en>Gilliam: Reverted edits by 165.138.103.28 (talk) to last version by Loraof

2014-12-17T14:30:11Z

Reverted edits by 165.138.103.28 (talk) to last version by Loraof

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en>Wcherowi: /* The isosceles triangle theorem */ added section on partitions (from triangle article)

2014-03-04T22:55:28Z

The isosceles triangle theorem: added section on partitions (from triangle article)

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2014-02-02T09:34:56Z

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{{ for|A kind of [[skipper (butterfly)]]|Caprona ransonnetti}}

[[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 section]]; that is, into two [[Arc (geometry)|arc]]s 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

:<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 137.508°, or about 2.39996 [[radian]]s.

The name comes from the golden angle's connection to the [[golden ratio]] ''φ''; 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.

== Derivation ==
The golden ratio is equal to ''φ'' = ''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 ''φ''<sup> 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:Goldener Schnitt Blattstand.png|thumb|right|300px|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 [[floret]]s on a [[sunflower]].

== References ==
{{refbegin}}
*{{Cite journal
| last =Vogel
| first =H
| title =A better way to construct the sunflower head
| journal =Mathematical Biosciences
| issue =44
| pages =179–189
| year =1979
| doi =10.1016/0025-5564(79)90080-4
| volume =44
}}
*{{cite book
| last =Prusinkiewicz
| first =Przemysław
| authorlink =Przemysław Prusinkiewicz
| coauthors =[[Aristid Lindenmayer|Lindenmayer, Aristid]]
| title =The Algorithmic Beauty of Plants
| publisher =Springer-Verlag
| date =1990
| location =
| pages =101–107
| url =http://algorithmicbotany.org/papers/#webdocs
| doi =
| isbn = 978-0-387-97297-8 }}
{{refend}}

== External links ==
* [http://mathworld.wolfram.com/GoldenAngle.html Golden Angle] at [[MathWorld]]

[[Category:Elementary geometry]]
[[Category:Golden ratio]]
[[Category:Angle]]

Isosceles triangle - Revision history

en>Gilliam: Reverted edits by 165.138.103.28 (talk) to last version by Loraof

en>Wcherowi: /* The isosceles triangle theorem */ added section on partitions (from triangle article)

112.207.1.119: /* See also */