The Mandelbrot fractal has all it's detail on the boundary, which is nice and all but there are other formulas (or settings for formulas) that instead seem to fill up the whole plane with detail everywhere as if you could zoom in to any location and see more details! Mandelbrot gets smooth if you zoom somewhere that isn't on the boundary.. which I see as a bit of a shortcoming personally.

Even the generalised Mandelbrot fractal formula starts to do this plane-filling via compounding 'branch cuts' in the 'lake'/inside areas when the exponent (a complex number) is a negative non-integer like -1.5+0.0i instead of the usual 2.0+0.0i.

Here's an example I tried to add to Wikipedia but unfortunately it was rejected:

https://commons.m.wikimedia.org/wiki/File:AlanTIsVeryHelpful...

Could be an illusion that it's actually filling the plane, but if so it's an illusion I'm rather fond of!