Eversion of the sphere

In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space. (The word means "turning inside out".) Remarkably, it is possible to smoothly and continuously turn a sphere inside out in this way (with possible self-intersections) without cutting or tearing it or creating any crease. 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.

More precisely, let

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

such that ƒ₀ = ƒ and ƒ₁ = −ƒ.

An existence proof for crease-free sphere eversion was first created by Stephen Smale (1957). 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. 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 (Levy 1995). His reasoning was that the degree of the Gauss map must be preserved in such "turning"—in particular it follows that there is no such turning of S¹ in R². But the degree 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 degree of the Gauss map of all immersions of S² in R³ 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 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.

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