Diana Davis and Samuel Lelièvre

Instructor in Mathematics; Maître de Conférences
Mathematics department, Phillips Exeter Academy; Mathematics department, Université Paris-Saclay
Exeter, NH, USA; Orsay, France

We are research mathematicians studying billiards and flat surfaces. For the past six years, we have been working together to study periodic billiard paths on regular polygons, starting with the regular pentagon. We wrote a program in Sage to draw pictures of the paths, which turn out to be really beautiful. We have made our pentagon pictures into jewelry and T-shirts that people can wear, to bring the beauty of mathematics to more people. This year, we expanded our Sage program to draw periodic billiard paths on any regular polygon with an odd number of sides, and we're just starting to explore the vast world of new billiard pictures that we've unearthed.

Computer graphics

This is an artistic depiction of the COVID-19 virus, made out of periodic billiard paths. We designed it to look like the microscope picture of the virus that was circulating widely in spring 2020. We created it while both of us were under lockdown in France, collaborating remotely. Both the central regular polygon with 19 sides, and each small equilateral triangle, contains a periodic billiard path: the path of a ball that bounces around inside the polygon, and eventually returns to where it started and repeats the path. For the large path, there were hundreds of choices; we chose this one because we think it's beautiful. For the 19 small triangles, we chose the 19 simplest paths that have mirror symmetry, from shortest (top) to longest.