3-D fractals

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.

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

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

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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