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# Mandelbrot Set

Robert P. Munafo, 2013 Apr 14.

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.

Definition :

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

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

Z0 = 0 + 0 i

Zn = Zn-12 + C for n > 0

C is a member of the Mandelbrot set if and only if sqrt(a2 + b2) remains "within a limited size" for all values of n. sqrt(a2 + b2), where a is the real component and b the imaginary component of Zn, is called the "magnitude of Zn", and is the distance of Zn from the origin. The "limited size" is an arbitrary constant called the escape radius, and can be anything greater than 2.

Other mathematical properties :

The Mandelbrot Set is connected. Its boundary has Hausdorff dimension 2.0.