If It Looks Like a Sphere...
Exploring the newly proposed solution to a famous problem about
three-dimensional shapes
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.
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DISTORTED ORB. Each of these stone and bronze objects, created
by artist Allen Linder, can be viewed topologically as a distorted version of a
sphere.
Linder |
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: Available to subscribers at http://www.sciencenews.org/20030426/fob1.asp).
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 Poincar? 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 Poincar? 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 Poincar? conjecture have involved intricate,
hard-to-check arguments. "This one feels 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 Poincar? wondered whether there's an
easy way to tell when a given closed three-dimensional space is a distorted
version of the hypersphere. Poincar? 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.
Poincar?'s conjecture is one of the simplest possible questions to ask about
three-dimensional spaces, yet it has stumped mathematicians from Poincar?'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 a way to tame the menagerie of three-dimensional
spaces?an idea that gave mathematicians a roadmap for proving the Poincar?
conjecture. The key, Thurston suspected, was in an analogy between the geometry
of three-dimensional spaces and that of two-dimensional surfaces.
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KNOTTED GEOMETRY. Helaman Ferguson created this marble
sculpture to celebrate William Thurston's powerful idea that there is a precise
way to classify the geometry of three-dimensional spaces, no matter how tangled
or distorted.
Ferguson |
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 Poincar? 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
Poincar? conjecture," Cheeger says.
Erasing the bar
If Thurston's conjecture can be proved, the Poincar? conjecture will follow
automatically. The logic goes more or less like this: In a closed
three-dimensional space, if all loops of thread can be pulled tight to a point,
mathematicians know that the only one of the eight geometries that can fit the
space is spherical geometry. That means that no matter how convoluted the space
appears, it must simply be a distorted version of the hypersphere.
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WILD SPHERE. In the 1920s, Princeton mathematician J.W.
Alexander imagined a wildly distorted sphere, which sprouts two arms that reach
out to each other yet never touch. These arms, in turn, each sprout a pair of
fingers, and the fingers each sprout a pair of even smaller extensions, and so
on. Despite this complexity, the object is still topologically a sphere. Artist
and mathematician Helaman Ferguson captured some of the intricacy of Alexander's
"horned sphere" in this bronze sculpture.
Ferguson |
After Thurston's work, mathematicians who wanted to prove the Poincar?
conjecture could focus on demonstrating that Thurston's vision of
three-dimensional spaces is correct. By the early 1990s, Richard Hamilton of
Columbia University had proposed a technique that he hoped would do just
that?show that each three-dimensional space can be smoothed out into Thurston's
special pieces. He defined a method, called the Ricci flow, for changing the
shape gradually at each point to make the space more uniform. His equation
resembles the physics equation that describes how heat spreads through a
material.
"If you take a body where parts are hot and parts are cold and you 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 in some places,
maybe flat or negatively curved in other places, and you just let the curvature
flow and try to even itself out."
For instance, the Ricci flow would make an egg-shaped surface gradually
flatten out on the ends and bulge even more in the middle, getting closer and
closer to a perfect sphere.
Hamilton was aware, however, that the flow would not always produce a uniform
geometry. At any point in the space, the flow is determined mainly by the local
geometry, not by the overall shape of the space. So, sometimes the geometry of
one part of the space might change much faster than that of another part,
producing a highly uneven geometry overall.
For example, picture a dumbbell?two weights connected by a thin bar?each
portion of which is flowing with a mind of its own. The bar wants to even out
its geometry with the weights to turn the whole thing into a nicely rounded
sphere. Each weight, on the other hand, wants to make itself as spherical as
possible. In the three-dimensional version of the dumbbell, depending on the
initial geometry, the weights may predominate, growing rounder and rounder while
the bar stretches into a long, thin neck.
Hamilton's idea for dealing with this difficulty was simply to snip out the
neck at some appropriate point, continue the Ricci flow on the pieces, and glue
the neck back in at the end. The resulting shape would have the right kinds of
building blocks for Thurston's conjecture. But for more complicated shapes than
the dumbbell, he couldn't show that these necks were the only extreme geometric
forms the flow would produce. Other extremities, such as awkward protrusions he
called cigars, might result.
What's more, perhaps every time the flow evened out one portion of the space,
that portion's extreme shape would have moved somewhere else, like bulges in a
rug that is being fit into a room too small for it. Extreme geometric features
might cycle around and around, without the whole space ever growing uniform.
