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Amidst the heated discussion of Mandelboxenbulb-related, modern fractals, I sadly have another old-fashioned topic on my mind.

Here's the problem. I would like if you could recommend a software package and the procedure for 3D object manipulation. The first step could be to define an ordinary or semi-transparent sphere with the given radius. The second step is to transform the sphere according to a 3D mapping such as f(x,y,z) = .... The third step is to repeat the transformation recursively and display several steps (iterations) in high quality.

In 2D, the repeated mapping tends to produce a Julia set in the form of morphed initial 2D object (or an image). In 3D, the expected result is a 3D Julia set.

This should be the opportunity to review the software packages. One of the first that comes to mind is the PovRay. I have a minimal experience with this program and I am aware of affine transformations of a selected, built-in object. We have worked-out examples of Quaternion Julias and IFS fractals, but morphing an object represents the additional challenge. For this, I would like to find documentation. Processing interpreter could contain scripts for defining and manipulating objects. In the same group, we have the StructureSynth, ContextFree and general java applets (if you know any). I'd rather avoid Mathematica, but there could be a notebook for these transformations...  

The challenges of 3D object transformations are the voxel interpolation, handling of different types of functions, vector vs. non-vector objects...

that is a really interesting topic, i wanted to play with segmented fractals ( i call them metameric cos it sounds cool) , where a shape like a cube/cilinder is replicated in space in many sizes and shapes in patterns, sometimes it can split into 10-20 100 of itself, mix different shapes in series etc.

dont know a prog that can do that though. best make it yourself if you know c++
i cant even pogram that well

Probably possible using a script in a 3d package.. or just write in C++. I don't see a need for voxelisation.

If you want fractals then you only need to consider five transformations. Affine transforms are too much, they include shear which would cause infinite stretch on the small parts.
Just translation, rotation, reflection, scale and inversion (distance from 0,0,0 becomes 1/distance).
In 3d (or higher dimensions) any other transformation is either a combination of the above, or it will cause unbounded stretching.

This might be of interest. K3dSurf is a free program that does 3D mapping that uses f(x,y,z), it also exports objects to .obj

http://k3dsurf.sourceforge.net/

The developer of K3dSurf also scripted a Menger Sponge using f(x,y,z)

http://k3dsurf.s4.bizhat.com/k3dsurf-ftopic61.html

i bumped into structure synth, it does what you say...

http://www.google.co.uk/images?um=1&hl=en&rlz=1T4SKPT_enFR413FR413&biw=1436&bih=809&tbs=isch%3A1&sa=1&q=structure+synth&aq=f&aqi=g1&aql=&oq=

Interesting find thanks!  I can confirm the default example exports to a 30MB obj file, which also imports into glc_player.exe
and a lot faster into Meshman for further processing.

Have you guys seen Groboto ?

http://kuzy62.deviantart.com/gallery/583136 (http://kuzy62.deviantart.com/gallery/583136)

(http://www.sockrateaze.com/stuff/lissahdritestalpha.jpg)

3D Obj file exported from Groboto and rendered in Carrara - Using HDRI Lighting from ...  Guess...  An HDRI Image of Lissa!  (At least I'm consistent  O0 )

Nice rendering! I have to checkout Groboto. From the you tube videos, it looks like it uses similar fractals generating code that Xfrog uses to generate
3d plant meshes.

Thank you all for your answers. The recursed geometrical transformations are in the heart of all fractals in some way, but we usually do not ask what do they transform or what can they transform with what consequences. That is why I was wondering... so, great. :)