We discussed the results from iterating yesterday’s seeds in the function z², finding 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 (Fatou names the complement of the Julia Set; in the case of C = 0, the Fatou set is the entirety of the rest of the complex plane except for the unit circle).
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 instructions 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 we will be using the ones found on Complex Paint Worksheet 2. We started this today and will finish it tomorrow. For each, note the type of Julia Set you get (area, string, or dust), any symmetry you observe, and the destination of orbits inside any area Julia Sets.