This is the Golden Section, which equals 2/(sqrt(5) -1). The ratio of two succesive numbers of the Fibonacci sequence (F(n)/F(n-1)) tends to the Golden Section.
This is one of those magic numbers, like pi, which occur at odd places in nature. An article in the Guardian on the special properties of this strange number.
Update: Corrected. It is 2/(sqrt(5)-1), not sqrt(5)-1