In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the golden angle is the smaller of the two angle

Angle

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s created by sectioning the circumference of a circle according to the golden section; that is, into two arc

Arc (geometry)

In geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of the circumference of a circle...

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 c be the circumference of a circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

, divided into a longer arc of length a and a smaller arc of length b such that


and


The golden angle is then the angle subtended by the smaller arc of length b. It measures approximately 137.51°, or about 2.399963 radian

Radian

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

s.

The name comes from the golden angle's connection to the golden ratio

Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

 φ; the exact value of the golden angle is

or

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 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.


But since

it follows that


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:


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


or in radians as:

Golden angle in nature

The golden angle plays a significant role in the theory of phyllotaxis

Phyllotaxis

In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem .- Pattern structure :...

. Perhaps most notably, the golden angle is the angle separating the florets on a sunflower

Sunflower

Sunflower is an annual plant native to the Americas. It possesses a large inflorescence . The sunflower got its name from its huge, fiery blooms, whose shape and image is often used to depict the sun. The sunflower has a rough, hairy stem, broad, coarsely toothed, rough leaves and circular heads...

.