Golden Angle
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Golden Angle
The golden angle is the angle subtended by the smaller (red) arc when two arcs that make up a circle are in the golden ratio

In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two arcs 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, 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

The golden angle is then the angle subtended by the smaller arc of length b. It measures approximately 137.5077640500378546463487 ...° or in radians 2.39996322972865332 ... .

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

or

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.

But since

it follows that

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

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

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

or in radians as :

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; for example, the golden angle is the angle separating the florets on a sunflower.[1]

References

  1. ^ Jennifer Chu (2011-01-12). "Here comes the sun". MIT News. Retrieved .

External links


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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