Few steps behind perfect 3D Mandelbrot

[Alef]

There are site dedicated to it http://www.skytopia.com/project/fractal/mandelbrot.html. The most 3D mandelbrot are mandelbulbs, however even if mandelbulbs generate gorgeous pictures, fine details of zoom in do not resemble what you get by zooming in 2D mandelbrot set. In quaternion numbers formula z=z^2+c in produce just nice julia sets and formless rotation surface instead of Mandlebrot set with infinite details.

Mandelbrot set is generated becouse in complex numbers x^2 = real^2 >< y^2 = i^2. So mandelbrot set is property of numbers and not a mix of square function parabolas of z=z*z+c.

In quaternion numbers still x^2 >< y^2. But I think z^2 (= j^2) = y^2 (= i^2). So correct z^2  allways will be eather x^2 or y^2 or n*y^2. Square have just two posibilities + or -. So by having equal values at z and y axis it generate rotation surface. Unless numbers are being manipulated so that x^2 >< y^2 >< z^2

A bitt modified previous quaternion formula so increasing iterations fractal woun't degrade set to black screen as the previous equation. Cardoid body reminiscent of 2D mandelbrot, just with some problems around x=0. The head must be the proper feature of the 3D set. More important, zoom reveals the same fine features as 2D set. These 3D pictures realy resemble fractal details of Mandelbrot set. So the result looks as going to the right direction.

Iterate
z= quaternion ( real(z) , imag(z), part_j(z) + part_k(z), 0.5*part_k(z) - part_j(z) );
z= z*z+C;

Could be that this thing could quaternionise most mandelbrot type sets just right coefitients and signs must be found.
p.s.
I would be lucky if this would add my 5 cents to understanding of universe;)

[Alef]

Tried to make  quaternion number manipulation equation so that it would produce good results in all z^3, z^4, z^5, but with increasing power fractal detiorates to rotation surface and z^3 fails to reveal fine details even if it do looks like 2D set. An it have some problems around real=0 axis.

[Alef]

Well there could be some criterias, is a fractal 3D mandelbrot. Probably:
1.) Do it looks like Mandelbrot. There is no proof that 3D mandelbrot couldn't have letters google on the upper surface, but expectation is that it would have something to explore on its back.
2.) Do it have the same cross section of Mandelbrot in z=0 or in some other cross section.
3.) Do it reveals small scale Mandelbrot like fine structures in 3 dimensions.
4.) Is it analytical.

Mathematicly correct z=z*z+c rotation surface quaternion fulfills second and 4th criteria but fails at others, mandelbulb based triplex mandelbrots fulfills first two criterias but fails to reveal branching stalks going to all 3 dimensions. This set mostly fulfills all the criterias, it have right cross cection at z=0 (the same formula generates Mandelbrot in 2D), zoom in reveals swirls and stalcks going to all dimensions. But 3D mandelbrot is expected to have some sort of stalcks on its back and rear, it is expected that main set would consist of spheres, it is expected to have spirals everywhere, wouldn't have hole and j+k 0.5*j-k is pure try and error so not so analitycal.

1.) Fullfilled by half
2.) Fullfilled
3.) Fullfilled
4.) Fulfilled by half

Added different projections and zoom in head area. Chaos Pro were unable to zoom deeper.

[Alef]

Re- rendered the old picture


Here is the same 3D mandelbrot formula rewriten using 3 parameter vector with one additional hidden variable (it's not real 4D becouse 4th and 3rd variables are interdependant ):

zzx = zx*zx - zy*zy -2*zz*zz -zz*zw -1.25*zw*zw +Cx;
 zzy = 2*zx*zy            +Cy;
 zzz = 2*zx*(zz+zw)         +Cz;
 zzw = zx*(zw - 2*zz);

After this there were a lots of candidates, but this have features expected of mandelbrot expanded to 3 dimensions. If there are 3D mandelbrot, it should be something very simmilar, as stalks perfectly corresponds to mandelbrot stalks expanded to 3D space. So far no 3D mandelbrot have stalks. Alsou julia set are the same as quaternionic.
 There are no bulbs on it's back, but very probably that with arithmetic formulas it is impossible to generate fractal with all around bulbs, or z must have two self contradictory features.

