Rational billiards

The mathematics of billiards might be considered an abstraction of the game of billiards, but the sense in which this is true is (sadly) guaranteed not to improve your pool game. First, we remove the pockets and consider a single ball's motion to be modeled by a point moving in straight lines. Then, we replace the rectangular boundary of the table by a Euclidean polygon. And of course, we neglect friction and spin. What we preserve is the boundary rule: angle of incidence equals angle of reflection for trajectories on this idealized "table."

from a summer course for undergraduates: "The Mathematics of Billiards," by me.

The subject of rational billiards is about billiard trajectories in polygons where all angles are rational multiples of pi. The questions are about the orbit structure under these simple rules. Perhaps surprisingly, though the questions can be framed in such an elementary way as this, the most successful ways to get answers have gone through topological surfaces and Teichmuller theory, lattices in locally compact groups, and ergodic theory. The most useful contemporary tool is the SL(2,R)-action on the space of flat structures, for which an intimate understanding of how that group acts on the hyperbolic plane is indispensable.

A succinct survey of results:

Samuel Lelievre on Veech surfaces and rational billiards

A thorough survey, taking it from the top:

Masur-Tabachnikov from Handbook of Dynamical Systems

(Photo credit: me, from a sportsbar near Gansbaai, South Africa. Email me if you want to use it.)