Golden angle

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In geometry, the golden angle is the angle created by dividing the circumference c of a circle into a section a and a smaller section b such that

c=a+b \,

and

\frac{c}{a}=\frac{a}{b}

and taking the angle of arc subtended by the length of circumference equal to b as the golden angle. It measures approximately 137.51°, or about 2.399963 radians.

The name comes from the golden angle's connection to the golden ratio (φ), its numerical equivalent.

[edit] Derivation

The golden ratio is defined as \frac{a}{b} given the conditions above. This provides an interesting relationship.

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.

f=\frac{b}{c}
f=\frac{b}{\frac{a^2}{b}}
f=\frac{b^2}{a^2}
f=\frac{1}{\frac{a^2}{b^2}}=
f=\frac{1}{\left (\frac{a}{b} \right)^2}

Hence, we see that

f=\frac{1}{\phi^2}

This is equivalent to saying that φ2 golden angles can fit in a circle. It can also be shown that

\frac{1}{\phi ^2}=2-\phi
f=2-\phi \,
\phi \approx 1.6180

Therefore,

f=0.381966 \,

A third expression for f can be derived algebraically, without needing to know phi.

c=a+b \,

and

\frac{c}{a}=\frac{a}{b} \,

by definition of the golden angle. We get

\frac{a}{c}=\frac{b}{a} \,

by taking the reciprocal of both sides of the second equation. Then,

\frac{a^2}{c}=b \,.

Subtracting b from both sides of the first equation yields

a=c-b \,

We can substitute that in and simplify to get

b=\frac{(c-b)^2}{c} \,
b=\frac{c^2-2bc+b^2}{c} \,
b=\frac{b^2-2bc+c^2}{c} \,
b=\frac{1}{c}\times b^2-2b+c \,
0=\frac{1}{c}\times b^2-3b+c \,

The quadratic formula gives us

b=\frac{3\pm\sqrt{9-4}}{\frac{2}{c}}

We simplify to get

b=\frac{c(3\pm\sqrt{5})}{2}
\frac{b}{c}=\frac{3\pm\sqrt{5}}{2}
f=\frac{3\pm\sqrt{5}}{2}

Because \frac{3+\sqrt{5}}{2} is greater than 1, and \frac{b}{c} should be a proper fraction, we choose the other solution.

f=\frac{3-\sqrt{5}}{2}

Thus we show again that:

f\approx 0.381966

Regardless of how we get f, a very simple calculation lets us get the actual measurement of the golden angle.

Let g be the golden angle and t the total angular measurement of the circle.

g=ft \,

In degrees,

t=360^\circ
g\approx 360 \times 0.381966
g\approx 137.51^\circ

In radians,

t=2\pi \,
g\approx 2\pi \times 0.381966 \,
g\approx 2.399963 \,

[edit] 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.

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