geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the golden angle is the smaller of the two

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...

s 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 A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

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

\frac = \frac

The golden angle is then the angle

subtend In geometry, an angle is subtended by an arc, line segment or any other section of a curve when its two rays pass through the endpoints of that arc, line segment or curve section. Conversely, the arc, line segment or curve section confined wi ...

ed by the smaller arc of length ''b''. It measures approximately 137.5077640500378546463487 ...° or in

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...

s 2.39996322972865332 ... . The name comes from the golden angle's connection to the golden ratio ''φ''; the exact value of the golden angle is :

360\left(1 - \frac\right) = 360(2 - \varphi) = \frac = 180(3 - \sqrt)\text

or :

2\pi \left( 1 - \frac\right) = 2\pi(2 - \varphi) = \frac = \pi(3 - \sqrt)\text,

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.

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. :

f = \frac = \frac.

But since :

= \varphi^2,

it follows that :

f = \frac

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 :

f \approx 0.381966. \,

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

g \approx 360 \times 0.381966 \approx 137.508^\circ,\,

or in radians as : :

g \approx 2\pi \times 0.381966 \approx 2.39996. \,

Golden angle in nature

The golden angle plays a significant role in the theory of phyllotaxis; for example, the golden angle is the angle separating the

floret This glossary of botanical terms is a list of definitions of terms and concepts relevant to botany and plants in general. Terms of plant morphology are included here as well as at the more specific Glossary of plant morphology and Glossary o ...

s on a

sunflower The common sunflower (''Helianthus annuus'') is a large annual forb of the genus ''Helianthus'' grown as a crop for its edible oily seeds. Apart from cooking oil production, it is also used as livestock forage (as a meal or a silage plant), ...

. Analysis of the pattern shows that it is highly sensitive to the angle separating the individual

primordia A primordium (; plural: primordia; synonym: anlage) in embryology, is an organ or tissue in its earliest recognizable stage of development. Cells of the primordium are called primordial cells. A primordium is the simplest set of cells capable o ...

, with the Fibonacci angle giving the parastichy with optimal packing density. 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.

References

* *

External links

Golden Angle
at MathWorld {{Metallic ratios Elementary geometry Golden ratio Angle Mathematical constants

Derivation

Golden angle in nature

See also

References

External links