# What is phi, how was it discovered and are its uses?

## A calculator app allowed for the use of $e , \pi \mathmr{and} \phi$, but I wasn't sure what $\phi$ is mathematically. I think it had a value of roughly $1.6$.

Jun 30, 2017

A few thoughts...

#### Explanation:

$\phi = \frac{1}{2} + \frac{\sqrt{5}}{2} \approx 1.6180339887$ is known as the Golden Ratio.

It was known and studied by Euclid (approx 3rd or 4th century BCE), basically for many geometric properties...

It has many interesting properties, of which here are a few...

The Fibonacci sequence can be defined recursively as:

${F}_{0} = 0$

${F}_{1} = 1$

${F}_{n + 2} = {F}_{n} + {F}_{n + 1}$

It starts:

$0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , 233 , 377 , 610 , 987 , \ldots$

The ratio between successive terms tends to $\phi$. That is:

${\lim}_{n \to \infty} {F}_{n + 1} / {F}_{n} = \phi$

In fact the general term of the Fibonacci sequence is given by the formula:

${F}_{n} = \frac{{\phi}^{n} - {\left(- \phi\right)}^{- n}}{\sqrt{5}}$

A rectangle with sides in ratio $\phi : 1$ is called a Golden Rectangle. If a square of maximal size is removed from one end of a golden rectangle then the remaining rectangle is a golden rectangle.

This is related to both the limiting ratio of the Fibonacci sequence and the fact that:

phi = [1;bar(1)] = 1+1/(1+1/(1+1/(1+1/(1+1/(1+...)))))

which is the most slowly converging standard continued fraction.

If you place three golden rectangles symmetrically perpendicular to one another in three dimensional space, then the twelve corners form the vertices of a regular icosahedron. Hence we can calculate the surface area and volume of a regular icosahedron of given radius. See https://socratic.org/s/aFZyTQfn

An isosceles triangle with sides in ratio $\phi : \phi : 1$ has base angles $\frac{2 \pi}{5}$ and apex angle $\frac{\pi}{5}$. This allows us to calculate exact algebraic formulae for $\sin \left(\frac{\pi}{10}\right)$, $\cos \left(\frac{\pi}{10}\right)$ and ultimately for any multiple of $\frac{\pi}{60}$ (${3}^{\circ}$). See https://socratic.org/s/aFZztx8s