Editing Sphere eversion (section)

==History==
An [[existence proof]] for crease-free sphere eversion was first created by {{harvs|txt|authorlink=Stephen Smale|first=Stephen|last= Smale|year= 1958}}.
It is difficult to visualize a particular example of such a turning, although some [[computer animation|digital animations]] have been produced that make it somewhat easier. The first example was exhibited through the efforts of several mathematicians, including [[Arnold S. Shapiro]] and [[Bernard Morin]], who was blind. On the other hand, it is much easier to prove that such a "turning" exists, and that is what Smale did.

Smale's graduate adviser [[Raoul Bott]] at first told Smale that the result was obviously wrong {{harv|Levy|1995}}. 
His reasoning was that  the [[Degree of a continuous mapping|degree]] of the [[Gauss map]] must be preserved in such "turning"—in particular it follows that there is no such ''turning'' of '''S'''<sup>1</sup> in '''R'''<sup>2</sup>. But the degrees of the Gauss map for the embeddings ''f'' and &minus;''f'' in '''R'''<sup>3</sup> are both equal to 1, and do not have opposite sign as one might incorrectly guess. The degree of the Gauss map of all immersions of '''S'''<sup>2</sup> in '''R'''<sup>3</sup> is 1, so there is no obstacle. The term "veridical paradox" applies perhaps more appropriately at this level: until Smale's work, there was no documented attempt to argue for or against the eversion of '''S'''<sup>2</sup>, and later efforts are in hindsight, so there never was a historical paradox associated with sphere eversion, only an appreciation of the subtleties in visualizing it by those confronting the idea for the first time.

See [[h-principle|''h''-principle]] for further generalizations.