Can Differential Equations Be Used to Create Art?

[alexkiddo]

Hello all,

I have been assigned a project to create artwork using differential equations, the obvious idea would be to use Mathematica or Maple to obtain general solutions and replace values to obtain different level curves and then fill in the space between them.

However, I'm wanting to go a little deeper than that; so I was wondering if anyone out there has any other ideas for creating visual art using differential equations.

Thanks!

 

[jackmell]

You could look into strange attractors:

http://en.wikipedia.org/wiki/Attractor

Those are often created by coupled non-linear DEs. For example, the Lorenz attractor, the owl-eye icon of Chaos Theory, is created by the system:

$$x'=-ax+cy$$
$$y'=rx-y-xz$$
$$z'=-bz+xy$$

That's easy to draw in Mathematica:

mysol = NDSolve[{x'[t] == -3 (x[t] - y[t]), 
   y'[t] == -x[t] z[t] + 26.5 x[t] - y[t], 
   z'[t] == x[t] y[t] - z[t], 
   x[0] == z[0] == y[0] == 1}, 
   {x, y, z}, {t, 0, 20}, MaxSteps -> 3000]
ParametricPlot3D[Evaluate[{x[t], y[t], z[t]} /. mysol],
 {t, 0, 20}, 
 PlotPoints -> 1000]

but that's kinda' bland-looking. Try improving it with some color variation:

Remove[x, y, z]
mysol = NDSolve[{x'[t] == -3 (x[t] - y[t]), 
   y'[t] == -x[t] z[t] + 26.5 x[t] - y[t], z'[t] == x[t] y[t] - z[t], 
   x[0] == z[0] == y[0] == 1}, {x, y, z}, {t, 0, 200}, 
  MaxSteps -> 30000]
ParametricPlot3D[Evaluate[{x[t], y[t], z[t]} /. mysol], {t, 0, 200}, 
 PlotPoints -> 1000, 
 ColorFunction -> Function[{x, y}, ColorData["NeonColors"][y]]]

It's a start. The attractor at the top of the Wikipedia is nice-looking. Not sure how they generated that. Maybe you can find out and make others nice-looking also.