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If we accept that a 3d equivalent of the mandelbrot would want to be a conformal mapping that maps the 3d space onto itself, then we run into this: http://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings)

It states that in 3d you can only generate conformal mappings out of scale, rotation, translation, reflection and sphere inversion which means replacing the magnitude of all vectors with r*r/magnitude for some r.
So a conformal mapping might be: invert, translate by (1,0,0), invert, scale by 2, rotate around y axis 90 degrees.

So that's the set we have, no doubling of angles, or squaring the magnitude.
My tests show that any combination of these produce completely smooth boundaries.. which would suggest there is no _direct_ 3d analogy of a mandelbrot.

But there are a couple of loose ends:
1. non-smooth transformations seem to allow more variation
2. Maybe if there are no helpful conformal mappings then 3d mandelbrot would be the single mapping that is closest to conformal.

Maybe something related to this: http://en.wikipedia.org/wiki/Quasiconformal_mapping, but I don't know anything about it.

Maybe I didn't understand some things but doubling a rotation to me is pretty much equivalent to rotation (actually it's equal. Only the amount of rotation changes, depending on where the point is found) and sqaring the radius is replacing the magnitude... (though I'm not sure if that could be written as "r*r/magnitude for some r")

>> doubling a rotation to me is pretty much equivalent to rotation (actually it's equal. Only the amount of rotation changes, depending on where the point is found)
doubling a rotation stretches out any shape, just rotating the whole coordinates doesn't stretch anything.
This is what is so special about the 2d mandelbrot, the Z^2 doesn't stretch any shape (that is small enough) no matter where it is in the complex plane.

BradC, quasi-conformal maps sound like they might be way to go. It seems to mean that stretching is allowed but it is finite / bounded.

ah, I see...
That makes sense :)

So the set of operations listed above seems unlikely to be able to produce interesting fractals as it doesn't have two attractors, everything seems to converge on one point (a single branch point I guess).
The other option is to find the *most* conformal 3d fractal, such as http://www.fractalforums.com/3d-fractal-generation/a-new-3d-mandelbrot-like-fractal/?action=post;quote=9827;num_replies=9;sesc=d3a594eba608ec813a9dad70a2e9fdf2
which is probably about '2 quasi conformal' meaning small spheres map to ellipsoids with eccentricity (long radius / short radius) no greater than 2.
Does anyone know of any 3d mandelbrot-like fractal with conformality < 2 ? (1 is fully conformal like 2d mandelbrot, mandelbulb is infinity-quasi conformal).
Even so, the above link shows that it needs to be more than just nearly conformal, the mapping probably needs to be smooth (infinitely differentiable)... and I'm not sure that it is even possible to have a continuous 3d mapping with conformality < 2.

So what other options? Perhaps in 3d one shouldn't look for it being conformal in euclidean space, but conformal in some other space.

@ infinitely differentiable...

x² can be derived twice
and each iteration doubles the differentiability...

so at iteration infinite, just as the standard mandelbrot, the whole set should become infintely differentiable.
Or not differentiable at all as the graph has infinite k along the border of the set.
But that then should be the same for the standard mandelbrot.

there, you quoted it!

aaaaaaaaaaargh  :-X

don't see what you mean  :pray: