Mandelbrot set (nonfiction): Difference between revisions
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[[File:Mandelbrot_set_command_line_depiction.png|thumb|Command-line 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. | [[File:Mandelbrot_set_command_line_depiction.png|thumb|Command-line 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. | ||
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. | ||
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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. | ||
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. | ||
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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. | ||
== | == In the News == | ||
<gallery mode="traditional"> | <gallery mode="traditional"> | ||
File:Benoit Mandelbrot.jpg|link=Benoit Mandelbrot (nonfiction)|[[Benoit Mandelbrot (nonfiction)|Benoit Mandelbrot]]. | File:Benoit Mandelbrot.jpg|link=Benoit Mandelbrot (nonfiction)|[[Benoit Mandelbrot (nonfiction)|Mandelbrot]] is "pleased with his life's work," says [[Benoit Mandelbrot|articial intelligence]]. | ||
File:Mandelbrot set command line depiction.png|link=Mandelbrot set (nonfiction)|Command-line [[Mandelbrot set (nonfiction)|Mandelbrot set]] yearns for color, launches Kickstarter campaign to fund upgrade. | |||
</gallery> | </gallery> | ||
File:Mandelbrot set command line depiction.png|link=Mandelbrot set (nonfiction)|Command-line [[Mandelbrot set (nonfiction)|Mandelbrot set]] yearns for color, launches Kickstarter campaign to fund upgrade. | |||
== Nonfiction cross-reference == | |||
* [[Benoit Mandelbrot (nonfiction)]] | * [[Benoit Mandelbrot (nonfiction)]] | ||
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* [[Benoit Mandelbrot]] | * [[Benoit Mandelbrot]] | ||
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 | ||
Revision as of 18:25, 19 June 2016
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
Mandelbrot is "pleased with his life's work," says articial intelligence.
Command-line Mandelbrot set yearns for color, launches Kickstarter campaign to fund upgrade.
File:Mandelbrot set command line depiction.png|link=Mandelbrot set (nonfiction)|Command-line Mandelbrot set yearns for color, launches Kickstarter campaign to fund upgrade.
Nonfiction cross-reference
Fiction cross-reference
External links:
- Mandelbrot set @ wiki.karljones.com
- Mandelbrot set @ Wikipedia