A 3D Kleinian group

[JosLeys]

Here is something I've been working on: calcs in Ultrafractal and data exported to Povray.
This uses 3D Moebius transformations consisting of quaternions instead of complex numbers.
A bit long to explain in detail here. I plan to publish an article on it within the next couple of months

See also a small movie here http://www.josleys.com/gfx/Indra3D002_01.mov and http://www.josleys.com/gfx/Indra3D008_01.mov

[Tglad]

I think a 3d Mobius is equivalent to a combination of 3d translation, 3d rotation, scale and 3d inversion(+flip). i.e. the set of conformal transforms.

And you generate a limit set like so:
given two such transforms:
take a point, apply the two transforms on it, repeat on the resulting points.

Is this right?

[kram1032]

Pretty much, yes. Which is perfectly equivalent to Möbius transforms with quaternions, using only the imaginary parts.
Using the real part as well, you'll essentially get a 4D Möbius transform....

I like that spiny structure. Looks neat.

[JosLeys]

Yes, but here there are three such transforms that move spheres around. Such a Moebius transform will take a sphere to another sphere.

[KRAFTWERK]

Yes, this looks very Holy Grail:ish.

Very interesting work Jos!!!

[knighty]

Wow! There are not many 3D quasi fuchsian renders out there. Especially those which have spirals.
How do you find the mobius transformations? Is there any recipe like those described in indra pearles book?

[JosLeys]

Some of my wisdom comes from this article :http://arxiv.org/abs/0707.2427
This states the generators in geometric form : see expression 6.1 on page 29.

[knighty]

Thank you! Now I have a terrible headache... Just kidding, but those math articles make me feel stupid.

[Tglad]

Nice work Jos Leys.. do you have a distance estimate for them?
Here are some renders from the paper... notice the shape of the top left one.
(It is worth saying that these are in no way related to a 3d Mandelbrot set (if one existed). They are the 3d version of 2d Kleinian group fractals. The Mandelbrot is a completely different sort of transformation).
 

[JosLeys]

No I don't have a DE (and I wish I had !)
The way i draw them is to calculate the radii and position of spheres in the limit set, by transforming a base sphere, that I know is part of the set, by 'words' in the three generators and their inverses.
For a DE scheme, what one could do is to take a point on a ray, transform it by a set of words in the generators, and look at the closest distance to the base sphere. However, to get any sort of detail, we are talking easily about half a million words or more. The image below has 472.000 spheres, so to do this with DE, one would need to do this large number of calcs a couple of times on every ray, not to mention adjacent rays to find a normal...

...an all this has nothing to do with any attempt at a 3D Mandelbrot set!

[lycium]

~400k spheres is nothing for a modern rendering system though!

You could render some really good looking images with Indigo, if you saved out some XML for those spheres, eg:

   <sphere>
      <center>-0.5 0.0 0.31</center>
      <radius>0.3</radius>

      <material_name>simple2</material_name>
   </sphere>


If you'd like the full example XML for editing, you can download it here: https://dl.dropbox.com/u/3038174/simple_test.igs

[JosLeys]

Lycium, I'm afraid I don't understand what you mean...

[lycium]

For these sphere collections, you can use a much better renderer than POV-ray Indigo is very physically accurate, produces exceptionally high image quality, and I expect it will render some spheres really, really fast!

If you'd like help rendering at high resolution, I'd be happy to do this for you on our office quadcores

[knighty]

When there is no loxodromic transform in the generator set, it's already possible to have a DE or at least an implicit representation: The set of generators can be "factored" into a set of reflexions and inversions for which the limts set is the same. In this case one can use Mandelbrot method to get an implicit representation.

Vladimir Bulatov is using fragmentarium to render such reflexion groups limit sets. He must be using the non DE renderer  :.(http://www.youtube.com/user/bulatov2011)

It's much more difficult in the case where there are loxodromic generators. I believe that it's possible to obtain an implicit representation and a DE. The method I'm investigating needs three ingredients:
1- Determination of a fundamental domain ---> looks very difficult and involved.
2- An "escaping" algorithm that given a fundamental domain and a point finds a transformation sequence (an orbit) that escapes, that ends up inside the fundamental domain. This seems complicated but it is maybe the simplest part. It's even (I believe) possible to find an iterative algorithm instead of a recursive one.
3- A method to find some part of the convex hull of the limit set---> Well I really don't know how to obtain it. There is a slight simplification in the "Maaskit slice": The limit set is composed of circles (spheres in 3D) which are part of the convex hull boundary.

The only information on the subject that I could find are math articles. As such they are quite involved and mainly intended for mathematician comunity. They are not "eazily" (and I don't mean without effort) accessible to non mathematician. That explain why I couldn't hide my frustration in my previous post  .

Anyway, That would be a good collaborative project.

[JosLeys]

To Lycium : I have no intention to pay close to $600 for Indigo when Povray is free!
However if you could get us some complimentary copies, we could all try it out!