Revisiting the Riemann sphere (again)

[msltoe]

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

Code:

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=z1;

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

[hobold]

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.

[msltoe]

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.

[hobold]

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

[msltoe]

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

[msltoe]

Another version, with bounding spheres to provide some smooth/sharp contrast:

[trafassel]

Very nice finding.

[msltoe]

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

[Kabuto]

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

[msltoe]

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

[jehovajah]

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!

[msltoe]

jehovajah: Thanks! I love spheres for some reason

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

[trafassel]

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++;
}

[trafassel]

Other Variants

[kram1032]

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