Fractals & Chaos Recap for 12/16

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.

One of the quickest things we noticed is that the value of A seems to be the major factor in the type of fixed point we get, while B only seems to affect where the fixed point is (this may not 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). Furthermore, sometimes, the attracting pattern moved in a straight line (essentially the direct pattern we recall from the reals), and sometimes it looped around the fixed point (echoing the alternating pattern we saw before). But sometimes that alternating pattern produces clear “spokes” like with 4, 5, 6, and 8 from the sheet, and sometimes it produced more ambiguous spiral arms (like 7).

Understanding why this happens is what we’ll look at next.

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