View source for Burning Ship fractal

{{Short description|Complex plane fractal}}
{{More footnotes needed|date=November 2021}}
[[File:BurningShip 4.jpg|thumb|Detail of the Burning Ship fractal]]
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.<ref>{{cite journal |title=Inventive Burning Ship |last1=Agarwal |first1=Shafali |last2=Negi |first2=Ashish |journal=International Journal of Advances in Engineering & Technology |date=2013}}</ref> 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>

Virtually all images of the Burning Ship [[fractal]] are reflected vertically for aesthetic purposes, and some are also reflected horizontally.<ref>{{cite web | url=https://www.hpdz.net/StillImages/BurningShip.htm | title=HPDZ.NET - Still Images - Burning Ship }}</ref>

== Implementation ==
[[File:Burning Ship Fractal Zoom-out 64.gif|thumb|right|upright=2.0|Animation of a continuous zoom-out to show the amount of detail for an implementation with 64 maximum iterations]]
The below [[pseudocode]] implementation hardcodes the complex operations for Z. Consider implementing [[complex number]] operations to allow for more dynamic and reusable code.

 '''for each''' pixel (''x'', ''y'') on the screen, '''do''':
     ''x'' := scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))
     ''y'' := scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1, 1))
 
     ''zx'' := ''x'' ''// zx represents the real part of z''
     ''zy'' := ''y'' ''// zy represents the imaginary part of z'' 
 
     ''iteration'' := 0
     ''max_iteration'' := 100
<!-- The following line of this text must be empty (no blancs!). Then the next paragraph can expand wide enough. -->

     '''while''' (zx*zx + zy*zy < 4 '''and''' iteration < max_iteration) '''do'''
         ''xtemp'' := zx*zx - zy*zy + x 
         ''zy'' := abs(2*zx*zy) + y ''// abs returns the absolute value''
         ''zx'' := xtemp
         ''iteration'' := ''iteration'' + 1
 
     '''if''' ''iteration'' = ''max_iteration'' '''then''' ''// Belongs to the set''
         '''return''' ''INSIDE_COLOR''
 
     '''return''' ''(max_iteration / iteration)'' × ''color // Assign color to pixel outside the set''

==Gallery==

{{Gallery
|title=Burning Ship fractal renderings
|align=center
|height=125
|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 2.3·10<sup>−50</sup>
|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
|File:Ghost Ship fractal.png|Ghost Ship - The Burning Ship fractal rendered using the [[Nebulabrot]] technique
|File:A Corresponding Julia Set of Burning Ship Fractal 2.PNG|A corresponding Julia set of Burning Ship fractal
|File:A Corresponding Julia Set of Burning Ship Fractal.PNG|A corresponding Julia set of Burning Ship fractal
|File:Burning Ship Fractal 4800x3600.png|High resolution image of the burning ship fractal
|File:High quality rendering of the structure of the Burning Ship fractal's pattern.png|The structure of the Burning Ship fractal
|File:Burning Ship 20210818.png|High-quality overview image of the Burning Ship fractal
|File:Burning Ships.jpg|Chain of burning ships
|File:Burning Ship 0002 20210818.png|High-quality image of the large ship in the left antenna
|File:BurningShip full.jpg|Full fractal
}}

==References==
{{Reflist}}

==External links==
{{Commons}}
*[https://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,
*[https://web.archive.org/web/20110821220641/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]
*[https://7affer.github.io/fractalTS/ FractalTS] Mandelbrot, Burning ship and corresponding Julia set generator.

{{Fractal software}}
{{Fractals}}
{{Mathematics and art}}

[[Category:Fractals]]
[[Category:Articles with example pseudocode]]