Definition:Golden Mean

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.

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.

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.