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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
:
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 (); the exact value of the golden angle is
: degrees or
: radians,
where the equivalences follow from well-known algebraic properties of the golden ratio.

Contents

Derivation

Golden angle in nature

References

See also

Derivation

The golden ratio is equal to given the conditions above.
Let ''f'' 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 ,
:
This is equivalent to saying that 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. Perhaps most notably, the golden angle is the angle separating the florets on a sunflower.

References

★

★ The Algorithmic Beauty of Plants, , Przemyslaw, Prusinkiewicz, Springer-Verlag, , ISBN 978-0387972978

See also

★ Fermat's spiral