Chapter 11 Circle packing and discrete analytic function theory

Circle packings — configurations of circles with specified patterns of tangency — came to prominence with analysts in 1985 when Thurston conjectured that maps between such configurations would approximate conformai maps. The proof by Rodin and Sullivan launched a topic which has grown steadily as ever more connections with analytic functions and conformai structures have emerged. Indeed, the core ideas have matured to the point that one can fairly claim that circle packing provides a discrete analytic function theory . There are two related but distinct aspects vis-a-vis the classical model — analogy and approximation. This survey concentrates on the analogies, with a largely pictorial tour intended for the reader familiar with classical conformai geometry. The companion survey, K. Stephenson, Proceedings of the Third CMFT Conference, Vol. 11, treats approximation. Section 1 of the chapter covers circle packing basics and introduces discrete analytic functions as maps between circle packings. Representatives of the standard classes of analytic functions on the sphere, plane, and unit disc are illustrated and the discrete parallels to basic theorems in complex analysis are reviewed. Section 2 visits several geometric facets of conformai geometry in their discrete encarnations: extremal length, harmonic measure, conformai welding, and conformai structures, among others. The final topic there, conformai tiling, shows both the potential for synergy between the discrete and classical theories and the value of the experimental capabilities available with circle packing. Appendix A comments on computational aspects of circle packing and Appendix B summarizes the circle packing literature.