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IF IT LOOKS LIKE A SPHERE...Exploring the newly proposed solution to a famous problem

about three-dimensional shapesBY ERICA KLARREICH

Look around at the world, and the objects in it—buildings, trees, people, birds, insects—appearto come in an endless variety of shapes. At first,cataloging these diverse shapes may seemimpossible. But on closer inspection, relation-

ships emerge. The bumpy surface of a starfish, forexample, is simply a stretched and distorted version ofa sphere. The same goes for the surface of a table or a telephone pole. In contrast, a coffee cup is not a sphere but insteada distorted version of a doughnut, and a pretzel can be considereda doughnut with three holes instead of one.

What about more complicated shapes likea fishnet or a bicycle wheel? Amazingly,more than a hundred years ago, math-ematicians proved that every closedsurface in space is simply someversion of a sphere, a doughnutsurface—which they call a torus—or a torus with extra holes.

Even though spheres and torisit in three-dimensional space,mathematicians focus on their sur-faces and so view them as two-dimen-sional, unlike solid balls and filled-indoughnuts, which are three-dimensional. Asmall patch of a sphere or torus surface looks almost likea piece of a flat plane and has area rather than volume.

Mathematicians also study an analogous collection ofwhat they call closed three-dimensional shapes. Unlikeordinary three-dimensional objects, these shapes live infour-dimensional—or higher—space and curve in on themselvesas the sphere and torus do in three-dimensional space. Althoughsuch shapes are difficult to visualize, some cosmologists speculatethat our own universe may be of that form, rather than the infi-nitely extending space that most people envision.

For a century, mathematicians have wondered whether there’sa classification of three-dimensional shapes like the simple break-down of two-dimensional shapes into spheres and tori. Now, aRussian mathematician may finally have proved that the answeris yes (SN: 4/26/03, p. 259). Details are starting to emerge of hiswork, which gives a way to distort a three-dimensional object, lit-tle 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 thePoincaré conjecture, a 99-year-old question about three-dimen-sional shapes that’s one of the most famous problems in mathe-matics. After working for years in near seclusion and supportinghimself largely on personal savings, Grigory Perelman of the SteklovInstitute of Mathematics in St. Petersburg, Russia, announced thathe has proved the conjecture, which gives a way to identify whether

a complicated shape is a distorted version of a sphere. He alsoclaims to have proved the much broader Thurston geometrizationconjecture, which considers all closed three-dimensional shapes.

Over the years, dozens of mathematicians have mistakenlyclaimed to have proved the Poincaré conjecture. For this reason,mathematicians—including Perelman himself—are not rushingto judgment. Perelman has declined to talk to the press until col-leagues verify his proof.

It will take months, some mathematicians say, to dissect thedetails of Perelman’s densely written papers. But Perelman’s trackrecord makes many optimistic that his work will stand up toscrutiny. “He’s singularly brilliant,” says Jeff Cheeger of the CourantInstitute of Mathematical Sciences at New York University. What’s

more, Perelman’s colleagues note, the portions of hiswork that have already been verified are full

of groundbreaking ideas.“Whether or not he has a completeproof, he has clearly made very

important contributions to math-ematics,” says John Milnor, amathematician at the State Uni-versity of New York at StonyBrook who attended a series of

lectures Perelman gave there inApril and May. Many past attempts to prove the

Poincaré conjecture have involved intri-cate, hard-to-check arguments. “This onefeels like a much more natural, verypromising approach,” Milnor says. “Itseems like the right way to handle theproblem.”

RECOGNIZING THE HYPERSPHERE Even though a sphereand a torus are two-dimensional to mathematicians, there’s noway to fit them into a flat plane without squashing them. Similarly,some three-dimensional shapes can’t fit comfortably into ordinarythree-dimensional space.

For instance, just as the sphere is the two-dimensional bound-ary of the three-dimensional ball, mathematicians have defined thehypersphere as the three-dimensional boundary of the four-dimen-sional ball—a space that’s hard to visualize but that can neverthe-less be analyzed mathematically. Researchers have also discovereda three-dimensional analog of the torus, as well as an infinitelylarge family of more exotic three-dimensional spaces.

