The strange beauty of the Gumowski-Mira Attractor

The world of fractals and (strange) attractors have always had my interest; mathematical formulas which expose interesting visual patterns and/or chaos when mapped to the screen. I thought I’d seen most it, but was pleasantly surprised when I recently encountered some of the organic results of the Gumowski-Mira attractor. Seemed I had some programming to do and create them myself.

The Gumowski-Mira equation was developed in 1980 at CERN by I. Gumowski and C. Mira to calculate the trajectories of sub-atomic particles. It is in fact a formula to plot a 2-dimensional dynamic system, and the main equation is as follows:

[math]f(x) = ax + frac{2(1-a) x^2 }{1 + x^2}[/math]

Using that equation, we can iterate the following formulas to calculate sequential x,y locations:

[math]x_{n+1} = by_n + f(x_n)[/math]
[math]y_{n+1} = -x_n + f(x_{n+1})[/math]

So we have two parameters, a and b (b is usually kept at 1 for the most interesting results), and to get the system going we also need some initial values for both x and y. You can set all those values in the editor below using the sliders, with some of them having a finetune-slider as well.

When you change any value, the program gives you a quick render of the results (10.000 iterations). If you want to explore more iterations, click the render (and more) button repeatedly to increase the iterations by 50.000. The color-slider is used to set the increase in blackness that each pixel will get when a point is drawn; set it to a high value if you want to quickly check how your current settings will evolve, and set it to a low value for the best images (note: the quick-render isn’t affected by this slider). Below the program are some renders I made.

[SWF]http://www.petervandernoord.nl/swf/gumowski-mira/main.swf,640,780[/SWF]





















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