle packings give a discrete analogue of conformal maps, and fractal examples such as the Apollonian gasket.