Title: IFS fractals from Mobius transforms
Post by: fractalrebel on December 16, 2009, 08:43:28 PM
Iterated Function System (IFS) fractals are normally generated using affine transforms, which are linear transforms of the type:
xn+1 = a*xn + b*yn + e
yn+1 = c*xn + d*yn + f
Several affine transforms are used to generate an IFS fractal, with the choice at each iteration being made randomly. Apophysis is an example of software for generating IFS fractals using affine transforms.
Another type of transform, the Mobius transform
 = (a*z + b)/(c*z + d))
where z is complex
can also be used to generate IFS fractals. Mobius transforms are also known as fractional linear transforms. Several Mobius transforms are selected randomly at each iteration, analogous to affine transforms, to generate a fractal. The following image is a simple demonstration of this approach.
xn+1 = a*xn + b*yn + e
yn+1 = c*xn + d*yn + f
Several affine transforms are used to generate an IFS fractal, with the choice at each iteration being made randomly. Apophysis is an example of software for generating IFS fractals using affine transforms.
Another type of transform, the Mobius transform
can also be used to generate IFS fractals. Mobius transforms are also known as fractional linear transforms. Several Mobius transforms are selected randomly at each iteration, analogous to affine transforms, to generate a fractal. The following image is a simple demonstration of this approach.
Title: Re: IFS fractals from Mobius transforms
Post by: fractalrebel on December 16, 2009, 08:53:26 PM
Here is an IFS Mobius example which has been mapped to a Riemann sphere.
Title: Re: IFS fractals from Mobius transforms
Post by: kram1032 on December 16, 2009, 09:02:50 PM
very nice but the jpg compression kills them quite much...
Title: Re: IFS fractals from Mobius transforms
Post by: BradC on December 16, 2009, 09:03:11 PM
Wow, nice! I can almost see spiders crawling all over that first one. :)
I'm curious how many transforms are involved and how you chose them.
I'm curious how many transforms are involved and how you chose them.
Title: Re: IFS fractals from Mobius transforms
Post by: fractalrebel on December 16, 2009, 10:09:19 PM
Wow, nice! I can almost see spiders crawling all over that first one. :)
I'm curious how many transforms are involved and how you chose them.
I'm curious how many transforms are involved and how you chose them.
The images were created with IFS Mobius which is in the Ultrafractal database. The number of Mobius transforms can vary from 2 to 6. The first image uses two transforms and the second one uses 4 transforms. Rather than acting on points, as is the case with affine transforms. the Mobius transforms act upon circles and lines. In the first image the transforms operate repeatedly on 4 starting circles which were chosen randomly as long as they met the Kissing Schottky criteria (see Indra's Pearls). In the second they operate on 3 starting circles chosen randomly which met the criteria of no overlap.
Title: Re: IFS fractals from Mobius transforms
Post by: fractalrebel on December 16, 2009, 10:22:17 PM
Here is one of my favorites, called Cosmic Chicken:
Title: Re: IFS fractals from Mobius transforms
Post by: fractalrebel on December 16, 2009, 10:23:46 PM
The Cosmic Chicken (or any other IFS fractal) can easily be used as an orbit trap. Here is an example:
Title: Re: IFS fractals from Mobius transforms
Post by: kram1032 on December 16, 2009, 10:36:07 PM
cool :D
Title: Re: IFS fractals from Mobius transforms
Post by: Nahee_Enterprises on December 30, 2009, 01:11:54 AM
Interesting !! :)
Title: Re: IFS fractals from Mobius transforms
Post by: stijnw on January 23, 2010, 11:40:44 PM
This looks real nice! The first image reminds me of a popular flame-fractal I used as background image for quite some time...
Regards,
Stijn Wolters.
Title: Re: IFS fractals from Mobius transforms
Post by: paxinum on July 08, 2010, 05:23:58 PM
All fractals here can be generated by Möbius mappings:
http://en.wikipedia.org/wiki/Apollonian_gasket
There is a book called Indras Pearls using only Möbius mappings.
http://en.wikipedia.org/wiki/Apollonian_gasket
There is a book called Indras Pearls using only Möbius mappings.
Title: Re: IFS fractals from Mobius transforms
Post by: bib on July 08, 2010, 06:13:25 PM
Thanks paxinum for upping this thread.
Very interesting fractalrebel. I did not find any IFS or Mobius trap shape or formula in reb.ulb or in the public classes. Would you mind sharing the UF parameter set or give some explanations ?
The Cosmic Chicken (or any other IFS fractal) can easily be used as an orbit trap. Here is an example:
Very interesting fractalrebel. I did not find any IFS or Mobius trap shape or formula in reb.ulb or in the public classes. Would you mind sharing the UF parameter set or give some explanations ?
Title: Re: IFS fractals from Mobius transforms
Post by: KRAFTWERK on July 09, 2010, 10:21:44 AM
Whoah! O0
Can we lift this to the 3D KIFS?
