Outside In
(Image from The Geometry Center)
In 1957, Steve Smale proved that it is mathematically possible to turn a sphere inside out (sphere eversion) without tearing, puncturing or creasing the surface, as long as one allows the sphere to intersect itself. That is, the surface of the sphere can be pulled and twisted like a sheet of rubber and can also pass through itself. Given these properties it can be reshaped in such a way that the inside surface becomes the outside surface. This problem in differential topology is known as Smale's Paradox. "Outside In" is a 22 minute video created in 1994 by the Geometry Center at the University of Minnesota. It presents a possible implementation of Smale's solution based on the ideas of Bill Thurston. It is accompanied by the book Making Waves: Outside In which explains the mathematical concepts behind the visualization in greater detail. The video illustrates the rules that govern the behaviour of the sphere using two and three dimensional examples and then demonstrates how the sphere can be everted in compliance with these rules.
This video is one of the reasons I decided to go back to school and study Math Visualization. It is very effective at conveying a complex topic in mathematics without requiring the viewer to understand the symbols and language of differential topology.
Not Knot
(Images from The Geometry Center)
This one, also from the Geometry Center, is a little harder to follow but worth taking the time to understand. It describes knots and their complements (a complement is the space around the knot - hence the title Not Knot). The complements tend to have a hyperbolic geometry and thus the video has an interesting representation of hyperbolic space.
It might help to read the book Flatland: A Romance of Many Dimensions by Edwin A. Abbott. This book is both a commentary on Victorian social class structure and an exploration of three dimensions from the perspective of a person living in a two dimensional world.
