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Periodic Orbits of Billiards on an Equilateral Triangle

Andrew M. Baxter·Ron Umble
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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 $n$.

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Andrew M. Baxter, Ron Umble. 2007-04-01. Periodic Orbits of Billiards on an Equilateral Triangle. https://arxiv.org/abs/math/0509292

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