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Author Topic: Kaleidoscopic (escape time) IFS  (Read 34337 times)
Description: An interresing class of fractals
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David Makin
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« Reply #15 on: May 03, 2010, 09:45:19 PM »

Thanks.
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JosLeys
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« Reply #16 on: May 03, 2010, 10:47:15 PM »

Quote
The -2 is in reality the radius of a bounding sphere of the fractal..

I'm not sure I understand.
These things do not turn to dust as far as I can see.
I'm now doing something where I've zoomed to a magnification of about 1.5 million, using just DE=sqrt(R)/scale^i. I simply stop if DE<A, and my A is now about 1E-11. Must say I have max iters set at 100 and bailout at 1000.

Here it is (without anti aliasing) :


* KaleidoIFS_010.jpg (176.08 KB, 640x640 - viewed 277 times.)
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Buddhi
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« Reply #17 on: May 03, 2010, 10:49:08 PM »

These Kaleidoscopic fractal are amazing!!! I have to try with this.
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subblue
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« Reply #18 on: May 03, 2010, 11:26:12 PM »

I'm loving this new algorithm. Lots to explore smiley

Lots of windows:


And the inside:

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knighty
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« Reply #19 on: May 04, 2010, 02:08:56 PM »

Quote
The -2 is in reality the radius of a bounding sphere of the fractal..

I'm not sure I understand.
These things do not turn to dust as far as I can see.
I'm now doing something where I've zoomed to a magnification of about 1.5 million, using just DE=sqrt(R)/scale^i. I simply stop if DE<A, and my A is now about 1E-11. Must say I have max iters set at 100 and bailout at 1000.

Here it is (without anti aliasing) :
With a max iterations of 100 you don't need the -2. If scale=3, A=1E-11 corresponds roughly to 21 iteration. So with max iters set 100 you can (in principle) zoom much deeper before getting dust.

Here is a sequence for the last picture of the original post. this is done with rotate1()=rotate2(). the rotation is about the z axis.


Here is the algorithms I use for simple tetra-sierpinski and menger sponge. A nice feature of these is that you can control the recursion depth just like the classic "geometric" method:
Code:
sierpinski(x,y,z){
   scale=2;
   r=0;
   for(i=0;i<MI && r<1;i++){
      x1=rot[0][0]*x+rot[1][0]*y;
      y1=rot[0][1]*x+rot[1][1]*y;
      z1=z;
      x=x1;y=y1;z=z1;
     
      if(x-y<0){x1=y;y=x;x=x1;}
      if(x+y<0){x1=-y;y=-x;x=x1;}
      if(x-z<0){x1=z;z=x;x=x1;}
      if(x+z<0){x1=-z;z=-x;x=x1;}
      if(y-z<0){y1=z;z=y;y=y1;}
      if(y+z<0){y1=-z;z=-y;y=y1;}
     
      r=(x+y-z-1)*1/sqrt(3);
     
      x=scale*x-(scale-1);
      y=scale*y-(scale-1);
      z=scale*z-(scale-1);
   }
   return r*scale^(1-i);
}

menger(x,y,z){
   scale=3;
   r=0;
   for(i=0;i<MI && r<1;i++){
      x=abs(x);y=abs(y);z=abs(z);
      if(x-y<0){x1=y;y=x;x=x1;}
      if(x-z<0){x1=z;z=x;x=x1;}
      if(y-z<0){y1=z;z=y;y=y1;}
     
      x1=x-1;y1=y-1;z1=z-1;
      r=max(x1,max(y1,z1));
      /*nice alternative that gives rounded (external) edges.
      It's not necessary to calc the sqrt() at eache iteration
      if(x1<0 && y1<0 && z1<0) r=max(x1,max(y1,z1));
      else r=sqrt(max(0,x1)^2+max(0,y1)^2+max(0,z1)^2);
      */
     
      x=scale*x-(scale-1);
      y=scale*y-(scale-1);
      z=scale*z;
      if(z>0.5*(scale-1)) z-=(scale-1);
   }
   r*scale^(1-i)
}
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bib
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« Reply #20 on: May 04, 2010, 03:08:18 PM »

I like a lot the "mechanical" aspect of this cubes.

