PlanetMath: golden ratio

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golden ratio

(Definition)

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

$\displaystyle 1.61803398874989484820\ldots$

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 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 $\phi$ .

$\includegraphics[scale=.6]{golden.eps}$

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

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

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

The value

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

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

$\frac{1}{\phi} = - \phi'$
$1-\phi = \phi'$
$\frac{1}{\phi'} = - \phi$
$1-\phi' = \phi$

and so on. These give us

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

which implies

$\displaystyle \phi^{n-1} + \phi^n = \phi^{n+1}$

"golden ratio" is owned by Mathprof. [ full author list (3) | owner history (2) ]

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Other names:

golden number

Attachments:: values of $\frac{F_n}{F_{n - 1}}$ for (Example) by PrimeFan

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Cross-references: implies, Fibonacci sequence, recurrence relation, roots, diagram, square, premise, lengths, side, ratio, rectangle, number
There are 12 references to this entry.

This is version 12 of golden ratio, born on 2001-11-04, modified 2007-02-15.
Object id is 663, canonical name is GoldenRatio.
Accessed 12489 times total.

Classification:

AMS MSC:	11B39 (Number theory :: Sequences and sets :: Fibonacci and Lucas numbers and polynomials and generalizations)
	40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)

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