Periodic paths on the pentagon
helping to spread the word that math is beautiful

Samuel Lelievre and I are studying periodic trajectories on the regular pentagon. In the course of our investigations, we created a program (in Sage, using CoCalc) that draws pictures of these trajectories. They are so beautiful! So I decided to make them into jewelry and give them away for free -- to students, faculty and staff at my College, and at conferences and talks, to help people understand and appreciate my research specifically, and the beauty of mathematics in general.

There are countably infinitely many periodic paths on the regular pentagon. Here are the first 500 or so.

If you see me at a conference, I will be happy to give you some of my beautiful mathematics research, which you can then display on your body. How cool!

Here are several beautiful pentagon trajectories. In this picture I have made them into earrings, which I then gave away for free to spread the beauty of research mathematics to the world. Which one is your favorite?

Interestingly, 40%-50% of people seem to all choose the same one. It seems that aesthetics are universal.

How did I make them? In Swarthmore College's MakerSpace, on the laser cutter. Below is a video of the laser cutter engraving a trajectory. At the beginning, it seems "random," and then order emerges from chaos. The laser is drawing the lines in the same order as the path that a billiard ball would take in the pentagon, if it were shot in the direction of one of the lines on this pentagon. Video has sound (background noise of laser cutter and fan).

Diana Davis
Swarthmore College, Department of Mathematics and Statistics, 500 College Avenue, Swarthmore PA 19081