Robert Spann
I have always been fascinated with the behavior of dynamical systems -- be they mathematical, physical, or financial. Where do they end up and how do they get there? Do they reach an equilibrium or do they continue moving randomly forever? Computer graphics allows one to see both the numerical and aesthetic properties of these systems. I am particularly fascinated with iterations of a function of a complex variable, where the function has no attracting fixed or periodic points. Such functions exhibit chaotic behavior. Points initially close together move further apart with iterations. If the function has symmetries, iterations of such functions can be used to produce interesting symmetric images.
Vortex is produced by iterating the function f(z) = (sz^4+1)/(z^4+s) once with s = -.17 - .666i. This function has no attracting fixed or periodic points. It exhibits four-fold symmetry in the sense that f(z) = f(iz) = f(-z)= f(-iz). Letting Zf be the value of f(z) after one iteration, I compute (1-cos(t))/2 where t=arg(Zf) and use these values to color the image.
Ribbons and bows is produced by iterating the function f(z) = -1 + 2/z^2 six times. This function has no attracting fixed or periodic points. It does have the property that f(-z) = f(z). As such, it can be used to produce images that are symmetric with respect to a 180 degree rotation. Letting Zf be the value of f(z) after six iterations, I compute (1-cos(t))/2 where t=arg(Zf) and use these values to color the image. The coloring function adds vertical symmetry to the image. The combination of symmetry with respect to a 180 degree rotation and vertical symmetry means the resulting image will also have horizontal symmetry.