One of the important messages of the new science of chaos is that very simple mathematical systems can exhibit extremely complex behaviour. In this talk we aim to demonstrate this fact by considering some simple iterations. The mathematics involved is very simple, indeed it is possible to investigate the subject on any hand-held calculator. Though the basis of the subject is very easy to understand these iterative methods can lead to all kinds of fascinating phenomena such as period doubling, bifurcations and chaos itself. Furthermore, graphical representation of the results can lead to fractal images of stunning beauty and complexity.

We will demonstrate some of the properties of fractal images by investigating a family of fractals generated by a process called the Chaos Game. This process can lead to images which bear an uncanny likeness to many objects found in nature, such as trees and clouds.

Another characteristic of chaotic systems is their sensitive dependence upon initial conditions, also known as the Butterfly effect. We will also demonstrate this property with reference to simple iterative processes and briefly discuss its implications in other fields such as weather forecasting.