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From Create Your Own Story
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| - | + | [[Image:Mandel zoom 00 mandelbrot set.jpg|322px|right|thumb|Initial image of a Mandelbrot set zoom sequence with continuously coloured environment]]<!-- The sequence \\, is inserted in MATH items to ensure consistency of representation | |
| + | -- Please don't remove it --> | ||
| + | The '''Mandelbrot set''' is a set of [[Point (geometry)|points]] in the [[complex plane]] that forms a [[fractal]]. Mathematically, the Mandelbrot set can be defined as the set of complex ''c''-values for which the orbit of 0 under iteration of the [[complex quadratic polynomial]] ''x''<sup>2</sup> + ''c'' remains bounded. | ||
| - | + | Eg. c = 1 gives the sequence 0, 1, 2, 5, 26… which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. | |
| - | + | On the other hand, c = i gives the sequence 0, i, (-1 + i), –i, (-1 + i), -i… which is bounded, and so it belongs to the Mandelbrot set. | |
| - | + | ||
| - | + | When computed and graphed on the complex plane, the Mandelbrot Set is seen to have an elaborate boundary, which does not simplify at any given magnification. This qualifies it as a fractal. | |
| - | + | ||
| - | + | The Mandelbrot set has become popular outside [[mathematics]] both for its aesthetic appeal and for being a complicated structure arising from a simple definition. [[Benoît Mandelbrot]] and others worked hard to communicate this [[Areas of mathematics|area of mathematics]] to the public. | |
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Revision as of 23:44, 17 December 2007
The Mandelbrot set is a set of points in the complex plane that forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex c-values for which the orbit of 0 under iteration of the complex quadratic polynomial x2 + c remains bounded.
Eg. c = 1 gives the sequence 0, 1, 2, 5, 26… which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set.
On the other hand, c = i gives the sequence 0, i, (-1 + i), –i, (-1 + i), -i… which is bounded, and so it belongs to the Mandelbrot set.
When computed and graphed on the complex plane, the Mandelbrot Set is seen to have an elaborate boundary, which does not simplify at any given magnification. This qualifies it as a fractal.
The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and for being a complicated structure arising from a simple definition. Benoît Mandelbrot and others worked hard to communicate this area of mathematics to the public.
