The Corridor of Graphical Iterations

a=3.9969234

The “corridor” shown above is a little piece of generative art which I have created with Processing. It’s basically copying the idea of Graphical Iteration, but with colors.

So what is graphical iteration? It is a geometric tool used to analyze iterated functions. The iterated function which I used is x=ax(1-x), and the graph shown above has a=3.9969234. When we do graphical iterations, first we graph the function, for example, y=4x(1-x) and the line y=x.

Screen Shot 2015-03-11 at 7.43.16 PM

Then we take the original x value, plot it on the x axis, and make a line going from that point straight up meeting the curve of the function y=4x(1-x). From that meeting point, we make a line going across meeting the line y=x. Then we do another line going straight up meeting y=4x(1-x), then another one going across meeting y=x. We do that a bunch of times.

Screen Shot 2015-03-11 at 7.43.53 PM

Then we do that 20000 times.

Screen Shot 2015-03-11 at 7.44.09 PM

Then maybe we add some color.

Screen Shot 2015-03-11 at 7.41.22 PM

And that makes a “corridor”. But usually we don’t do graphical iterations for the sake of art, nor does the lines become all crazy like this. It only happens with certain values of a, like when a=4. You get different graphs with different values of a.

a=3.267317(a=3.267317)

a=7.118788(a=7.178788)

Enjoy!

Derek