Wednesday, June 5, 2013

Sending Order Through Chaos: Student examines "Rule 30" math problem

A Randolph College student is using his math and computer programming skills to help solve a puzzling mathematical problem.

Nam Hoang ’15, an engineering physics major, and Marc Ordower, a mathematics professor, are working with Stephen Wolfram’s “Rule 30” to explore ways to keep chaos from disrupting a pattern. When Hoang expressed interest in the Summer Research Program, Ordower suggested working on this problem because it has fascinated him for several years.

This computer simulation demonstrates how Rule 30 can be
applied to a pattern. After enough rows, the pattern (the alternating
on-and-off straight lines) disappears into chaos.
Wolfram’s Rule 30 demonstrates entropy—the move toward chaos. To apply the rule, mathematicians define a pattern in a row of cells that can be either alive or dead. On the next row, the status of each cell depends on the pattern of cells immediately above it. After a few rows, the entire system begins to demonstrate chaos.

Ordower wanted to find out if there is a way to create a pattern that can remain constant over an infinite number of new rows, even as chaos grows around the pattern. If such a pattern does exist, it could have implications for numerous situations, such as sending data electronically. “We transmit data all the time, and some of that data is lost,” Ordower said. “There are lots of people working on how robust you can make a message, and how many resources you need to make a message robust so it arrives correctly.”

Ordower and Hoang are spending most of their research time with pencil and paper trying to devise an algorithm and mathematical proof that would show that there is a pattern that could continue indefinitely. Hoang also has been writing computer programs to help dig deeper into the question. “At first it looked like it was so simple. We were trying to prove that we could send information through chaos,” Hoang said. “It turned out that it was very complicated. I’ve learned to analyze it and understand it.”

Hoang’s efforts have yielded some insight. His computer program discovered a pattern that disappears into chaos, but parts of the pattern emerge again.

Solving a complex problem like this is much larger than a summer project, so Ordower expects he and Hoang will continue seeking an answer even after the Summer Research program completes. He said it is great experience for Hoang, who hopes to become a computer engineer, in working with a mathematics problem that has never been solved, which is the real work of mathematics.

“The bigger mathematics gets, the more open problems there are,” Ordower said. “There are thousands of mathematicians working on millions of open problems around the world.”