Definition:Golden Mean

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

(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,
 * $$\phi = 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}.$$

The convergents to $$\phi$$ are given by the ratios of consecutive Fibonacci numbers.