Ingeometry, thegolden angle is the smaller of the twoangles created by sectioning the circumference of a circle according to thegolden ratio; that is, into twoarcs such that the ratio of the length of the smaller arc to the length of the larger arc is the same as the ratio of the length of the larger arc to the full circumference of the circle.

Algebraically, leta+b be the circumference of acircle, divided into a longer arc of lengtha and a smaller arc of lengthb such that

${\displaystyle \frac{a+b}{a}=\frac{a}{b}}$

The golden angle is then the anglesubtended by the smaller arc of lengthb. It measures approximately137.5077640500378546463487...°OEIS: A096627 or inradians2.39996322972865332...OEIS: A131988.

The name comes from the golden angle's connection to thegolden ratioφ; the exact value of the golden angle is

${\displaystyle 360\left(1-\frac{1}{\phi }\right)=360(2-\phi )=\frac{360}{{\phi }^2}=180(3-\sqrt{5})\text{ degrees}}$

or

${\displaystyle 2\pi \left(1-\frac{1}{\phi }\right)=2\pi (2-\phi )=\frac{2\pi }{{\phi }^2}=\pi (3-\sqrt{5})\text{ radians},}$

where the equivalences follow from well-known algebraic properties of the golden ratio.

As itssine andcosine aretranscendental numbers, the golden angle cannot beconstructed using a straightedge and compass.^[1]

The golden ratio is equal toφ = a/b given the conditions above.

Letƒ be the fraction of the circumference subtended by the golden angle, or equivalently, the golden angle divided by the angular measurement of the circle.

${\displaystyle f=\frac{b}{a+b}=\frac{1}{1+\phi }.}$

But since

${\displaystyle 1+\phi ={\phi }^2,}$

it follows that

${\displaystyle f=\frac{1}{{\phi }^2}}$

This is equivalent to saying thatφ ² golden angles can fit in a circle.

The fraction of a circle occupied by the golden angle is therefore

${\displaystyle f\approx \mathrm{0.381966.}}$

The golden angleg can therefore be numerically approximated indegrees as:

${\displaystyle g\approx 360\times 0.381966\approx {137.508}^{\circ },}$

or in radians as :

${\displaystyle g\approx 2\pi \times 0.381966\approx \mathrm{2.39996.}}$

The golden angle plays a significant role in the theory ofphyllotaxis; for example, the golden angle is the angle separating theflorets on asunflower.^[2] Analysis of the pattern shows that it is highly sensitive to the angle separating the individualprimordia, with theFibonacci angle giving theparastichy with optimal packing density.^[3]

Mathematical modelling of a plausible physical mechanism for floret development has shown the pattern arising spontaneously from the solution of a nonlinear partial differential equation on a plane.^[4][5]

• 137 (number)
• 138 (number)
• Golden ratio
• Fibonacci sequence

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