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Where Chaos Meets Order

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Submitted: September 28, 2012
Image Size: 10.6 MB
Resolution: 4096×4096
Submitted with: Sta.sh

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Cosbrot Convergenceby =pifactorial

Digital Art / Fractal Art / Raw Fractals

Inspired by Ampersand by

(who has some fabulous mathematically inspired art), I realized that, in an escape-time fractal, the reciprocal harmonic sum of the sequence at any given point ought to be almost always convergent. The magnitude and complex argument of the convergent value can be used for simultaneous "in-colouring" and "out-colouring" of the fractal. (Points inside the fractal set will converge to zero, but you can get around this by using a small number of iterations and a logarithmic scale for the magnitude, as I have here.)

While in the case of an ordinary Mandelbrot, this colouring scheme produces colourful but not terribly exciting result, I decided to try applying it to more of my Mandelbrot variants. Cosbrot, which in the manner of Sincbrot uses the cosine function to approximate z^2 (see Sincbrot Galaxies for a somewhat confusing explanation), gave some pretty cool rainbow lightning when I tried it out.

z = 2*(1-cos(z)) + c
x = -3 ... 1
y = -2 ... 2
(x and y are swapped in the image)