[pfedak]

As others have pointed out, a generalization of this method (iterated function systems) can generate a large class of fractals. There's a some fun software for generating these ( see https://en.wikipedia.org/wiki/Fractal_flame) which also adds coloring based on the sequence of function applications, resulting in a pattern that respects the fractal structure.

Going a step further, changing the "functions" over time can create smoothly animated fractals. Electric Sheep (https://electricsheep.org/) is a screensaver version of this which doubles as a distributed computing network to generate them. It also mixes in non-linear transformations which makes for some impressive, mesmerizing patterns.