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It is an open problem in dynamical systems if every triangle has a periodic billiard orbit. So far it has been proven that equilateral triangles, isosceles triangles, right triangles, and obtuse triangles with no angle greater than 100 degrees have periodic billiard orbits.

I am wondering if there are any other broad or specific cases I am missing? For example, in this paper, it is shown that any triangle with angles $\alpha$, $\beta$ satisfying $k\alpha=l\beta$ have a perpendicular periodic trajectory, where $k, l \in \mathbb{N}$ (page 13). There is a similar finding in this paper stating that every triangle with angles $\alpha$, $\beta$ satisfying $k\alpha + n\beta = \pi$, where $k, n \in \mathbb{N}$ admits a perpendicular periodic trajectory.

Does anyone know of any other general cases? I am particularly interested in the existence of cases where triangles have periodic trajectories when its angles are given by some function, that is if $\alpha$ is an angle in triangle $T$, then the second angle used to define $T$ is $\beta = f(\alpha)$ for example. My department doesn't have people specializing in this area, so I apologize in advance if my question does not belong here or is unclear, give any comments and I will try my best to reciprocate.

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    $\begingroup$ Masur proved that billiards where all the angles are rational multiples of pi have periodic trajectories. $\endgroup$
    – Anthony Quas
    Commented Apr 9, 2019 at 5:48
  • $\begingroup$ @AnthonyQuas thanks yes that is one I forgot to mention in my post, I don’t believe I am aware of any past those however $\endgroup$
    – user918212
    Commented Apr 9, 2019 at 5:58
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    $\begingroup$ All acute triangles have periodic billiard paths. $\endgroup$
    – JRN
    Commented Apr 9, 2019 at 6:06
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    $\begingroup$ @JoelReyesNoche : this is contained in the OPs statement that if all angles are below 100 degrees... $\endgroup$
    – Anthony Quas
    Commented Apr 9, 2019 at 6:15
  • $\begingroup$ You mean the one that says "obtuse triangles with no angle greater than 100 degrees"? $\endgroup$
    – JRN
    Commented Apr 9, 2019 at 6:17

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This paper in 2018 showed that all obtuse triangles with angles at most 112.3 degrees have periodic trajectories. It's also a computed assisted proof but not the same method as Richard Schwarz's. https://arxiv.org/abs/1808.06667

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