The images in this project depict the trajectories of a ball on perfect billiards in the shape of regular polygons with N sides, for N between 3 and 15. A point near the center of the billiard and a direction is chosen randomly for each pictures, and the trajectory is plotted for 1000 rebounds against the edges of the billiard.

Some very interesting emergent patterns arise from this simple procedure. While most trajectories appear ergodic (i.e. seem to cover the whole figure), some choices of starting point and angle lead to periodic or almost periodic trajectories.

Python code adapted from John Mangual's script.

A recent reference with pointers to older literature.

Here are some pictures showing the evolution of the trajectories after 25, 50, 100, 200 and 500 rebounds

Dall-e's interpretation of dynamical billiards on regular polygons.