Fractals

A fractal is a pattern of shapes, either human-created or occurring naturally in the universe. In a fractal pattern each consecutive, reduced shape is a fraction of the whole, a concept called self-similarity. Fractals occur in nature often, in crystalline structures, seashells (the chambered Nautilus, in fact, forms a perfect Fibonacci spiral), plants, and many other places. In fact, according to the Fractal Foundation, fractal folding is key in the structure of DNA in the nucleus of every living cell. For more examples and images of fractals in the natural world, follow this link: http://webecoist.com/2008/09/07/17-amazing-examples-of-fractals-in-nature/

Not only are fractals vital to life forms and found in the natural world, they can also be applied to many practical tasks in the mathematical, scientific, and even artistic fields. They are also a lot of fun, as demonstrated in this video: http://www.youtube.com/watch?v=DK5Z709J2eo
The xkcd drawing at the top of this post is a Sierpinski triangle, a form of iterated pattern and also a fractal.