In differential topology, Smale's paradox states that it is possible to turn a sphere inside out in 3-space with possible self-intersections but without creating any crease, a process often called sphere eversion. More precisely, let In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... Robert Boyles self-flowing flask fills itself in this diagram, but perpetual motion machines do not exist. ...

be the standard embedding; then there is a continuous one-parameter family of immersions In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...

such that f₀ = f and f₁ = − f.

This 'paradox' was discovered by Stephen Smale in 1958. It is quite hard to visualize a particular example of such a turning, but there are now some movies to help you. The first example was exhibited through the efforts of several mathematicians, including one who was blind, Bernard Morin. On the other hand, it is much easier to prove that such a "turning" exists and that is what was done by Smale. Stephen Smale (born July 15, 1930) is an American mathematician and winner of the Fields Medal in 1966. ... Bernard Morin is a French mathematician, especially a topologist, born in 1931, who is now retired. ...

The legend says that when Smale was trying to publish this result the referee's report stated that although the proof is quite interesting the statement is clearly wrong 'due to invariance of degree of the Gauss map'. Indeed, 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. In fact the degree of the Gauss map of all immersions of a 2-sphere in R³ is 1; so there is in fact no obstacle. This article is about the term degree as used in mathematics. ... In differential geometry, the Gauss map (named, like so many things, after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere . ...

See h-principle for further generalizations. The homotopy principle (h-principle) is a very general way to solve partial differential equations PDE (and more generally partial differential relations PDR). ...

References

• Max Nelson, "Turning a Sphere Inside Out", International Film Bureau, Chicago, 1977 (video)
• Anthony Phillips, "Turning a surface inside out, Scientific American, May 1966, pp. 112-120.

External links

• Optiverse video, available online.
• A History of Sphere Eversions
• A brief history of sphere eversions
• Sphere Eversion by Erik de Neve.