# golden ratio

The “Golden Ratio”, or $\phi$, has the value

 $1.61803398874989484820\ldots$

This number gets its rather illustrious name from the fact that the Greeks thought that a rectangle with ratio of side lengths equal to $\phi$ was the most pleasing to the eye, and much of classical Greek architecture is based on this premise. In , an aesthetically pleasing aspect of a rectangle with this ratio, from a mathematical viewpoint, is that if we embed and remove a $w\times w$ square in the below diagram, the remaining rectangle also has a width-to-length ratio of $\phi$.

Above: The golden rectangle; $l/w=\phi$.

$\phi$ has plenty of interesting mathematical , however. Its value is exactly

 $\frac{1+\sqrt{5}}{2}$

The value

 $\frac{1-\sqrt{5}}{2}$

is often called $\phi^{\prime}$. $\phi$ and $\phi^{\prime}$ are the two roots of the recurrence relation given by the Fibonacci sequence. The following hold for $\phi$ and $\phi^{\prime}$ :

• $\frac{1}{\phi}=-\phi^{\prime}$

• $1-\phi=\phi^{\prime}$

• $\frac{1}{\phi^{\prime}}=-\phi$

• $1-\phi^{\prime}=\phi$

and so on. These give us

 $\phi^{-1}+\phi^{0}=\phi^{1}$

which implies

 $\phi^{n-1}+\phi^{n}=\phi^{n+1}$
Title golden ratio GoldenRatio 2013-03-22 11:56:02 2013-03-22 11:56:02 Mathprof (13753) Mathprof (13753) 17 Mathprof (13753) Definition msc 40A05 msc 11B39 golden number ProportionEquation ConstructionOfCentralProportion DerivationOfPlasticNumber