Drumming Up Answer to Math Mystery
Christopher Sogge and Steven
Zelditch at the blackboard. "It's pure thought research,"
Sogge says. "We do it by thinking and
brainstorming."
PHOTO BY HPS/WILL KIRK

Profs aim to solve 200yearold puzzle about vibrating
shapes
By Lisa De Nike Homewood
A $975,398 National Science Foundation grant is
allowing a team including mathematicians at Johns Hopkins'
Krieger School of Arts and Sciences to tackle a
200yearold question: How does the shape of drums
influence the frequency and geometry of their vibrations
when they are struck?
The $402,000 portion of the grant coming to Johns
Hopkins is the largest ever awarded to faculty in the
Department of
Mathematics, according to Christopher Sogge, department
chair, who has teamed up with Professor Steven Zelditch for
the threeyear investigation.
The Johns Hopkins team is sharing the grant with
fellow mathematicians Daniel Tataru and Maciej Zworski at
University of California, Berkeley, and Hart Smith at the
University of Washington, who are taking on the same
challenge. According to Sogge, the purpose of the grant is
to get likeminded researchers collaborating.
"At the heart of our project is a 200yearold
question that remains largely unanswered today," Sogge
says. "Mathematicians are interested in what can be known,
and even better, proven, about vibrating shapes. This is
important in mathematics and physics because it deals with
what we call 'modes of vibration' of objects ranging from
drums to atoms and molecules to the whole universe."
The question dates back to the early 19th century,
when a German scientist named Ernst Chladni impressed
Emperor Napoleon Bonaparte with his ability to make sound
waves "visible" using a simple experiment. When Chaldni
sprinkled sand onto metal plates and drew a violin bow
across the edges, the grains scattered and settled into
intricate geometric designs on the portions that were not
shaking with sound. Intrigued by those results, the emperor
offered a reward to anyone who could explain how the sand's
patternsand the invisible sound waves that produced
themrelated to the shape of the metal surface upon which
they settled.
Sogge and Zelditch have chosen to study various drum
shapes because their contours provide a simple model for
any vibrating object, including atoms and molecules. The
general question of how the frequencies and shapes of
vibration reflect the shape of the object is the same,
whether one is talking about drums or atoms, they say.
"The shapes we are dealing with range from the kind a
drummer would use in a rock band to drums of any dimension
and shape," Zelditch says. "We are especially interested in
what we call 'extreme drums': those capable of making
extreme sounds at a given frequency. Extreme to most people
would mean 'loud.' But we think geometrically."
It's important to note that no actual drums are
involved in the team's inquiry. Sogge and Zelditch are
operating entirely in the realm of the theoretical: Their
only handson work involves putting chalk to blackboard or
pen to paper in the quest for the formula or proof that
solves the question at hand. And though some mathematicians
and physicists are tackling this challenge with the help of
computers, the Johns Hopkins pair is using oldfashioned
brainpower.
"It's pure thought research; we do it by thinking and
brainstorming," Sogge says. "The best way to visualize what
we are doing is to imagine playing billiards on a drumhead
and using math to predict and describe the trajectory of
the ball when you hit it. For instance, the usual billiard
table is rectangular, and the path of a ball hit upon it
is, therefore, predictable. But if one makes a pool table
in the shape of a football stadium, it turns out that the
ball, no matter how you hit it, will move around in a
chaotic way. It's as if we are playing an imaginary game of
billiards atop drums of various shapes to determine what
their vibrations will look like."
Though the Johns Hopkins team anticipates having some
answers by the end of the threeyear grant, Sogge and
Zelditch say they expect to be grappling with similar
problems and theories for the rest of their careers.
"What we're doing is of interest to the math world
because the properties we are talking about are very basic
to lots of areas of mathematics, from number theory to
partial differential equations to differential geometry,"
Sogge says. "But it also is of interest to physicists who
study dynamical systems and nanostructures, such as quantum
dots and electron corals. In both cases, the concentration
patterns of waves and excited states play a fundamental
role in the systems they study."
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2004
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