This article is from the Fractal FAQ, by Ermel Stepp firstname.lastname@example.org 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 tracing.
View Frank Rousells hyperindex of clickable/retrievable 3D images:
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.