«All that glitters: a review of psychological research on the aesthetics of the golden section Christopher D Green Department of Psychology, York ...»
Perception, 1995, volume 24, pages 937-968
All that glitters: a review of psychological research
on the aesthetics of the golden section
Christopher D Green
Department of Psychology, York University, North York, Ontario M3J 1P3, Canada
Received 10 February 1994, in revised form 20 March 1995
Abstract. Since at least the time of the Ancient Greeks, scholars have argued about whether the
golden section—a number approximately equal to 0.618—holds the key to the secret of beauty.
Empirical investigations of the aesthetic properties of the golden section date back to the very origins of scientific psychology itself, the first studies being conducted by Fechner in the 1860s.
In this paper historical and contemporary issues are reviewed with regard to the alleged aesthetic properties of the golden section. In the introductory section the most important mathematical occurrences of the golden section are described. As well, brief reference is made to research on natural occurrences of the golden section, and to ancient and medieval knowl- edge and application of the golden section, primarily in art and architecture. Two major sections then discuss and critically examine empirical studies of the putative aesthetic properties of the golden section dating from the mid-19th century up to the 1950s, and the empirical work of the last three decades, respectively. It is concluded that there seems to be, in fact, real psychological effects associated with the golden section, but that they are relatively sensitive to careless methodological practices.
1.1 What is the golden section?
Imagine that you are asked to divide a line so that the ratio of the shorter segment to the longer segment is the same as the ratio of the longer segment to the whole line.
That is, if a represents the length of the shorter segment and b represents the length of the longer segment, find values for a and b such that a_ b ~b " a + b' where a + b = 1. T h e division of a line that answers to this requirement has come to be known as the 'golden section' (1) of a line, and it is shown in figure 1.
It has often been claimed, since the time of the Ancient Greeks, and perhaps much earlier, that the golden section is the most aesthetically pleasing point at which to divide a line. Consequently, it has been incorporated, in a variety of ways, into many artworks and architectural designs through the ages. Establishing empirically the claims of aesthetic primacy for the golden section was among the very first topics of scientific psychological research as the new discipline emerged in the 19th century.
l +' Figure 1. A line divided in the golden section. If the short segment is taken as 1, the long segment is § in length. If the long segment is taken as 1, the short segment is in length.
(' (1 It is variously called t
Fechner first fixed his analytical gaze upon this task as early as the 1860s. Since that time it has been the focus of a number of research programs, winning the attention of structuralists, gestaltists, behaviorists, social psychologists, psychiatrists, and neuroscientists at various points in time.
The exact value of the golden section of a line is given by an irrational number approximately equal to 0.618. That is, with a line divided at slightly more than 60/40, the short segment will bear the same mathematical relation to the long segment as the long segment does to the whole line. By adding 1 to the golden section— 1.618—one gets a closely related number widely known as. This number stands | alongside of jt and e as the most important irrational constants in mathematics.
Constants such as Jt, e, and gain their importance from being the solutions to basic | mathematical problems. In addition, they perhaps gain their special mystique from being irrational numbers; their exact values never being known. Just as jt is the ratio of the circumference of any circle to its diameter, and e is the solution to the expression lim[l +(l/n)]n, the solution to the equation x2 = x + l is another of these long-standing problems in mathematics. Its exact solution is (1 ± /5)/2.
The positive solution is (^-approximately 1.618—and the negative solution is approximately -0.618 (exactly the negative of the golden section described above).
For convenience, I shall denote the negative solution as -|', and its negative (the golden section itself) as |'.(2) A number of interesting relations hold between these two numbers. To begin with, §' is equal to both |-1 and to 1/|, or §~l. Thus, ty —ty'= §x§' = 1. Just as (j)"1 is equal to |-1, so 2 is equal to |+1, or c^ + c))0.
| Similarly, §3 is equal to (p + t))1, and so on; §n = tn_1 + (|n~2, for any value of n. A number of other interesting mathematical and geometrical properties are discussed below.
The origins of the names of these two numbers are matters of some dispute.
Berlyne (1971) claims that the reference to gold "was adopted in large measure, because of vague associations with the 'golden mean'" (page 229). Kepler, however, referred to jt and as the mathematical equivalents of gold and precious gems as far | back as 1596. Fowler (1982) claims that the first use of "goldne Schnitt" appears in an 1835 mathematical text by Martin Ohm, and that its first titular appearance was in an 1849 book by AWiegang. It was the publications of Adolf Zeising (1854, 1855, 1884), however, that did the most to popularize the name widely. Fowler also claims that the first English-language use of the term golden section was in James Sully's article on aesthetics in the 1875 edition of the Encyclopaedia Britannica.
