Revisiting the Riemann sphere (again)

[KRAFTWERK]

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

[hobold]

Here is msltoe's object in tumbling motion:

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!

[trafassel]

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));
  }

[trafassel]

... and the corresponding inner view.

[Adam Majewski]

Great

[mclarekin]

Interesting and a cool find.

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/

[M Benesi]

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));
  }

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

[trafassel]

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.

[trafassel]

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

[trafassel]

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

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.

[trafassel]

... 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.

[jehovajah]

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