Golden ratio

:This article is about the mathematical ratio. For the Aristotelian concept of "golden mean" see Nicomachean Ethics.

Alternate forms

The formula phi = 1 + 1/phi can be expanded recursively to obtain a continued fraction for the golden ratio:

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:phi = = 1 + rac{1}{1 + rac{1}{1 + rac{1}{1 + cdots}}}

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and its reciprocal:

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:phi^{-1} = = 0 + rac{1}{1 + rac{1}{1 + rac{1}{1 + cdots}}}.

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Note that the successive convergents of these continued fractions are ratios of Fibonacci numbers.

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The equation phi^2 = 1 + phi likewise produces the continued square root form:

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:phi = sqrt{1 + sqrt{1 + sqrt{1 + sqrt{1 + cdots}}}}.

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Also

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:phi=2cos(pi/5)=2cos 36^circ.,

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which is a consequence of the fact that the length of the diagonal of a regular pentagon is φ times the length of its side.

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