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 degrees, or 2.4000 radians.

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

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 $φ 2$ 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 experssion 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,

$t=2\pi \,$

$g\approx 2\pi \times 0.381966 \,$

$g\approx 2.4000 \,$

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

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Category: Elementary geometry

Golden angle

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Derivation

Golden angle in nature

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