In a previous post I explored complex functions from the perspective of domain coloring. I used the paradigm I found on Wikipedia, which is to assign a color to each point in the complex plane based on it’s phase, and a shading based on it’s modulus.

Now I want to mix it up a bit; let’s assign color based on modulus, and shading based on phase. The color cyan represents when the modulus is near an integer value, and the shading is darkest when the phase approaches 0 or pi (as it decreases towards those values).

Now, as before, we can start applying different functions.

On the left is the previous image transformed with the function z = f(z) = z^2. On the right is the classic complex plane under the same transformation. On the right – the colors cycle twice as one circles the origin (as opposed to once for the untransformed plane), and the dark circles get closer together as one moves away from the origin. On the left the shading pattern repeats itself twice as one circles the origin, and the colored circles get closer together as one moves further away from the origin.

Here we have the transformation z = f(z) = z^3. Again with the new way on the left, and the old way on the right.

This is z = f(z) = sin(z).

z = f(z) = tan(z).

z = f(z) = tan(sin(a))

z = f(z) =tan(exp(z))

z = f(z) = sin(tan(z))

z = f(z) = tan(tan(z))

I like the pictures that are produced as a result of the original paradigm better. They have a cartoon-like property that I find quite appealing.

The real fun begins when I try doing a Newton’s Method type of picture, but fiddle around with the derivative. Here is the highlights from about two hours of exploration.