The Golden Ratio

The golden ratio is an irrational number. It is related to many functions; the most notable of them being the Fibonacci Sequence. The golden ratio connects to the Fibonacci series in many different ways. The most striking feature of the relation of the golden ratio and Fibonacci series is that as the Fibonacci series progresses, the ratio between two consecutive terms approaches the Golden Ratio . The golden ratio is represented by \(\phi \), the greek letter phi.

The golden ratio is defined as the positive root of the Polynomial \(x^2 - x -1=0\)

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Using the quadratic formula, \(x = \dfrac {1 + \sqrt 5}{2} = \phi\). The negative root has been omitted.

The golden ratio can be expressed as a continued fraction, as \(\left[ 1,1,1,1\cdots \right] \)

\( = 1+\dfrac { 1 }{ 1+\dfrac { 1 }{ 1+\dfrac { 1 }{ \ddots } } } \)

In geometry, a golden spiral is a logarithmic spiral whose growth factor is \(\phi\), the golden ratio. (Wikipedia)

The Golden Spiral