====== Fractals ======
{{tag>Unknowable}}

The word 'Fractal' was coined in 1975 by the mathematician [[https://en.wikipedia.org/wiki/Benoit_Mandelbrot|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 [[https://en.wikipedia.org/wiki/Romanesco_broccoli|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 [[https://en.wikipedia.org/wiki/Julia_set|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 [[https://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf|How long is the coast of Britain? Statistical self-similarity and fractional dimension]] {{:oa_padlock_grn.png?16}}//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).



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Also see :[[content:mathematics:chaos_theory]] and [[content:mathematics:random_numbers]]