golden ratio
The “Golden Ratio^{}”, or $\varphi $, has the value
$$1.61803398874989484820\mathrm{\dots}$$ 
This number gets its rather illustrious name from the fact that the Greeks thought that a rectangle with ratio of side lengths equal to $\varphi $ 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 widthtolength ratio of $\varphi $.
Above: The golden rectangle; $l/w=\varphi $.
$\varphi $ 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 ${\varphi}^{\prime}$. $\varphi $ and ${\varphi}^{\prime}$ are the two roots of the recurrence relation given by the Fibonacci sequence^{}. The following hold for $\varphi $ and ${\varphi}^{\prime}$ :

•
$\frac{1}{\varphi}={\varphi}^{\prime}$

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$1\varphi ={\varphi}^{\prime}$

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$\frac{1}{{\varphi}^{\prime}}=\varphi $

•
$1{\varphi}^{\prime}=\varphi $
and so on. These give us
$${\varphi}^{1}+{\varphi}^{0}={\varphi}^{1}$$ 
which implies
$${\varphi}^{n1}+{\varphi}^{n}={\varphi}^{n+1}$$ 
Title  golden ratio 

Canonical name  GoldenRatio 
Date of creation  20130322 11:56:02 
Last modified on  20130322 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 