Mandelbrot set (nonfiction): Difference between revisions

From Gnomon Chronicles
Jump to navigation Jump to search
(Created page with "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...")
 
 
(15 intermediate revisions by the same user not shown)
Line 1: Line 1:
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.
[[File:Mandelbrot_set_command_line_depiction.png|thumb|ASCII-art depiction of the Mandelbrot set.]]In [[mathematics (nonfiction)|mathematics]], 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.
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.
Line 8: Line 6:


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 ==


Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications.
Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications.
Line 32: Line 13:
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.
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 ==
== In the News ==


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, and is one of the best-known examples of mathematical visualization.
<gallery mode="traditional" widths="200px" heights="200px">
File:Lanfranc-canterbury-mandelbrot.jpg|link=Canterbury scrying engine|The [[Canterbury scrying engine]] computes and displays a simple text-based Mandelbrot set.
File:Benoit Mandelbrot.jpg|link=Benoit Mandelbrot (nonfiction)|[[Benoit Mandelbrot (nonfiction)|Mandelbrot]] is "pleased with his life's work," says [[Benoit Mandelbrot|artificial intelligence]].
File:Mandelbrot-AI-interview.jpg|link=Benoit Mandelbrot|Artist-Engineers prepare to interview famed artificial intelligence [[Benoit Mandelbrot]].
</gallery>


== Nonfiction cross-reference ==
== Nonfiction cross-reference ==


* [[Benoit Mandelbrot]]
* [[Benoit Mandelbrot (nonfiction)]]
* [[Mathematics (nonfiction)]]
* [[Mathematics (nonfiction)]]


== Fiction cross-reference ==
== Fiction cross-reference ==


* [[Benoit Mandelbrot (nonfiction)]]
* [[Benoit Mandelbrot]]


== External links ==
External links:


* [http://wiki.karljones.com/index.php?title=Mandelbrot_set Mandelbrot set] @ wiki.karljones.com
* [http://wiki.karljones.com/index.php?title=Mandelbrot_set Mandelbrot set] @ wiki.karljones.com
* [https://en.wikipedia.org/wiki/Mandelbrot_set Mandelbrot set] @ Wikipedia
* [https://en.wikipedia.org/wiki/Mandelbrot_set Mandelbrot set] @ Wikipedia


[[Category:Fiction (nonfiction)]]
[[Category:Nonfiction (nonfiction)]]
[[Category:Mandelbrot set (nonfiction)]]
[[Category:Mandelbrot set (nonfiction)]]
[[Category:Mathematics (nonfiction)]]
[[Category:Mathematics (nonfiction)]]

Latest revision as of 23:50, 17 December 2016

ASCII-art depiction of the Mandelbrot set.

In mathematics, 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.

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.

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.

In the News

Nonfiction cross-reference

Fiction cross-reference

External links: