Fractals as Attractors of IFSs
One of the most important properties of fractals is that they are
self-similar, meaning that any portion of the given set is equivalent to
the whole through a certain set of transformations. In fact this mapping is a
terrific way of generating fractals: just start with a set, like a line segment
or a triangle, and apply a series of transformations to it, over and over again.
These transformations are called an IFS (Iterated Function System).
Fractals are seen as attractors of these IFSs, what you end up with
after many many iterations of the IFS.
Below are two "sets" (images), and by clicking on the image you can iterate the
IFS one step. The important point to realize here is that no matter what "set"
one starts with, the end result (in the limit of course) is the same, the
fractal generated by the IFS specified.
equals(?)...