Q20: How can 3-D fractals be generated?  
A20: A common source for 3-D fractals is to compute Julia sets with  
quaternions instead of complex numbers. The resulting Julia set is four  
dimensional. By taking a slice through the 4-D Julia set (e.g. by fixing one  
of the coordinates), a 3-D object is obtained. This object can then be  
displayed using computer graphics techniques such as ray tracing.  
  
View Frank Rousells hyperindex of clickable/retrievable 3D images:  
ftp://ftp.cnam.fr/pub/Fractals/3D/Index.gif  
  
The papers to read on this are:  
  
1. J. Hart, D. Sandin and L. Kauffman, Ray Tracing Deterministic 3-D  
Fractals, _SIGGRAPH_, 1989, pp. 289-296.  
  
2. A. Norton, Generation and Display of Geometric Fractals in 3-D,  
_SIGGRAPH_, 1982, pp. 61-67.  
  
3. A. Norton, Julia Sets in the Quaternions, _Computers and Graphics,_  
13, 2 (1989), pp. 267-278. Two papers on cubic polynomials, which can  
be used to generate 4-D fractals:  
  
1. B. Branner and J. Hubbard, The iteration of cubic polynomials, part I.,  
_Acta Math_ 66 (1988), pp. 143-206.  
  
2. J. Milnor, Remarks on iterated cubic maps, This paper is available from  
anonymous ftp: math.sunysb.edu:/preprints/ims90-6.ps.Z . Published in  
1991 SIGGRAPH Course Notes #14: Fractal Modeling in 3D Computer  
Graphics and Imaging.  
  
Instead of quaternions, you can of course use other functions. For instance,  
you could use a map with more than one parameter, which would generate  
a higher-dimensional fractal.  
  
Another way of generating 3-D fractals is to use 3-D iterated function  
systems (IFS). These are analogous to 2-D IFS, except they generate points  
in a 3-D space.  
  
A third way of generating 3-D fractals is to take a 2-D fractal such as the  
Mandelbrot set, and convert the pixel values to heights to generate a 3-D  
"Mandelbrot mountain". This 3-D object can then be rendered with normal  
computer graphics techniques.