Sphere Eversions

If you take a loop of string in the plane and place an arrow along it pointing clockwise, is it possible to deform the string, keeping it in the plane, so that the arrow points counterclockwise, without causing any kinks in the string?

A moment’s reflection seems to indicate that this is impossible. Is it? Can you prove it?

What about a sphere in 3-space? Is it possible to turn the sphere “inside out”, allowing self-intersections but not allowing any sharp kinks, creases, or tearing of the surface?

Surprisingly, in 1957, Steve Smale proved that this is in fact possible! Such an operation is called a sphere eversion. And later on, several people constructed explicit methods for doing so, among them William Thurston.

The reference contains a video that shows the Thurston eversion of a sphere!

The Math Behind the Fact:
Requiring that loops in the plane have no kinks is equivalent to giving them unit speed parametrizations and requiring that the parametrizations are continuously differentiable, i.e., their rates of change vary continuously. The study of differentiable structures on geometric objects is called differential geometry and the study of smooth deformations of such objects is often called differential topology.

How to Cite this Page: 
Su, Francis E., et al. “Sphere Eversions.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

The Geometry Center video: Outside In, available here.

Fun Fact suggested by:
Francis Su

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