Файл:Golden Angle.svgThe 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
<math> \frac{a + b}{a} = \frac{a}{b} </math>
The golden angle is then the angle subtended by the smaller arc of length b. It measures approximately 137.5077640500378546463487 ...° Шаблон:OEIS2C or in radians 2.39996322972865332 ... Шаблон:OEIS2C.
The name comes from the golden angle's connection to the golden ratioφ; the exact value of the golden angle is
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.
<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 2 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 degrees as:
<math> g \approx 2\pi \times 0.381966 \approx 2.39996. \,</math>
Golden angle in nature
Файл:Phyllotaxis golden angle.svgThe angle between successive florets in some flowers is the golden angle.Файл:Sunflower seed pattern animation.gifAnimation 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 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]
