Fractals and Complex Variables
Fractals and Complex Variables This Graphics Perspective shows a fractal being generated in slow motion. The important input variables are the complex variables, z=x+iy, its square z^2, and k. First zz+k is calculated. Suppose that the result is z1z1. This goes back to the input and becomes z1z1+k. This is calculated with the result z2z2.This goes back to the input and becomes z2z2+k. This iteration continues for an arbitrary number of times. This is in the Spreadsheet Perspective. The red hyperbola is x^2-y^2. This is the real part of z^2. The blue hyperbola is 2xy. This is the imaginary part of z^2. The small blue point moving around the circle is z. It is at the intersection of two hyperbolas. The small purple circle represents z^2 . The k plane is scanned by using two sliders.The z plane is scanned by two additional sliders. One slider is for the real part, and one slider is for the imaginary part. Each iteration is assigned a color. Reflections and filtering add additional traces and ignore other traces. A fractal is whatever the author wants it to be. The animation can be stopped by turning off the animation in the Free Objects. You can add traces to anything that moves. This will generate interesting animations. You can use any function of z that you know how to convert into x and y. Notes. Remember that the traces are ephemeral. If you zoom them then you lose them. You might try a screen shot. Squared variables tend to diverge. I have adjusted for this arbitrarily. This demonstration is conceptually based on the work of Mandelebrot and Julia. A Mandelbrot fractal keeps z constant. A Julia fractal keeps k constant. This demonstration is neither. Change things cautiously to begin with. These iterations change rapidly.