We spent some time reviewing the results from Complex Paint Worksheet 2. We have observed a few things, the most obvious of which are that all Julia Sets have 180-degree rotational symmetry and the only Julia Sets that have x- and y-axis symmetry have a parameter that is a real number. It turns out that both of these observations are provable facts about Julia Sets, which we proceeded to start, and will finish tomorrow.
We saw yesterday that a sequence of geometric transformations can illustrate the same “fixed point attractor” behavior that we’ve seen when iterating a mathematical expression. We explained this by illustrating that such geometric transformations are analogous to iterating a linear function in the Complex number system. Specifically, adding two complex numbers is analogous to translating, and multiplying two complex numbers is analogous to a combined dilation/rotation.
The conventional rectangular form of a complex number a + bi tells us the horizontal and vertical components of the translation achieved by adding the complex number, but this form is not useful for when the number is being used as a multiplication factor. For that, we need the number’s Polar Form. Once finding it, the value of r tells us the dilation factor and the value of θ tell us the rotation factor.
We’ll dig in to this some more tomorrow, but for now please watch this video recapping polar vs rectangular coordinates (which the video refers to as “Cartesian coordinates”) and how to convert between the two.
We wrapped up our conversation about numbers of cycles findable in the Feigenbaum Plot, noting again that for every attracting cycle we find, there is an accompanying “evil twin” repelling cycle. These “evil twin” repelling cycles serve a purpose. As the members of the attracting cycle bifurcate, their patterns branch out further and further. When this pattern of branching reaches one of the values of the repelling cycle – indeed it will reach all members of the repelling cycle simultaneously – the branching stops and the system dissolves into chaos once more. This explains why these spontaneously generating cycles appear in “windows” in the Feigenbaum Plot, with an abrupt end to chaos on the left, and an abrupt resuming of chaos on the right.
We wrapped up today with a recap of the work we’ve done so far, occupying our time exclusively within the Real numbers. The last portion of the course will now branch into the Complex numbers, effectively repeating the analyses we’ve done with iterating functions into the complex plane. Our first stop will be to understand how to represent geometric transformation with operations on complex numbers. We modeled this in class by showing how the repetition of a series of geometric transformations can result in the same “fixed point attracting” behavior as we saw with the real numbers.