A new twist on outside in.

It's a regal but ghostly transformation: A golden ball turns itself inside out to reveal its purple inner surface without suffering any rips or creases along the way. Mathematicians call this process sphere eversion. The sphere acts as if it were made of a stretchy though delicate material that readily passes through itself but self-destructs if punctured or sharply pinched.

Until 1957, mathematicians were unsure whether it was possible to turn a sphere inside out without making a hole. Then, Stephen Smale of the University of California, Berkeley, proved that such an operation is feasible, although his proof furnished no clear picture of how to do it.

In subsequent years, mathematicians developed a number of different ways to visualize sphere eversion, gradually simplifying the steps to make it easier to follow the process (SN: 5/13/89, p.299; 6/20/92, p.404). The latest version comes from Silvio Levy, Delle Maxwell, and Tamara Munzner of the Geometry Center at the University of Minnesota in Minneapolis, who have created a dramatic computer animation that shows a sphere eversion in its full glory.

Levy and his coworkers based their visualization on a geometric technique developed by William P. Thurston of the Mathematical Sciences Research Institute in Berkeley, Calif., to help understand certain smooth curves and surfaces known as immersions. Thurston found it useful to imagine such curves and surfaces as springy, meaning they could be moved and bent at will. This strategy allowed him to introduce corrugations--wavy bends--to make these shapes extremely pliable, gaining insights into how immersions maintain their smoothness during transformations such as eversions.

In an eversion, a sphere's initially unwrinkled surface develops a symmetric set of bulges, or corrugations (see illustration). The poles push part way through each other, creating loops at the equator and revealing patches of the sphere's purple inside. The two polar caps then twist in opposite directions to undo the loops, and the equatorial region collapses and pushes through itself. Finally, the corrugations disappear, and the eversion is complete.

Why turn a sphere inside out? "The short answer is that it is a mathematical puzzle that is interesting and counterintuitive, and therefore, challenging to solve," Maxwell explains. This exercise and the corrugation technique also help to elucidate various aspects of the mathematical classification of surfaces.