Is the Golden Ratio's purported aesthetic appeal supported by scientific evidence?
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I found a little discussion of the issue in Russell (2000), where he summarises some of the views of the scientific literature:
Green (1995) argues that "[t]here seems to be, in fact, real psychological effects associated with the golden section, but they are relatively sensitive to careless methodological practices."
Russel (2000) explored the height to width ratio of a large database of famous paintings and did not find any particular support for a preference for the golden-section. Russel suggests that in the real world functional factors tend to constrain ratios.
McManus (1980, PDF) provides an interesting discussion of the methodological issues and the findings of research that has empirically studied aesthetic judgements of simple figures. McManus advocates the use of the paired-comparison method whereby participants make aesthetic judgements on which of series of pairs of objects is more aesthetically attractive. McManus observed that there were substantial individual differences in such preferences. McManus felt that existing empirical research was unable to adequately differentiate preference for the golden-section versus other similar ratios such as 1.5, 1.6 or 1.75.
other answers are helpful but what is missing from them so far is that there seems to be basic golden ratios built into the natural human body proportions eg relative dimensions of body parts, including esp in the face dimensions, which is highly oriented with perception of beauty (but dont have an immediate authoritative/scientific ref for this). not sure but possibly DaVinci was interested in this connection eg in the iconic Vitruvian Man diagram but whether DaVinci used the golden ratio intentionally is subject to intense scholarly debate. it is possible that his artwork was just so lifelike that it naturally mimicked the golden ratio in natural human proportions.
it is natural to speculate whether these proportions emerged from maybe evolutionary constraints, or some kind of feedback loop between psychology and biology ala evolutionary psychology. but in any case the golden ratio has emerged in many other biological organism morphology/dimensions, so it is not outlandish to observe it naturally evolving in humans also.
so the appearance of the golden ratio in geometric designs as "intrinsically beautiful to humans" might have marginal scientific support, but it seems to be encoded into natural human dimensions. also it is now well established in the field of evolutionary psychology that human [sexual] attraction is highly oriented around ideal physical features, esp in the face. this overall topic could easily fill a thesis or book. for now heres at least one ref
To add a small neuroscientific point to excellent @JeromyAnglim answer - there has been an interesting study by Rizzolatti group (guy who 'discovered' mirror neurons) published in PLoS ONE. Di Dio, et al. (2007) looked at the brain responses to Classical and Renaissance sculptures, but they manipulated the proportion of sculptures' features by violating the golden ratio, as showed on the figure below (source: Di Dio et al., 2007):
During the experiment participant's were asked to either observe sculptures as if they were in a museum, or give an aesthetic/proportion judgements. fMRI scans were taken during the task. Following the results from brain activity, authors concluded that:
Therefore, the presence of the golden ratio in the sculpture determined brain activations different to those where this parameter was violated. It's quite interesting result that indirectly supports the association between positive aesthetic perception and the presence of golden ratio.