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Author Topic: Quasi-Fuchsian fractal goodness  (Read 12781 times)
Description: Yes that old chestnut
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twinbee
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« on: January 22, 2010, 07:16:36 PM »

I still love the look of this 3D fractal type. Despite its IFS nature, in many ways it's much closer the the 'true' 3D Mandelbrot, than the Mandelbulb is.

However, nobody apart from the original creators has ever rendered it, or even spoke much about it. There is information out there to create this object, but it's pretty scarce and opaque (well for me anyway wink ).

It is of course the 'Quasi-Fuchsian' fractal by Dr. Kazushi Ahara and Dr. Yoshiaki Araki. Here is their gallery: http://www.math.meiji.ac.jp/~ahara/quasisphere/Gallery.html

... and here is a video:
<a href="http://www.youtube.com/v/3lcO9zRCv-4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/3lcO9zRCv-4&rel=1&fs=1&hd=1</a>

Anybody had a crack at recreating this?
« Last Edit: January 22, 2010, 07:18:09 PM by twinbee » Logged
JosLeys
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« Reply #1 on: January 22, 2010, 08:09:14 PM »

The technique consists of repeatedly mirroring the space enclosed between a set of spheres in all the spheres that make up the set. The 'mirroring' is mirroring in a sphere, so in other words, a sphere inversion. Sphere inversions preserve spheres, so that the tiniest detail of the objects in the film are still pieces of spheres.
Due to this, to me this is not anywhere near the 'Holy Grail', although it generates very interesting pictures.

The trick to making this work is to start with the right set of spheres, which is not trivial. It is a lot like finding the right parameters for 'Kleinian Groups' as explained in 'Indra's Pearls', and this is only 2D.
 I can find no publicly available articles where the principles of the right constellation of spheres is explained. I guess we need a new, 3D version, of Indra's Pearls to start playing with this..
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kram1032
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« Reply #2 on: January 22, 2010, 08:36:36 PM »

what would happen if you took the total spherical inverse of this? Just a huge set of nearly-spacefilling spheres with some gaps in the center?
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JosLeys
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« Reply #3 on: January 22, 2010, 10:08:12 PM »

The spheres that form the basic buiding block will necessarily overlap, so no sphere packing I'm afraid...
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kram1032
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« Reply #4 on: January 23, 2010, 02:17:33 PM »

That's not what I meant.

Those Quasi-fuchsian fractals have gaps in them where no spheres show up.
So, if you take the inverse, those gaps would still be somewhere...
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twinbee
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« Reply #5 on: January 23, 2010, 07:14:24 PM »

They make it sound easy here:
http://www.math.meiji.ac.jp/~ahara/quasisphere/index.html

I wonder what a 2D equivalent of this set would look like.
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Aexion
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« Reply #6 on: January 24, 2010, 02:44:05 AM »

I still love the look of this 3D fractal type. Despite its IFS nature, in many ways it's much closer the the 'true' 3D Mandelbrot, than the Mandelbulb is.

However, nobody apart from the original creators has ever rendered it, or even spoke much about it. There is information out there to create this object, but it's pretty scarce and opaque (well for me anyway wink ).

It is of course the 'Quasi-Fuchsian' fractal by Dr. Kazushi Ahara and Dr. Yoshiaki Araki. Here is their gallery: http://www.math.meiji.ac.jp/~ahara/quasisphere/Gallery.html

... and here is a video:
<a href="http://www.youtube.com/v/3lcO9zRCv-4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/3lcO9zRCv-4&rel=1&fs=1&hd=1</a>

Anybody had a crack at recreating this?



Oh..That thing... grin
After some intense search over the net, related how that was done, I think that I got some clues from this webpage (created by Kentaro Ito):
http://www.math.nagoya-u.ac.jp/~itoken/index.html

You can start by downloading the following paper (pdf), that appears on the page:
# (with Y. Araki) An extension of the Maskit slice for 4-dimensional Kleinian groups,
Conform. Geom. Dyn. 12 (2008), 199-226.


Take a look at the gallery:
http://www.math.nagoya-u.ac.jp/~itoken/3d-maskit.html

With the parameters from the article, go to this applet and play with them (some very interesting kleinians will appear):
http://www.math.nagoya-u.ac.jp/~itoken/java/masinv3.htm

Replace the points with spheres and you will probably get that fractal.
I have tested some and I think that I have hit very near.
Also at the end of the article there are several 3d kleinians that looks like that fractal.

