Sphere turned inside out

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How can I turn a sphere inside out? I saw this video on YouTube and I didn't understood how can i turn a sphere inside out. any help will be appreciated.

http://www.youtube.com/watch?v=R_w4HYXuo9M

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Video, of course, is not a proof. You can find references to several rigorous proofs (including the original one) at Smale's Paradox ("References" section).