We closed out our work with the Feigenbaum Plot today and took a moment to reflect. With our work in the real numbers, we started with linear functions (mx+b) and quickly discovered nearly everything there was to find. Linear functions are easy and non-chaotic. When we moved to non-linear functions, things started getting interesting. We looked at a few examples of the form x^2 + c, but did a deep dive with the logistic map of ax(1-x). For a single parameter, we could make a cobweb diagram to show the behavior of that specific function, but the really interesting things happened when we made our catalog of behavior for all parameters, creating the Feigenbaum Plot.
We will follow the same path through the forest of Complex Numbers. We will start with linear functions of the form Az+B, and understand what we can find there. We’ll then move on to non-linear functions, specifically of the form z^2 + C, and look at the behavior of single, specific values of C. Eventually, we will move to a catalog view there and see what we find.
Today was our first step towards that goal, with a discussion of how arithmetic on complex numbers can mimic geometric transformations. Complex numbers have two coordinates, a real part and an imaginary part, so their operations provide a convenient way of movie around the coordinate plane. Specifically, adding complex numbers produces translations and multiplying complex numbers produces a dilation/rotation. Exactly how the dilation/rotation is understood requires another way of referring to these complex numbers: Polar Form. We will discuss this on Monday.