Around 1900, French mathematician Henri Poincaré wonderedwhether there’s an easy way to tell when a given closed three-dimensional space is a distorted version of the hypersphere. Poin-caré made a daring conjecture. To recognize a hypersphere, heguessed, all that’s needed is information about one-dimensionalcurves in the space. If every closed loop of thread in the space canbe drawn in to a single point, then the space is a hypersphere indisguise, he hypothesized. On a torus, by contrast, a loop that goes

DISTORTED ORB — Each ofthese stone and bronzeobjects, created by artistAllen Linder, can be viewedtopologically as a distortedversion of a sphere.

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around the hole can’t be pulled tight to a single point.Poincaré’s conjecture is one of the simplest possible questions

to ask about three-dimensional spaces, yet it has stumped math-ematicians from Poincaré’s time to the present. Surprisingly, higher-dimensional spheres turn out to be more amenable to analysis.Decades ago, mathematicians proved the corresponding conjec-tures for spheres of four dimensions and higher.

GEOMETRIC BUILDING BLOCKS In the late 1970s, mathe-matician William Thurston, now at the University of California,Davis, envisioned a way to tame the menagerie of three-dimensionalspaces—an idea that gave mathematicians a roadmap for provingthe Poincaré conjecture. The key, Thurston suspected, was in ananalogy between the geometry of three-dimensional spaces and thatof two-dimensional surfaces.

Every closed surface can be distorted intoa particular shape with an especially uni-form geometry. For starfish, tables, andtelephone poles, that most uniform shapeis simply the sphere, which looks the sameat every point.

Among tori, the doughnut surface ismore homogeneous than the coffee cup,but it is not perfectly uniform. Points onthe outer ring are positively curved, like asphere, while points on the inner ring arenegatively curved, like a saddle’s centralpoint. However, mathematicians havefound a way to conceptualize a completelyuniform torus, in which each small patchof the torus has the same geometric struc-ture as a flat piece of paper.

All other two-dimensional surfaces—thetori with multiple holes—can be given what’scalled hyperbolic geometry, which makesthe surfaces negatively curved at all points.

Among closed surfaces, spherical, flat,and hyperbolic geometry are mutuallyexclusive. Breaking down these surfacesinto geometric types thus gives a way todistinguish two-dimensional spheres, forexample, from other surfaces. A similarbreakdown for three-dimensional spaces,Thurston realized, would give mathemati-cians a useful tool for distinguishing hyper-spheres from other shapes, the goal of thePoincaré conjecture.

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

In pioneering work, Thurston proposed that there is neverthe-less a precise way to classify the geometry of three-dimensionalspaces. Each closed space, he conjectured, can be given a specialgeometric structure built from components selected from eightgeometric types. Three of the eight are spherical, flat, and hyper-bolic geometry; the other five are slightly more complicated butstill uniform geometries. Thurston, who proved large portions ofhis conjecture, was awarded a Fields Medal—mathematics’ versionof a Nobel prize—in large part for this body of work.

“What Thurston proposed was a revolutionary idea that wentwell beyond the Poincaré conjecture,” Cheeger says.

ERASING THE BAR If Thurston’s conjecture can be proved, thePoincaré conjecture will follow automatically. The logic goes moreor less like this: In a closed three-dimensional space, if all loops ofthread can be pulled tight to a point, mathematicians know that the

only one of the eight geometries that can fit the space is sphericalgeometry. That means that no matter how convoluted the spaceappears, it must simply be a distorted version of the hypersphere.

After Thurston’s work, mathematicians who wanted to provethe Poincaré conjecture could focus on demonstrating thatThurston’s vision of three-dimensional spaces is correct. By theearly 1990s, Richard Hamilton of Columbia University had pro-posed a technique that he hoped would do just that—show thateach three-dimensional space can be smoothed out into Thurston’sspecial pieces. He defined a method, called the Ricci flow, forchanging the shape gradually at each point to make the spacemore uniform. His equation resembles the physics equation thatdescribes how heat spreads through a material.

