Cutting Sequences on the Double Pentagon, explained through dance from Diana Davis on Vimeo.
This article has been updated to include results from the contest.
Imagine walking in a straight line on a torus, a bagel-like geometric shape. It would be possible to go through the hole and end up where you started, walk around the perimeter and end up back at the beginning, or to walk in spirals and zig-zags. These are the surfaces that serve as the subject of thesis research for Diana Davis GS, who is studying in the Department of Mathematics.
But for Davis, writing a thesis on these surfaces was not enough.
"My research is so visual, so I thought I could do a really good job making it into a dance," Davis said.
When a friend of Davis' told her about the annual "Dance Your Ph.D." contest, sponsored by the journal Science and TEDx, Davis thought it was a perfect opportunity to "show what math research is," she said. Many people have the misconception that math research is looking up formulas in books, when really it involves questions and solving a puzzle without a known outcome, she said. Winners of the contest were announced Monday, and Davis won a $500 prize for best video in the physics category.
The path of a person walking on a bagel could be recorded by tracking which edge of the bagel the person hits with each pass. Davis' interests lie in what would happen to this sequence of edges if the bagel were to be twisted, chopped up and glued back together, she said.
"People figured out what happens (to these sequences) for the octagon, so I suggested she try this for other shapes," said Professor of Mathematics Richard Schwartz, Davis' PhD adviser. Davis' research expanded these findings to show that when the double pentagon - a shape made by two pentagons sharing a common edge - is sheared in vertical lines and reassembled, the original pattern can be obtained by using rules of substitution, he said. The substitution works like a code - the act of shearing the double pentagon encrypts the pattern, and Davis' theorem explains how to decode the encrypted pattern and restore the original sequence.
Davis proved this substitution rule works for many more shapes than just the double pentagon, including one shape comprising a long sequence of polygons strung together, Schwartz said. Davis' video features a dancer leaping across a double pentagon, and Davis said she would love to expand her video in the future to include additional shapes.
The dancer in the video is Libby Stein '15, who took the section of MATH 0180: "Intermediate Calculus" taught by Davis last year.
Davis came to one of Stein's dance performances when she was in her class and later contacted Stein to see if she would be interested in working on the video, Stein said.
The video was filmed in the organ room of Sayles Hall last April. Davis set the stage by placing a picture of a double pentagon made out of butcher paper on the floor and hanging a camera suspended from the ceiling. Stein improvised a dance across the double pentagon, though she was constantly interrupted by tour groups coming into the room.