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I asked a vague question about torus eversion earlier, with no hard math, so while I'm at it, how about this one, which may involve hard math:

"Everybody knows" that Stephen Smale showed us how to evert a sphere without tearing or creasing:

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

Has anyone done the same with a torus?

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  • $\begingroup$ Do you mean invert a sphere? $\endgroup$
    – Git Gud
    Sep 20, 2013 at 22:31
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    $\begingroup$ sphere eversion $\endgroup$
    – Dan Rust
    Sep 20, 2013 at 22:33
  • $\begingroup$ @DanielRust No entry at Cambride Dictionaires for evert. Thanks. $\endgroup$
    – Git Gud
    Sep 20, 2013 at 22:34
  • $\begingroup$ I was just wondering this the other day. Here's the related video. youtube.com/watch?v=kQcy5DvpvlM $\endgroup$
    – Jemmy
    Sep 20, 2013 at 23:24
  • $\begingroup$ @GitGud : No. I meant "evert". $\endgroup$
    – Michael Hardy
    Sep 21, 2013 at 0:27

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This seems to suggest the answer is yes. Séquin goes in to some detail in this document and I'll add the image he references, made by Cheritat.

enter image description here

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    $\begingroup$ This is a beautiful picture. My father (Derek Hacon; a student of Zeeman) had shown me the same picture when I was a kid. He also observed that this gives a sphere eversion. Just push the south pole upwards, past the equator, then a cross section looks like two circles (just as with the torus; but you have to glue two disks and an annulus instead of two annuli). Then follow the above sequence of cross-sections. I believe he tried to publish it in some expository / general audience journal, but it was rejected as too elementary. $\endgroup$
    – user110111
    Nov 20, 2013 at 3:30

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