The word 'Fractal' was coined in 1975 by the mathematician Benoît Mandelbrot - but the study of self-repeating mathematical systems dates back several centuries.

Mandelbrot provided a definition of a fractal as : "A rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole".

There are many naturally-occurring examples e.g. crystals, tree structures, electrical discharges, frictional surfaces, bio-rhythms, mountain ranges, coastline shapes, sea-surface waves, etc etc.

The purely mathematical representations are often an infinite series, but the 'real-world' examples are bound by limits - even so, one of the key features of systems which have a fractal component is that they are hard (or sometimes impossible) to accurately quantify.

See, for example, this now-famous paper by B. B. Mandelbrot How long is the coast of Britain? Statistical self-similarity and fractional dimension Science: 156, 1967, 636-638

To specify the conditions for the existence of a similarity dimension is not a fully solved mathematical problem. In fact, a number of conceptual problems familiar in other uses of randomness in science are also raised by the idea that a geographical curve is random."

In other words :

Geographical curves are so involved in their detail that their lengths are often infinite or more accurately, undefinable."

The implications for any system with a fractal component (not just geographical ones) is that a completely accurate mathematical description is often impossible. The amount of 'fractal-ness' - called the 'Fractal Dimension' ( D ) - can however often be specified (or estimated).

Also see :Chaos Theoryplugin-autotooltip__plain plugin-autotooltip_bigChaos Theory

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Chaos Theory is the concept that the behaviour of some complex dynamical systems (e.g. global weather patterns) can be extremely sensitive to tiny changes in initial conditions.

Any large-scale system which has a complex set of interacting feedback and feed-forward loops can become chaotic - thus making accurate and specific long-term predictions about the system unreliable, if not impossible. and Random numbersplugin-autotooltip__plain plugin-autotooltip_bigRandom numbers

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"We can never decide for sure that a number is random, but what we can do is apply an increasing number of tests and treat our sequence of numbers as innocent until proved guilty."

Source : Prof. Colva Roney-Dougal, senior lecturer in Pure Mathematics at the University of St Andrews, speaking in