Equations can help computers solve geometric problems

CONTACT: B.J. Almond
PHONE:
(713) 348-6770
EMAIL: balmond@rice.edu

EQUATIONS CAN
HELP COMPUTERS SOLVE GEOMETRIC PROBLEMS
Mathematician at Rice University
is optimistic his research will have practical applications

A Rice University
mathematician is on the hunt for equations that represent countless geometric
problems.

When these equations are
identified, computers can be used to find solutions to complicated problems that
might be encountered in engineering and other industries.

“To understand the
geometry of something, you need ways to analyze it systematically and learn more
about the properties it might have,” said Brendan Hassett, the Edgar Odell
Lovett Assistant Professor of Mathematics at Rice in Houston. “Trying to
visualize the abstract properties of something can be a struggle, but if you can
represent those properties with equations, you can use a computer to perform an
analysis.”

Hassett specializes in
translating geometric problems into algebraic equations. Mathematicians use
coordinates to study lines, circles and other geometric objects, just as sailors
use longitude and latitude to map a coastline. The relations among the
coordinates of the points on a line or circle are expressed concisely as
equations.

The geometric problems
that Hassett is most interested in involve the fourth dimension, in which
objects are identified by four coordinates, or numbers. That increases the
complexity of the equations as well as the number of possible
solutions.

“For a computer to solve
an equation, it needs clear, step-by-step instructions on how to calculate the
definitive answer,” Hassett said. “So it’s important to come up with a system
for identifying all the possibilities and expressing each with a
formula.”

The complexity of such
problems, which can entail thousands of equations, means that their solutions
might not find immediate uses in industry. But the vast quantity of information
coursing through today’s digital world means that intricate mathematics arises
more and more in everyday life.
Hassett cites the compact disc as an example,
where sophisticated algebraic formulas are used to safeguard the integrity of
the music.

“If you scratch a music
CD, the disc will keep playing,” he said. “Some redundancy has been built into
the disc. The extra information allows the CD player to reconstruct the music
when some of the information is lost.

“Mathematically, you
want to repeat information so it can be recovered in more than one way,” Hassett
said.

The geometric
inspiration for the algorithms used to store information on CDs is more than 120
years old. But that’s par for the course in mathematics, he said.

“When you work in
mathematics, you need to be confident that your findings will continue in the
future,” said Hassett, whose research is supported by the National Science
Foundation. “The long-term development of science, the things you study out of
pure curiosity, will lead to very useful applications, even if you can’t predict
120 years from now how your work will benefit society. The historical trend is
that mathematics tends to be useful, even in areas where it wasn’t
expected.”