Sphere eversion

Arnaud Chéritat, C.N.R.S.

Illustrating Mathematics
Providence, July 2016

Setting: deformable surfaces.

To evert a sphere is to turn it inside-out.

A practical way is to poke a hole an pull the surface through it.

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Differential geometers are interested in a trickier way:

on one hand, the surface is allowed not only to distort but also to self-intersect during the process,
on the other hand, you are not allowed to tear nor crease it: it must remain smooth.
The movie Outside-In explains the rules very well.

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It is very hard to evert the sphere with these rules.

In the late 1950s, when Smale discovered that it was possible he surprised everyone.

But his proof was non-constructive:

This posed an irresistible puzzle to geometers.

1966: A. Phillips
Scientific American

1960: A. Shapiro
1979: Francis and Morin

?1966 B. Morin
1979: J.P. Petit and Morin
Pour la science

1970: chickenwire models
(C. Pugh)
1976: Movie (N. Max & co)

19××: D. Hakon
unpublished

1974: B. Thurston
1994: Movie Outside In
Geometry Center

1995: D. Sullivan
1998: Movie Optiverse

2010: Movie Holiverse
I. Atchinson

1996: de Neve
2015: Movie by C. Hills

I learned of the sphere eversion theorem in 1995, when I was 20. I tried several times to evert the sphere without help of a theory, by pure trial and error, but never succeeded. I remember spending many hours that year, sketching things.

Curiously, everting a torus is much easier. I managed with an indication from a friend. After simplification, I drew this sketch:

I was presented the movie Outside-In a couple years later. It was realized by The Geometry Center in 1994, the main idea is due to Bill Thurston. [Show extracts]
I would have preferred to be given more time to think about the puzzle before seeing a solution, but so it is.

The movie is excellent. There is in particular a very well done part on the analog problem in one dimension less. However, I could not understand the central moment. It reinforced my belief that

everting a sphere must be complicated.

Later, I saw a short extract of The Optiverse (John Sullivan, George Francis & Stuart Levy, 1998). [Show extracts]

It is computed automatically by following the flow of a surface energy functionnal. Watching it revealed that eversions can be much simpler than Thurston's. This encouraged me to try again everting the sphere.
... More sketches ... No success.

Much later, I saw Outside-In again.

It was projected during the memorial conference for Bill Thurston.

I could understand more of the process, but there remained a step at the core that is not really visually readable and goes too fast: you have to believe it works.

It is all summed-up in the narrative by the second sentence below:

[him] “That looks complicated...”
[her] “Yes, but the corrugation is just following the guide strip that you saw before.”

Obviously there is a "you can follow" theorem that they did not want to elaborate on.

This motivated me to try again everting the sphere on my own.

To come home from the conference I had a plane at JFK. I decided to take a bus from Cornell Univ to NYC. The ride takes 6h.

The bus was pretty uncomfortable and there was no wifi. I was quite tired but could not really sleep. I decided to spend the 6 hours trying to evert the sphere by merely closing my eyes...

... This was an interesting experience.

The track I explored in the bus was trying to convert a torus eversion into a sphere eversion.

It is indeed easier to evert a torus.
I made a movie in 2008, using POV-Ray.

Almost everything is a piece of simple primitives : cylinders, tori. In some moments in the movie there are joints made of pieces of algebraic surfaces (show source extracts).

Just after the bus drive, I had holidays with family, we decided to go to the beach near Lisboa. I continued thinking hard about the eversion, during the long drives and the quiet moments when basking in the sun. I made a lot of progress but did not get the eversion. The holidays finished and I started sketching what I had come to.

surf inter

cut sur int

Still couldn't evert the damn thing. I had two configurations with a small variation, one of which I could deform into the sphere and the other into the everted sphere. But no way to pass.

Then I watched again Outside-In. I tried to figure-out what it would give if the 8-fold symmetry became 1-fold. In the middle of this effort, came a movement that unlocked my method.

animate

With at last a way to evert the sphere that I could understand and believe, I stopped the 1-fold project. I started to draw a complete sketch of the eversion.

To be able to convince others and myself, I modified again and again the way things were presented, so that it can be sliced and the hard parts become easier to follow in the slices.
Then came a realization: "click".

I realized that the one dimension less analogue, which Outside-In spends half of its time explaining, could help to simplify my proof. Below we define the winding number.

The Withney-Graustein theorem states the winding number is invariant under deformation conversely two curves with the same winding number can be deformed into another.

I just needed to improve a little bit the Whitney-Graustein theorem: the set of curves* with a given winding number is not only connected, but also contractible.

How it works (use blackboard)

Time:
Correct:

My first proof of contractibility used a single corrugation. However, Whitney's genialistically simple formula directly gives an interpolation, quite easy to program.

Untie:

Move loop:

Model

Slices

Smooth surfaces, self-intersecting or not, are called immersed.

The tangent plane at the highest/lowest point of a surface is necessarily horizontal.

Theorem : Assume a closed surface is immersed in space so that its tangent space is horizontal at only two points. Then it can be untied into a Euclidean sphere by the procedure above.

surf

The end