If it looks like a sphere…Exploring the newly proposed solution to a famous problem about three-dimensional shapes

Science News, June 14, 2003 by Erica Klarreich

Poincare's conjecture is one of the simplest possible questions to ask about three-dimensional spaces, yet it has stumped mathematicians from Poincare's time to the present. Surprisingly, higher-dimensional spheres turn out to be more amenable to analysis. Decades ago, mathematicians proved the corresponding conjectures for spheres of four dimensions and higher.

GEOMETRIC BUILDING BLOCKS In the late 1970s, mathematician William Thurston, now at the University of California, Davis, envisioned away to tame the menagerie of three-dimensional spaces--an idea that gave mathematicians a roadmap for proving the Poincare conjecture. The key, Thurston suspected, was in an analogy between the geometry of three-dimensional spaces and that of two-dimensional surfaces.

Every closed surface can be distorted into a particular shape with an especially uniform geometry. For starfish, tables, and telephone poles, that most uniform shape is simply the sphere, which looks the same at every point.

Among tori, the doughnut surface is more homogeneous than the coffee cup, but it is not perfectly uniform. Points on the outer ring are positively curved, like a sphere, while points on the inner ring are negatively curved, like a saddle's central point. However, mathematicians have found a way to conceptualize a completely uniform torus, in which each small patch of the torus has the same geometric structure as a flat piece of paper.

All other two-dimensional surfaces--the tori with multiple holes--can be given what's called hyperbolic geometry, which makes the surfaces negatively curved at all points.

Among closed surfaces, spherical, flat, and hyperbolic geometry are mutually exclusive. Breaking down these surfaces into geometric types thus gives a way to distinguish two-dimensional spheres, for example, from other surfaces. A similar breakdown for three-dimensional spaces, Thurston realized, would give mathematicians a useful tool for distinguishing hyperspheres from other shapes, the goal of the Poincare conjecture.

Mathematicians have known for decades that three-dimensional spaces can't be categorized as neatly as two-dimensional surfaces can. Some spaces, for instance, consist of a hyperbolic chunk and a flat chunk sewn together. Other spaces have geometric structures that don't match any of spherical, flat, or hyperbolic geometry.

In pioneering work, Thurston proposed that there is nevertheless a precise way to classify the geometry of three-dimensional spaces. Each closed space, he conjectured, can be given a special geometric structure built from components selected from eight geometric types. Three of the eight are spherical, flat, and hyperbolic geometry; the other five are slightly more complicated but still uniform geometries. Thurston, who proved large portions of his conjecture, was awarded a Fields Medal--mathematics' version of a Nobel prize--in large part for this body of work.

"What Thurston proposed was a revolutionary idea that went well beyond the Poincare conjecture," Cheeger says.