Fractured Branched Circle Packings on the Plane

William Thurston first proposed that real circles could be used to approximate the underlying infinitesimal circles of conformal maps in 1985. Inspired pioneers developed Circle Packing into a very rich and deep field that can be used as a method for constructing discrete conformal maps of surfaces on different types of geometries. Offering the advantages of a computational method that lends itself to experimentation and the easy creation of visual models, Circle Packing has proven itself as valuable new tool in approaching both old and new problems.

In particular, Circle Packing has been used to make discrete analogues of continuous functions; however existing methods are inadequate for certain classical functions. As a solution to this problem, Kenneth Stephenson has suggested using a branched circle packing where the extra angle sum is distributed amongst more than one circle. The purpose of this paper is to investigate the behavior of such circle packings on the plane. The result is the revelation of a subject worthy of interest beyond its potential aide to other problems.

Normally, maps are made in Circle Packing are created by laying out circles adjacently to each other like a group of coins laid out on a table. Taking a group of circles similar to this, we can cut a "slit" from the exterior to a point in the center called the branch point. We can then wrap the cut edges around like a spiraling staircase by a multiple of 2π, creating a branched map. Branched maps are mostly similar to non-branched packings with the exception that they are necessarily globally non-univalent. Adding fractured multiples of 2π to more than a single point does not necessarily result in a map that makes any sense. Regardless of how complicated or simple our original map may be, most of these questions can be answered by surprisingly simple geometry. Furthermore, despite the difficulty that these unfamiliar terms may cause the non-mathematician, the visual nature of circle packing provides models and pictures that bring the concepts to life, making these ideas accessible to most anyone with a high school level understanding of geometry.