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

<math>c=a+b \,</math>

and

<math>\frac{c}{a}=\frac{a}{b}</math>

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 2.4000 radians.

The name comes from the golden angle's connection to the golden ratio (<math>\phi</math>), its numerical equivalent.

Derivation

The golden ratio is defined as <math>\frac{a}{b}</math> 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.

<math>f=\frac{b}{c}</math>

<math>f=\frac{b}{\frac{a^2}{b}}</math>

<math>f=\frac{b^2}{a^2}</math>

<math>f=\frac{1}{\frac{a^2}{b^2}}=</math>

<math>f=\frac{1}{\left (\frac{a}{b} \right)^2}</math>

Hence, we see that

<math>f=\frac{1}{\phi^2}</math>

This is equivalent to saying that <math>\phi^2</math> golden angles can fit in a circle. It can also be shown that

<math>\frac{1}{\phi ^2}=2-\phi</math>

<math>f=2-\phi \,</math>

<math>\phi \approx 1.6180</math>

Therefore,

<math>f=0.381966 \,</math>

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

<math>c=a+b \,</math>

and

<math>\frac{c}{a}=\frac{a}{b} \,</math>

by definition of the golden angle. We get

<math>\frac{a}{c}=\frac{b}{a} \,</math>

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

<math>\frac{a^2}{c}=b \,</math>.

Subtracting b from both sides of the first equation yields

<math>a=c-b \,</math>

We can substitute that in and simplify to get

<math>b=\frac{(c-b)^2}{c} \,</math>

<math>b=\frac{c^2-2bc+b^2}{c} \,</math>

<math>b=\frac{b^2-2bc+c^2}{c} \,</math>

<math>b=\frac{1}{c}\times b^2-2b+c \,</math>

<math>0=\frac{1}{c}\times b^2-3b+c \,</math>

The quadratic formula gives us

<math>b=\frac{3\pm\sqrt{9-4}}{\frac{2}{c}}</math>

We simplify to get

<math>b=\frac{c(3\pm\sqrt{5})}{2}</math>

<math>\frac{b}{c}=\frac{3\pm\sqrt{5}}{2}</math>

<math>f=\frac{3\pm\sqrt{5}}{2}</math>

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

<math>f=\frac{3-\sqrt{5}}{2}</math>

Thus we show again that:

<math>f\approx 0.381966</math>

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.

<math>g=ft \,</math>

In degrees,

<math>t=360^\circ</math>

<math>g\approx 360 \times 0.381966</math>

<math>g\approx 137.51^\circ</math>

In radians,

<math>t=2\pi \,</math>

<math>g\approx 2\pi \times 0.381966 \,</math>

<math>g\approx 2.4000 \,</math>

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.fr:Angle d'or sk:Zlatý uhol zh:黃金角