Oh well, if you want to see it in the fast way, iterate the following code:
Select the transformation (tr) randomly or in a structured fashion and (get the parameters from the article):

const   float p=0;
const   float q=1.9;
const   float r=0.04;
const   float l=2;

switch(tr){
case 0:
   w=(x*x+y*y+z*z);
   X1=x/w+p;
        Y1=-y/w+q;
        Z1=z/w+r;
break;
case 1:
      w=((x-p)*(x-p)+(y-q)*(y-q)+(z-r)*(z-r));
      X1=(x-p)/w;
      Y1=(-y+q)/w;
      Z1=(z-r)/w;
       break;
case 2:
      X1=x-l;
      Y1=y;
      Z1=z;    
      break;
case 3:
      X1=x+l;
      Y1=y;
      Z1=z;
break;

}

x=X1;
y=Y1;
z=Z1;

Now, with the iteration values 3d point, map it to the spherical function:
double rr=sqrt(q*q+r*r);
double xn=X1;
double yn=Y1+2*r/rr;
double zn=Z1-2*q/rr;
Xp=4*rr*xn/(xn*xn+yn*yn+zn*zn);
Yp=4*rr*(yn/(xn*xn+yn*yn+zn*zn)-r/(4*rr));
Zp=4*rr*(zn/(xn*xn+yn*yn+zn*zn)+q/(4*rr));

and plot the point (Xp,Yp,Zp)..

That set of parameters will not produce something like that fractal, but it will produce something very very interesting.. smiley



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twinbee
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« Reply #7 on: January 27, 2010, 05:38:16 PM »

Nice one, let's see some pics! smiley

Also found these videos (first found by Paul Nylander a while back) which have the same style:

<a href="http://www.youtube.com/v/yY1-D13KICw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/yY1-D13KICw&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/dkBNc2WqdrI&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/dkBNc2WqdrI&rel=1&fs=1&hd=1</a>
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Aexion
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« Reply #8 on: January 29, 2010, 07:21:24 PM »

Nice one, let's see some pics! smiley

Also found these videos (first found by Paul Nylander a while back) which have the same style:



 shocked

The first one is the real 3D Kleinian!!.. all of my kleinian renders just pale in front of that beauty!
I have tried several 3d kleinians using quaternionic mobius transformations, but have failed because thanks to the noncommutative quality of the quaternions, there are eight possibilites in every transformation..  huh?
Heres a good link for those quaternionic mobius transformations if anyone is interested: https://faculty.digipen.edu/~jhanson/geometry/moebius/qmoebius_demo1.html

The second one looks like sphere inversion fractals, based on quaternionic mobius transformations too.. those are easier to render (and I think that i have one somewhere in my hdd), but is not the first one..

Just for a curiosity, anyone have tried a triplex mobius transformation..??

ps. How I upload an image to the forums.. 


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xenodreambuie
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« Reply #9 on: February 01, 2010, 09:46:09 AM »

Aexion, many thanks for posting the 3D Maskit links and code. I only understand half the math in the paper but it's a big help. Until I get Indra's Pearls I won't really know the difference between Kleinian, Fuchsian and Schottky groups.

I also did some experiments using only 2 generators: the case 0 sphere inversion and a rotation (180, 120, 90, etc), which makes some nice spirals, especially with a base shape.
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KRAFTWERK
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« Reply #10 on: February 01, 2010, 10:50:49 AM »

ps. How I upload an image to the forums.. 

You have to clic "Additional Options" to get a browse-button.

J


* Namnlost.png (48.65 KB, 518x609 - viewed 658 times.)
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Aexion
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« Reply #11 on: February 04, 2010, 01:44:44 PM »

ps. How I upload an image to the forums.. 

You have to clic "Additional Options" to get a browse-button.

J

thanks!!
The equations that I have posted, produce something like the the following image..
(If the overall shape reminds some other fractal, its just an intentional coincidence..  grin )


* Maskit.jpg (112.74 KB, 600x600 - viewed 638 times.)
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KRAFTWERK
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« Reply #12 on: February 04, 2010, 02:21:58 PM »

Hmmm reminds me of one in this very thread  afro
Beautiful!!!
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kram1032
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« Reply #13 on: February 04, 2010, 09:37:45 PM »

wow, nice cheesy
Well done smiley
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Tglad
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« Reply #14 on: February 17, 2010, 12:46:11 AM »

This website is a superb interactive description of the indra's pearls type fractals- http://www-m10.ma.tum.de/bin/view/MatheVital/IndrasPearls/WebHomeEn
It shows you exactly how they work.

Its 2d, but since mobius transforms also all exist in 3d, it is clear that you could make 3d versions with the same method. Or 4d ones.

Some of the patterns look Mandelbrot-like, but they are of course not, since Z^2 is a different transformation altogether to the mobius transformations.
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