3.1.4 through the first half of page 72, due October 10

One the "proof" involving the semicircle, I understood the actually math of each step and where they started from but I don't think I quite got the conclusion that they were drawing from it all. I also still don't exactly understand how the fibbonacci numbers are related to the golden ratio, it said that they're related but I never saw how. I'm interested to how the fibbonacci numbers are useful. I've seen them before and I know what they are, but it is such a simple pattern that I've never really seen how they would be useful.