Golden angle

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 2.4000 radians.

The name comes from the golden angle's connection to the golden ratio ($\phi$), 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 $\phi ^{2}$ 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 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,

$t=2\pi \,$

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

$g\approx 2.4000\,$

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.