Golden Ratio Approximations

Nice approximations for the golden ratio phi are given by

phi approx sqrt((5pi)/6)
 approx (7pi)/(5e),

the last of which is due to W. van Doorn (pers. comm., Jul. 18, 2006) and which are accurate to 1.2×10^(-5) and 1.6×10^(-5), respectively. An even more amazing approximation uses Catalan's constant K and the Feigenbaum constant alpha is given by

 phi approx -[alpha+(7/8)^K],

which is accurate to within 1.4×10^(-8) (D. Ross, cited in Pegg 2005).

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

 phi approx 1/2K^(gamma-19/7)pi^(2/7+gamma),

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

See also

Almost Integer, Golden Ratio

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Pegg, E. Jr. "Math Games: Keen Approximations." Feb. 14, 2005.

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Golden Ratio Approximations

Cite this as:

Weisstein, Eric W. "Golden Ratio Approximations." From MathWorld--A Wolfram Web Resource.

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