Golden Ratio Approximations

Nice approximations for the golden ratio are given by

			(1)
			(2)

the last of which is due to W. van Doorn (pers. comm., Jul. 18, 2006) and which are accurate to and , respectively. An even more amazing approximation uses Catalan's constant and the Feigenbaum constant is given by

(3)

which is accurate to within (D. Ross, cited in Pegg 2005).

A curious (although not particularly useful) approximation due to D. Barron is given by

(4)

where is Catalan's constant and is the Euler-Mascheroni constant, which is good to two digits.