Welcome to Fractal Forums

Hello everyone.
Looking forward to some interesting discussions with you!

About me: I initially studied studied as an architect, but over the years have been drawn strongly towards geometry and mathematics.
I document some of my explorations on my blog : SpaceSymmetryStructure (http://spacesymmetrystructure.wordpress.com/)
There I've looked at stuff like:

 Sphere Inversion (http://spacesymmetrystructure.wordpress.com/category/inversion/), (and inverse perspective (http://spacesymmetrystructure.wordpress.com/2007/07/28/view-from-the-vanishing-point/))
 Rheotomic Surfaces (http://spacesymmetrystructure.wordpress.com/rheotomic-surfaces/) generated from functions of complex numbers
 and 4d rotation of the 3-sphere with stereographic projection (http://spacesymmetrystructure.wordpress.com/2008/12/11/4-dimensional-rotations/)

It is this last one that is particularly relevant to my current question.

I want to look at the stereographic projection of the intersection of a 3-sphere with the Mandelbrot/Julia set in 4 dimensions.

Of the renderings of 4d fractals I have seen, there seem to be lots of variations on the iteration technique, but as far as I understand they all take a planar section of the set (that is to say, its intersection with some flat 3d space) as the final 3d form.

We know the set has rich structure and variation in all directions, so the intersection with the sphere should be non-trivial, and I'm hoping that maybe because of the conformal nature of stereographic projection, something different and interesting could be preserved in this way that is lost in the typical 'whipped cream' look of planar sections of 4d fractals.

I have equations for 4d stereographic projection that I am confident in from my rotation experiments, and it is quick and easy to get a 4d coordinate on the 3-sphere corresponding to each point in a 3d space.

My questions are:

(a) Has this been tried already ?
(b) How might I get started ?
I have done a little programming, but never any fractals or raytracing, and don't want to start from scratch if I can avoid it, so I want to find existing code to adapt.

Which of the fractal programs out there might be adjustable to my purpose ?

If possible I don't want to change from the usual iteration method, or the way the rays are calculated, only the choice of mapping between 4D and 3D coordinates.

Any thoughts much appreciated!

Dan

hi there, welcome to the forums

have you read the 3d mandelbrot thread here on the forum ?
http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/

i think that ultrafractal is the most amazing program for extendibility, although it is not really sufficent to do 3d rendering, even if david makin and others do it quite succsesfully, i can also advice to take a look at incendia

Thanks Trifox,

I will definitely look into ultrafractal and incendia.
and I did ask about this in the 3D Mandelbrot thread, but it got ignored in the (entirely understandable) excitement over recent developments

lol, yes, they are just talking about weird stuff, like differentiating pseudo-3d formulas ;)
but i love it !

Greetings, and a belated Welcome to this particular Forum !!!     :D

First of all, UF is really not designed for 3-D fractals, nor can it produce "real objects" that may be exported to true 3-D rendering applications.  And for that matter, if you are wanting to do your own coding, none of the purchase-ware will be suitable for your purposes, since the source code is not available.

But there are plenty of locations where you can get as much source code as you can handle, and allow you to play with it to your heart's content.     ;)     The question is, what programming language do you prefer working in ??