Today we had the opportunity to play with our next Very Important Program: Complex Paint. This software was created by former F&C students Devon Loehr and Connor Simpson, and will be invaluable to us going forward. We’ll discuss all the features of the program as they become relevant, but after downloading/unzipping the folder listed in the Google Drive folder linked, read the README file for help on how to use the software. We started our analysis with the top half of a new worksheet, exploring and developing summary ideas of iterations of complex linear functions of the form Az+B. We saw that A is the major factor in the type of fixed point we get, while B only seems to affect where the fixed point is. This didn’t come as a significant surprise considering we made a similar observation about the slope and y-intercept of the linear functions we iterated in the real numbers.
We also finally developed a Polar Form for a linear function. Instead of always converting A from polar to rectangular in order to enter it into Complex Paint, we can instead use the form A = Rcos(2πT) + i*Rsin(2πT), where T represents the fraction of a full turn we are attempting to rotate our iterated points by (e.g., if we want a 180° rotation, T = 1/2; if we want a 45° rotation, T = 1/8). This eliminates the need to consider degree vs. radian mode for measuring angles.
Your homework over the weekend: continue the analysis we were doing in class on the Complex Paint worksheet, completing Part II up through T = 0.32. Do not go past that point just yet!