Sphere eversion

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A Morin surface seen from "above" MorinSurfaceFromTheTop.PNG
A Morin surface seen from "above"
Sphere eversion process as described in [1]
Paper sphere eversion and Morin surface Eversion flat.jpg
Paper sphere eversion and Morin surface
Paper Morin surface (sphere eversion halfway) with hexagonal symmetry Eversion six flat.jpg
Paper Morin surface (sphere eversion halfway) with hexagonal symmetry

In differential topology, sphere eversion is a theoretical process of turning a sphere inside out in a three-dimensional space (the word eversion means "turning inside out"). It is possible to smoothly and continuously turn a sphere inside out in this way (allowing self-intersections of the sphere's surface) 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.

Contents

More precisely, let

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

such that f0 = f and f1 = f.

History

An existence proof for crease-free sphere eversion was first created by StephenSmale  ( 1958 ). 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 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 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.

See h-principle 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. 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:

Minimax sphere eversion; see the video's Wikimedia Commons page for a description of the video's contents
Sphere eversion using Thurston's corrugations; see the video's Wikimedia Commons page for a description of the video's contents

Variations

Surface plots
Ruled model of halfway with quadruple point
Hw-a.png
top view
Hw-b.png
diagonal view
Hw-c.png
side view
Closed halfway
Morin-a.png
top view
Morin-b.png
diagonal view
Morin-c.png
side view
Ruled model of death of triple points
Ttw-b.png
top view
Ttw-b1.png
diagonal view
Ttw-b2.png
side view
Ruled model of end of central intersection loop
Ddw-b.png
top view
Ddw-b1.png
diagonal view
Ddw-b2.png
side view
Ruled model of last stage
Ww-b.png
top view
Ww-b1.png
diagonal view
Ww-b2.png
side view
Nylon string open model
Q-point.jpg
halfway top
Q-point2.jpg
halfway side
T-point.jpg
triple death top
T-point2.jpg
triple death side
D-point.jpg
intersection end top
D-point2.jpg
intersection end side

See also

References

  1. 1 2 Bednorz, Adam; Bednorz, Witold (2019). "Analytic sphere eversion using ruled surfaces". Differential Geometry and Its Applications. 64: 59–79. arXiv: 1711.10466 . doi:10.1016/j.difgeo.2019.02.004. S2CID   119687494.
  2. "Outside In: Introduction". The Geometry Center. Retrieved 21 June 2017.
  3. Goryunov, Victor V. (1997). "Local invariants of mappings of surfaces into three-space". The Arnold–Gelfand mathematical seminars. Boston, Massachusetts: Birkhäuser. pp. 223–255. ISBN   0-8176-3883-0.

Bibliography