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 3space? Is it possible to
turn the sphere "inside out", allowing selfintersections
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.
<http://www.math.hmc.edu/funfacts>.
