Alex's Adventures in Numberland - Alex Bellos [123]
What is surprising is how a tiny change in the angle can cause such a huge variation in the positions of the seeds. At the golden angle, the seed head is a mesmerizing pattern of interlocking logarithmic spirals. It is the most compact arrangement possible. Nature chooses the golden angle because of this compactness – the seeds are bound together more closely and the organism will be stronger because of it.
In the late nineteenth century the German Adolf Zeising most forcefully put forth the view that the golden proportion is beauty incarnate, describing the ratio as a universal law ‘which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form’. Zeising was the first person to claim that the front of the Parthenon is in the shape of a golden rectangle. In fact, there is no documentary evidence that those in charge of the architectural project, who included the sculptor Phidias, used the golden ratio. Nor, if you look closely, is the golden rectangle a precise fit. The edges of the pedestal fall outside. Yet it was Phidias’s connection to the Parthenon that, in around 1909, inspired the American mathematician Mark Barr to name the golden ratio phi.
Despite the eccentric tone of Zeising’s work, he was taken seriously by Gustav Fechner, one of the founders of experimental psychology. In order to discover if there was any empirical evidence that humans thought the golden rectangle more beautiful than any other sort of rectangle, Fechner devised a test in which subjects were shown a number of different rectangles and asked which they preferred.
Fechner’s results appeared to vindicate Zeising. The rectangle closest to a golden one was the top choice, favoured by just over a third of the sample group.ven though Fechner’s methods were crude, his rectangle-testing began a new scientific field – the experimental psychology of art – as well as the narrower discipline of ‘rectangle aesthetics’. Many psychologists have conducted similar surveys on the attractiveness of rectangles, which is not as absurd as it sounds. If there were a ‘sexiest’ rectangle, this shape would be of use to the designers of commercial products. Indeed, credit cards, cigarette packets and books often approach the proportions of a golden rectangle. Unfortunately for phi-philes, the most recent and detailed piece of research, by a team led by Chris McManus of University College London, suggests that Fechner was wrong. The 2008 paper stated that ‘more than a century of experimental work has suggested that the golden section actually plays little normative role in subjects’ preferences for rectangles’. Yet the authors did not conclude that analysing rectangle preference is a waste of time. Far from it. They claimed that while no one rectangle is universally preferred by humans, there are important individual differences in the aesthetic appreciation of rectangles that merit further investigation.
Lesser scientists than Fechner were also inspired by Zeising’s theories. Frank A. Lonc from New York measured the height of 65 women compared to the height of their belly buttons and discovered that the ratio was 1.618. He had excuses when it wasn’t: ‘Subjects whose measurements did not fall within this ratio testified to hip injuries or other deforming accidents in childhood.’ The French architect Le Corbusier created ‘Modulor man’ in order to use proper proportions in architecture and design. In his man, the ratio of the height of the man to the height of his belly button is 1829/1130, or 1.619, and the ratio of the distance between his belly button and his upheld right hand and his belly button and his head is 1130/698, again 1.619. Le Corbusier’s intention for the Modulor was to use these proportions in architecture and design.
Gary Meisner is a 53-year-old business consultant from Tennessee. He calls himself the Phi Guy and on his website sells merchandise