Not 3D mandelbrot but 3D tricorn

Alef

Fractal Supremo

Posts: 1174

Not 3D mandelbrot but 3D tricorn

« on: June 01, 2017, 04:40:13 PM »

Hi all.

In 2D mandelbrot set have property of universality. You can make some random formula, zoom and somewhere you could find a mandelbrot. In 3D it is impossible.
The only analog of that property is 3D tricorn fractal conj(z^2) +c . In 3D conj( quaternionPower2(z)) +c generates very ugly but tricorn fractal.
However in Mandelbulb3D there is a trigonometry based tricorn with tricorn and mandelbrot sets in cutouts. Hypercomplex multiplication based tricorn fractal is just like that, with tricorns on side and mandelbrot set on top.
Hence there is some universality.

I think all of the versions of 3D mandelbrots is generated becouse of symmetry brake. There is x dimension. And then for all "normal" numbers y = z = w. Conj function induces symmetry brake and the fractal could be generated.
Maybe all of the trigonometric based mandelbulbs are also created becouse of symmetry brake at the local level. Mandelbulb surface realy could be just the result of repeating trigonometric functions.

Aka there is no real 3D mandelbrot. And all of the fractals are carved out from featurless 3D rotation or square surfaces.

Formula of this alredy are here. http://www.fractalforums.com/mandelbulb-3d/em-formulas-for-mandelbulb3d/ Actualy I found it seeking more methods of multiplication. It just a conjugated hypercomplex power 2.

HypercomplexTricorn_2_cutoutYZ.jpg (12.55 KB, 500x320 - viewed 177 times.) HypercomplexTricorn_2.jpg (36.41 KB, 600x480 - viewed 3 times.)
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Alef

Fractal Supremo

Posts: 1174

Re: Not 3D mandelbrot but 3D tricorn

« Reply #1 on: June 12, 2017, 01:48:31 PM »

In fractal generation crucial part should play the
https://en.wikipedia.org/wiki/Symmetry_breaking

This hasn't been mentioned a lot when speaking about fractals. The concept is popularn in cosmology and particle physics. The iterations should produce fractals becouse of symmetry breaking.
Field of numbers is gradient. Then iterating ducks fractals (in fractalforums - kaliset) small changes on values produce hudge difference of outcome.

The conjugate function breakes symmetry between dimensions y and z. I noticed that most of the good looking 3d mandelbrot formulas used as raising in power produces non-symetric folds or there are certain other non- symetry. Well, most of folks pretend they are more specialist on fractals that they are.

Anyway, this 3D mandelbrot beyond mandelbulb is an illusory target. All of them are more or less 3D mandelbrots without any true.

In 3D people realy are interested in characteristics no 3D mandelbrot could have. In 3D threads just don't look good. Hence in 3D grail is Mandelbox. Or mayby grail is Folding Int Power, since its is one of the most used formula. Or maybe there is something even better than these, say 3D fractal of seashells, ferns and trees.


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Alef

Fractal Supremo

Posts: 1174

Re: Not 3D mandelbrot but 3D tricorn

« Reply #2 on: July 04, 2017, 05:06:24 PM »

I found that in fractalforums threads in old Mandelbulber hypercomplex number mandelbrot have different formula:
http://www.fractalforums.com/index.php?topic=3926.msg36463#msg36463
(I tried it instead and it generated whiped cream tricorn and ugly mandelbrot)

So here is this hypercomplex number power + conj function ( conj(x,y)=(x,-y) ) :

Code:

x1:= x*x - y*y - z*z + w*w;
y1:= 2*z*w - 2*x*y;
z1:= 2*x*z - 2* y*w;
w1:= 2*x*w + 2* y*z;

So here hypercomplex power 2 is:

Code:

x1:= x*x - y*y - z*z + w*w;
y1:= 2*x*y - 2*z*w;
z1:= 2*x*z - 2* y*w;
w1:= 2*x*w + 2* y*z;

Not shure whitch is correct in mathworld sense or then I should recalculate power 2.

Realy 3D mandelbrot is owerexpectations. Power 8 Mandelbulb is OK, and mandelbox have qualities 3D requires. Visual qualities of 2D mandelbrot if transfered into 3D must not be astonishing. 3D is a new world. Searching for it was more important than the result. It's dao.


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youhn

Fractal Molossus

Posts: 696

Shapes only exists in our heads.

Re: Not 3D mandelbrot but 3D tricorn

« Reply #3 on: July 04, 2017, 07:53:31 PM »

You are - most probably - right!

At least the part "most of folks pretend they are more specialist on fractals that they are" which could apply on me, as I do read a lot of subjects (science, art, fractals, music, computers, open source, etc) I do not have the time to fully dive into those subjects. This gives me a broader view with lots of general feelings of connectivity, but makes it hard to explain details clearly and correctly. The "real" 3D Mandelbrot/Mandelbulb always seemed an illusion to me, not searching the unknown but looking for the non-existing.

Thanks for referencing the Symmetry Breaking wikipedia page. This is just 3 clicks away from fractal related webpages:

Symmetry breaking --> Pattern formation --> Patterns in nature --> Trees, fractals

* youhn have to read more on entanglement, symmetry breaking and the structure of time-space.


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Alef

Fractal Supremo

Posts: 1174

Re: Not 3D mandelbrot but 3D tricorn

« Reply #4 on: July 05, 2017, 02:57:01 PM »

It's just like scientists of old (classical) times. When there was just one all encompassing discipline.


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Kalter Rauch

Iterator

Posts: 198

Re: Not 3D mandelbrot but 3D tricorn

« Reply #5 on: July 19, 2017, 08:50:09 AM »

Another thing I don't see talked about are fractional power Mandel/Juliabulbs.
Some 3D formulas in ChaosPro allow that, as does the Joukowski transform which I use a lot.
So far, the effect of a fractional power seems to be a kind of "mitosis" or splitting of features (eg. "petals", etc.)
along a seam.
This visibly annoying crease can be overcome by using Perlin FBM and other transforms.
It's possible to zoom onto the tip of the developing power increase and see the distortion before actually splitting.


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