Periodic Orbits of Billiards on an Equilateral Triangle

Abstract

We give a complete solution of the following problem: Find, classify and count the (classes of) periodic orbits on an equilateral triangle. We prove that Fagnano's period 3 orbit is the only periodic orbit with odd period. A periodic orbit is either prime or some d-fold iterate thereof. We count prime and iterate periodic orbits of period $2n$ via a bijection with a certain partition of $n$, then count only prime orbits using the prime factorization of $n$.

Version 7: Minor edits made, some figures redrawn, Table 1 moved from appendix to body of text.