This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
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
Frank Rousell's hyperindex of 3D images
4D Quaternions by Tom Holroyd
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
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 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 hypercomplex number such
as in "FractInt", or other functions. For instance, you could use a
map with more than one parameter, which would generate a
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.
POV-Ray 3.0, a freely available ray tracing package, has added 4-D
fractal support. It takes a 3-D slice of a 4-D Julia set based on an
arbitrary 3-D "plane" done at any angle. For more information see the
POV Ray web site at http://www.povray.org/ .