Fractal types

There are many different fractal types which are not covered when explaining coloring algorithms because they are either not common in fractal explorations or because they are not present in the popular fractal software packages.

But basically, we can classify fractal types in six main groups:

a) Fractals derived from standard geometry by using iterative transformations on an initial common figure like a straight line (the Cantor dust or the von Koch curve), a triangle (the Sierpinski triangle), or a cube (the Menger sponge). The first fractal figures invented near the end of the 19th and early 20th centuries belong to this group.

b) IFS (Iterated Function Systems). This is a type of fractal introduced by Michael Barnsley. The structure of these fractals is described by a set of affine (linear) functions that compute the transformations undergone by each point by homothety, translation, and rotation. The functions introduced into the system are selected randomly, but the final set is fixed and shows fractal structure.

c) Strange attractors. These sets can be considered as the representation of a chaotic movement (in no place and no time identical). These attractors are very complex and composed by a line of infinite length drawing tightly intertwined loops that never crosses its own trajectory.

d) Plasma fractals. Created with techniques like the fractional Brownian motion (fBm) or the midpoint displacement algorithm, these fractal type produce beautiful textures with fractal structure, like clouds, fire, stone, wood, etc. widely used in CAD programs. Skilled fractal artists love plasma to create textures or backgrounds in their images.       

e) L-Systems, also called Lindenmayer systems, were not invented to create fractals but to model cellular growth and interactions. A L-System is a formal grammar that recursively applies its rules to an initial set. As a result, sometimes, a fractal structure is created.

f) Fractals created by the iteration of complex polynomials: perhaps the most famous fractals (Julia, Mandelbrot…). Only this group of fractals has been widely experimented with different coloring algorithms by the fractal art community.

Many fractals sets can be considered subgroups from this types, for example fractal terrains are a three dimensional representation of a plasma fractal. Fractal music is the sonic representation of a chaotic attractor movement. Other fractals, like quaternionic or (recently) hypernionic fractals may be considered as an extension to higher dimensions of polynomial fractals iterated in the complex plane.