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Calculated Learning

By Kristen Cole

In math, rendering geometrical shapes from formulas typically involves the use of a computer, not a welding tool. But one Smith mathematician invites his students to take math into their own hands.

A few years ago, Associate Professor Pau Atela challenged 16 students to create the three-dimensional sculpture of a mathematical theory. The computer performed the task in seconds, the students in about six months.

In doing so, Atela’s students learned equations, turned the equations into a three-dimensional image using computer software, and then teased the image apart and mathematically recreated it with 50 pounds of wire and welding tools.

“What we did inch by inch -- even for me it was an incredible experience,” says Atela, of the first time he taught the course Sculptures in Mathematics. “It’s so hard to create something that looks pure on the screen.”

In late July, at a North American conference about art and mathematics, Atela described how his students created a sculpture of the theory, called “Boy’s surface,” which describes a surface with no boundary or edge. For years, the problem had plagued mathematicians who were delving into questions about how humans perceive the world around them.

On paper, the theory looks like a bunch of numbers and letters but as a three-dimensional form it has a sphere at the top -- imagine a north pole -- and twines around itself to undulate in three swells toward the bottom.

Atela’s students essentially “measured” every square centimeter of the shape using formulas before sculpting it, a task that could be compared to an artist sculpting a human body by first measuring every square centimeter of a person on a computer screen. Working from pages and pages of numbers, the students then approached 60 pieces of steel wire as thick as yarn and marked off the points on every wire where it would intersect with another wire.

Slowly the students created the sculpture in the machine shop within McConnell Hall with the assistance of Smith technician Greg Young.

“It represented an enormous accomplishment by the students,” says James Callahan, professor of mathematics, who watched the seven-by-six-foot sculpture emerge. “They had their hands on mathematics and it made a difference in how they understood it.”

For one student who had a background in art, the experience was life changing. Marylea Ryan ’03 says she had never considered becoming a mathematician because she is learning disabled and has difficulty remembering numbers. One day, she discovered Atela’s Web site full of colorful geometrical shapes. “I saw it and said, ‘that’s beautiful,’” says Ryan, who then approached Atela about working with the class on the project.

“I just learned the math by default -- because I had to,” says Ryan, who is now studying for her doctorate in applied math at Brown University, a decision, she adds, “shaped by the idea that math can be fun.”

Since teaching that first course of Sculptures in Mathematics, Atela has assigned subsequent classes to work on other mathematical theories.

The sculpture of Boy’s surface remained in the basement of McConnell Hall until this summer when building renovations necessitated that the class project be cut in half and moved. It will soon be installed in Burton Hall, where a plaque will describe its mathematical significance and a new group of passersby will be introduced to the idea of creating mathematical theories.

“One of the features of a sculpture is that you can walk around it -- you don’t view it from one vantage point,” says Callahan. “You look at it and you understand something that you wouldn’t with words.”

“Geometric objects have a beauty,” he adds.

But how someone interprets the sculpture can depend on the extent of that individual’s knowledge of mathematics. “When I look at this, I see math,” says Atela, running his finger over one of the wires. “It’s the math that guides the object.”

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