Life's a Beach

A sample of images from the interactive Sandpile Galleries (link below)

Wes Pegden

Every child who's ever played in a sandbox or on the beach knows a little something about a deep mathematical truth: Pour sand out of a bucket a little at a time, it will build up into a pile. If the pile gets too steep, grains of sand will tumble away from the peak in a mini-avalanche.

In 1987 physicists developed a mathematical model inspired by this behavior, which captures the statistical properties of natural phenomena as diverse as forest fires, earthquakes and avalanches, which, like the sandpiles on the beach, are the result of sudden changes from gradual buildup.

In this model, individual grains of sand even themselves out to a stable organization by continually moving sand away from the steepest areas. Model this on a computer and you get very intricate images that look almost like fractals.

"You have this crazy, complex picture that's been around for a long time, and we didn't have any explanation, mathematically," said Wes Pegden, assistant professor of mathematical sciences in Carnegie Mellon's Mellon College of Science. "Now we understand some of what's going on here."

For the past few years Pegden has been working on the sandpile problem with CMU alumnus Charles Smart (S'02, CS'02), an assistant professor of mathematics at Cornell University. They met as postdoctoral researchers at New York University and were intrigued by the problem.

"For me, the most interesting thing about it is that you can have a simple rule that produces incredible complexity," Pegden said.

The simple rule is this: Imagine an infinite chessboard. At one of the vertices you place 1,000 grains of sand (or chips). When there are more than four chips at a vertex, they topple, with one chip sliding down each edge to a neighboring vertex. Then a neighboring vertex might spill over, causing a chain reaction. Whether you start with 1,000 chips or 1 billion, they topple and spread out to form the same configuration.

"One of the striking things about the sandpile is that it looks like there's some limit of this process. When I increase the number of chips, I still get basically the same picture. I just get a higher resolution version of it," Pegden said.

Pegden and Smart work in different fields — combinatorics and partial differential equations — but they combined their expertise to prove that there is a limit to the toppling process. They also developed a mathematical explanation for why there are triangular shapes in the image and why they're arranged in the way that they are.

"Having a simple rule that produces complexity is already a mystery," Pegden said. "We had no idea what was going on with this for 25 years. Now, we can say a lot about it mathematically."

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