The Burning Ship Set

Arthur C. Clarke’s documentary on Fractals:

A basic set with fractal behaviour is the set of Complex numbers (C):

No matter how much one zooms in or out, the set is self-similar with infinite detail.

The typical fractal sets (Mandelbrot, Julia, Fatou) follow a pattern of 3 infinities: an infinite number of points is run an infinite number of times through a recursive polynomial and it will/will not reach infinity:

To make the step to 3d, the major issue is that the 2d rules cannot be generalized because there is no corresponding set of numbers for 3d space. 1d space has Real numbers, 2d space has Complex numbers, but there is no 3d equivalent. However, Quaternions (hypercomplex numbers) are a theoretical set of points corresponding to a 4d space. Therefore there are two possible approaches: a) Define a three dimensional set of points in polar coordinates and switch them back to a cartesian coordinates in order to build it in computer space. b) Build a theoretical 4d fractal using hypercomplex numbers and cast its 3d shadow in 3d.

Either way, results are similar:

Dealing with infinite numbers, infinite iterations and infinite sets of points, computation times become an issue. One way around this is to build ray-traced images estimating distances to a virtual fractal (not physically storing the points of the fractal in memory):

The image above is magnified ~3.10e13 times. In other words, presuming the size of the sectional model is 1m, it scaled up to roughly the size of the Solar System.

Building 3d models in computer space is slightly trickier because of the huge number of points involved to define even a limited section of a fractal. The issue is to define an algorithm for the correct order of the points in order to build a mesh. A rather neat solution is to ray-trace consecutive sections through a fractal (ray-tracing involves a Z-buffer anyway) and work from there. Here is an example (a 38,000,000 face-mesh obtained from 1,000 sections):

An important tool in exploring 3d fractals is building Julia sets (the only difference is that they use a constant increment at each iteration rather than the initial step):