Fractals & Chaos Recap for 12/21

After the weird results we observed yesterday with Phi and the golden ratio, and its ubiquity in nature, we doubled-down by showing that we can use Phi to derive an explicit formula for the Fibonacci Sequence. Yes, the most famous recursively defined set of integers can be derived explicitly using an irrational number. Wild. If you missed it, check out this video from Mathologer.

I also demonstrated another Fractal-generating software: IFS Construction Kit. Selecting “rotation” mode in this software allows you to define a piecewise-linear function in complex numbers, one that allows a greater amount of flexibility by using different dilation or rotation factors for the real and imaginary components of the complex number you are iterating. Use the default settings when the software opens up, and you get a very familiar image…

This break, please think about the back of Complex Paint Worksheet 1. It is a summary of the work we’ve been doing with complex linear functions, and completing this will allow us to move on to the final stage of the course: iterations of complex quadratics.