golden ratioGoldenRatio2001-11-04 07:11:512007-02-15 17:41:17Definition\usepackage{amssymb}
\usepackage{amsmath}
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\usepackage{psfrag}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 \PMlinkescapetext{addition}, 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 $\phi$.
\begin{center}
\psfrag{l}{$\;\;\;\;\;\;\;\;l$}
\psfrag{w}{$\!\!\!w$}
\includegraphics[scale=.6]{golden.eps}
{\tiny Above: The golden rectangle; $l/w = \phi$. }
\end{center}
$\phi$ has plenty of interesting mathematical \PMlinkescapetext{properties}, however. Its value is exactly
$$ \frac{1+\sqrt{5}}{2} $$
The value
$$ \frac{1-\sqrt{5}}{2} $$
is often called $\phi'$. $\phi$ and $\phi'$ are the two roots of the recurrence relation given by the Fibonacci sequence. The following \PMlinkescapetext{identities} hold for $\phi$ and $\phi'$ :
\begin{itemize}
\item $\frac{1}{\phi} = - \phi' $
\item $1-\phi = \phi'$
\item $\frac{1}{\phi'} = - \phi $
\item $1-\phi' = \phi $
\end{itemize}
and so on. These give us
$$ \phi^{-1} + \phi^0 = \phi^{1} $$
which implies
$$ \phi^{n-1} + \phi^n = \phi^{n+1} $$