Creating Fractals using Iterated Function Systems

Suppose you had no ideas what fractals were, and you were asked to model objects like clouds, grass, plants, etc.. What else could you use to define the geometries of these objects? Although polynomials can easily define objects with smooth geometry, they are pretty useless if you want to model complex objects like grass or plants, since those possess infinitely non smooth, highly structured geometries. In the early 1980s, mathematicians have become concerned with non-smooth sets, that is, sets where the method of classical calculus couldn't be applied. It was the fundamental work of Mandelbrot that opened up a new way to model natural phenomena. What is a fractal? Many definitions exists, and mathematicians have not yet agreed on one. Benoit Mandelbrot refers to the word "fractals" as objects who possess self-similarity. For the rest of this paper, we will adopt this as our definition of a fractal.