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way coool, hello and welcome to the forums

you can assume that we guys want to play with your algorithm for now you cen elaborate some inner mechanics, are you using a massive hybrid to create this shape fitting formula ? after the shape search is finished what is the result ? a bunch of numbers that do not slow down the final rendering process? it looks like you do it in realtime, but hard to tell from a video

It's inspired by the analytic 2D solution from a few years ago, and uses a lot of the same formulas. But, that approach used a bunch of conformal theory that don't appear to generalize trivially to 4D (I asked), so I formulated a numerical approach.

The paper goes into gruesome detail on how this is all done. It is much, much slower than the usual quaternion Julia sets, because it doesn't use a quadratic polynomial map. Instead it uses a rational function with hundreds of roots, which is why it can fit things, but it then takes a long time to evaluate, e.g. an hour per frame. The video is a result of deploying the render to a cluster. Definitely not real-time.

The GPU isn't as helpful as usual on this front either, because it needs to remember hundreds of root locations instead of just one "c" constant to add every time, so it runs out of registers quicker than usual.

So, it looks like this is possible  but not yet fast  .