The Burning Ship fractal, first described and created by Michael Michelitsch and

Otto E. Rössler Otto is a masculine German given name and a surname. It originates as an Old High German short form (variants '' Audo'', '' Odo'', '' Udo'') of Germanic names beginning in ''aud-'', an element meaning "wealth, prosperity". The name is recorded fr ...

in 1992, is generated by iterating the function: :

z_ = (, \operatorname \left(z_n\right), +i, \operatorname \left(z_n\right), )^2 + c, \quad z_0=0

in the

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

\mathbb

which will either escape or remain bounded. The difference between this calculation and that for the

Mandelbrot set The Mandelbrot set () is a two-dimensional set (mathematics), set that is defined in the complex plane as the complex numbers c for which the function f_c(z)=z^2+c does not Stability theory, diverge to infinity when Iteration, iterated starting ...

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 In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin-Louis Cauchy, Augustin Cauchy and Bernhard Riemann, consist of a system of differential equations, system of two partial differential equatio ...

. Virtually all images of the Burning Ship

fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...

are reflected vertically for aesthetic purposes, and some are also reflected horizontally.

Implementation

The below

pseudocode In computer science, pseudocode is a description of the steps in an algorithm using a mix of conventions of programming languages (like assignment operator, conditional operator, loop) with informal, usually self-explanatory, notation of actio ...

implementation hardcodes the complex operations for Z. Consider implementing

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

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 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

References

External links

About properties and symmetries of the Burning Ship fractal
featured by Theory.org
Burning Ship Fractal
Description and C source code.
Burningship
Video,

includes the first representations and the original paper cited above on the Burning Ship fractal.
3D representations of the Burning Ship fractalFractalTS
Mandelbrot, Burning ship and corresponding Julia set generator. {{Mathematics and art Fractals Articles with example pseudocode