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

Look around at the world, and the objects in it--buildings, trees, people, birds, insects--appear to come in an endless variety of shapes. At first, cataloging these diverse shapes may seem impossible. But on closer inspection, relationships emerge. The bumpy surface of a starfish, for example, is simply a stretched and distorted version of a sphere. The same goes for the surface of a table or a telephone pole. In contrast, a coffee cup is not a sphere but instead a distorted version of a doughnut, and a pretzel can be considered a doughnut with three holes instead of one.

What about more complicated shapes like a fishnet or a bicycle wheel? Amazingly, more than a hundred years ago, mathematicians proved that every closed surface in space is simply some version of a sphere, a doughnut surface--which they call a torus--or a torus with extra holes.

Even though spheres and tori sit in three-dimensional space, mathematicians focus on their surfaces and so view them as two-dimensional, unlike solid balls and filled-in doughnuts, which are three-dimensional. A small patch of a sphere or torus surface looks almost like a piece of a flat plane and has area rather than volume.

Mathematicians also study an analogous collection of what they call closed three-dimensional shapes. Unlike ordinary three-dimensional objects, these shapes live in four-dimensional--or higher--space and curve in on themselves as the sphere and torus do in three-dimensional space. Although such shapes are difficult to visualize, some cosmologists speculate that our own universe may be of that form, rather than the infinitely extending space that most people envision.

For a century, mathematicians have wondered whether there's a classification of three-dimensional shapes like the simple breakdown of two-dimensional shapes into spheres and tori. Now, a Russian mathematician may finally have proved that the answer is yes (SN: 4/26/03, p. 259). Details are starting to emerge of his work, which gives a way to distort a three-dimensional object, little by little, to make its shape more uniform.

A few years ago, the Clay Mathematics Institute in Cambridge, Mass., offered a $1 million bounty to anyone who could settle the Poincare conjecture, a 99-year-old question about three-dimensional shapes that's one of the most famous problems in mathematics. After working for years in near seclusion and supporting himself largely on personal savings, Grigory Perelman of the Steklov Institute of Mathematics in St. Petersburg, Russia, announced that he has proved the conjecture, which gives a way to identify whether a complicated shape is a distorted version of a sphere. He also claims to have proved the much broader Thurston geometrization conjecture, which considers all closed three-dimensional shapes.

Over the years, dozens of mathematicians have mistakenly claimed to have proved the Poincare conjecture. For this reason, mathematicians--including Perelman himself--are not rushing to judgment. Perelman has declined to talk to the press until colleagues verify his proof.

It will take months, some mathematicians say, to dissect the details of Perelman's densely written papers. But Perelman's track record makes many optimistic that his work will stand up to scrutiny. "He's singularly brilliant," says Jeff Cheeger of the Courant Institute of Mathematical Sciences at New York University. What's more, Perelman's colleagues note, the portions of his work that have already been verified are full of groundbreaking ideas.

"Whether or not he has a complete proof, he has clearly made very important contributions to mathematics," says John Milnor, a mathematician at the State University of New York at Stony Brook who attended a series of lectures Perelman gave there in April and May.

Many past attempts to prove the Poincare conjecture have involved intricate, hard-to-check arguments. "This one fells like a much more natural, very promising approach," Milnor says. "It seems like the right way to handle the problem."

RECOGNIZING THE HYPERSPHERE Even though a sphere and a torus are two-dimensional to mathematicians, there's no way to fit them into a flat plane without squashing them. Similarly, some three-dimensional shapes can't fit comfortably into ordinary three-dimensional space.

For instance, just as the sphere is the two-dimensional boundary of the three-dimensional ball, mathematicians have defined the hypersphere as the three-dimensional boundary of the four-dimensional ball--a space that's hard to visualize but that can nevertheless be analyzed mathematically. Researchers have also discovered a three-dimensional analog of the torus, as well as an infinitely large family of more exotic three-dimensional spaces.

Around 1900, French mathematician Henri Poincare wondered whether there's an easy way to tell when a given closed three-dimensional space is a distorted version of the hypersphere. Poincard made a daring conjecture. To recognize a hypersphere, he guessed, all that's needed is information about one-dimensional curves in the space. If every closed loop of thread in the space can be drawn in to a single point, then the space is a hypersphere in disguise, he hypothesized. On a torus, by contrast, a loop that goes around the hole can't be pulled tight to a single point.