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HEADERS_END

Popularised by Benoit Mandelbrot, the Mandelbrot Set is a fractal,

one of the first, if not the first, ever discovered and understood

as such.

[[[> http://www.solipsys.co.uk/images/MB_Set.png ]]]

* Choose a constant, /c./

* Start with a complex number (in fact 0), EQN:z_0=0

* Start with the complex number 0, EQN:z_0=0

* Repeatedly apply the function EQN:z_{i+1}=f(z_i)=z_i^2+c.

* Ask, do the values go off to infinity, or do they remain nearby.

* If they remain nearby, then the initial point /c/ is in the Mandelbrot set.

* If the values go off to infinity, then /c/ is not in the set.

Of course, it's hard to know for sure that the values will remain nearby,

and they can exhibit some quite interesting behaviour. However, we can

know for sure that if the absolute value gets large then the points will

definitely go off to infinity. We can also plot the orbits of points,

and have a "good guess(tm)" as to whether they appear to be "closed".

Points outside the set can be coloured according to just how quickly they

go off to infinity, and points inside the set can be coloured according

to the number of points in the orbit to which they converge.

See also Julia sets.

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* http://en.wikipedia.org/wiki/Mandelbrot_set

* http://mathworld.wolfram.com/MandelbrotSet.html

* http://www.google.co.uk/search?q=mandelbrot+set