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Smale's Paradox

  1. Sep 28, 2014 #1
    I caught a video online released by UIUC entitled "The Optiverse." Very cool video! Anyways, the idea is that a sphere can be turned inside-out under the premises that 1.) you cannot tear, puncture, or crease the edges, and 2.) that the sphere can pass through itself.

    While I understand that you cannot tear or puncture the sphere -- that would defeat the point -- and that the sphere must be able to pass through itself, I do not understand why you cannot crease it. I would also imagine that creasing would fall under the list of things that would "defeat the point," but I'm just wondering if there is something unique about the crease.
  2. jcsd
  3. Sep 29, 2014 #2


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    I think that creasing is non differentiable.

    Allowing creases also makes the theorem trivial Just push the sphere through itself along a great circle. This creates a loop which ties itself off.
  4. Sep 29, 2014 #3
    That makes sense to me! Yes, it can't be differentiable if an increasingly tight loop ends up becoming a point. And obviously, points aren't differentiable. Thanks!
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