There are two widely used approaches to discretizing holomorphic maps and Riemann surfaces, based on the infinitesimal circle preservation and length cross-ratio preservation properties. The circle-preserving approach leads to William Thurston’s circle packing theory, while the length cross-ratio preserving approach gives rise to the vertex-scaling conformal geometry of polyhedral surfaces. In this talk, we survey several developments in this active area of research, including recent progress on discrete analogues of the uniformization theorem.