If I were a mathematician, I could give a formula and then upload mathematical proof to arxiv.org, but since I 'am not, this is only visual. And it's not a mandelbox, or it alredy would expanded. So probably this means that Quest for Holly Grail is closed.

Zooming at the side bulbs:



Closer:



Stalk:



Here is arrow shaped Perpendicular Mandelbrot of this (XZ plane on y=0). It's streched, smoohed in certain areas and have a satelites, but probably that is the reason why you can see swirled stalks in 3D and not just an one mass of mess. Fractal features of this should be in the same position as bulbs in 3D.

[Alef]

Another render of a bulb right from the previous. IMHO from geometric point of view it's so far closer candidate for a 3D mandelbrot. But not from a aesthetical point of view, too much smooth spirals and rings.

[KRAFTWERK]

Lots of nice images here Alef, glad you keep up the search.

I got a bit interested in this one:



What is it?  An "ordinary" mandelbulb pow2? looks interesting to me...

[cbuchner1]

Another render of a bulb right from the previous.

Looks like a delicate sculpture made out of play-doh. It's the way the colors mix, creating this impression.

IMHO way too whippy-creamy to be a holy grail candidate.

Christian

[cKleinhuis]

Looks like a delicate sculpture made out of play-doh. It's the way the colors mix, creating this impression.

IMHO way too whippy-creamy to be a holy grail candidate.

Christian

dont confuse whipped cream with low iterations, but it looks quaternion style to me as well,
although some features are really nice and a good function overall!

[LMarkoya]

I agree about it looking like a quat, but beautifly fluid and more airy, love the look and indeed it may represent  all that is fluid dynamics, or even the newly discovered breath fingerprint.
Great work.....what are you using to generate the images?

[Alef]

Yeah, its the same pow2. Just can't find that parameter to re- render.
It's still rather outdated Chao Pro but with increased parameters and Render jobs -> standart antialiasing. Only advanced thing is wave trichrome colouring.
Whiped cream or I think it's more like plastic is becouse of iterations, or else some parts are too noisy.  

Maybe I was a bitt skeptical about any better Holy Grail. But this is simple extension of 3D in quaternion numbers using just multiplication and addition so that z are extended to j and k and with not much math behind. It looks like quaternion julia set, and thats way better than quaternion mandelbrot set, and it's julia set looks as quaternion julia as I think it should by.
4. cons.
- Structures loose size too quickly, so that or larger structure looks whiped cream or a smaller structures dissapears.
- Too many extruded rotations and a not enought a real 3D structures.
- Interesting zooms are in negative x range, in real positive there are stretchings as with error - brots.
- All interesting thing apears around y~0.

So 3D space extension to 4D numbers with z coordinate extended to two left variables is a possible solution for 3D mandelbrot, aka the hidden variable. Formula still is not the right, it weren't found by analytical means, but looks that a 3D mandelbrot is possible. I think, it must be extended to 4 variables so that coordinates |(0,0,m)*(0,0,m)|=m^2 and not anything else. Just that don't expect too much as 2D mandelbrot is made of threads and slower escaping fields possible only with Levi render.

Same spot rendered with larger iteration number. No whipped cream, but too small structures created noise.


Another braid. If 3D mandelbrot represents some physical process that is more probable than seashells.


Zoom at anteriour bulbs. Would be better without stretching at posteriours of 2nd order minubulbs.


Zoom at the seashell or cornucopia of opper left of the previous image.

[Alef]

Zoom at the first anteriour bulb. More clear mandelbrot pattern. Somehow forward looking stalks are more clear than backward looking (here it's look from bottom, hence its oriented in opposite). Maybe with some math this formula could be tweaked to more balanced thing.



Then I zoomed to the ballon at the top right corner.


I think my next contest entry would be based on this. I 'm alredy rendering and tweaking it.

