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 fractalsâ€”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:

- It has a fine structure at arbitrarily small scales.
- It is too irregular to be easily described in traditional Euclidean geometric language.
- It is self-similar (at least approximately or stochastically – probably this is the most important characteristic to me).
- It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).
- It has a simple and recursive definition.