Think I will skip my vacation and hang here... :)
Can we lift this to the 3D KIFS?
Think I will skip my vacation and hang here... :)
Title: Re: IFS fractals from Mobius transforms
Post by: kram1032 on July 09, 2010, 01:54:21 PM
A 3D variant would be from just adding another variable...
I wonder about a mixed variation:
(a*x+b*y[+c*z]+d)/(e*x+f*y[+g*z])+h)
Or even further extended... You could do that from a polynomial of any degree, basically...
That would be a mix of a Möbius transform and a general 3-cone section. :)
You could extend that even further by allowing permutations....
x*x=x² y*x=x*y z*x=x*z
x*y y*y=y² z*y=y*z
x*z y*z z*z=z²
So... a*x²,b*y²,c*z²,d*2xy,e*2xz,f*2yz
I wonder about a mixed variation:
(a*x+b*y[+c*z]+d)/(e*x+f*y[+g*z])+h)
Or even further extended... You could do that from a polynomial of any degree, basically...
That would be a mix of a Möbius transform and a general 3-cone section. :)
You could extend that even further by allowing permutations....
x*x=x² y*x=x*y z*x=x*z
x*y y*y=y² z*y=y*z
x*z y*z z*z=z²
So... a*x²,b*y²,c*z²,d*2xy,e*2xz,f*2yz
Title: Re: IFS fractals from Mobius transforms
Post by: KRAFTWERK on July 09, 2010, 02:26:33 PM
Wahooooo here we go!!! 8-)
Sorry, holiday... O0
Sorry, holiday... O0
Title: Re: IFS fractals from Mobius transforms
Post by: Thunderwave on July 15, 2010, 05:06:54 AM
A 3D variant would be from just adding another variable...
I wonder about a mixed variation:
(a*x+b*y[+c*z]+d)/(e*x+f*y[+g*z])+h)
Or even further extended... You could do that from a polynomial of any degree, basically...
<Quoted Image Removed>
That would be a mix of a Möbius transform and a general 3-cone section. :)
You could extend that even further by allowing permutations....
x*x=x² y*x=x*y z*x=x*z
x*y y*y=y² z*y=y*z
x*z y*z z*z=z²
So... a*x²,b*y²,c*z²,d*2xy,e*2xz,f*2yz
I wonder about a mixed variation:
(a*x+b*y[+c*z]+d)/(e*x+f*y[+g*z])+h)
Or even further extended... You could do that from a polynomial of any degree, basically...
<Quoted Image Removed>
That would be a mix of a Möbius transform and a general 3-cone section. :)
You could extend that even further by allowing permutations....
x*x=x² y*x=x*y z*x=x*z
x*y y*y=y² z*y=y*z
x*z y*z z*z=z²
So... a*x²,b*y²,c*z²,d*2xy,e*2xz,f*2yz
Could we be talking dimensions beyond the usual, or am I stupid about this? I mean a fractal dimension above 3, like 3.23 etc. Is that even possible? Maybe I'm not really understanding this. Sorry if I sound stupid. I keep trying to learn more. ;)
Title: Re: IFS fractals from Mobius transforms
Post by: paxinum on July 15, 2010, 09:46:06 AM
A 3D variant would be from just adding another variable...
I wonder about a mixed variation:
(a*x+b*y[+c*z]+d)/(e*x+f*y[+g*z])+h)
Or even further extended... You could do that from a polynomial of any degree, basically...
<Quoted Image Removed>
That would be a mix of a Möbius transform and a general 3-cone section. :)
You could extend that even further by allowing permutations....
x*x=x² y*x=x*y z*x=x*z
x*y y*y=y² z*y=y*z
x*z y*z z*z=z²
So... a*x²,b*y²,c*z²,d*2xy,e*2xz,f*2yz
I wonder about a mixed variation:
(a*x+b*y[+c*z]+d)/(e*x+f*y[+g*z])+h)
Or even further extended... You could do that from a polynomial of any degree, basically...
<Quoted Image Removed>
That would be a mix of a Möbius transform and a general 3-cone section. :)
You could extend that even further by allowing permutations....
x*x=x² y*x=x*y z*x=x*z
x*y y*y=y² z*y=y*z
x*z y*z z*z=z²
So... a*x²,b*y²,c*z²,d*2xy,e*2xz,f*2yz
Could we be talking dimensions beyond the usual, or am I stupid about this? I mean a fractal dimension above 3, like 3.23 etc. Is that even possible? Maybe I'm not really understanding this. Sorry if I sound stupid. I keep trying to learn more. ;)
You can have any positive number as fractal dimension. We mathematicians deals with n-dimensional (integer dimensional) spaces on a daily basis, but they are quite complicated to visualize.
You can easily generalize the sierpinski trianlge to any dimension, as an example.