3D Kleinian

[JosLeys]

3D Kleinian



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=12924

Utrafractal+Povray

[Alef]

Wow, cool. It looks like nereus http://en.wikipedia.org/wiki/Nereididae throught if not fisshing enthusiast you probably woun't like comparision;)

How this is made? Ultra Fractal don't have much 3D capabilities unless you do them by yourself;)

[vector]

the picture is great, great, great...

i wonder, why in another thread you and others wrote, that these kleinian groups would not be related to Julia- or Mandelbrot-sets.
As far as i understood(some time before i read Indra´s pearls and did some 2d-pictures of loxodromic circles and their images), the Kleinian group consist of mirrored images of a starting sphere(correct?), the mirroring determined by Moebius transformations on quaternions(or only the three imaginary components).
A Julia set consists as well on the images of the circle(including the interior of the circle) with radius(in most programs as parameter called:) bailout. We only calculate it different: We take any point of the complex plane(or corresponding 3d) and test, whether its orbit by taking the coordinates to a power n(mostly squaring)and iterating it, will remain within this radius. We can, as well known, get the same Julia sets by taking the pictures of the original circle with radius bailout by the opposite projection, calculating the square roots of all points of this circle and adding the addition(or now subtraction) of vector c. The same you do in Kleinian groups, but somehow more elegantly: you calculate the center and radius of a sphere, or two(three, four?)spheres as images of the original sphere. This by following several sequences of projective calculations. The same we get in 2d, we get first two square-root-images of the original circle, then 4..8...and so on, ending up in the shape of the Julia set. Therefore i guess it´s the grail, the only problem is, it is not so easy to say, how you get the Mandelbrot-set by this repetitive-images-method.
nevertheless..great work   

[fractalrebel]

There is a persistent myth on this forum that Ultrafractal cannot be used to create 3D objects, in spite of numerous comments from Jos Leys, David Makin and myself. Here is a 3D Kleinian example that was created over 6 years ago using Using Ultrafractal.

[fractalrebel]

Here is another 3D Kleinian example.