Mathematical Billiards (2024)

Inner billiards study trajectories of a frictionless point ball on a convex table, such as an ellipse, where each bounce follows the law of reflection and the billiard map transforms directed chords. Mathematicians ask whether a table admits a periodic trajectory, a question still open for all triangles. Outer billiards moves a point outside a shape by reflecting it across a tangent point, producing a distinct dynamical system unrelated to physical collisions.