Mandelbrot Set

Robert P. Munafo, 2000 Feb 7.

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Major Features : names and pictures of the largest features.

Exploring : an overview of what types of things you'll find when you start exploring the Mandelbrot Set on your own.

Pixel-Counting : the latest results on the calculation of the Mandelbrot Set's area.

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Definition :

The Mandelbrot Set is a set in the domain of complex numbers (see Point).

For each complex number C, a sequence of iterates Z_n is defined as follows :

Z₀ = 0 + 0 i

Z_n = Z_n-1² + C for n > 0

C is a member of the Mandelbrot set if and only if mag(Z_n) is finite for all values of n. Here, mag(Z_n) indicates the magnitude of Z_n, that is,

mag(Z_n) = sqrt(a² + b²)

where a is the real component and b the imaginary component of Z_n.

See also: algorithms, iteration, Mu map.

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Other mathematical properties :

The Mandelbrot Set is connected. Its Boundary has Hausdorff dimension 2.0. Its Area is about 1.5065916...

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From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo.     Mu-ency index

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