What is it:

Originally called the extreme and mean ratio it makes one of its earliest appearances in Euclid. Usually denoted   (pronounced "fee", but usually spelled "phi"), the golden ratio can be derived geometrically by finding a subdivision of a line into two pieces such that the ratio of the whole to the longe.r is the ratio of the longer to the shorter, or .

Some History:

It is almost certainly the case that    is one of the first numbers to have been proven to be irrational, with   the other likely candidate. The existence of irrational numbers provided some philosophical problems for the Pythagoreans, and numerous (largely unverifiable) stories indicate that the existence of irrational numbers was something of a secret, one that members of the sect may have even murdered over [Cho80, Liv02]. 

The first use of a more sensational name than the extreme and mean ratio was by Luca Pacioli who renamed it the divine proportion for reasons that would only be convincing to a rennaisance or medieval mystic. Da Vinci illustrated a manuscript published in 1509 for Pacioli with the title De Devina Proportione late in his life, but there is little evidence to indicate that he ever used the ratio in his work outside of the realm of geometry [Liv02, Mar05]. Use of the name golden ratio appears to date from the 19th century (1835 is the earliest date I'm aware of, see [Mar05]), as well as a significant increase in the number of erroneous claims about its role in nature and the arts.

Some Standards:

Markowsky suggests a very useful acceptance range for investigation of claims about the golden ratio, namely that we should reject any claim of use of the golden ratio if the ratio falls outside the range of 1.58 to 1.66 (corresponding to ). This does not however mean that any ratio which does fall into the range constitutes convincing evidence, merely that the claim is worthy of investigation.

Some Useful Facts:

When measuring ratios one must exercise care not to bias the result. Suppose we have two numbers 0<m<M. Many authors will use as evidence of the presence of a golden ratio a proportion   interchangably with   and compare with . The problem with this is that   is always closer to the golden ratio than   and shows significantly less variability [mar05, Put95]. An equally problematic fact is that the expected value (the number you would expect to get on average over a large number of samples) of the ratio   with m varying uniformly over the interval [(m+M)/4,(m+M)/2] (and there are lots of contexts in which such a range of values forms a reasonable constraint on the class of objects being investigated) is 4ln(3/2)-1  or approximately 0.6219, which is within 0.6% of the (reciprocal of the) golden ratio [Put95]. The point is, we need more than averages and rough measurements to support a claim. 

References:

[Cho80] James R. Choike, The pentagram and the discovery of an irrational number, The Two-Year college Mathematics Journal 11 (1980), no. 5, 312–316.

[Liv02] Mario Livio, The golden ratio, Broadway Books, New York, 2002, The story of phi, the world’s most astonishing number. MR MR1938220 (2003k:11025)

[Mar05] George Markowsky, The golden ratio, Notices of the AMS 52 (2005), no. 3, 344–347.

[Put95] John F. Putz, The golden section and the piano sonatas of mozart, Mathematics Magazine 68 (1995), no. 4, 275–282.

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