# 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 Z_{n}
is defined as follows :

Z_{0} = 0 + 0 i

Z_{n} = Z_{n-1}^{2} + C for n > 0

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

See also: algorithms, iteration, Mu map.

Other mathematical properties :

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

Its Area is about 1.5065916...

revisions: 20000207 oldest on record; 20130414 rewrite to avoid use of the word 'finite'

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2013 Apr 15. s.27