Mandelbrot set (nonfiction): Difference between revisions

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Treating the real and imaginary parts of each number as image coordinates, pixels are colored according to how rapidly the sequence diverges, if at all.
Treating the real and imaginary parts of each number as image coordinates, pixels are colored according to how rapidly the sequence diverges, if at all.
More precisely, the Mandelbrot set is the set of values of ''c'' in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial:
z_{n+1}=z_n^2+c
remains bounded.
That is, a complex number ''c'' is part of the Mandelbrot set if, when starting with ''z''<sub>0</sub> = 0 and applying the iteration repeatedly, the absolute value of ''z''<sub>''n''</sub> remains bounded 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'' = −1 gives the sequence 0, −1, 0, −1, 0,…, which is bounded, and so −1 belongs to the Mandelbrot set.


== Self-similarity ==
== Self-similarity ==

Revision as of 09:54, 6 June 2016

Command-line depiction of the Mandelbrot set.

In mathematics (nonfiction), the Mandelbrot set is the set of complex numbers 'c' for which the sequence (c, c² + c, (c²+c)² + c, ((c²+c)²+c)² + c, (((c²+c)²+c)²+c)² + c, ...) does not approach infinity.

Description

The set is closely related to Julia sets (which include similarly complex shapes) and is named after the mathematician Benoit Mandelbrot (nonfiction), who studied and popularized it.

Mandelbrot set images are made by sampling complex numbers and determining for each whether the result tends towards infinity when a particular mathematical operation is iterated on it.

Treating the real and imaginary parts of each number as image coordinates, pixels are colored according to how rapidly the sequence diverges, if at all.

Self-similarity

Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications.

The "style" of this repeating detail depends on the region of the set being examined.

The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

Outside mathematics

The Mandelbrot set has become popular outside mathematics (nonfiction) both for its aesthetic appeal, and as an example of a complex structure arising from the application of simple rules.

The Mandelbrot set is one of the best-known examples of mathematical visualization.

Nonfiction cross-reference

Fiction cross-reference

External links

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