I wonder if i ever could have learnt this without the visual representation. I can imagine it would have taken years to learn if you didn't have any illustration at all (straight number crunching). And now, 20 minutes, and you still get a bit annoyed by the overly pedagogic narration.
http://en.wikipedia.org/wiki/Smale's_paradox
the math behind it is pretty complex. i'm a math major going into my senior year of undergrad, and i don't understand a word of that wikipedia article.
it's a result from a branch of math called topology, which deals with surfaces and maps between them in different dimensions. pretty cool, but it requires a lot of background (that i don't have) to really grasp.
the videos are just a way to visualize the result. proving it is much more difficult, i would imagine.
Sooooooo, it can theoretically happen but has absolutely no rational use, at least that I can think of. They basically just said "here is how you turn a sphere inside out, but you need this super sphere that cannot exist" Fuuuuuuuuu