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\u27 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

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

Erratum to: 36th International Symposium on Intensive Care and Emergency Medicine

[This corrects the article DOI: 10.1186/s13054-016-1208-6.]

Reconfiguring the Omweso board game: performing narratives of Buganda material culture

My artwork titled Nakulabye, which is 4 meters long and weighs 440 pounds, is an intimidating sculptural replica of the Omweso game board (Fig. 1). The wooden sculpture, twenty times larger than an average Omweso game board, includes four cane stools to sit on during play. Its composition is derived from a human face, and it has thirty-two pits (8 × 4) in the configuration of a mancala board. This sculpture was inspired by my engagement with a group of men that I visited in July 2016 in Nakulabye, a town in an urban area of Kampala City, Uganda. At the Nakulabye Omweso Club, a shop veranda in Nakulabye Town, these men play Omweso and chat against the backdrop of a small television that mostly screens British Premiere Leagues. Observing their exchanges, which seem to be informed by moves on the Omweso board and reveal strong, clearly gendered power dynamics, I became curious about the performative place of Omweso as a cultural artifact of the Baganda people