These questions dogged Hamilton and his followers for more than a decade.
Then last November, Perelman sent several mathematicians an e-mail, saying only
that he had posted a paper on the Internet that might be of interest to them. In
the paper, he writes that his work "removes the major stumbling block in
Hamilton's approach to geometrization." Although the posted paper makes no
reference to the Poincar? conjecture, experts in the field immediately realized
what he was driving at.
Music of the spheres
In the early 1990s, working in the United States, Perelman had emerged as a
major player in Riemannian geometry, which studies subjects such as curvature.
"In that domain he was considered a phenomenon at that time, incredibly
brilliant," recalls Cheeger.
Then abruptly, Perelman all but vanished from the mathematical scene. In
1995, he turned down job offers from several top universities and returned to
Russia. When U.S. mathematicians asked Perelman's colleagues at the Steklov
Institute what he was working on, they generally replied that they had no clue.
Some mathematicians speculated that Perelman had quit mathematics. Every now
and then, however, one or another mathematician would receive an e-mail from
Perelman with probing, insightful questions. "All of a sudden, there would be
concrete evidence that he was following certain developments," Cheeger says.
Once Perelman's first paper on the Ricci flow appeared on the Internet in
November 2002, rumors started flying that he had proven the Poincar? conjecture
and Thurston's geometrization conjecture. On March 10, Perelman posted a second
paper that developed the ideas in his first paper and explicitly claimed a proof
of the two conjectures. He has promised a third paper with a few remaining
details.
This spring, Perelman visited the United States to present lectures on his
work in Cambridge, Mass., and Stony Brook. So far, he has answered all the
questions raised about his work, 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-dimensional space. Roughly, he
described what the pitch of a space would be if someone could ring the space
like a bell. Perelman then proved that as the space slowly morphs under the
Ricci flow, its pitch gets higher and higher.
Perelman's result immediately shows that the geometry of a space can't cycle
around under the Ricci flow?if it did, its pitch would be unchanged after each
cycle. Perelman claims that the result about pitch, together with other ideas
that he develops in his papers, also does away with the possibility of cigars
and other potential obstacles to carrying out Hamilton's program.
"Perelman's results are as spectacular as the Poincar? conjecture," says
Dennis Sullivan, a mathematician at Stony Brook. "In just a few pages of work,
he puts a hand grenade in the brick wall Hamilton had run into and blows a hole
through it. Whether that has enabled him to crawl through to the meadow on the
other side remains to be seen."
Many mathematicians have accepted the correctness of Perelman's result about
the pitch of a space, but they have not finished studying the portions of
Perelman's papers that explore the ramifications of the result. Once Perelman's
papers have been published, if no one exposes a hole in his work within 2 years,
he will be eligible for the Clay Institute's prize.
For many mathematicians, however, the appeal of the Poincar? conjecture lies
beyond the million-dollar prize and accompanying fame. "It's important for the
same reason Beethoven's Ninth Symphony is important," Sullivan says. "It's
great."
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References:
Perelman, G. Preprint. The entropy formula for the Ricci flow
and its geometric applications. Available at http://xxx.lanl.gov/abs/math.DG/0211159.
______. Preprint. Ricci flow with surgery on three-manifolds.
Available at http://xxx.lanl.gov/abs/math.DG/0303109.
Further Readings:
Klarreich, E. 2003. Spheres in disguise: Solid proof offered
for famous conjecture. Science News 163(Apr. 26):259. Available to
subscribers at http://www.sciencenews.org/20030426/fob1.asp.
Information about the $1 million prize offered by the Clay
Mathematics Institute and the Poincar? conjecture can be found at http://www.claymath.org/Millennium_Prize_Problems/.
Examples of Helaman Ferguson's mathematical artworks can be
seen at http://www.helasculpt.com/.
Sources:
Jeff Cheeger
Department of Mathematics
Courant
Institute of Mathematical Sciences
New York University
251 Mercer
Street
New York, NY 10012
Richard Hamilton
Department of Mathematics
Columbia
University
2990 Broadway
509 Mathematics Building
Mail Code:
4406
New York, NY 10027
John W. Milnor
Department of Mathematics
SUNY Stony
Brook
Stony Brook, NY 11794-3651
Dennis Sullivan
Department of Mathematics
SUNY Stony
Brook
Stony Brook, NY 11794-3651
Bill Thurston
Mathematics Department
University of
California, Davis
One Shields Avenue
Davis, CA 95616
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