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The Mandelbrot Universe

Mandelbrot Set fractal

The Mandelbrot Set and its Meaning

Category: ScienceMathematicsSacred GeometryComputer GraphicsComputer Aided DesignPatterning

Date: 15 May 2011


The Mandelbrot Set

In mathematics the Mandelbrot set, named after Benoît Mandelbrot, is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically the Mandelbrot set can be defined as the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded. That is, a complex number, c, is in the Mandelbrot set if, when starting with z0=0 and applying the iteration repeatedly, the absolute value of zn never exceeds a certain number (that number depends on c) however large n gets.

For example, letting 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 (where i is the square root of -1) gives the sequence 0, i, (−1 + i), −i, (−1 + i), −i..., which is bounded and so i 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 the boundary as a fractal.

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