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Conway found a closed billiard-ball trajectory in a regular tetrahedron:

^{Image: Izidor Hafner}

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.

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?

https://i.sstatic.net/UJCBD.jpg

Conway found a closed billiard-ball trajectory in a regular tetrahedron:

^{Image: Izidor Hafner}

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.

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?

Conway found a closed billiard-ball trajectory in a regular tetrahedron:

^{Image: Izidor Hafner}

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.

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?

Conway found a closed billiard-ball trajectory in a regular tetrahedron:

^{Image: Izidor Hafner}

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.

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?

https://i.sstatic.net/UJCBD.jpg

Conway found a closed billiard-ball trajectory in a regular tetrahedron:

^{Image: Izidor Hafner}

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.

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?

Conway found a closed billiard-ball trajectory in a regular tetrahedron:

^{Image: Izidor Hafner}

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.

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?