geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the golden angle is the smaller of the two

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

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 summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

; 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 point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

, 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 subtended (from Latin for "stretched under") by a line segment at an arbitrary vertex is formed by the two rays between the vertex and each endpoint of the segment. For example, a side of a triangle ''subtends'' the op ...

ed by the smaller arc of length ''b''. It measures approximately ...° 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. It is defined such that one radian is the angle subtended at ...

s ... . 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 In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...

and

cosine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...

are

transcendental numbers In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are and . T ...

, 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 ...

s on a

sunflower The common sunflower (''Helianthus annuus'') is a species of large annual forb of the daisy family Asteraceae. The common sunflower is harvested for its edible oily seeds, which are often eaten as a snack food. They are also used in the pr ...

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

Fibonacci Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''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 ''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 ...

{{Metallic ratios Golden ratio Angle Mathematical constants Real transcendental numbers

Derivation

Golden angle in nature

See also

References

External links