Kleinian Limit Sets

[gandreas]

Now that quadrium | flame 1.2 is finally available, time to show off a bit.  These are kleinian limit sets - basically, if you are familiar with how you can make Sierpenski's triangle using the limit set of three affine transforms (where one scales by 50%, the second scales by 50% and offsets x+0.5, and the third scales by 50% and offsets y+0.5), these are the same sort of thing only done with two bilinear transformations* (plus their inverses):

 
http://www.deviantart.com/deviation/59895968/

 
http://www.deviantart.com/deviation/59970373/

 
http://www.deviantart.com/deviation/60004297/

* bilinear transformations are of the form f(x) = (a * x + b) / (c * x + d) where all the values are complex numbers.  Bilinear transforms like this are often called Moebius Transformations or fractional linear transformations, and preserve angles and circles (unlike affine transformations which can distort circles into ellipses).

[lycium]

for those interested, there are quite a few sites with images and info on these fractals:

(artists)
http://www.josleys.com/galleries.php?catid=7 (probably the most famous kleinian limit set artist)
http://jsp7707.deviantart.com/gallery/ (friend of mine from deviantart)
http://lyc.deviantart.com/gallery/ (my gallery)

(mathematics)
http://en.wikipedia.org/wiki/M%C3%B6bius_transformation
http://www.hiddendimension.com/KleinianGroup_Fractals_Main.html
http://klein.math.okstate.edu/IndrasPearls/ (website related to the book indra's pearls)

[fractalwizz]

How do you create these Kleinian Shapes in Ultra Fractal?
I have tried and I have got weird results.

[lycium]

you keep insisting that it be done in ultrafractal, why?

personally i read up in indra's pearls about them and implemented it in my fractal rendering applications, almost the others would be done in apophysis and finally jos leys' are done in pov-ray with some custom scripts to set up the scene.

[fractalwizz]

1. I use Ultra Fractal more than any other program out there.
2. I don't have the admin password to my computer to install the software.
Otherwise I would have Apophysis exe
Is there a stand-alone program of Apophysis? If it just runs off the .and not the installer, can use it.
Sorry if  I thought your comment was a little rude. That's how I react to things like that.

[lycium]

i was being cheeky (explicitly labelled with ), but the basic point is that if you can't find any formulae for it in the uf db, then you'll probably have to code it yourself.

i'd recommend reading indra's pearls as it exposes simple group theory in a visual manner that's really easy to follow, plus they give good coverage of those magical complex numbers -- from whence our beloved chaos springs -- specifically studying nested circle inversions. great bedtime reading! 

[David Makin]

If I remember correctly there should be something in the "reb" collections but I can't remember which file or if it's written as a colouring or a main formula.

[fractalwizz]

I will have to check the reb files for them. I briefly looked at the txt versions on the site, and I have seen klein all over the place.
I will check back when I have looked at it.

[Dinkydau]

1. I use Ultra Fractal more than any other program out there.
2. I don't have the admin password to my computer to install the software.
Otherwise I would have Apophysis exe
Is there a stand-alone program of Apophysis? If it just runs off the .and not the installer, can use it.
Sorry if  I thought your comment was a little rude. That's how I react to things like that.

The latest beta versions don't require installation.
http://sourceforge.net/projects/apophysis/

Only thing is, you will have to create a folder for scripts yourself, if you want to use them.

[_db_]

In UF there are Indra's Pearls Workshop and Kleinian Group Raytrace colouring algorithms in reb.ucl.

There are some sample uprs and short discussions in my UltraFractalWiki which can be downloaded from:  http://www.mediafire.com/ultrafractalwiki

follow links:  root page >  [ConceptsAndTerms] > [Raytracing]

cheers,

db