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In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden section; that is, into two arcs such that the ratio of the length of the larger arc to the length of the smaller arc is the same as the ratio of the full circumference to the length of the larger arc.

Algebraically, let a+b be the circumference of a circle, divided into a longer arc of length a and a smaller arc of length b such that

\( \frac{a + b}{a} = \frac{a}{b}. \)

The golden angle is then the angle subtended by the smaller arc of length b. It measures approximately 137.508°, or about 2.39996 radians.

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

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

or

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

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

Derivation

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

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

\( f = \frac{b}{a+b} = \frac{1}{1+\varphi}. \)

But since

\( {1+\varphi} = \varphi^2, \)

it follows that

\( f = \frac{1}{\varphi^2} \)

This is equivalent to saying that φ2 golden angles can fit in a circle.

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

\( f \approx 0.381966 \,

The golden angle g can therefore be numerically approximated in degrees as:

\( g \approx 360 \times 0.381966 \approx 137.508^\circ,\, \)

or in radians as:

\( g \approx 2\pi \times 0.381966 \approx 2.39996. \, \)

Golden angle in nature
The angle between successive florets in some flowers is the golden angle.

The golden angle plays a significant role in the theory of phyllotaxis. Perhaps most notably, the golden angle is the angle separating the florets on a sunflower.
References

Vogel, H (1979). "A better way to construct the sunflower head". Mathematical Biosciences 44 (44): 179–189. doi:10.1016/0025-5564(79)90080-4.
Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0387972978. http://algorithmicbotany.org/papers/#webdocs.

See also

Fermat's spiral

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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