Diana Davis and Samuel Lelièvre
We are research mathematicians studying billiards and flat surfaces. Since 2014, 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, to bring the beauty of mathematics to more people. We just (in February 2022) added the ability to draw paths on tables with an even number of sides, such as this 32-gon, so we're especially excited about exploring the paths on such tables and understanding their properties.
Imagine that you have a billiard table in the shape of a regular
polygon with 32 sides. The picture shows 16 different periodic
(repeating) paths of a billiard ball on such a table. Each
horizontal row has a different symmetry: dihedral symmetries of
order 2, 4, 8 and 16, respectively. The paths in each row are
members of what we call a "family" of periodic trajectories: They
get longer and longer, but they make a similar pattern. We get
subsequent family members by repeatedly "twisting" the surface
associated to this billiard table. In the limit, you can't see the
lines anymore; it has become a shading pattern. We have found that
these families are visually appealing; people seem to like the
contrast between light and dark regions.