Golden angle

In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden section; that is, into two arcs such that the ratio of the length of the larger arc to the smaller is the same as the ratio of the full circumference to the larger.

Algebraically, let "c" be the circumference of a circle, divided into a longer arc of length "a" and a smaller arc of length "b" such that :$c=a+b\; ,$

and

:$frac\{c\}\{a\}\; =\; frac\{a\}\{b\}.$

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

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

: $360left(1\; -\; frac\{1\}\{varphi\}\; ight)\; =\; 360(2\; -\; varphi)\; =\; frac\{360\}\{varphi^2\}\; ext\{\; degrees\}$

or

: $2pi\; left(\; 1\; -\; frac\{1\}\{varphi\}\; ight)\; =\; 2pi(2\; -\; varphi)\; =\; frac\{2pi\}\{varphi^2\}\; ext\{\; radians\},$

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.

:$f\; =\; frac\{b\}\{c\}\; =\; frac\{b\}\{b+a\}\; =\; frac\{1\}\{1+varphi\}.$

But since

: $\{1+varphi\}\; =\; varphi^2,$

it follows that

:$f\; =\; frac\{1\}\{varphi^2\}$

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:

:$g\; approx\; 360\; imes\; 0.381966\; approx\; 137.51^circ,,$

or in radians as:

:$g\; approx\; 2pi\; imes\; 0.381966\; approx\; 2.399963.\; ,$

Golden angle in nature

The golden angle plays a significant role in the theory of phyllotaxis. Perhaps most notably, the golden angle is the angle separating the florets on a sunflower.

References

*Citation
last =Vogel
first =H
title =A better way to construct the sunflower head
journal =Mathematical Biosciences
issue =44
pages =179–189
year =1979
*cite book
last =Prusinkiewicz
first =Przemyslaw
authorlink =Przemyslaw Prusinkiewicz
coauthors =Lindenmayer, Aristid
title =The Algorithmic Beauty of Plants
publisher =Springer-Verlag
date =1990
location =
pages =101–107
url =http://algorithmicbotany.org/papers/#webdocs
doi =
id = ISBN 978-0387972978

ee also

*Fermat's spiral