Lebesgue's density theorem: Difference between revisions
en>Helpful Pixie Bot m ISBNs (Build KE) |
"For almost every point x of A the density exists and is equal to 1" is the correct statement of the theorem, "is equal to 0 or 1" is to weak. Apparently there was a contamination with the next line of the article that refers to x in R^n. |
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[[Image:Burning Ship Fractal Overview.jpg|thumb|High-quality overview image of the Burning Ship fractal]] | |||
[[Image:Burning Ship Left.jpg|thumb|High-quality image of the large ship in the left antenna]] | |||
The '''Burning Ship [[fractal]]''', first described and created by Michael Michelitsch and [[Otto E. Rössler]] in 1992, is generated by iterating the function: | |||
:<math>z_{n+1} = (|\operatorname{Re} \left(z_n\right)|+i|\operatorname{Im} \left(z_n\right)|)^2 + c, \quad z_0=0</math> | |||
in the [[complex plane]] <math>\mathbb{C}</math> which will either escape or remain bounded. The difference between this calculation and that for the [[Mandelbrot set]] is that the real and imaginary components are set to their respective absolute values before squaring at each iteration. The mapping is non-analytic because its real and imaginary parts do not obey the [[Cauchy–Riemann equations]].<ref>Michael Michelitsch and Otto E. Rössler (1992). "The "Burning Ship" and Its Quasi-Julia Sets". In: ''Computers & Graphics'' Vol. 16, No. 4, pp. 435–438, 1992. Reprinted in [[Clifford A. Pickover]] Ed. (1998). ''Chaos and Fractals: A Computer Graphical Journey — A 10 Year Compilation of Advanced Research''. Amsterdam, Netherlands: Elsevier. ISBN 0-444-50002-2</ref> | |||
{{Gallery | |||
|title=Burning Ship Fractal renderings | |||
|align=center | |||
|height=125 | |||
|lines=5 | |||
|File:Burning Ship 3e-9.jpg|High-quality deep-zoom image of a small ship in the armada in the left Western antenna of the main ship structure | |||
|File:Burning Ship Deep Zoom 2e-50.jpg|Burning Ship Deep Zoom to 2.3e-50 | |||
|File:Burning Ship Fractal.png|The Burning Ship fractal | |||
|File:Burning Ship Fractal Zoom.png|A zoom-in to the lower left of the Burning Ship fractal, showing a "burning ship" and self-similarity to the complete fractal. | |||
|File:Burning Ship Fractal Zoom 2.png|A zoom-in to line on the left of the fractal, showing nested repetition (a different colour scheme is used here). | |||
|File:Burning_Ship_144x.jpg|High quality image of the Burning Ship fractal. | |||
|File:Jentererforetrukket.jpg|The Burning Ship Fractal featured in the 1K intro "JenterErForetrukket" by Youth Uprising; a demoscene production. | |||
}} | |||
==See also== | |||
* [[Mandelbrot set]] | |||
* [[Julia set]] | |||
* [[Fractal]] | |||
==References== | |||
{{reflist}} | |||
==External links== | |||
{{commons}} | |||
*[http://www.theory.org/fracdyn/burningship/ About properties and symmetries of the Burning Ship fractal], featured by Theory.org, | |||
*[http://paulbourke.net/fractals/burnship/ Burning Ship Fractal], Description and C source code. | |||
*[http://www.graphicandfractalworld.com/Me.html Burning Ship with its Mset of higher powers and Julia Sets] | |||
*[http://jet.ro/video_burningship Burningship], Video, | |||
*[http://michelitsch-fractals.webs.com/Michelitsch_Fractals.htm Fractal webpage] includes the first representations and the original paper cited above on the Burning Ship fractal. | |||
*[http://www.fractalforums.com/index.php?action=gallery;sa=view;id=4057 3D representations of the Burning Ship fractal] | |||
[[Category:Fractals]] | |||
{{Fractals}} | |||
{{geometry-stub}} | |||
Revision as of 19:01, 28 January 2014
The Burning Ship fractal, first described and created by Michael Michelitsch and Otto E. Rössler in 1992, is generated by iterating the function:
in the complex plane which will either escape or remain bounded. The difference between this calculation and that for the Mandelbrot set is that the real and imaginary components are set to their respective absolute values before squaring at each iteration. The mapping is non-analytic because its real and imaginary parts do not obey the Cauchy–Riemann equations.[1]
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See also
References
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External links
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- About properties and symmetries of the Burning Ship fractal, featured by Theory.org,
- Burning Ship Fractal, Description and C source code.
- Burning Ship with its Mset of higher powers and Julia Sets
- Burningship, Video,
- Fractal webpage includes the first representations and the original paper cited above on the Burning Ship fractal.
- 3D representations of the Burning Ship fractal
Template:Fractals
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- ↑ Michael Michelitsch and Otto E. Rössler (1992). "The "Burning Ship" and Its Quasi-Julia Sets". In: Computers & Graphics Vol. 16, No. 4, pp. 435–438, 1992. Reprinted in Clifford A. Pickover Ed. (1998). Chaos and Fractals: A Computer Graphical Journey — A 10 Year Compilation of Advanced Research. Amsterdam, Netherlands: Elsevier. ISBN 0-444-50002-2