Monday, September 23, 2013

Golden Ratio - 1.6180339887...

This article is about the number. For the pop music album, see The Golden Ratio (album). For calendar dates, see Golden number (time).
Line segments in the golden ratio
A golden rectangle with longer side aand shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship  \frac{a+b}{a} = \frac{a}{b} \equiv \varphi.
In mathematics and the arts, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to their maximum. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b witha > b,
\frac{a+b}{a} = \frac{a}{b} \ \stackrel{\text{def}}{=}\ \varphi,
where the Greek letter phi (\varphi) represents the golden ratio. Its value is:
\varphi = \frac{1+\sqrt{5}}{2} = 1.61803\,39887\ldots.
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