We spent some time reviewing the results from Complex Paint Worksheet 2. We have observed a few things, the most obvious of which are that all Julia Sets have 180-degree rotational symmetry and the only Julia Sets that have x- and y-axis symmetry have a parameter that is a real number. It turns out that both of these observations are provable facts about Julia Sets, which we will investigate tomorrow!
Secondly, we observed that many Julia Sets are connected (areas and strings) but many are not connected. This distinction – between connected and unconnected Julia sets – will form the basis for the formation of the Mandelbrot Set. Be sure you are in class tomorrow, as it will be an important day!
If you’re here to watch the video Dr. Yerky is instructing you to view in class, click here (it is also linked below). You’ll want to refer to Complex Paint Worksheet 2.
After a recap of what we learned with complex linear functions, we have started work with complex quadratic functions. We will be replicating the process we did with real numbers: analyzing a family of quadratics where we only adjust a single parameter value and investigate the behavior of iterations for specific values of the parameter. Eventually, we will find a way to categorize and catalog these behaviors.
The family of functions we will be analyzing is z^2 + C, and the first parameter we looked at is C = 0. We found that some seeds will attract to zero, some will spiral off to infinity, but others seem trapped in a unit circle around the origin, falling into a cycle or landing on (1,0). This unit circle forms a boundary between seeds that diverge and seeds that do not, and that boundary is called a Julia Set, named after French mathematician Gaston Julia, and developed by Julia and Pierre Fatou.
Not all Julia Sets (in fact pretty much none of them) look as simple as this, and Complex Paint is a great tool to help us see them. Watch this video for another take on the derivation of Julia Sets, plus instructions on how to use Complex Paint to create and understand this new class of mathematical object. Feel free to explore whatever parameter values you would like, but in particular I want you to use the ones on Complex Paint Worksheet 2.