When I was in high school, I was astounded by the golden ratio and all of its connections. I asked my teachers, and I even asked my first math professor in college. I didn’t really get good answers to my questions. Years later, I pieced together a coherent story of many of the connections that arise with Fibonacci. And the mysterious connection with nature? Well, in nature, things build on previous things like how the Fibonacci relation builds new from old.
Here’s a sketch of my favorite way to understand some of the connections of the golden ratio Phi and the Fibonacci sequence:
I’d LaTeX it up, but I’m too lazy.