Smale's Paradox

In differential topology, Smale's paradox states that it is possible to turn a sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease, a process often called sphere eversion (eversion means "to turn inside out"). This is surprising, and is hence deemed a veridical paradox. More precisely, let

be the standard embedding; then there is a regular homotopy of immersions

such that ƒ0 = ƒ and ƒ1 = −ƒ.

Read more about Smale's Paradox:  History, Proof

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