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

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

1 Answer
Jun 30, 2017

A few thoughts...

Explanation:

phi = 1/2+sqrt(5)/2 ~~ 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,...

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

lim_(n->oo) F_(n+1)/F_n = phi

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

F_n = (phi^n - (-phi)^(-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 (2pi)/5 and apex angle pi/5. This allows us to calculate exact algebraic formulae for sin(pi/10), cos(pi/10) and ultimately for any multiple of pi/60 (3^@). See https://socratic.org/s/aFZztx8s