Swarthmore College mathematician Diana Davis, whom I am lucky enough to call a friend, is multitalented. Her combined love of math and running led to an earlier appearance in this blog, where she investigated the question of whether, if you run a marathon at, say, a 7:00 minute per mile pace, must you have run any mile in exactly 7 minutes? Recently, she has been making jewelry, clothing, coasters, and other objects based on her research involving billiard trajectories in pentagons. I asked her about her research and the beautiful things she has made from it. You can find her creations in person at conferences including the Joint Mathematics Meetings or explore her website.

First, introduce yourself. Where are you from, where did you go to school, and where do you work now? Apart from math, what are you passionate about?

I'm originally from New Hampshire. I went to Williams College for undergrad and to Brown University for my Ph.D. I'm currently a visiting assistant professor at Swarthmore College. Apart from math, I'm passionate about good teaching, fairness in all things, and long-distance running.

What is your research area, and what are some of the math questions you like to think about?

My research is in dynamical systems, primarily in billiards, which is the study of a ball (or point) bouncing around in a shape (usually a polygon, and often a pentagon). People have understood billiards on the square for over a hundred years, and that's about it: as humans, we don't understand billiards on any other shape, except maybe a couple of special triangles (equilateral, isosceles right, that sort of thing). So I want to know what happens with billiards on tables that are other polygonal shapes.

The question that drove me for many years was: suppose you have a ball on a regular pentagon billiard table, and you choose a direction to hit your ball, and you know that the ball's path is going to be periodic: the ball will bounce around for a while, and then return to where it started and repeat. How many times will it bounce before it repeats — how can you get this information from the direction? For the square, if the slope of the path (assuming the edges of the table are horizontal and vertical) is rational, then you know the path will be periodic, and if the slope is p/q in lowest terms, then the ball will bounce 2(p+q) times before it repeats. It's a beautiful result, and I wanted to generalize it to the regular pentagon.

What's so great about pentagons? Did you love them before they got to be such a big part of your work, or did your love for pentagons grow as you researched them more?

Five has always been my favorite number. This is not why I studied the pentagon. The main reason to study it is that it's, in some sense, the "next-simplest" regular polygon after the square — after all, a square has 4 sides and a pentagon has 5. Arguably, though, the regular octagon is a little simpler, because it's just a square with the corners cut off. Indeed, John Smillie and Corinna Ulcigrai had done some work on the regular octagon, and my Ph.D. advisor told me to read their paper about the octagon, and see if I could use the same strategies to understand the regular pentagon. Indeed I could! So that's how I got started with the pentagon. In that case, I was actually studying a surface made from two pentagons, which you can see in my viral dance video.

Then I was at a conference in Oberwolfach in spring 2014, and Samuel Lelièvre came up to me. He told me that he liked my video about the double pentagon surface, and that we should work together to understand billiards on the regular pentagon billiard table. We've been working on it ever since.

How did you get the idea to start making pretty things based on your work, and what pretty things do you make right now?

In summer 2017, I was working in Moon Duchin's research cluster at Tufts University. She let us take time out to get trained on the equipment in Tufts's Makerspace. Before that, I had no interest in laser cutters or any kind of Makerspace activities, but after I learned how to use them, suddenly I wanted to try all kinds of things. The first things I made were coasters -- plastic pentagons about 5 inches across, engraved with periodic billiard trajectories. But although coasters are easy to make, there is really low demand for coasters. I imagine that many people have about five or ten times as many coasters as they actually use.

One day, I had the idea to make some earrings, probably just for my roommate — I don't have pierced ears, myself — and I made a couple of tester pairs. As soon as I had that idea, and the test pairs turned out well, I realized I was really onto something. You see, almost everyone wears earrings. I didn't realize that when I started this project — again, I don't have pierced ears — but I've found that about 90% of (women) mathematicians have pierced ears. And, unlike with coasters, people are always happy to have another pair of earrings.