Periodic Orbits for Billiards on an Equilateral Triangle

1. INTRODUCTION. The trajectory of a billiard ball in motion on a frictionless billiards table is completely determined by its initial position, direction, and speed. When the ball strikes a bumper, we assume that the angle of incidence equals the angle of reflection. Once released, the ball continues indefinitely along its trajectory with constant speed unless it strikes a vertex, at which point it stops. If the ball returns to its initial position with its initial velocity direction, it retraces its trajectory and continues to do so repeatedly; we call such trajectories periodic. Nonperiodic trajectories are either infinite or singular; in the later case the trajectory terminates at a vertex. More precisely, think of a billiards table as a plane region bounded by a polygon π. A nonsingular trajectory on π is a piecewise linear constant speed curve α : R → , where α(t) is the position of the ball at time t. An orbit is the restriction of some nonsingular trajectory to a closed interval; this is distinct from the notion of " orbit " in discrete dynamical systems. A nonsingular trajectory α is periodic if α(a + t) = α(b + t) for some a < b and all t ∈ R; its restriction to [a, b] is a periodic orbit. A periodic orbit retraces the same path exactly n ≥ 1 times. If n = 1, the orbit is primitive; otherwise it is an n-fold iterate. If α is primitive, α n denotes its n-fold iterate. The period of a periodic orbit is the number of times the ball strikes a bumper as it travels along its trajectory. If α is primitive of period k, then α n has period kn. In this article we give a complete solution to the following billiards problem: Find, classify, and count the classes of periodic orbits of a given period on an equilateral triangle. While periodic orbits are known to exist on all nonobtuse and certain classes of obtuse triangles [5], [8], [11], [14], existence in general remains a long-standing open problem. The first examples of periodic orbits were discovered by Fagnano in 1745. Interestingly, his orbit of period 3 on an acute triangle, known as the " Fagnano orbit, " was not found as the solution of a billiards problem, but rather as the triangle of least perimeter inscribed in a given acute triangle. This problem, known as …