Editing Golden angle

{{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>

== Derivation ==
The golden ratio is equal to ''&phi;''&nbsp;=&nbsp;''a''/''b'' given the conditions above.

Let ''&fnof;'' be the fraction of the circumference subtended by the golden angle, or equivalently, the golden angle divided by the angular measurement of the circle.

:<math> f = \frac{b}{a+b} = \frac{1}{1+\varphi}.</math>

But since

: <math>{1+\varphi} = \varphi^2,</math>

it follows that

:<math> f = \frac{1}{\varphi^2} </math>

This is equivalent to saying that ''&phi;''<sup>&nbsp;2</sup> golden angles can fit in a circle.

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

:<math>f \approx 0.381966. \,</math>

The golden angle ''g'' can therefore be numerically approximated in [[Degree (angle)|degrees]] as:

:<math>g \approx 360 \times 0.381966 \approx 137.508^\circ,\,</math>

or in radians as :

:<math> g \approx 2\pi \times 0.381966 \approx 2.39996. \,</math>

== Golden angle in nature ==
[[File:phyllotaxis_golden_angle.svg|thumb|upright|The angle between successive florets in some flowers is the golden angle.]]
[[File:Sunflower seed pattern animation.gif|thumb|upright|Animation simulating the spawning of sunflower seeds from a central meristem where the next seed is oriented one golden angle away from the previous seed.]]
The golden angle plays a significant role in the theory of [[phyllotaxis]]; for example, the golden angle is the angle separating the [[floret]]s on a [[sunflower]].<ref>{{cite web
| url        = https://news.mit.edu/2012/sunflower-concentrated-solar-0111
| title      = Here comes the sun
| author     = Jennifer Chu
| work       = MIT News
| date       = 2011-01-12
| access-date = 2016-04-22
}}</ref> Analysis of the pattern shows that it is highly sensitive to the angle separating the individual [[primordia]], with the [[Fibonacci]] angle giving the [[parastichy]] with optimal packing density.<ref>{{Cite journal|last=Ridley|first=J.N.|date=February 1982|title=Packing efficiency in sunflower heads|journal=Mathematical Biosciences|language=en|volume=58|issue=1|pages=129–139|doi=10.1016/0025-5564(82)90056-6}}</ref>

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.<ref>{{Cite journal|last1=Pennybacker|first1=Matthew|last2=Newell|first2=Alan C.|date=2013-06-13|title=Phyllotaxis, Pushed Pattern-Forming Fronts, and Optimal Packing|url=https://www.math.arizona.edu/~anewell/publications/196newell.pdf|journal=Physical Review Letters|language=en|volume=110|issue=24|pages=248104|doi=10.1103/PhysRevLett.110.248104|pmid=25165965|arxiv=1301.4190 |bibcode=2013PhRvL.110x8104P |issn=0031-9007|doi-access=free}}</ref><ref>{{Cite web|title=Sunflowers and Fibonacci: Models of Efficiency|url=https://thatsmaths.com/2014/06/05/sunflowers-and-fibonacci-models-of-efficiency/|date=2014-06-05|website=ThatsMaths|language=en|access-date=2020-05-23}}</ref>

==See also==
*[[137 (number)#Psychology and mysticism|137 (number)]]
*[[138 (number)#In mysticism|138 (number)]]
*[[Golden ratio]]
*[[Fibonacci sequence ]]

== References ==
{{Reflist}}
{{refbegin}}
*{{Cite journal
  | last =Vogel
  | first =H
  | title =A better way to construct the sunflower head
  | journal =Mathematical Biosciences
  | issue =3–4
  | pages =179–189
  | year =1979
  | doi =10.1016/0025-5564(79)90080-4
  | volume =44
}}
*{{cite book
  | last =Prusinkiewicz
  | first =Przemysław
  | author-link =Przemysław Prusinkiewicz
  | author2 =Lindenmayer, Aristid
  | author-link2 =Aristid Lindenmayer
  | title =The Algorithmic Beauty of Plants
  | publisher =Springer-Verlag
  | date =1990
  | pages =[https://archive.org/details/algorithmicbeaut0000prus/page/101 101&ndash;107]
  | url =https://archive.org/details/algorithmicbeaut0000prus/page/101
  | isbn =978-0-387-97297-8
  }}
{{refend}}

== External links ==
{{commons category|Golden angle}}
* [http://mathworld.wolfram.com/GoldenAngle.html Golden Angle] at [[MathWorld]]

{{Metallic ratios}}

[[Category:Golden ratio]]
[[Category:Angle]]
[[Category:Mathematical constants]]
[[Category:Real transcendental numbers]]