What do 3-Manifolds Look Like?

The Geometrization Theorem of Thurston and Perelman provides a roadmap to understanding topology in dimension 3 via geometric means. Specifically, it states that every closed 3-manifold has a decomposition into geometric pieces, and the zoo of these geometric pieces is quite constrained: each is built from one of eight homogeneous 3-dimensional Riemannian model spaces (called the Thurston geometries). In this talk, we approach the question of 'what does a 3-manifold look like' from the perspective of geometrization. Through animations of simple examples in dimensions 2 and 3 we review what it means to put a geometric structure on a manifold, and construct an example admitting each of the Thurston geometries. Using software written in collaboration with Remi Coulon, Sabetta Matsumoto and Henry Segerman, we explore these manifolds 'from the inside' by raytracing along geodesics, then re-assemble these geometric pieces to understand an inside view of general 3-manifolds.