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What are the properties of the mandelbrot set?
         Diversity: Each area is different.
                      The mandelbox does pretty well. The mandelbulb and tetrabrot seems better though.
         History: When you find a baby mbrot, you see patterns remiscent of everything you passed earlier.
                      I believe that the mandelbox fails this criteria (are there baby mandelboxes), the mandelbulb and tetrabrot passes.
         Not stretched: Detail is found in all directions
                     The mandelbox is perfect here, but the mandelbulb has whipped cream regions. The tetrabrot has long "tubes".
         Connected (less important in my opinion):
                    The mandelbulb/tetrabrot passes the mandelbox sometimes fails.
There may be no perfect grail because of you-know-what theorem, however, why can't the transformations' non-conformity cancel each other out? Or nearly so; for example, the mandelbar/tricorn has in most places a tolerable distortion.

Can there be a way that a score system be developed and the formula "evolve" better and better? Or could we improve our human-based scoring system by looking more closely at these.

There must be some judging system to recognise HIM.
But mandelbox is not 3D mandelbrot, becouse 3D mandelbrot must be Z squared what are first requirement for 3D mandelbrot. Mandelbox is something another a pattern formula like 2D ducks/ kalisets.

I don't think HIM will appear suddenly, it will take time and effort or else it would have been found, a true HIM is noble prize material! However, we could tweak the formula incrementally to get closer.

Surely there's more variety in Mandelbox than the tetrabrot- https://sites.google.com/site/mandelbox/negative-mandelbox  :D

I've got a pet theory that black holes are the grail, and that, rather than being spherical they are somehow fractal-like with infinite surface area.
That is based on zero knowledge of black holes, just a guess!
i.e. rather than it existing in 3d euclidean space, it exists in the massively warped 4d space-time of general relativity.

There are couple z^2 varieties in this thread. (http://www.fractalforums.com/index.php?topic=6206.msg51761#msg51761)

(https://lh6.googleusercontent.com/-UsZSY594LBM/UE2KtG50R6I/AAAAAAAABh0/IdAIBUKpIzg/s400/2nd%2520order%2520stalk.jpg)

Nice image Matt. What section of z^2 mandelbulb is this?

When you refer to history, that does not mean that you find small copies of the complete set. Is that right?

Important condition is that "Holly Grail" should be mathematically meaningfull. Mandelbrot set are, it corresponds to bifurcation diagramm of of the logistic map (throught I have no idea what it means). But some of visualy appealing formulas probably just are not very mathematical.

(http://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Verhulst-Mandelbrot-Bifurcation.jpg/220px-Verhulst-Mandelbrot-Bifurcation.jpg)

Anyway I had posted about "Arc of Covenant", 3D fractal what would have natural elements such as fern fronds, coral reefs, trees or seashells. Rotbox hybrid seems somewhat like that item.

Hey, missed your reply.  It's the "stalk" section- the part aligned with the negative x-axis.  The thread is over in the Mandelbulb theory section.  Don't want to hijack this thread, but this current formula has certain features that others lack.