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.

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 $\hat \phi$:
 * $\phi' = \dfrac {1 - \sqrt 5} 2 \approx -0.61803\ 39887 \ldots$

Geometrical Interpretation
Let $\Box ADEB$ be a square.

Let $\Box ADFC$ be a rectangle such that:
 * $AC : AD = AD : BC$

where $AC : AD$ denotes the ratio of $AC$ to $AD$.


 * GoldenMean.png

Then if you remove $\Box ADEB$ from $\Box ADFC$, the sides of the remaining rectangle have the same ratio as the sides of the original one.

Thus if $AC = \phi$ and $AD = 1$ we see that this reduces to:


 * $\phi : 1 = 1 : \phi - 1$

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. It is said to produce the most pleasing proportions, and as a consequence many artists have used this ratio in their works.

Euclid called it the extreme and mean ratio:

That is:
 * $A + B : A = A : B$

Also see

 * Equivalence of Definitions of Golden Mean