# Sphere eversion movie

Date started: 2014
Leads: Arnaud Chéritat, Jos Leys, Jean-François Barraud

### Abstract

In the late 1950’s Steve Smale proved a theorem that implies among other things that one can evert the sphere, i.e. that there is a continuous path in the space of smooth maps from $S^2$ to $R^3$ from the canonical immersion (the identity if $S^2 =$ the unit Eudlidean sphere) to the antipodal one (minus the identity), such that these maps are all immersions.

However, Smale’s theorem was quite general and complicated and did not provide an explicit way to perform such an eversion. Many people, since, proposed such ways, and several movies have been made about them. A common feature is that they all are hard to understand.

I would like to make an addition to this list, by realizing a 10 minute movie that explains and shows a way to evert the sphere that I discovered in 2014. (I already have a 1 minute animation without words, but it is hard to figure out what is going on in it.)

### Media

Here is a one minute video I made of my sphere eversion (cut in half):

Video I made explaining contour lines on maps:

Tomography of an orange (A. Chéritat and JF Barraud):

Tomography of a Klein bottle:

1 minute video I made of the sphere eversion: