After observing yesterday that the type of fixed point we get depends more on A than on B, we focused our efforts on analyzing the behavior of iterations for different values of A. To do so, 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.
We worked on most of the second part of the Complex Paint Worksheet and made a few observations. First, the value of R determines whether the fixed point is attracting or repelling:
- If R < 1, the fixed point is attracting
- If R > 1, the fixed point is repelling
- If R = 1, the fixed point is neutral
This wasn’t overly surprising, as it lines up with what we noticed before about the slope of linear functions as we iterated in the real numbers.
What’s new is that we have a more sophisticated idea of what counts as an “alternating” pattern. If T = 0 (so there is no rotation in our composition of transformations), we could call the pattern “direct.” But let T be anything else and we get either spokes or spirals. The denominator of T appears to govern how many spokes we get, but what else can we find?