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Turning a sphere inside-out: http://www.youtube.com/watch?v=BVVfs4zKrgk

And part 2: http://www.youtube.com/watch?v=x7d13SgqUXg

This is really, really cool. They describe things in really simple terms though (like referring to curvatures with smiles and frowns) and sometimes I wish I could connect it to the math that I know if relevant terms were brought up. The animations are simply amazing, and I can't believe this was made in 1994!

I was wondering if anyone knows if these people made other videos like this using their techniques... So cool!

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Related: math.stackexchange.com/questions/18843/… - videos for Mobius transformations, origami, the Poincare conjecture, and some other geometrically flavored topics. –  anon Aug 29 '11 at 22:08
    
I have converted the question to community wiki, as it's asking for a list of examples and there is no single right answer. –  Zev Chonoles Aug 30 '11 at 0:41
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up vote 3 down vote accepted

Check out this Math Overflow thread! For instance, you may enjoy this video about the Möbius band.

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Niles Johnson at UGA recently posted this video he made illustrating the Hopf fibration: http://www.youtube.com/watch?v=AKotMPGFJYk

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