# زاوية ذهبية

في الهندسة، الزاوية الذهبية Golden angle هي الزاوية التي نصنعها عندما نقسم محيط الدائرة إلى قطاع a و قطاع صغير b بحيث يتحقق :

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
${\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}}$

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

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

or

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

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

As its sine and cosine are transcendental numbers, the golden angle cannot be constructed using a straightedge and compass.[1]

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## الاشتقاق

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.

${\displaystyle f={\frac {b}{a+b}}={\frac {1}{1+\varphi }}.}$

But since

${\displaystyle {1+\varphi }=\varphi ^{2},}$

it follows that

${\displaystyle f={\frac {1}{\varphi ^{2}}}}$

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

${\displaystyle f\approx 0.381966.\,}$

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

${\displaystyle g\approx 360\times 0.381966\approx 137.508^{\circ },\,}$

${\displaystyle g\approx 2\pi \times 0.381966\approx 2.39996.\,}$

## الزاوية الذهبية في الطبيعة

الزاوية بين بتيلات في بعض الأزهار هي الزاوية الذهبية.

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.[2] 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.[3]

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.[4][5]

## المراجع

1. ^ Freitas, Pedro J. (2021-01-25). "The Golden Angle is not Constructible" (in الإنجليزية). arXiv:2101.10818v1. Bibcode:2021arXiv210110818F – via arXiv. Cite journal requires |journal= (help)
2. ^ Jennifer Chu (2011-01-12). "Here comes the sun". MIT News. Retrieved 2016-04-22.
3. ^ Ridley, J.N. (February 1982). "Packing efficiency in sunflower heads". Mathematical Biosciences (in الإنجليزية). 58 (1): 129–139. doi:10.1016/0025-5564(82)90056-6.
4. ^ Pennybacker, Matthew; Newell, Alan C. (2013-06-13). "Phyllotaxis, Pushed Pattern-Forming Fronts, and Optimal Packing" (PDF). Physical Review Letters (in الإنجليزية). 110 (24): 248104. doi:10.1103/PhysRevLett.110.248104. ISSN 0031-9007. PMID 25165965.
5. ^ "Sunflowers and Fibonacci: Models of Efficiency". ThatsMaths (in الإنجليزية). 2014-06-05. Retrieved 2020-05-23.