## Iterated Function Systems

Fractals reproducing realistic shapes, such as mountains, clouds, or plants, can be generated by the iteration of one or more affine transformations. An affine transformation is a recursive transformation of the type

Each affine transformation will generally yield a new attractor in the final image. The form of the attractor is given through the choice of the coefficients a through f, which uniquely determine the affine transformation. To get a desire shape, the collage of several attractors may be used (i.e. several affine transformations). This method is referred to as an Iterated Function System (IFS).

An example of an iterated function system is the black spleenwort fern. It is constructed through the use of four affine transformations (with weighted probabilities):

The resulting image is:

This image is infinitely complex — it is a self-similar fractal on all scales. What is truly amazing is that only 28 numbers are necessary to generate this infinitely complex image: four 2 x 2 transformation matrices, four 2 x 1 translational vectors, and four weighted probabilities for the transformations (each attractor).