Definition:Golden Mean

Definition
The golden mean is the unique positive real number $\phi$ satisfying
 * $\phi = \dfrac 1 {\phi - 1}$

It is also known as the golden ratio or golden section.

Euclid called it the extreme and mean ratio.

A geometric interpretation of this condition is as follows: if we draw a rectangle of sidelengths $\phi$ and $1$, and remove from this a square of sidelength $q$, then the sides of the remaining rectangle have the same relation as the sides of the original one.

Equivalently, $\phi$ is the real number
 * $\phi = \dfrac{1 + \sqrt 5} 2$

This follows from the Quadratic Formula.

Its approximate value is:
 * $\phi \approx 1.61803\ 39887 \ldots$

Note also that:
 * $1 - \phi = - \dfrac 1 \phi$

which follows directly from taking reciprocals of the definition.

This number $1 - \phi$ is often denoted $\phi'$ or $\hat \phi$:
 * $\phi' = \dfrac {1 - \sqrt 5} 2 \approx -0.61803\ 39887 \ldots$

Continued Fraction Expansion
The golden mean has the simplest possible continued fraction expansion, namely $[1, 1, 1, 1, \ldots]$. That is:
 * $\phi = 1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots}}}$

As demonstrated here, the convergents to $\phi$ are given by the ratios of consecutive Fibonacci numbers.

Historical Note
The symbol $\phi$ originates from the Greek artist Phidias who was said to have used it as a basis for calculating proportions in his sculpture.