More videos

The Geometry Supercomputer Project/The Geometry Center http://www.geom.uiuc.edu/ created several interesting geometric animations:

The video "Not Knot" (1991) is concerned with 3-dimensional hyperbolic geometry. A still picture from this video appears on the cover of Bill Thurston's "Three-Dimensional Geometry and Topology". The title refers to the hyperbolic structure on the complement of a figure-eight knot in S^3.

The video "Outside In" (1994) shows how to turn a sphere inside-out. Self-intersections must be allowed, but no cuts or creases.  The fact that the standard embedding of S^2 in R^3 can be deformed, through so-called immersions, so that the outward unit vectors end up pointing inward, was first proven by Steve Smale.

A concrete realization of this deformation was first found by the blind mathematician Bernard Morin.  An energy-minimizing realization (1998) was visualized by John M. Sullivan, George Francis and Silvio Levy. (If no video appears when you click on the link, try downloading the file and opening with a video viewer.)