Mathematical Billiards (2024)

This explains several idealized billiard games and what their trajectories reveal about underlying geometry. Inside a convex table the billiards map sends a directed chord xy to yz following the law of reflection; trajectories are sequences of such chords and central questions include existence of periodic trajectories (e.g., it remains open whether every triangle admits one). Outside the table, outer billiards replaces collisions with a reflection construction: from a point x outside a shape, reflect through a chosen tangency point to get y and iterate. For polygons this map has singularities along rays through edges and more points whose forward orbits hit those rays; computer-generated singularity diagrams show that equilateral triangle, square and hexagon tables produce plane-tiling singularity patterns with every nonsingular point periodic, while other polygons (notably the pentagon) exhibit fractal structures and nonperiodic orbits. Regions with no singularities - “periodic islands” - act as tiles whose interior points all share periodic behavior.

Exploring outer billiards in hyperbolic geometry and via computation reveals richer phenomena. Varying side lengths in hyperbolic regular polygons produces cases where the singularity set matches uniform tilings: for any integers n,k≥3 with 1/n+1/k<1/2 one can find side lengths so singularities tile the hyperbolic plane by n- and k-gons (e.g., triangles paired with k-gons), yielding infinitely many regular-table shapes with all trajectories periodic. These discoveries grew from interactive software experiments that guided rigorous analysis. The write-up also introduces a related Euclidean variant called outer length billiards, which uses a circle tangent to the table and its tangents to define each step.