“If you take a body where parts are hot and parts are cold andyou let it stand, heat tends to flow by itself until the temperature is

even,” Milnor says. “In Hamilton’s process,you have a manifold that is very curved insome places, maybe flat or negatively curvedin other places, and you just let the curva-ture flow and try to even itself out.”

For instance, the Ricci flow would makean egg-shaped surface gradually flattenout on the ends and bulge even more in themiddle, getting closer and closer to a per-fect sphere.

Hamilton was aware, however, that theflow would not always produce a uniformgeometry. At any point in the space, theflow is determined mainly by the localgeometry, not by the overall shape of thespace. So, sometimes the geometry of onepart of the space might change much fasterthan that of another part, producing ahighly uneven geometry overall.

For example, picture a dumbbell—twoweights connected by a thin bar—each por-tion of which is flowing with a mind of itsown. The bar wants to even out its geom-etry with the weights to turn the wholething into a nicely rounded sphere. Eachweight, on the other hand, wants to makeitself as spherical as possible. In the three-dimensional version of the dumbbell,depending on the initial geometry, theweights may predominate, growingrounder and rounder while the barstretches into a long, thin neck.

Hamilton’s idea for dealing with this difficulty was simply tosnip out the neck at some appropriate point, continue the Ricci flowon the pieces, and glue the neck back in at the end. The resultingshape would have the right kinds of building blocks for Thurston’sconjecture. But for more complicated shapes than the dumbbell,he couldn’t show that these necks were the only extreme geomet-ric forms the flow would produce. Other extremities, such as awk-ward protrusions he called cigars, might result.

What’s more, perhaps every time the flow evened out one por-tion of the space, that portion’s extreme shape would have movedsomewhere else, like bulges in a rug that is being fit into a roomtoo small for it. Extreme geometric features might cycle aroundand around, without the whole space ever growing uniform.

These questions dogged Hamilton and his followers for morethan a decade. Then last November, Perelman sent several math-ematicians an e-mail, saying only that he had posted a paper on theInternet that might be of interest to them. In the paper, he writesthat his work “removes the major stumbling block in Hamilton’sapproach to geometrization.” Although the posted paper makes noreference to the Poincaré conjecture, experts in the field immedi-ately realized what he was driving at.

WILD SPHERE — In the 1920s, Princetonmathematician J.W. Alexander imagined awildly distorted sphere, which sprouts twoarms that reach out to each other yet nevertouch. These arms, in turn, each sprout a pairof fingers, and the fingers each sprout a pairof even smaller extensions, and so on.Despite this complexity, the object is stilltopologically a sphere. Artist and mathemati-cian Helaman Ferguson captured some ofthe intricacy of Alexander’s “horned sphere”in this bronze sculpture.

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MUSIC OF THE SPHERES In the early 1990s, working in theUnited States, Perelman had emerged as a major player in Rie-mannian geometry, which studies subjects suchas curvature. “In that domain he was considereda phenomenon at that time, incredibly brilliant,”recalls Cheeger.

Then abruptly, Perelman all but vanishedfrom the mathematical scene. In 1995, he turneddown job offers from several top universities andreturned to Russia. When U.S. mathematiciansasked Perelman’s colleagues at the Steklov Insti-tute what he was working on, they generallyreplied that they had no clue.

Some mathematicians speculated that Perel-man had quit mathematics. Every now and then,however, one or another mathematician wouldreceive an e-mail from Perelman with probing,insightful questions. “All of a sudden, therewould be concrete evidence that he was follow-ing certain developments,” Cheeger says.

Once Perelman’s first paper on the Ricci flowappeared on the Internet in November 2002,rumors started flying that he had proven thePoincaré conjecture and Thurston’sgeometrization conjecture. On March 10, Perel-man posted a second paper that developed theideas in his first paper and explicitly claimeda proof of the two conjectures. He has prom-ised a third paper with a few remaining details.

This spring, Perelman visited the United Statesto present lectures on his work in Cambridge, Mass., and StonyBrook. So far, he has answered all the questions raised about hiswork, several mathematicians told Science News.

