Interesting post! Also, I enjoy reading this blog a lot!
About the pi approximation at the end, I did a little research on it and here is what I’ve found:
The last equation can be written as pi = 4*sqrt(phi-1) , which can also be written as 4/pi = sqrt(phi)
According to Wikipedia, this relation is coincidental: http://en.wikipedia.org/wiki/Golden_ratio#Mathematical_pyramids_and_triangles