If I missed something that allows period 3 with height 4, you can probably ignore this submission, as you'd have seen much cooler things during the "Height > Period > Cycles" round. If not, here's my nonsense:

I wanted to prove to myself that a period 3 solution was possible for this puzzle, so I made this! I wasn't able to get it to 4 height, so it doesn't beat my 4/4/272 solution, but it felt like an elegant proof of concept, so I want to share it anyway.

Enjoy