From the old fractalland.com FAQ page:

A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Fractals are generally self-similar and independent of scale.

There are many mathematical structures that are fractals; e.g. Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and Lorenz attractor. Fractals also describe many real-world objects, such as clouds, mountains, turbulence, and coastlines, that do not correspond to simple geometric shapes.

According to Mandelbrot, who invented the word: “I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means “to break:” to create irregular fragents. It is therefore sensible – and how appropriate for our needs! – that, in addition to “fragmented” (as in fraction or refraction), fractus should also mean “irregular,” both meanings being preserved in fragment.” (The Fractal Geometry of Nature, page 4.)

Natural objects that are approximated by fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, snow flakes, various vegetables (cauliflower and broccoli), and animal coloration patterns. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics; for instance, it is regular enough to be described in Euclidean terms.

A fractal often has the following features:

Related Images: