Phi in Psychology
"In the 19th century, interest in the golden section began to turn experimental." -Christopher D. Green, All That Glitters: A Review of Psychological Research on the Aesthetics of the Golden Section
In the late 1800's, Gustav Fechner set about systematically studying the golden ratio in an attempt to build the concept of aesthetics and human preference from the ground up. He presented subjects with a set of ten rectangles that ranged in side-length ratio from 1:1 to 2.5:1, all with equal area. The orientation of the rectangles remains unknown. The golden rectangle was the seventh largest, and the order in which the rectangles were shown to each subject was randomized and different for each trial. (Green 943)
First, Fechner had the subjects choose which rectangle they found most aesthetically pleasing, and allowed for multiple choices if necessary. 35.0% of subjects expressed preference for the golden rectangle, 20.6% preferred the 1.5:1 rectangle, and 20.0% chose the 1.77:1 rectangle. None of the subjects selected the golden rectangle as their least-preferred choice. Fechner took these results to mean that the golden rectangle is most preferred (Green 943).
However, Fechner could not explain the preference psychologically. His contemporary psychologist Oswald Külpe, however, suggested a psychophysical account: "We have in the pleasingness of the golden section simply the pleasingness of apparently equal differences. It represents, so to speak, a symmetry of a higher order" (qtd. in Green 945). Fechner also collected data from some 20,000 paintings to analyze the golden ratio in art as part of his "method of use," which sought to explore phi in society and its products. The golden ratio did not characterize the height-to-width ratio of the paintings (Green 944).
Throughout the 20th century, a large number of studies were conducted to analyze preference for the golden ratio in aesthetics, and the overall result is inconclusive. Green suggests that "there has been a tendency among many psychologists to discount the golden section a priori as a 'numerological fantasy.'" One specific study, carried out by Angier in 1903, led to a criticism of averaging across subjects' rankings, as he claimed that "such a total average may fall wholly without the range of judgment of any one" (qtd. in Green 947). In the study, nine participants were asked to divide a horizontal line at the most pleasing place 72 times each. The mean proportion was about 0.600, but only two subjects regularly chose the golden section. However, it is possible that these results were biased due to Angier's possibly having an agenda favoring symmetry over phi.
Many more studies on this issue have been conducted, and the results are detailed in the three tables below from Green 962-964:
Overall, the golden ratio has been found to be significant, insignificant, or somewhere in between by countless studies, each offering a different conclusion. Also, methodological problems have plagued the research, specifically in the late 20th century in the research of Nakajima and Ohta (1986) and Davis and Jahnke (1991), and so their positive results are brought into question. The prevalence of the golden ratio in aesthetics is very on-again, off-again: one study confirms the preference only to be refuted by evidence to the contrary a year or two down the line. Green makes the conclusion that "the traditionally held aesthetic effects of the golden section may well be real, but if they are, they are fragile as well. Repeated efforts to show them to be illusory have, in many instances, been followed up by efforts that have restored them, even when taking the latest round of criticism into account" (966). There is a lack of evidence either way, and so the question of the golden ratio and aesthetics still remains one to be discussed. For a more thorough treatment of this topic, I highly recommend a reading of Green's paper. Documentation may be found here, or by clicking any of the citations above.