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The Shape of Greens

Honors College math major Jeff Gwaltney looked no further than his adviser’s lunch for a senior thesis project.

“I went to talk to him about the possibilities. He knew that I was interested in art and photography as well as non-Euclidean geometry,” Gwaltney said, referring to Chaim Goodman-Strauss, professor and chair of mathematical sciences department. “He had a bag of lettuce in his lunch and it went from there.”

Gwaltney began exploring the geometry of lettuce using the camera and mathematics. The Russellville native bought green leaf lettuce and cabbage at the grocery store, took it home and started shooting. At the same time, he began exploring the field of differential geometry, which combines aspects of geometry with calculus, to study the shape of both vegetables.

Green leaf lettuce and cabbage sport distinctively different shapes. Cabbage leaves fold firmly one over another, forming a tightly layered ball. Green leaf lettuce has wrinkled, undulating edges that when photographed close up form a landscape.

“To us the wrinkles indicate that the lettuce will hold the salad dressing well. But there’s a lot more to it than that,” Gwaltney said.

Many plant leaves do not fold or tear when pressed on paper. But cabbage and green leaf lettuce leaves behave differently. Cabbage leaves bend in with a positive curvature similar to a dome or bowl, while green leaf lettuce leaves sprawl out in waves with negative curvature.

The meaning of the differences in shape becomes clearer when trying to flatten the leaves on a sheet of paper. The curved-in cabbage leaf must be cut to fit flat, because there is not enough surface area at the edge of the leaf for it to lie flat. The wrinkled edges of the green leaf lettuce, however, bunch up and spill out everywhere.

“There’s too much leaf to conform to flat space,” Gwaltney said.

Gwaltney looked at ways to mathematically calculate the curvature of the leaves. He started with the Gauss-Bonnet theorem, which relates curvature to the topology of an object. Then he looked at ways to characterize the curvature of the leaf using only edge of the leaf.

“An analogy would be how far you have to turn the steering wheel when driving on a curve,” Gwaltney said.

To characterize the curvature, he cut a thin strip along the boundary of the leaves – no thicker than a few millimeters to a centimeter. Then he forced it to lie on a flat surface by pressing it between two sheets of glass.

When the strip is cut and laid flat, it forms circles, partial circles or multiple circles. A flat leaf forms one whole circle. The cabbage leaf forms a partial circle, because the surface area of the boundary is less when curved than when flat. And the curvy leaf of the green leaf lettuce forms multiple circles because the wrinkles allow for more boundary than would normally fit in that amount of space.

While examining the geometry, Gwaltney found a mathematical limitation on how green leaf lettuce forms its distinctive shape.

“The curvature has to be introduced along the boundary of the leaf,” he said.

The story with cabbage leaves remains less clear, but it seems that these leaves may start with the curvature they stay with, forming at the molecular level with a set curvature engrained, and that this curvature may govern the size of the cabbage head.

“The leaves serve a structural role,” he said. In the cabbage, that means the leaves form a tight ball. In the green leaf lettuce, the wrinkled structure keeps leaves open, allowing for more sunlight and water to get in. Geometry plays a role in both cases. “What brings the form to the leaf is the curvature,” he said.

Gwaltney’s study has provided a greater understanding of the mathematics behind the structure of leaves and how they grow. The images help people understand the mathematical concepts, and the paper serves as an artist’s statement.

“I used math in a different way to study this beautiful thing,” he said.

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