To put it simply, fractal is any formation constructed of a theme(s) that is/are repeated within itself in several generations. In a pure fractal this repetition happens infinitely.
Though fractal shapes are most familiar to us, fractal geometry can be applied to or exist in anything else like sound, movement, light or even thought!
The animated images of simple fractal patterns on these pages demonstrate effectively what this is all about!
A very simple fractal where a circle is extended with a smaller circle in the end of an "arm". The smaller circle is then extended in the same way, and then the next circle etc.
Most people associate fractals to computer generated images of complex geometries and colours. The most well known of these fractals is the Mandelbrot set, named after it's discoverer, a Swiss mathematician. Below is a typical image of the Mandelbrot set.
Computer fractals are usually generated with fairly simple mathematical instructions that define a "set". A 2-dimensional (or 3-dimensional!) area made of small points (pixels) is scanned and each point is tested; does it belong to the set or not. If the pixel belongs to the set, it is coloured according to it's location in the set. Alternatively, instead colour this information can use to define height to make a 3D fractal.