Two further spherical coordinate systems

[Mircode]

Hi there!

I want to suggest two (I hope) new spherical coordinate systems for dealing with the rotation.

The colors are X, Y, Z, phi1, phi2 and the yellow point is the point of interest P.
The 3rd component R is still the distance between P and the origin.

I hope it is understandable. The second method even preserves the original Mandelbrot fractal in the x-y-plane.

As soon as I find some time I can implement this, I just wanted to create some eagerness with this post

Greetings,
  Mirko

[cKleinhuis]

this looks scary
we are excited
 

[hobold]

I think both these constructions lead to variants of the Riemann sphere. Probably with different parameterization than what I used for the Riemandelettuce, but that would not be a bad thing. I tried to preserve a 2D Mandelbrot at least in a specific cutting plane, but that might not actually be a good choice for a 3D fractal.

[reesej2]

Looks like fun! Also looks challenging to code. Good luck!

[Timeroot]

Trippy! I suppose there should be some cool results coming from these... can't wait to see!

Hobold: I wonder if anyone has tried to create a 3D Mandelbrot, deliberately destroying any connection with it? So far, all of the most popular contenders are, in my opinion, just spherical-ized versions of the 2D Mandelbrot, with nothing really new to offer. I would like to see some that are intentionally trying to achieve the diversity that we originally wanted with this, not some copycat of the Mset...   

[Mircode]

just spherical-ized versions of the 2D Mandelbrot, with nothing really new to offer

I thought the goal was not just to find a diverse 3D fractal, but a 3D version of the Mandelbrot...
Isnt it a good idea then to try to transform the 2D Mandelbrot principles into 3D Space? And even to preserve the original Mandelbrot set in one cutting plane is totally retro and back to the roots from my point of view

I would like to know if twinbee tried all the coordinate systems he posted here http://www.physicsforums.com/showthread.php?t=331883, at least the working ones. Are the results somewhere?

[hobold]

Different people are looking for different things. Some of us have a specific idea how a voluminous Mandelbrot set was supposed to look - msltoe's newest juliabulb comes very very close.

Some of us are looking more abstractly for a 3D fractal based on a "simple" formula, that exhibits a large variety of structures in all three dimensions. Something that is analogue to the Mandelbrot set in spirit. The original power 8 Mandelbulb is the prime example, with three dimensional seahorse valleys, spirals, and all kinds of beautiful structures. The simplicity of the formula is an aesthetic requirement that would justify such a fractal as being fundamental, or otherwise of singular significance in a sea of more arbitrary fractals.

And some of us don't even care much about the formula, or similarities to the Mandelbrot set, but just want to be blown away by sights never before seen. I guess Tglad's Mandelbox is an example of such "constructed" fractals, and later got topped by the merged formulas that manage to blend completely different kinds of fractals into one single shape. (Mind blowing really, but words failed me, so I never commented in any of those threads.)

And then there are people like me, who explore different approaches mostly for the sake of exploration; question the unquestioned stuff, such as "Okay, zero poles isn't possible - but do we really need two?". I didn't have any further goal, just wanted to know.

[Mircode]

What a very nice response. I really love the spirit here.

Could you support your references with links?
"msltoe's newest juliabulb"
"Tglad's Mandelbox is an example of such "constructed" fractals"
"merged formulas that manage to blend completely different kinds of fractals into one single shape"
I guess it is easy to find with the forum search but it would be nice if I just had to click

"Okay, zero poles isn't possible - but do we really need two?". I didn't have any further goal, just wanted to know.

Yeah I know that drive
And I dont know if this is just an example or if you are still looking for an answer, but I think the first spherical coordinate system I suggested is one. It has only one... kind of double pole. If you construct the iso lines it looks like the attached pic. And I didnt try to support this mathematically, but it looks like all red and blue lines are even perpendicular.

I hope I could brighten your view of the world a little

[hobold]

Links ... allright ...

the "monopolar" spherical coordinates:
http://www.fractalforums.com/theory/alternate-co-ordinate-systems/msg11438/#msg11438

the Juliabulb:
http://www.fractalforums.com/theory/choosing-the-squaring-formula-by-location/msg14246/#msg14246

the Mandelbox, initially known as "The Amazing Fractal"
http://www.fractalforums.com/3d-fractal-generation/amazing-fractal/

and finally some of the hybrid formulas:
http://www.fractalforums.com/3d-fractal-generation/mandelbulb-and-mandelbox-mix/
http://www.fractalforums.com/images-showcase-(rate-my-fractal)/bulbbox-an-amazing-fractal-extension/

[Mircode]

Thanks, hobold, for all the links.

And its nice to see ones exact idea already realised

Here now the result of the transformation suggested in the second pic. Its frus-tra-ting! I also zoomed in a little, its just a mess.

If someone is interested in the formula anyway, here it is.

Code:

R=sqrt(zx*zx+zy*zy+zz*zz);
phi1=atan2(flip(zz) + R-zx);
l1=cos(phi1)*R;
l2=sqrt(zz*zz + (zx-R)*(zx-R));
phi2=atan2(flip(zy) + (l1-l2));

R=R^nR;
phi1=phi1*nphi1;
phi2=phi2*nphi2;

l1=cos(phi1)*R;
l2=(1-cos(phi2))*l1;
zx=R-cos(phi1)*l2;
zy=sin(phi2)*l1;
zz=sin(phi1)*l2;

zx=zx+cx;
zy=zy+cy;
zz=zz+cz;

[reesej2]

As I recall, the standard order-2 Mandelbulb has a similar "unsatisfying" look at first. Have you tried higher orders? I'd like to see what the order-8 version of this looks like.

[Mircode]

Well, I can handle setbacks. I can't expect to have a breakthrough after so few experiments. But seriously? Does it have to look THAT ugly?

[reesej2]

Interesting "claw"-like effect in the bulbs. Also, it's fascinating that, with the only change being the spherical coordinate system, the extra bulbs disappear entirely. All of the bulbs are on a single plane (unless there are others hidden on the other side). Also, the "whipping" effect resembles quaternions... hm...

[jehovajah]

Not as disappointig as you think. You have a bit of tinkering to do and i would suggest starting in the step size and the bailout value. Your scheme seems to highlight a bilateral symmetry so you got some time savings there. However check atan2 to ensure right quadrant evaluation for your scheme, as it is non standard.

Tender hearted as you are you got to accept praise due!

[Mircode]

starting in the step size

Which step size? You mean along the view ray? That was already pretty small I think.

However check atan2 to ensure right quadrant evaluation for your scheme, as it is non standard

I have a slight idea what this means, but I thought atan2 ensures right quadrant evaluation in contrast to atan. Maybe I get you wrong.

Tender hearted as you are

No I'm not, I'm an orange!