3-D fractals

How can 3-D fractals be generated?

A modern software called Mandelbox can be used. Also, 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 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.