# Ivars Peterson's MathTrek

February 9, 2004

## Turning a Snowball Inside Out

Turning a sphere inside out without allowing any sharp creases along the way is a tricky mathematical maneuver. Carving an intricate snow sculpture depicting a crucial step in this twisty transformation presents its own difficulties.

This was the challenge facing a team led by mathematician Stan Wagon of Macalester College in St. Paul, Minn., last month at the 14th International Snow Sculpture Championship in Breckenridge, Colo. In the end, after 5 days of arduous labor, the team managed to shape a 12-foot-high block of snow into a daring, prize-winning creation.

Turning a Snowball Inside Out: An award-winning snow sculpture. Photo courtesy of Carlo Séquin

Turning a sphere inside out is harder than it sounds. In principle, a determined beachgoer can do it to a beach ball by deflating the ball, pulling the empty bag through its opening, and pumping up the ball again. The task for mathematicians is more difficult: The perfect sphere that they work with has no orifice, and the rules are different.

Imagine a ball made of a ghostly membrane that can stretch, bend, and pass through itself. The idea is to turn such a sphere inside out without puncturing, ripping, or creasing it.

You could try simply pushing the poles of a sphere toward each other, as if to make them pass through each other and change places. At some point, however, the distorted surface would develop a sharp kink, and that's not allowed, according to the mathematician's rules. Avoiding such a kink makes the task of exchanging a sphere's inner and outer surface—called an eversion—a challenging mathematical puzzle.

No one knew if a sphere eversion was even possible until 1959, when Stephen Smale proved a theorem that indirectly leads to the proposition that it could be done. However, Smale's step-by-step path for accomplishing a sphere eversion was so complicated that no one could visualize his procedure.

Gradually, visual answers to the sphere-eversion problem began to emerge, and mathematicians continued to look for increasingly simple ways of describing and displaying how the change occurs.

In the 1970s, the task fell to French topologist Bernard Morin, who is blind. Morin put together what can be thought of as a set of architectural plans for a sequence of three-dimensional constructions showing the essential steps in a sphere eversion. It was the halfway point in Morin's famous representation that ended up as a snow sculpture.

This was Wagon's sixth entry in the prestigious Breckenridge competition. He was joined this time by software engineer Dan Schwalbe, computer science student Alex Kozlowski of the University of California, Berkeley, and mathematician John M. Sullivan of the University of Illinois at Urbana-Champaign. Berkeley computer scientist Carlo H. Séquin designed the sculpture and served as team representative.

In Séquin's design, a lattice structure of struts and bands represented a sphere's inside surface and solid material represented its outside surface. The resulting form elegantly captured both the tension and the swirled flow evident at the midpoint of a complex transformation.

Out of the 14 entries in this year's competition, Turning a Snowball Inside Out earned an honorable mention for "most ambitious" sculpture. That's a fine showing for an intricate, challenging piece of mathematics!

References:

Peterson, I. 2003. A graceful sculpture's showy snow crash. MAA Online (Feb. 10).

______. 2002. A snowy twist. MAA Online (Feb. 18).

______. 2001. White narcissus. MAA Online (Feb. 12).

______. 2000. A minimal winter's tale. MAA Online (Feb. 7).

______. 1999. Minimal snow. MAA Online (March 8).

______. 1998. Surreal films. Science News 154(Oct. 10):232-234. Available at http://www.sciencenews.org/sn_arc98/10_10_98/Bob1.htm.

______. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York: W.H. Freeman.

______. 1889. Inside moves. Science News 135(May 13):299.

Mathematician Stan Wagon has a Web site at http://www.stanwagon.com/, with links to pages devoted to the annual Breckenridge International Snow Sculpture competition.

John Sullivan's account of the snow sculpting competition can be found at http://torus.math.uiuc.edu/jms/Snow/.

Carlo H. Séquin has a Web site at http://www.cs.berkeley.edu/~sequin/. His illustrated account of the team's snow-sculpting effort is available at http://www.cs.berkeley.edu/~sequin/SCULPTS/SnowSculpt04/.

Additional information about sphere eversions can be found at http://mathworld.wolfram.com/SphereEversion.html and http://torus.math.uiuc.edu/optiverse/descr.html.