To understand the behavior of the Ricci flow, Perelman devised

a way to capture a specific characteristic of any three-dimensionalspace. Roughly, he described what the pitch of a space would be if

someone could ring the space like a bell. Perel-man then proved that as the space slowly morphsunder the Ricci flow, its pitch gets higher andhigher.

Perelman’s result immediately shows that thegeometry of a space can’t cycle around under theRicci flow—if it did, its pitch would be unchangedafter each cycle. Perelman claims that the resultabout pitch, together with other ideas that hedevelops in his papers, also does away with thepossibility of cigars and other potential obsta-cles to carrying out Hamilton’s program.

“Perelman’s results are as spectacular as thePoincaré conjecture,” says Dennis Sullivan, amathematician at Stony Brook. “In just a fewpages of work, he puts a hand grenade in thebrick wall Hamilton had run into and blows ahole through it. Whether that has enabled himto crawl through to the meadow on the other sideremains to be seen.”

Many mathematicians have accepted the cor-rectness of Perelman’s result about the pitch ofa space, but they have not finished studying theportions of Perelman’s papers that explore theramifications of the result. Once Perelman’spapers have been published, if no one exposes ahole in his work within 2 years, he will be eligi-ble for the Clay Institute’s prize.

For many mathematicians, however, the appeal of the Poincaréconjecture lies beyond the million-dollar prize and accompanyingfame. “It’s important for the same reason Beethoven’s Ninth Sym-phony is important,” Sullivan says. “It’s great.” � FE

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KNOTTED GEOMETRY — Hela-man Ferguson created this mar-ble sculpture to celebrate WilliamThurston’s powerful idea thatthere is a precise way to classifythe geometry of three-dimen-sional spaces, no matter howtangled or distorted.

ASTONOMY

Galactic RAVE

Viewing galaxies so distant that the lightnow reaching Earth reveals what theylooked like billions of years ago isn’t theonly way to learn about how galaxies form.Astronomers can examine a much closerspecimen—our own Milky Way.

In contrast to previous Milky Way proj-ects, which measured the motions of some2.5 million stars as they march across thesky, a survey that began in April will trackthe movement of stars toward or awayfrom Earth. By measuring this componentof motion, which is currently known foronly 20,000 stars, astronomers plan toreconstruct more details about how theMilky Way formed.

The survey, known as RAVE (RadialVelocity Experiment), uses a 1.2-meter tel-escope in Coonabarabran, Australia. By2005, scientists expect RAVE to have meas-ured the radial motion of 100,000 stars.With these data, astronomers will identifydozens, perhaps hundreds, of star group-ings that appear to be streaming coher-ently, says RAVE leader Matthias Stein-metz of the Astrophysical Institute inPotsdam, Germany.

Since these streams represent the rem-nants of small satellite galaxies that weresnared by the Milky Way billions of yearsago, they’ll indicate how the galaxy’s com-ponents assembled. —R.C.

BEHAVIOR

Toddlers ride rail to tool use

At 16 months of age, many children adaptthe way they use a handrail as they walkacross a perilously narrow bridge to reach

their parents on the other side. These on-the-fly changes that keep them from fallingrepresent an early example of tool use, ahallmark of human intelligence, concludetwo psychologists in the May Developmen-tal Psychology.

Sarah E. Berger of Adelphi University inGarden City, N.Y., and Karen E. Adolph ofNew York University studied 24 boys and24 girls. Most had been walking for about4 months. Each 16-month-old had a seriesof chances to walk from one platform toanother across a 29-inch-long woodenboard that was either narrow (5 or 7 inchesacross) or wide (from 14 to 28 inchesacross). On half the trials, a handrail wasplaced on one side of the bridge. An exper-imenter followed alongside children toensure their safety.

Toddlers always tried to walk across widebridges, rarely touching the handrail whenit was available. Most reached their desti-nation on their own.

In contrast, toddlers often stayed off nar-row bridges that lacked a handrail but usu-ally attempted to cross those that had one.

OFNOTE

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