Approaching the holy grail with a scoring system of the like

[kjknohw]

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.

[Alef]

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.

[kjknohw]

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.

[Tglad]

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

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.

[M Benesi]

There are couple z^2 varieties in this thread.

[jehovajah]

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

[mfg]

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

[Alef]

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.



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.

[M Benesi]

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

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.