Nice approximations for the golden ratio ϕ are given by ϕ | ≈ | sqrt((5π)/6) | ≈ | (7π)/(5e), the last of which is due to W. van Doorn 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 α is given by ϕ≈ - [α + (7/8)^K], which is accurate to within 1.4×10^(-8) (D. Ross, cited in Pegg 2005).
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