Spiral hexagonal circle packings in the plane

We discuss an intriguing geometric algorithm which generates infinite spiral patterns of packed circles in the plane. Using Kleinian group and covering theory, we construct a complex parametrization of all such patterns and characterize those whose circles have mutually disjoint interiors. We prove that these coherent spirals, along with the regular hexagonal packing, give all possible hexagonal circle packings in the plane. Several examples are illustrated.The last two authors gratefully acknowledge support of the National Science Foundation and the Tennessee Science Alliance.