Sphere eversion

This is surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while being true, on first glance seems false.

It is difficult to visualize a particular example of such a turning, although some 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.

But the degrees of the Gauss map for the embeddings f and −f in R3 are both equal to 1, and do not have opposite sign as one might incorrectly guess.

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 S2, 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.