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mandelbrot_set.md
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# Mandelbrot Set
Mandelbrot set is a famous two dimensional [fractal](fractal.md), a [set](set.md) of [points](point.md) in two dimensional plane that are defined by a specific very simple equation. It turns out this set has an infinitely complex border (i.e. its shape is a fractal) and the whole thing is just [beautiful](beauty.md) to look at, especially when we draw it with colors and start zooming in to various interesting places -- patterns keep emerging down to infinitely small scales so we may keep zooming in forever and still discover new and new things; some patterns show self similarity, some not. Applying tricks to further add colors to inside and outside of the set increases the visual beauty yet more -- rendering Mandelbrot set is in fact a quite popular activity among programmers as it's very easy to [program](programming.md) such visualizations (at least until we reach the limits of [floating point](float.md) precision, then some more cleverness has to be applied; and yes, Mandelbrot can also be rendered only using [fixed point](fixed_point.md)). The origins of exploring this set are somewhere around 1905 when Fatou and Julia explored the equations related to it, however due to the lack of [computers](computer.md) the set couldn't very well be drawn -- this was only achieved much later, the first rendering of the set seems to be from 1978, albeit of very poor resolution. The set is named after Benoit Mandelbrot who is often considered the father of the field of fractal geometry and who researched this particular set a lot. Of course, Mandelbrot set is awesome, it's a like a whole infinite world to explore, hidden in just one simple formula.
Mandelbrot set (also M-set) is a famous two dimensional [fractal](fractal.md), a [set](set.md) of [points](point.md) in two dimensional plane that are defined by a specific very simple equation. It turns out this set has an infinitely complex border (i.e. its shape is a fractal) and the whole thing is just [beautiful](beauty.md) to look at, especially when we draw it with colors and start zooming in to various interesting places -- patterns keep emerging down to infinitely small scales so we may keep zooming in forever and still discover new and new things; some patterns show self similarity, some not. Applying tricks to further add colors to inside and outside of the set increases the visual beauty yet more -- rendering Mandelbrot set is in fact a quite popular activity among programmers as it's very easy to [program](programming.md) such visualizations (at least until we reach the limits of [floating point](float.md) precision, then some more cleverness has to be applied; and yes, Mandelbrot can also be rendered only using [fixed point](fixed_point.md)). The origins of exploring this set are somewhere around 1905 when Fatou and Julia explored the equations related to it, however due to the lack of [computers](computer.md) the set couldn't very well be drawn -- this was only achieved much later, the first rendering of the set seems to be from 1978, albeit of very poor resolution. The set is named after Benoit Mandelbrot who is often considered the father of the field of fractal geometry and who researched this particular set a lot. Of course, Mandelbrot set is awesome, it's a like a whole infinite world to explore, hidden in just one simple formula.
{ Pretty amazing ASCII rendering of the Mandelbrot set can be found at http://www.mrob.com/pub/muency/asciigraphics.html. ~drummyfish }
@ -55,6 +55,7 @@ The following are some **attributes** of the Mandelbrot set:
- **area**: approximately 1.5052; this is a current best estimate, the area is not easy to calculate (it may be estimated e.g. with [Monte Carlo](monte_carlo.md) methods).
- It is **symmetric** along the *x* axis.
- It's proven the set is **connected**, i.e. it's just a single "island".
- The number [pi](pi.md) is embedded in the shape of the set in a hugely mysterious way which was discovered by mistake by David Bolle in 1991 who tried to measure the width of the gap touching the point [-0.75,0] -- as he increased precision of his iterative algorithm, the number of iterations started to approximate digits of pi.
- ...
**How to explore Mandelbrot set?** There are about billion programs for this, but a quite nice FOSS one is e.g. [Xaos](xaos.md).