The Pentagon Tells it all. The Golden Ratio and Fibonacci Numbers

The golden cut divides a line so that the ratio between the long and the short leg equals the ration between the sum of the two legs and the long leg. Surprisingly, the value for this ratio can be approached by dividing increasing neighboring Fibonacci numbers, numbers where the next in row is the sum of the two previous figures. Here we show how Fibonacci numbers occur as you construct pentagons where the side length equals the diagonal in the previous pentagon, and how such pentagons can be used to prove that dividing increasing Fibonacci numbers oscillate around the golden ratio.