This article is from the Fractal FAQ, by Ermel Stepp stepp@muvms6.mu.wvnet.edu 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:

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.

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