Visualize the Eversion of a Sphere

It is famously possible to continuously deform a sphere until it’s inside out. You have to imagine the sphere is something like a bubble, which can ‘pass through’ itself. This transformation is called an eversion.

The fact is established by a theorem due to Stephen Smale (1959 – JSTOR),
which is not constructive — Smale was not able to tell us exactly how we can carry out this deformation. However, recent computer-minimization techniques have allowed us to describe many (constructive) ways to turn a sphere inside out.

The process is difficult for most of us to visualize. Fortunately, three University of Illinois mathematicians have made a 7-minute computer animation that illustrates how it’s done. (Warning: this video is trippy.)



Smale’s theorem says that any two immersions of a sphere a real n-dimensional manifold are regularly homotopic. Thus, in particular, it follows that a sphere sitting in 3-space according to the immersion,



is regular homotopic to the same sphere turned inside out (that is, to inverse immersion). Smale is also well known for having proven higher-dimensional versions of the Poincaré conjecture in the 60’s.


Soul Physics is authored by Bryan W. Roberts. Thanks for subscribing.
Want more Soul Physics? Try the Soul Physics Tweet.

Leave a Reply

Your email address will not be published. Required fields are marked *

You can use standard LaTeX enclosed in dollar signs, e.g., $\sqrt{2}^2=2$ gets rendered as .
You can also use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>