Next challenge : can anyone do a fractal that will look like the Eiffel tower ? cheesy

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knighty
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« Reply #21 on: May 04, 2010, 09:39:36 PM »

That should be possible. We already have one that looks a little like "l'arche de la Défense".  grin

I'm loving this new algorithm. Lots to explore smiley

Lots of windows:
<Quoted Image Removed>

And the inside:
<Quoted Image Removed>

The inside looks promising. It would be useful to have a cutting plane at hand for those that have more interresting features inside.

You can add colouring as in this post http://www.fractalforums.com/index.php?topic=2526.msg12794#msg12794. I use this one for the menger sponge family:
Code:
Mengercolor(x,y,z){
   r=x*x+y*y+z*z;
   c=0;
   for(i=0;i<MI;i++){//It is better to use a fixed iterations number to avoid shift when zooming.

      rotate1();

      c1=0;
      c1+=0.5*(1-sgn(x));c1+=0.5*(1-sgn(y));c1+=0.5*(1-sgn(z));
      x=abs(x);y=abs(y);z=abs(z);
      if(x-y<0){x1=y;y=x;x=x1;c1+=1;}
      if(x-z<0){x1=z;z=x;x=x1;c1+=1;}
      if(y-z<0){y1=z;z=y;y=y1;c1+=1;}

      rotate2(); 
 
      x=scale*x-stc[0]*(scale-1);
      y=scale*y-stc[1]*(scale-1);
      z=scale*z;
      if(z>0.5*stc[2]*(scale-1)) {z-=stc[2]*(scale-1);c1+=1;}
      c+=c1*0.5;
      r=x*x+y*y+z*z;
   }
   return c;//this will be used as an index into a palette
}
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Hamilton
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« Reply #22 on: May 05, 2010, 12:14:14 AM »

That's great shapes you got there, Knighty.  smiley
It sounds like a very interesting algorithm, indeed.
Did you use your own renderer to get thoses images?
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knighty
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« Reply #23 on: May 05, 2010, 02:17:21 PM »

It is a little, badly written, full of bugs script that runs with evaldraw (a nice program by ken silverman that includes a just in time c-like language compiler).  grin
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Nahee_Enterprises
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« Reply #24 on: May 05, 2010, 02:43:49 PM »

I'm loving this new algorithm.  Lots to explore  smiley
   ..........
And the inside:

I like this cut-away view!!   smiley   
Reminds me of a four barrel carburetor that gave me problems once while I was trying to give it a new kit.
 
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Jesse
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« Reply #25 on: May 05, 2010, 05:24:14 PM »

Have only done some basics without rotation yet, but it gives also funny alternating hybrids:


Yes, it is a hybrid of vegetables and mint bonbons...
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knighty
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« Reply #26 on: May 05, 2010, 10:09:17 PM »

Nice rendering! (as usual  tongue stuck out) Is it a mix of juliabulb and tetra-sierpinski?
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JosLeys
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« Reply #27 on: May 05, 2010, 10:22:26 PM »

Knighty, do you have some background information on how you constructed the generator code?
I mean, how does does
if(x-y<0){x1=y;y=x;x=x1;}
    if(x+y<0){x1=-y;y=-x;x=x1;}
    if(x-z<0){x1=z;z=x;x=x1;}  etc etc

generate the Sierpinsky ? ..and the code for the Menger sponge is also a bit of magic in my eyes..

I was trying to generate an octahedral Sierpinsky, but so far all my efforts have failed..
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knighty
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« Reply #28 on: May 05, 2010, 11:22:45 PM »

These are the symmetry planes of the tetrahedra...etc. I have to go now. I'll try to give more info tomorrow.
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msltoe
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« Reply #29 on: May 06, 2010, 12:46:09 AM »

Good stuff here...

If you remember some of my escape-time Sierpinskis, what you can do in general is define a set of vertices (e.g., those for an octahedron). Then for a given "z", find the nearest vertex. Reflect off the vertex, etc. using 2*(vertex)-point or something like that.

http://www.fractalforums.com/3d-fractal-generation/sierpinski-like-fractals-using-an-iterative-function/

-mike
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