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Author Topic: Riemann fractals  (Read 4731 times)
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msltoe
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Posts: 187


« on: June 27, 2011, 03:33:03 AM »

 Inverted density Mandelboxes have some features that look like Riemann spheres. It follows that it might be possible to build a similar fractal directly from the stereographic projection/Riemann equations. There's a lot of possible ways to do this. Remember the one hobold invented? http://www.fractalforums.com/theory/alternate-co-ordinate-systems/15/

 Here's another variation using the stereographic projection transformations from Wikipedia.

 a=0;b=0;c=1;
 while ((norm<8 )&&(iter<imax)) {

    r=sqrt(x*x+y*y+z*z);
    x=x/r;y=y/r;z=z/r;
    s = x/(1-z);
    t = y/(1-z);
    w1=1+s*s+t*t;
  
    s=fabs(sin(pi*s));
    t=fabs(sin(pi*t));
    
    r=-0.25+pow(r*r,w1);
  
    w=1+s*s+t*t;
    x=r*2*s/w;y=r*2*t/w;z=r*(-1+s*s+t*t)/w;
    
    x+=a;y+=b;z+=c;
    norm = x*x+y*y+z*z;
  
    iter++;
 }


* pollen2.jpg (137.16 KB, 720x686 - viewed 427 times.)
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KRAFTWERK
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« Reply #1 on: June 27, 2011, 07:37:01 AM »

Wow, that is one beautiful fractal msltoe!  A Beer Cup
Seem to have lots of "holy Grail:ish" details... do you have a zoomed in image?  afro
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DarkBeam
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Posts: 2512


Fragments of the fractal -like the tip of it


« Reply #2 on: June 27, 2011, 10:55:34 AM »

Wow, that is one beautiful fractal msltoe!  A Beer Cup
Seem to have lots of "holy Grail:ish" details... do you have a zoomed in image?  afro

Well, some zones are strongly stretched, great formula anyway! It's provocative wink wink
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DarkBeam
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Posts: 2512


Fragments of the fractal -like the tip of it


« Reply #3 on: June 27, 2011, 07:14:57 PM »

I tried to resist... But simply can't grin

I have coded it, but some unwanted details show up; and he pretends a negative radius power huh? to work

But the image looks identical to msltoe's! I added many extra-goodies, hope he don't mind. grin
« Last Edit: February 10, 2015, 12:43:47 PM by DarkBeam, Reason: riemann m3f is incl in mb3d now » Logged

No sweat, guardian of wisdom!
msltoe
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Posts: 187


« Reply #4 on: June 27, 2011, 07:54:58 PM »

DarkBeam and Kraftwerk : Thanks for the encouragement.
DarkBeam: I'm sure there are better variants out there. This is just to get your curiosity started. I'm pretty sure we should be able to make something like Romanesco broccoli with a little bit of ingenuity and luck.

-mike
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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #5 on: June 27, 2011, 08:44:52 PM »

The problem of this formula is that uses "unstable" functions like sin and exp. The critical area is x<0 & y2+z2<something, that should be cutted off... Anyway I have swapped x and z to make it more hybridable smiley a good variant is with a neg scale...
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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #6 on: June 28, 2011, 10:15:11 AM »

If you shift by pi/2 s and t before the sin(), you obtain a set where the empty slots are filled in by new "cactuses" cheesy cheesy cheesy





Can it be done in automatic please? agree


Mandelbulb3Dv17{
P.....S....O/...w....2...............1.......s1E................................
........................................kz1........A./..................y.2.....
................/ME//....6Eo....U.....E2.....omEQgisq6rD/..........c./...w1....U
z.....kD32../..........wz..................................2.....y1...sD...../..
.w1...sDYsAIxzzzjz1..........WAbUdisqcljRRxgCKosEwnYRydIKa5uzwPN9SaPRGyDFHQlSHA1
Iw1iNMIOip7tzqF/G0ZZtVyD......YP..............kD.2....sD..kz....................
.............................UTRU4.Yq/O.cQ5c/.wRU4.Ks/O.kX5c/UMSU4..............
.....................wzzz1.U..6.P....M4...EB....W....61....F....8/...I1....UJl32
...U.iVFwxDE./ozPM2Tzz7lz16.mc..zXCc.El18XGQeGyDjvIRhrVAkz1............28.kFrA0.
FWb96aAIVzvh1se7Umvxz0........../6U0.wzzz1................................E.0c..
zzzz.................................2U.8.kzzzD.................................
/6U0.wzzz1...................................2CcN/UvNP6.eeWCNq0.yRii.EJJUk1f..XR
S1.mx3CcN/UvNP6.QsLsUa3.ibhV..bTV1OK.sSq40.ly3CcN/UvNP6.MwLsUa3.ibhV.kqTV1OK.sSq
40.kz3CcN/UvNP6...EsUa3.eeWCNq0.IJ36wk8.wyLsUa3.................................
E6...6....E.....I....c....UIdJKPVtaPm.kPrJaQ.........................MU/4.......
...................................................wz..........kTIsuFVf5tzn53iSI
suFyz.........zD................................................................
.....................2.....3....8....6JOZpKMitaA................................
4MU/..........................................................zD..........A.....
........................kz1.....................................................
................................}

my "optimized" version of your formula with the "goodies"...

    (rotate xyz)
    r=scale/sqrt(z*z+y*y+x*x);
    x=x*r;z=z*r;y=y*r;r=fabs(r);
    s = z/(1-x);
    t = y/(1-x);

    w=1+s*s+t*t;
   
    s=fabs(sin(pi*s+ssh));
    t=fabs(sin(pi*t+tsh));
   
    r=-0.25+pow(r,rp*w);w=2/(1+s*s+t*t); // It pretends rp=-2 and not +2, why? I've found the answer, since I use 1/r instead of r cheesy
    // originally; w=1+s*s+t*t;z=r*2*s/w;y=r*2*t/w;
    // x=r*(-1+s*s+t*t)/w=r*(1-2/w); optimized & subst...
    x=r*(1-w);
    y=r*t*w;z=r*s*w;
   
    x+=xsh;y+=ysh;z+=zsh;
    norm = z*z+y*y+x*x;
« Last Edit: June 28, 2011, 08:30:51 PM by DarkBeam » Logged

No sweat, guardian of wisdom!
msltoe
Iterator
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Posts: 187


« Reply #7 on: June 30, 2011, 03:52:01 AM »

DarkBeam: What if every iteration was filled? Would it get too fuzzy?

-mike
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