[Alef]

A bitt explored perpendicular brot (If mandelbrot set is at z=0, then perpendicular brot is at y=0).
At the start it don't looked anywere good, bhea distorted, but then it turned out unexpectedly revealing and a thing in itself.
Perpendicular brot:
zzx=zx*zx -2*zy*zy -zy*zw -1.25*zw*zw + Cx
zzy=2*zx*(zy+zw) + Cy
zzw = zx*(zw - 2*zy)

These endless (herman?) rings should corespond to swirls in 3D. It's pretty zoomable, throught I don't went beyon fast extended precision with deepest is 9.1E14 (scarecrow), it stretches to one side or another but then zooming deeper it streches back.


Julia set is with the seed real=0.4625; imag=0.275



And zooms in mandelbrot set in about the same area.





And some other mandelbrot zooms.





UF parameter, update formulas.

Code:

perpend_grail_1 {
fractal:
  title="perpend_grail_1" width=640 height=480 layers=1
  credits="Edgar;8/18/2013"
layer:
  caption="Background" opacity=100
mapping:
  center=0.4570065989755775/0.2741015669120785 magn=157510.55
formula:
  maxiter=200 percheck=off filename="em.ufm"
  entry="PerpendicularGrail" p_bailout=10000000000000 p_function=None
  p_talisfactor=0.25 p_settype=Mandelbrot p_switchsettype=Julia
  p_julia=-0.48098958252/0.72421874965 p_inverted=no p_fishstrenght=1
  f_func_post=ident p_vectype=None p_vfactor=-0.3/0
inside:
  transfer=none
outside:
  transfer=linear filename="em.ucl" entry="Pauldebrots_Smooth"
  p_convergent=no p_esm=no p_alt=no p_perfix=1.0 p_zmax=100 p_power=2
  p_fit=yes p_fittimes=1.0 p_fitminit=27 p_fitmaxit=200 p_transfer=Log
  p_transpower=3.0 p_bailout=10000000000000
gradient:
  smooth=yes rotation=1 index=0 color=6303744 index=64 color=12085789
  index=168 color=16777197 index=257 color=33023 index=343 color=512
opacity:
  smooth=no index=0 opacity=255
}

[Alef]

Here I noticed one thing.
3D mandelbrot are expected to have a bride like stalks. Stalks in 3D space must have some finite volume, or else we wouldn't be able to observe them. That means, every cutout of a stalk would be a 2D object with a finite area.

Paralel thing (cutout in XZ plane) have finite size objects. Mandelbrot set on another hand have finite size cardoid, finite size side bulbs, finite size (but very small) small mandelbrots and INFINETELY SMALL STALKS. Thus, if mandelbrot set is a cutout of any 3D fractal, then at the side of this cutout there woun't be any observable stalks. 
In another words, IT'S IMPOSIBLE to have expected 3D mandelbrot with all around stalks. Only spheres are possible. Or there should be a formula (in XY plane) so that when z=0 there is mandelbrot set but when z=n (with n being any finite number) there are set with properties of above (finite 2D stalks). Maybe it's possible only with formula involving infinity, or you can have invisible grailness. But in the side wre a mandelbrot set is a cutout stalks are pretty imposible.

Throught even if there is no 1:2 conformal transfarmations, if there alsou aren't proof that it's impossible, it could mean that there so far is not invented one.

Simple english speak me not I'm yoda.

[bib]

Stalks could be visible with iteration threshold. Same as in 2D: you're not supposed to see stalks if you do not use any coloring method for example to make gradient around them.

http://krzysztofmarczak.deviantart.com/art/Real-shape-of-Mandelbulb-381908457

[Alef]

Stalks could be visible with iteration threshold. Same as in 2D: you're not supposed to see stalks if you do not use any coloring method for example to make gradient around them.

http://krzysztofmarczak.deviantart.com/art/Real-shape-of-Mandelbulb-381908457

Wow, impressive render. It's beautifull!
Sounds as the same method as Levi's shot at Grail http://www.fractalforums.com/new-theories-and-research/a-shot-at-the-holy-grail/ 
But that didn't disproves that solid all around stalks are impossible with a mandelbrot set at z=0.