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Indexed under : Mathematics

# Fractals

The word 'Fractal' was coined in 1975 by the mathematician BenoĂ®t Mandelbrot - but the study of self-repeating mathematical systems dates back several centuries.

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

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.â€ś

In other words :

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

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).

Descriptions of fractal systems fall into the special category of Known Unknowables

Also see :Chaos Theoryplugin-autotooltip__plain plugin-autotooltip_bigChaos Theory

unknowable

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 feeâ€¦
and Random numbersplugin-autotooltip__plain plugin-autotooltip_bigRandom numbers

unknowable

"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.â€ś

Source Colva Roney-Dougal, Senior Lecturer in Pure Mathematâ€¦

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