Definition:Golden Mean

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

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
 * $$\phi = \frac{1+\sqrt{5}}{2}$$.

This follows from the Quadratic Formula.

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

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

which follows directly from taking reciprocals of the definition.

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

Continued Fraction Expansion
The golden mean has the simplest possible continued fraction expansion, namely $$[1,1,1,1,\dots]$$. 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.