November 13, 2012

Robert Strichartz Has Fun Figuring Out Fractals

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Prof. Robert Strichartz, mathematics, researches fractal analysis, or the application of calculus to fractals, to probe questions in mathematics.

Prof. Robert Strichartz, mathematics, researches fractal analysis, or the application of calculus to fractals, to probe questions in mathematics. When represented as an image, fractals are mathematical patterns in which each part is a copy of the whole that repeats itself infinitely.  These repeating geometric images can be found in trees, snowflakes, and are often portrayed in art.

The Sierpinski carpet, which is a pattern of cut-out squares, is a well-known example of fractal art. In addition to artwork, fractal analysis can also be applied to the examination of less geometrically rigid objects, according to Strichartz.

“Sunlight hits the top of the cloud – what happens to that heat?” said Strichartz. “If you can model a cloud as a pillow, then differential equations can let you tackle this question.”

But according to Strichartz, a cloud is more of a fractal than a pillow because it is a complicated mixture of water vapor and air.  Differential equations, which are relations between a function and its rates of change, can be used to help explain properties of an object such as vibration and heat dissipation.

Strichartz is interested in developing “analogs” of differential equations for fractals. One fractal he studies is the “Magic Carpet,” a variation on the regular Sierpinski carpet where edges of the cut-out squares are “stitched” together to create something that is more surface-like than the original carpet.

According to Strichartz, no one has ever proved that differential equations exist on the “Magic Carpet.”  But, he said that his team is building up experimental evidence showing that differential equations do exist on the Magic Carpet.  His experimental analysis on fractals and functions of fractals could yield useful information for understanding mathematical behaviors such as vibrational motion.

This experimental evidence is key to Strichartz’s research. “Experimental math” is a computational method of investigating mathematical questions.

“The term ‘experimental math’ is something of a misnomer,” Strichartz explained. “It’s really just messing around.”

Strichartz has worked with hundreds of students to employ experimental math to analyze fractals, especially through the Math Department’s Research Experience for Undergraduates (REU) program. As the director of the REU program since 1994, he has guided students from all over the globe in working through computations and understanding the theory behind the numerical analysis.

“It’s a terrific experience,” he said. By using experimental math, Strichartz and his students have found unexpected mathematical patterns that they could not have found any other way.

According to Strichartz, his students also find other ways to demonstrate their enthusiasm for math – such as through building a massive 3-D Sierpinski triangle out of straws and string that then placeing a garden gnome in the center of the sculpture.

“My students think I look like that,” Strichartz said with a laugh, referring to the gnome. “I get the advantage of working with talented, hard-working students. We have a lot of fun.”

Strichartz holds a wide variety of other hobbies, including gardening, writing, reading, playing the piano, and watching his grandchildren.

“It’s part of my holistic view. When I was growing up, there was the idea of the ‘Renaissance man,’being broadly interested in everything,” he said.   explaining the importance he places on expanding one’s boundaries through a variety of interests and activities.

“I try to be down to earth, since mathematics is kind of high in the sky,” he said.

Strichartz often combines his additional interests and math in unexpected ways. He recently wrote a choral piece entitled “The Mathemadrigals,” which puts excerpts from famous mathematical literature, such as Euclid’s Parallel Postulate and sections from Newton’s Principia Mathematica, to music.

He also draws parallels between math and dance. In choreography, he said, the challenge is for the choreographer to realize his or her vision through the physical movements of the dancers. Similarly in math, “there’s a tremendous tension between what you’d like to work out and what you can in fact discover,” he said.

Mathematics can be frustrating at times, Stricharz said. “I have a little twist on Lao Tzu’s famous quote: ‘A journey of a single step begins with a detour of a thousand miles.’”

Nonetheless, Strichartz said that he finds the unknown aspects of mathematics as well as its problem-solving process, appealing.

“You’re facing the challenges of math and trying to understand them doing it your own way,” he said. “You are completely free to decide what you want to work on and think about.”

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Original Author: Jacqueline Carozza