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Another variant of a Riemann sphere fractal that occurred to me yesterday.

(http://nocache-nocookies.digitalgott.com/gallery/15/803_15_01_14_1_28_06.png)

Looking through the gaps, there seem to be some interesting structures beneath the outer shell. Overall the shape appears to be fairly regular, though. More like a Julia set, with the same "theme" everywhere, and less like a Mandelbrot set, with different kinds of shape in different areas.

Yes. Julia-like. Perhaps, we could add a bit of "spin" or spiral to it. The one thing I didn't incorporate which I originally envisioned are bounding spheres to cover up the inside. This means that with too large an iteration count, this object disappears into foam. I think the inside may have some interesting locations, but it's also where things get non-conformal.

I am unable to render this fractal. What value did you choose for maximum iterations? What size is the object?

Give the system a cubic extent of (-4,-4,-4) to (4,4,4). Total number of iterations = 8.

Another version, with bounding spheres to provide some smooth/sharp contrast:
(http://farm4.staticflickr.com/3778/11964421373_5b109c3fa3_o.png)

Very nice finding.

trafassel: Thanks for replicating. Looks nice! There's some flexibility in the parameters. Perhaps, there's a romanesco hiding in there...

GLSL attempt: http://glsl.heroku.com/e#13664.1

It was a bit tricky to tweak the formula to emit a proper distance and it's still not 100% perfect and there are some strange "spiderwebs" inside now :)

Cool! I think when 1-z is close to zero, you can get infinite lines.

This reminds me of that cloud phenomenon when a volcano erupts! Although too regular at the moment it also reminds me of soap bubbles in the bath before they get like dry foam.

The sphere is such a remarkable form, especially because it does not tessellate. In terms of physicality , bubbles do have contact surface forces that shape them, so to be able to bound these spheres is interesting.

Very nice work!

jehovajah: Thanks! I love spheres for some reason :)

BTW, when you flip this fractal on its side, it becomes an impostor to the Mandelgrail:

Here is the variant:

while ((norm<40960)&&(iter<imax)) {

  (*nfunc)++;
 
   r = sqrt(x*x+y*y+z*z);

   r1 = 1.0/r;

   x=x*r1;y=y*r1;z=z*r1;

   x1 = x/(1-z);
   y1 = y/(1-z);
   z1 = (r-1.5)*(1+x1*x1+y1*y1);

   x1 = x1 - floor(x1+0.5);
   y1 = y1 - floor(y1+0.5);
   
   x=4*x1;y=4*y1;z=-2.0*z1;

  r = x*x+y*y+z*z;
   
  norm = r;
  iter++;
}

Other Variants

That colorful variant looks awesome! It's a flower fractal :D
The other two are great too :)

Nice to see someone trying out new ideas, and it looks beautiful too, great work guys!

Here is msltoe's object in tumbling motion:

http://www.vectorizer.org/rmdltc/msltrie001.mkv (http://www.vectorizer.org/rmdltc/msltrie001.mkv)  (25MB, 13sec)

It is a lot less solid than what I expected from the first pictures. Fluffy, almost, as far as most of the outer shell is concerned. The animation more clearly reveals the cool trick that msltoe has used: the poles are deep inside, so the object's surface does not suffer much from the usual defects that go with non-conformal mappings.

This looks like a promising new technique IMHO. Keep the variants coming! :)

In my tests the Riemann Sphere always dissolve at high iterations. So most images are created with 5 or 6 iterations. But if you add z=Math.Pow(z,0.999999) the structure becomes solid. Inner shape changes too. This image is created with 55 iterations.


 for (int n = 1;n < _cycles;  n++)
  {
    double xx = x*x;
    double yy = y*y;
    double zz = z*z;
    double r = Math.Sqrt(xx+yy+zz);
    if (r>5) return false;

    z = z/r;
    x = x/(1.0 - z)/r;
    y = y/(1.0 - z)/r;
    z = (r - 1)*(1 + x*x + y*y);
    z = Math.Pow(z,0.999999);
    x = 2.5*(x - Math.Floor(x + 0.5));
    y = 2.5*(y - Math.Floor(y + 0.5));
  }

... and the corresponding inner view.

Great

Interesting and a cool find. O0

BTW This reminds me that user gannjondal made a M3D jit  with various tiling methods for the Riemann  sphere (I have yet to try them out.)
http://www.fractalforums.com/new-theories-and-research/another-way-to-make-my-riemann-sphere-'bulb'-using-a-conformal-transformation/

  Didn't code yet, but you think maybe trying "if (z==0) then {z=1;}" would do the same thing? 

I tried, but it does not the same thing.

The same with if (z==1) then {z=0;}  .... sorry.

The answer is much simpler:

I have to replace
z = Math.Pow(z,0.999999);
with
if( z1<0) return true;

(Math.Pow(z,0.999999) gives NaN for negative numbers and this affects the following tests.)

Bye the way, perhaps we have to go back into the inside.

... and if you use
 if (z1<-10) return true;
you get spheres surrounded by spheres. This explains much. I will not pretend to understand all IFS formulas, but I begin to get a "feeling" of how all this works.

At first sight I thought I was looking at a picture of the known universe !

Spheres while perfect forms are for me the basis for curvilineal reference frames, and our best chance at modelling dynamics in natural systems at all scales, especially if connected intuitively to magnetic behaviour in a fluid paradigm