The GoldenRatio is
* the ratio (sqrt(5)+1)/2 : 1, which is approximately 1.61803398874989484820458683436564:1
* the limit of the ratio of successive terms of the FibonacciSequence
* the number that is least approximable by rationals
* found everywhere in nature, because of being difficult to approximate in rationals
* easily shown to be irrational
* the ratios between various segments of a pentagram
* the solution to the following question:
		 :	"What is the ratio between two quantities such that the whole is to the larger as the larger is to the smaller?"
		 :	In symbols, (a+b):a = a:b.
	 :	This has various simple geometrical interpretations, which explains why the OldGreeks liked it.
* the square root of its increment (g^2 = g + 1)
* the inverse of its decrement (1/g = g - 1)

It is often called "phi" or "tau", depending on which side of the pond you are on. Note that 1/phi = phi-1, And there's this really nice series to o with integral powers of phi.
* ''Can someone interpret this comment about the series for me? Thanks''
** Perhaps it's that phi = sum phi^(-n) for n>=1

A rectangle that has sides in the GoldenRatio was in the era before print and TV felt to be more pleasing to the eye than other ratios. The studies, however, generally compared 1:1 with 1:phi and 1:2, and people generally preferred the intermediate ratio. Later studies show that people can't really choose between 100:150, 100:155, 1:160, 1:165 and 1:170. The page at http://plus.maths.org/issue22/features/golden/ says a lot more about this.

Far more information available at Wikipedia

* http://en.wikipedia.org/wiki/Golden_ratio

and any Google search will turn up more than you can read in a year.
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''I find that 1:sqrt(2) is pleasing, simply because I am accustomed to metric-sized paper. But when putting a JFrame on the screen, I always centre it horizontally and move it up the screen vertically by the golden ratio.''

 JFrame f = new MyJFrame();
 // initialize the content of f
 f.pack();
 Dimension d = Toolkit.getDefaultToolkit().getScreenSize();
 f.setLocation((d.width-f.getWidth())/2,
     (d.height-f.getHeight()) * (1000-618) / 1000);

SmellsLikeJava

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''Phi is also the ratio of all voters to "No" voters, and of "No" voters to "Yes" voters in the recent Dutch referendum on the proposed EU Constitutional Treaty.''
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More pleasing, in the same sense that different musical modes instill different emotions.
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CategoryMath