Mandelbrot's Abyss 4K (improved)
15Here's a different take on the the famous Mandelbrot Set fractal I haven't seen before.
The set is defined as the points c in the complex plane such that the infinite recursive map z = z² + c converges. Usually, one counts how many iterations before the point diverges and colors the fractal based on that number. If it doesn't diverge after a set limit, you assume it's in the set, so you paint the associated pixel black.
I had the idea of shading it not by iteration count alone, but also by the length of the orbits, that is, the cumulative difference as one steps the iterations. What you get is a soft "shadow" around the edges of each iteration. This gives the Mandelbrot Set the appearance of an infinite abyss, with holes on many layers stacked on top of one another. Each layer is a "floor", corresponding to a given iteration step diverging.
This suggested a cool intuitive interpretation of the Mandelbrot Set: an infinite abyss, and the set is the locations in the complex plane one would "fall forever" if you jumped in. Otherwise, you hit one of the layers. I thought this was a pretty cool take on it.
Inspired by this new perspective, I figured why not try to render this "Mandelbrot's Abyss" with parallax? This is the result of a quick ShaderToy code to do it. You can run it in real time here (needs good GPU):
https://www.shadertoy.com/view/cdBXDy
The "lumpiness" comes from a mistake I did when computing the divergence. I ended up using the L1 norm not L2. I think this does not exclude any point from the set, though. In my opinion, the abyss visuals look better when you have a lot of little circular platforms going down. With P=2.0 in the ShaderToy, things are too smooth and the structure only gets interesting in lower levels.
Audio is just something shitty/creepy for the fun of it. :P