hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su

The Math Fun Facts App!

List All : List Recent : List Popular
About Math Fun Facts / How to Use
Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Advanced level:

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.

Su, Francis E., et al. "Sphere Eversions." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:
The Geometry Center video: Outside In, available here.

Keywords:    analysis, differentiability, topology
Subjects:    geometry, calculus, analysis, topology
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
4.16
current
rating
Click to rate this Fun Fact...
*   Awesome! I totally dig it!
*   Fun enough to tell a friend!
*   Mildly interesting
*   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!