Alright, here we go! We've swapped to calculating the checkerboard pattern after bouncing the point around, then calculating its polar coordinate and using that to map the board. You see the board isn't quite aligned, and thats because of the cartoid shape the board is twisted into. Luckily the cartoid's equation is known
Its typically defined in terms of complex exponentials, but we can just convert those to sines and cosines if we want to enough. Once the distortion is corrected I can scale the boards down based on board size, which is directly tied to their orbits. Since we already iterate hundreds of times anyway, I'll bet there's a way to track the points and calculate the period iteratively in a way thats gpu friendly.
Bulbs behave in way such that their orbits travel along paths of fixed length dependent on size. This bulb has 3 points it orbits along. It seem at least some (if not all) bulbs contain a shared point on the set at 0. However as you see here, the bulbs don't always line up, and there's a strange distortion around the edges.
Since the orbits all have fixed repeating points, and they all start within their own bulbs, aligned at iteration 0, we can calculate when they'll line up perfectly in the center as (n! - 1) where n is the maximum number of orbits you care about being properly mapped. The higher iteration count causes the distortion to be less prominent as well. The top picture is taken at n = (6!-1) = (720 - 1) = 719 aligning all bulbs up to orbits of length 6.
n=7 offers some beautiful recursivity however, running the mandelbrot set 5039 iterations every frame is starting to slow my graphics card down a little bit lol
Unfortunately julia sets don't work quite the same way, so I will have to put aside the julia board design until I can figures something out for them. They require a bit more study