Editing Golden angle (section)

{{Short description|Angle created by applying the golden ratio to a circle}}
{{for|the butterfly|Abaratha ransonnetii{{!}}''Abaratha ransonnetii''}}

[[File:Golden Angle.svg|right|thumb|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 [[angle]]s created by sectioning the circumference of a circle according to the [[golden ratio]]; that is, into two [[Arc (geometry)|arc]]s 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

:<math> \frac{a + b}{a} = \frac{a}{b} </math>

The golden angle is then the angle [[subtend]]ed by the smaller arc of length ''b''. It measures approximately {{val|137.5077640500378546463487}}...°  {{OEIS2C|id=A096627}} or in [[radian]]s {{val|2.39996322972865332}}... {{OEIS2C|id=A131988}}.

The name comes from the golden angle's connection to the [[golden ratio]] ''&phi;''; the exact value of the golden angle is

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

or

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

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

As its [[sine]] and [[cosine]] are [[Transcendental number|transcendental numbers]], the golden angle cannot be [[Straightedge and compass construction|constructed using a straightedge and compass]].<ref>{{Cite arXiv|last=Freitas|first=Pedro J.|date=2021-01-25|title=The Golden Angle is not Constructible|class=math.HO |eprint=2101.10818v1}}</ref>