Workshop on Topology: Identifying Order in Complex Systems

I'll talk about two related projects, with two different groups, both aiming to see three-dimensional manifolds "from the inside". That is, we generate images assuming that light travels along geodesics in the geometry of the manifold. The first project, with Rémi Coulon, Sabetta Matsumoto, and Steve Trettel, uses ray-marching to generate the inside-view in all eight Thurston geometries. I'll explain the ray-marching technique, and some aspects of our implementation. The second project, with David Bachman, Matthias Goerner, and Saul Schleimer, visualizes "cohomology fractals" in hyperbolic three-manifolds. These images come from cohomology classes in the manifold, and are closely related to the sphere-filling curves discovered by Cannon and Thurston.