To evert a sphere means to turn it inside out with a continuous transformation where the sphere can pass through itself without puncturing, tearing, ripping, creasing, or pinching. To see how nontrivial this is just imagine trying to push the north pole through the south pole; a crease will develop along the equator. Stephen Smale proved that an eversion was possible in 1957 but there wasn’t a concrete example until a few years later. The video shown below uses a technique of corrugation developed by Bill Thurston, who’s geometrization conjecture, which implies the Poincare conjecture, was recently proved by Grigori Perelman.
The above video is an excerpt from a 20 minute video called “Outside In“, which involved Bill Thurston’s input. The video can be found at You Tube in two parts. Part 1 is here and part 2 is here.