Why do these patterns appear in custom fractals?

Question

Last year I programmed a fractal renderer and played around a lot with fractals such as the Julia set and Mandelbrot set. Eventually I got curious and inputted my own algorithms for the fractal, and got some very interesting results.

I thought I'd share some of these images here as well as ask if anyone knows why these shapes appear.

Their looks to be cutouts of the original Mandelbrot set with strings of patterns cutting through them. Additionally I wonder if anyone has some more advanced reading about fractals and why these patterns exist, as I'd like to take a deep dive on it. Thanks!

Custom Fractals:

Formula: Zn+1 = Zn3 + C + 0.15 + 0.3i

Formula: Zn+1 = (Zn2 + 0.6 + 0.5)2 + c - c

Some of the coolest results I found where when I created a video of the fractals as I gradually changed the formula (such as changing a constant from 0.6 to 0.62). If there's enough interest I may post those videos as well.

Answer 1

If you want to develop some intuition about the properties of Mandelbrot-like fractals, the Wikipedia page might be a good place to start. There are also web-based tools for exploring systems like this, which let you drag parameter values around on the complex plane and see the fractals update in real-time (and drag/zoom the fractals like a map, animate parameters, etc.). Here's your first system and something like the second one (you seem to have changed the initial iteration value $z_0$ but not posted the value you used).

Some of the stuff in your expressions doesn't do anything interesting to the fractal: replacing $c$ with $c+0.15+0.3i$ just translates the image in the complex plane, and replacing $c$ with $c-ci$ is equivalent to multiplying $c$ by $1-i$, which just rotates and scales the image.

The more interesting visual changes you've discovered come from changing the initial value $z_0$. The iterated function has attracting fixed points and cycles that vary smoothly with $c$, leading to the colored and striped regions inside the figures, whereas the initial value $z_0$ determines whether such an attractor is actually reached, resulting in an overlaid "window" in parameter space outside of which the system diverges. You can see this in the demos linked above by dragging the s crosshair around and watching the fractal-shaped boundary move while the interior pattern stays stationary. (Doing this helps "reveal" new interesting regions in the parameter space, which lead to interesting Julia set fractals, for example.)