Chaos Theory is a very important area of mathematics which can explain a lot of what we see in the real world. A pendulum with one mass is relatively easy to explain mathematically, and it behaves nicely. However if you put another mass in there, it behaves chaotically. Technically, this means that if you change the starting positions only slightly, the state of the system a short time later can change drastically. The weather is chaotic- a small error in measuring it today could be the difference between rain and no rain in a weeks days time. Watch these two pendulum systems quickly diverge, though they both start off with nearly the same settings. [more] [code]
In 1975 Mandelbrot published The Fractal Geometry of Nature, which became a classic of chaos theory.
Cool name, awesome area of math.