If you magnify any part of a fractal, you see a minature recreation of the overall fractal structure repeated on the small scale. Magnify a small piece of any part of the small structure and you see the overall structure repeated again and again, ad infinitum. The coastlines of continents and the structure of snowflakes are just two of the many examples of fractals found in nature. The most famous shape among the fractals is the Mandelbrot set shown in figure 8. You can find out more about fractals on the Web.
Chaos enters into cutting-edge research into phenomena like convection .
To see chaos for a system as simple as a pendulum, check out this applet of a pendulum with a driver for the support point and damping.