30 May 2017

Golden rectangles (Fechner)

The idea that the golden ratio might have some aesthetic meaning originated with Zeising (1854), who squeezed the Parthenon in a golden rectangle (which in fact was nothing of the sort, see here). We now turn to the second key figure in the story: the German physicist and philosopher Gustav Fechner. The Weber-Fechner law concerning sensations and stimuli is evidence of his achievements in the field of psychophysics. He also did experimental aesthetics, and was sceptical of Zeising's theses when he turned to investigate them. Nevertheless, his experiments seemed to confirm them. (Here an English translation of his 1876 paper.) Let's take a closer look.

What Fechner displays:

(Diagram from A. van der Schoot, De ontstelling van Pythagoras, p. 256.) On the x-axis are the dimensions of ten cards, among which subjects were asked to choose, and the height gives the respective numbers of choices. (The decimal fractions come from multiple choices being allowed.)

As we shall explain below, the experiment was biased towards choosing 34/21 (1.619...) which Fechner calls the golden section. Even so,  a convincing 2/3 majority (150 of 228) did not choose it.

What Fechner concluded: when asked to choose one of ten cards, 34.5% of the subjects preferred the card with golden proportions. Second most preferred choice is only some 22%.

What Fechner should have displayed:

(We inverted Fechner's proportions, so that they fit in the interval 0-1.)

What Fechner should have concluded: when asked to choose a proportion in one of nine intervals (of varying length), two thirds of the subjects chose a proportion in the interval 0.59-0.67, the interval 0.59-0.64 being slightly preferred over 0.64-0.67.


1. We eliminated the square card from the choices. To almost everybody, a square is perceived as being essentially different from a (non-square) rectangle, in the same way as a circle amongst non-circular ellipses. Further, you need not be a great psychologist to know in advance that people dislike distorted perfection, like an almost-square or an almost-circle. Hence, we have a grand total of 221.73 votes, while Fechner had 227.97.

2. Fechner's results are biased by the fact that only a limited number of choices are possible, namely the numbers in red below.

Imagine someone with a taste for the proportion 0.6. As there is no card with that proportion, he has no choice but to select the number closest to 0.6, i.e., 0.618. Fechner counts this as a choice for 0.618, while in fact the favoured number could be anything in the interval IV, containing the numbers closer to 0.618 than to its neighbours. Hence, we considered the cards chosen as being in fact choices situated somewhere in the intervals extending left and right halfway to the next. (Center, beginning and end of these intervals below.) These intervals have unequal lengths, so we divided the absolute numbers by the length of the interval, to see in which intervals a unit of length collects most votes. This we translated into percentages. (Final column below.) Our above histogram "What Fechner should have shown" summarises the results.

There is not much to be inferred from the results once they have been freed from the bias introduced by the investigator. "People" prefer (non-square) rectangles with proportions from roughly 1/2 to 2/3, with a peak roughly halfway. The golden proportion (21/34, rather) only turns up because Fechner included this strange proportion among his very few choices. Had he included 2/π (which is 0.636...) instead, he would have concluded that most people chose the rectangle whose proportions are the ratio of a semicircle (πR) to its diameter (2R). Commentators would not fail to remark that people, since time immemorial, are very acquainted with circles, and that 2/π, locating the center of gravity of a semicircle, even has a physical meaning. Think about the thrilling consequences!

Many other methodological objections against Fechner's experiment can be raised. First thing to be investigated should be the threshold by which a subject is capable to distinguish one proportion from the other. If, for instance, it had been established that the threshold is 0.01, then one could systematically look for a preference among, say, 0.60, 0.61, 0.62,..., 0.70. One should also verify that next day, week, month or year the preference is still the same, and that it does not depend on anything else than pure shape.

Once such stable individual tastes are available in big numbers, correlations could be sought with race, culture, sex, age, education, profession and so on. (Fechner himself distinguished between men and women; we have quoted his results for males.) I would be surprised if anything more meaningful turned up than generalities like "Japanse prefer squares more than Germans do". Most Fechner-like experiments are as sloppy as the original one, and if his results are confirmed, they are probably obtained in the same biased way. The golden ratio is spotted in a numerical cloud simply because the investigator gave the silly number (√5-1)/2 some prominent place in the experiment!

In 1997, Fechner's experiment has been reproduced as meticulously as was possible given his sloppy protocol. His results were not confirmed, and the paper (here) has the telling title

The Golden Section Hypothesis — its last funeral.


A lot more reliable that Fechner's 10 button experiments are those that allow subjects to freely design their own rectangle. Sure enough, these free choice experiments fail to reveal any golden taste. In 1908, Lalo (L'esthétique expérimentale, here) had 29 subjects freely shift two squares to obtain a rectangle of their choice, gathering the proportions in bands of width 0.1. We have translated the results in percentages, and have also added Φ (1.61803). Like so many, Lalo confuses this irrational number some with simple approximating fraction, speaking of la section d'or 5/8, though 8/5 is but a very crude 1.6. (This statement and the results obtained are on his p.54.)