[Conway found](https://mathoverflow.net/q/357197/6094) a closed billiard-ball trajectory in a regular tetrahedron:

<img src="https://i.sstatic.net/UJCBD.jpg" width="300" />

<sup>Image: [Izidor Hafner](https://demonstrations.wolfram.com/ConwaysBilliardBallLoop/)</sup>

Since then Bedaride and Rao

> Bedaride, Nicolas, and Michael Rao. "Regular simplices and periodic billiard orbits." *Proc. Amer. Math. Soc.* **142**, no. 10 (2014): 3511-3519.
[JSTOR link](https://www.jstor.org/stable/24507260).

proved this:

**Theorem**. In a regular simplex $\Delta^n \subset \mathbb{R}^n$ 
there exists at least two periodic orbits:

* One has period $n + 1$ and hits each face once.
* The other has period $2n$ and hits one face $n$ times and hits each other face once.

For $n=3$, Conway's path accounts for the first length-$4$ periodic path. 
I haven't yet figured out explicit coordinates for their second length-$6$ path. 
I'd be interested if anyone has drawn this $6$-path.
In any case, my question is:

***Q***. Is there a complete inventory of periodic billiard paths in
a regular tetrahedron?

Is it even known that there are only a finite number of
such paths?