Fractal long-term behavior Mathematics serves as the language with which we try to understand how nature works. Because of limited calculational abilities in the past, the focus of mathematical activity was on smoothly varying and easily calculable quantities and objects. Mandelbrot was the first mathematician to open his eyes to the world and see that the true geometry of nature is very different indeed, and he named this geometry fractal .
Nature generally exhibits patterns of extreme intricacy and detail.
One aspect of the fractal philosophy is that this intricacy is the result of the evolution of relatively simple dynamical systems over long periods of time. A similar idea called Iterated Function Systems leads to even greater freedom in modelling natural objects. It is possible that their work may lead to the most efficient video compression algorithms. We will study the basics of fractal geometry in this chapter .