Golden ratio

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

Mathematical uses

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The number φ turns up frequently in geometry, in particular in figures involving pentagonal symmetry.

Related Topics:
Geometry - Pentagonal symmetry

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For instance the ratio of a regular pentagon's side and diagonal is equal to φ, and the vertices of a regular icosahedron are located on three orthogonal golden rectangles.

Related Topics:
Icosahedron - Golden rectangle

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The explicit expression for the Fibonacci sequence involves the golden ratio and its conjugate. The limit of ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence) equals the golden ratio; therefore, when a number in the Fibonacci sequence is divided by its preceding number, it approximates φ. e.g., 987/610?1.6180327868852. Because of this ratio, the approximation to φ gets better as the Fibonacci numbers get higher.

Related Topics:
Fibonacci sequence - Limit

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Interestingly enough, if all the approximation errors are added up, they equal φ. Stated mathematically:

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:sum_{n=1}^{infty}|F(n)arphi-F(n+1)| = arphi.

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Furthermore, the successive powers of φ obey the Fibonacci recurrence:

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:φ−2 = 1 − 2φ,

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:φ−1 = φ − 1,

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:φ0 = 1,

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:φ1 = φ,

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:φ2 = φ + 1,

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:φ3 = 2φ + 1,

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:φ4 = 3φ + 2,

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:φ5 = 5φ + 3,

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:φn = F(n)φ + F(n − 1),

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:...

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From a mathematical point of view, the golden ratio is notable for having the simplest continued fraction expansion, and of thereby being the "most irrational number" worst case of Lagrange's approximation theorem. It has been argued this is the reason angles close to the golden ratio often show up in phyllotaxis (the growth of plants). It is also the fundamental unit of the algebraic number field mathbb{Q}(sqrt{5}) and is a Pisot-Vijayaraghavan number.

Related Topics:
Lagrange - Approximation theorem - Phyllotaxis - Pisot-Vijayaraghavan number

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The golden ratio has interesting properties when used as the base of a numeral system: see golden mean base.

Related Topics:
Numeral system - Golden mean base

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