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 Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...

s created by sectioning the circumference of a circle according to the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

; 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 const ...

, 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 subtended 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 tha ...

s 2.39996322972865332 ... . The name comes from the golden angle's connection to the

''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 In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes o ...

, 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 In botany, phyllotaxis () or phyllotaxy is the arrangement of leaf, leaves on a plant stem. Phyllotactic spirals form a distinctive class of patterns in nature. Leaf arrangement The basic leaf#Arrangement on the stem, arrangements of leaves ...

; 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), as ...

. 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 Parastichy, in phyllotaxy, is the spiral pattern of particular plant organs on some plants, such as areoles on cacti stems, florets in sunflower heads and scales in pine cones. These spirals involve the insertion of a single primordium. See al ...

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 ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...

{{Metallic ratios Elementary geometry Golden ratio Angle Mathematical constants

Derivation

Golden angle in nature

See also

References

External links