[dmurray]

I'm not sure your edit is right. No reason why there can't be a tiny but very complex thing which looks single-colour at the next 1 or 10 or 100 orders of magnitude.

But there are an infinite number of very interesting things in this fractal. If you constantly zoom in on interesting looking areas, you will find this kind of complexity with minimal need for backtracking.

[mike_hock]

I'm guessing most fractals don't have a smooth boundary anywhere, so if you know any boundary point, you can zoom in on it and it'll be "interesting."

[danwills]

Yeah I think you're right mike_hock.

On top of that I think that locations on (or near) the boundary, tend to stay on the boundary (and stay in the center of the image too) when zooming out.

While purely zooming (not translating) to a known boundary (or near) point, you won't ever see a move to another section-of-boundary, so if there are both 'inside' and 'outside' regions, corresponding to attractors at 0 and infinity, (the 2 main ones in these types of fractals) in the most-zoomed in state, then there will always be regions of both states contained in the final image when zoomed out (until you get to the 'top').

Maybe it would be possible for there to be formulae that don't hold to this? If the fractal had an incredibly sparse structure, say? To be honest I'm more interested in the opposite myself: Structures where the boundary (between N regions or behaviors) is so wiggly, it's almost 2 dimensional itself!. (If anyone wants to read more, I've called one particular interesting example of this: 'mandelfield' on UltraIterator)

[rob74]

...except when you run out of precision in an "unsophisticated" implementation as simias mentioned, then you'll eventually get smooth boundaries - which are just an artifact however.