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
Canonical name	GoldenRatio
Date of creation	2013-03-22 11:56:02
Last modified on	2013-03-22 11:56:02
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	17
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 40A05
Classification	msc 11B39
Synonym	golden number
Related topic	ProportionEquation
Related topic	ConstructionOfCentralProportion
Related topic	DerivationOfPlasticNumber