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Abstract

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 RĂ©mi 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.