# Types of triangles admitting periodic billiard orbits

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.

• Masur proved that billiards where all the angles are rational multiples of pi have periodic trajectories. Commented Apr 9, 2019 at 5:48
• @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 Commented Apr 9, 2019 at 5:58
• All acute triangles have periodic billiard paths.
– JRN
Commented Apr 9, 2019 at 6:06
• @JoelReyesNoche : this is contained in the OPs statement that if all angles are below 100 degrees... Commented Apr 9, 2019 at 6:15
• You mean the one that says "obtuse triangles with no angle greater than 100 degrees"?
– JRN
Commented Apr 9, 2019 at 6:17