# 44 How can 3-D fractals be generated?

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.

Frank Rousell's hyperindex of 3D images
http://www.cnam.fr/fractals/mandel3D.html

4D Quaternions by Tom Holroyd
http://bambi.ccs.fau.edu/~tomh/fractals/fractals.html

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 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
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.

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/ .

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