The crystal cavern is *cool*
"Tglad, what made you think of this fractal?"
It came from looking at the whipped cream issue on the mandelbulb. Z^2 is a conformal transform, any tiny circle will remain a circle when transformed. The triplex mandelbulb Z^2 is not conformal, a sphere gets squashed and stretched by the transform. Almost all nice looking fractals are conformal, e.g. indra's pearls, appollian gasket, sierpinski triangle etc.
But in 3d you can't do Z^2, or anything like it which is conformal. This is proven (http://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings)
It says that in 3 dimensions or higher, the only conformal transformations are translation, rotation by a fixed angle, scale and sphere-inverse-plus-reflection.
I wanted a fractal that covers itself just like the mandelbrot does. That is impossible to do with the above set. But if you allow the transform to also be anti-conformal then you also can allow reflections about any plane. This allows you to fold (or reflect) the space over itself. The box fold is an example, if you then scale by 2 afterwards then we have a mapping which keeps small vectors close to the origin, but large vectors fly off to infinity, this is just like the mandelbrot.
msltoe's fractals did just this http://www.fractalforums.com/3d-fractal-generation/sierpinski-like-fractals-using-an-iterative-function/
and look awesome. I tried to improve on this idea by preventing discontinuities (no jumps in the mapping) and making use of the sphere inversion as a way to add variety to the fairly dull box-fold-only fractal.
Here's some new pics. ThroneMountain entrance
The +C is used so the fractal changes with location. Unfortunately, even though this is true, It looks like the mandelbox (and any similar objects) is missing a very nice property of the mandelbrot. In a mandelbrot, wherever you are, you can always navigate to a different area by zooming in on a minibrot, this keeps the shapes all mixing up at all detail levels. In the mandelbox each area sticks with its shapes as you zoom in, for example zooming in on an archway creates shapes out of lots of archways, zooming in on the disconnected cubes in the corner creates shapes out of disconnected cubes. You can't get to the disconnected cubes from inside the archways area by zooming.
I think the reason is that the mandelbox changes shape if you scale it each iteration, so it can never have proper mini-boxes. It comes down to it not using a Z^2 type operation, so back to square 1
The mandelbulb should retain this nice property, at least partially if not fully. So, I guess the optimal 3d Mandelbrot-like fractal depends on which properties you choose to keep in 3d... its certainly fun searching