Generalized Branching in Circle Packing

Circle packings are configurations of circle with prescribed patterns of tangency. They relate to a surprisingly diverse array of topics. Connections to Riemann surfaces, Apollonian packings, random walks, Brownian motion, and many other topics have been discovered. Of these none has garnered more interest than circle packings' relationship to analytical functions. With a high degree of faithfulness, maps between circle packings exhibit essentially the same geometric properties as seen in classical analytical functions. With this as motivation, an entire theory of discrete analytic function theory has been developed. However limitations in this theory due to the discreteness of circle packings are shown to be unavoidable. This thesis explores methods to introduce continuous parameters for the purpose of overcoming these difficulties. Our topics include, packings with deep overlaps, fractional branching, and shift-points. Using the software package CirclePack, examples of some previously non-realizable discrete functions in circle packing are shown to computational exist using these techniques. Some necessary theory is developed including a generalization for overlapping packings and some results for expressing singularities associated with faces.