In differential topology

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

, Smale's paradox

Paradox

Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...

states that it is possible to turn a sphere inside out in a three-dimensional space

Three-dimensional space

Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

 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

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

; then there is a regular homotopy

Regular homotopy

In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions....

 of immersions


such that ƒ₀ = ƒ and ƒ₁ = −ƒ.

History

This 'paradox' was discovered by .
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 Shapiro and Bernard Morin

Bernard Morin

Bernard Morin is a French mathematician, specifically a topologist, born in 1931, who is now retired. He has been blind since age 6 due to glaucoma, but his blindness did not prevent him from having a successful career in mathematics....

 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

Raoul Bott

Raoul Bott, FRS was a Hungarian mathematician known for numerous basic contributions to geometry in its broad sense...

 at first told Smale that the result was obviously wrong .
His reasoning was that the degree

Degree (mathematics)

In mathematics, there are several meanings of degree depending on the subject.- Unit of angle :A degree , usually denoted by ° , is a measurement of a plane angle, representing 1⁄360 of a turn...

 of the Gauss map

Gauss map

In differential geometry, the Gauss map maps a surface in Euclidean space R3 to the unit sphere S2. Namely, given a surface X lying in R3, the Gauss map is a continuous map N: X → S2 such that N is a unit vector orthogonal to X at p, namely the normal vector to X at p.The Gauss map can be defined...

 must be preserved in such "turning"—in particular it follows that there is no such turning of S¹in R². But the degree of the Gauss map for the embeddings f, −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 a 2-sphere in R³ is 1; so there is no obstacle.

See h-principle

H-principle

In mathematics, the homotopy principle is a very general way to solve partial differential equations , and more generally partial differential relations...

 for further generalizations.

Proof

Smale's original proof was indirect: he identified (regular homotopy) classes of immersions of spheres with a homotopy group of the Stiefel manifold

Stiefel manifold

In mathematics, the Stiefel manifold Vk is the set of all orthonormal k-frames in Rn. That is, it is the set of ordered k-tuples of orthonormal vectors in Rn. It is named after Swiss mathematician Eduard Stiefel...

. Since the homotopy group that corresponds to immersions of in vanishes, the standard embedding and the inside-out one must be regular homotopic. In principle the proof can be unwound to produce an explicit regular homotopy, but this is not easy to do.

There are several ways of producing explicit examples and mathematical visualization

Mathematical visualization

Mathematical visualization is an aspect of geometry which allows one to understand and explore mathematical phenomena via visualization. Classically this consisted of two-dimensional drawings or building three-dimensional models , while today it most frequently consists of using computers to make...

:

• the method of half-way models: these consist of very special homotopies. This is the original method, first done by Shapiro and Phillips via Boy's surface
  Boy's surface
  In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901...
  
  , later refined by many others. A more recent and definitive refinement (1980s) is minimax eversion
  Minimax eversion
  In geometry, minimax eversions are a class of sphere eversions, constructed by using half-way models.It is a variational method, and consists of special homotopies ; contrast with Thurston's corrugations, which are generic.The original method of half-way models was not optimal: the regular...
  
  s, which is a variational
  Calculus of variations
  Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
  
   method, and consist of special homotopies (they are shortest paths with respect to Willmore energy
  Willmore energy
  In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is defined to be the integral of the mean curvature squared...
  
  ). The original half-way model homotopies were constructed by hand, and worked topologically but weren't minimal.

• Thurston
  William Thurston
  William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
  
  's corrugations: this is a topological method and generic; it takes a homotopy and perturbs it so that it becomes a regular homotopy.

External links

• Outside In, full video (short clip here)
• Optiverse video, portions available online
• A History of Sphere Eversions
• "Turning a Sphere Inside Out"
• Software for visualizing sphere eversion