We did some more work with Complex Paint today, in the hopes of “solving” Az+B. To make this easier, we derived the “Polar-Linear” form of this formula, where instead of referring to A in rectangular coordinates, we refer to it in a modified form of polar coordinates, where we use R and T, the fraction of a full turn that the angle of polar form refers to. This allowed us to make the following observations:
- B has no impact on the type of fixed point we get, just its location
- If R < 1, the fixed point is an attractor
- If R = 1, the fixed point is neutral
- If R > 1, the fixed point is a repeller
- The denominator of T (in lowest terms) tells us how many spokes we get in a pattern.
The only thing we haven’t figured out yet is where spirals come from, and why spirals turn into spokes when we use values of R closer to 1. We’ll explore that on Monday.