"Outside In" is the name of a math film, reviewed by the math movie reviewers.
Despite the obvious lacking of well-rouded characters, the two narrators of this film explained some pretty interesting stuff. They explained how one can topologically turn a sphere inside out. What this means is that a sphere can be turned inside out by deforming it, but without cutting or gluing it. In much the same way, a coffee cup can be seen to be topolocially equivalent to a donut because both are one-holed surfaces, known to mathematicians as tori (singular, torus). It's rather odd to think that a sphere can be turned inside-out in this way, because a plain old two dimensional circle cannot.
So how did they perform this amazing act of contortion? Well, first they showed that a circle cannot be turned inside out. This had to do with orientation and turning numbers, which were demonstrated using a monorail car making turns around one full trip. Understanding the sphere involved viewing it as a rounded cylinder with polar caps and rather funky looking edges. If you want to know what it looks like, see the film.