The symbol, on the other hand, derives from the initial letter of the name of the | great Greek architect and sculptor, Phidias. Phidias was a proponent of the aesthetic qualities of the golden section, going so far, according to legend, as to incorporate it into the basic dimensions of his most famous work, the Parthenon (Ogden 1937).
Huntley (1970), however, says that the appellation,, was not adopted until "the early | days of the present century" (page 25).
Whatever the truth of these various claims, it is certain that the golden section has been the focus of a great deal of interest for a very long time. It is the aim of this paper to review the results of that interest, with special emphasis on the reputed aesthetic properties of the golden section as explored in the empirical work of the last 130 years. There have been other, nonaesthetic, research programs associated with the golden section, however. Most developed of these is its relation to judgments of interpersonal relations (Adams-Webber 1985; Adams-Webber and Benjafield 1973;
Benjafield 1985; Benjafield and Adams-Webber 1976). Other recent ones include its
relation to the structure of ethical cognition (Lefebvre 1985); the cognitions of sufferers of anxiety, depression, and agoraphobia (Schwartz and Michelson 1987);
consumers' perceptions of products and of retail environments (Crowley 1991; Crowley and Wiliams 1991); and even the proportion of wins a sports franchise needs to maintain fan support (Benjafield 1987). Nevertheless, this review is restricted to an aesthetic focus. It is intended to fulfill the needs of both historical and scientific researchers.
There are several different reasons one might be interested in a review of goldensection research. One reason, relevant to the history of psychology, is that, as mentioned above, studies of the golden section were among the very first empirical studies conducted in psychology; they take their place alongside Weber's and Fechner's psychophysical studies, Ebbinghaus's memory studies, and Helmholtz's studies of tone and color perception. Moreover, since that time the golden section has captured the interest of some of the most illustrious names in the discipline. A second reason is that the psychological phenomena associated with the golden section, despite repeated attempts to show them to be nonexistent or mere methodological artifacts, simply refuse to go away. Although many researchers have concluded that the effects are illusory, the more carefully conducted studies have fairly consistently shown that there is, in fact, a set of phenomena that require explanation, though no one has yet produced an explanation both adequate and plausible that has been able to stand the test of time.
The paper is structured as follows. First, some of the mathematical occurrences of the golden section are briefly examined. Second is a major section in which the empirical research conducted between the mid-19th century and about 1960 on the reputed aesthetic properties of the golden section is reviewed. In a third section the more-recent findings and theoretical explanations of psychologists are examined, with the focus particularly on the concerted efforts to show that golden-section effects are nothing but artifacts and on the stubborn resilience of the phenomenon many have tried to make go away. Finally, in a concluding section, I summarize the findings of this review.
1.2 Instances of § in mathematical and natural worlds Imagine that one of the segments of a line divided at the golden section is 'folded' at a 90° angle to the other, and a rectangle is formed on this L-shaped base, as in figure 2a. This figure is called the 'golden rectangle'. The golden rectangle can be constructed quite simply. Starting with a square, swing an arc, centered on the midpoint of the base of the square, from an upper corner of the square to a point collinear with the base. Build a rectangle on the new, longer base (see figure 2b). Just as it has 1 * f <
There are many other geometrical instances of. The best-known reference is | Huntley (1970). Another interesting account that includes links to contemporary work on fractals and tiling is Kappraff (1991). Not all the mathematical occurrences of the ( are geometric, however. Some are arithmetic as well. Most importantly, is | ) | the limit of the ratio of any two sequential numbers in any Fibonacci series (ie any series of integers in which xn = JCW_1 + JCW_2) such as 1, 1, 2, 3, 5, 8, 13,... or, for that matter, - 2, 7, 5,12,17,29,....
The question remains, however, of what interest all this might be to the nonmathematician. One answer is that there appear to be many examples of § in the natural world as well. Because ) is so intimately connected with the pentagon, as ( discussed above, pentagonal symmetry in nature has often been taken as a starting point for investigations of the 'natural reality' of |). Flowers of many kinds exhibit pentagonal symmetry (Berlyne 1971, page 224; Ghyka 1946/1977, page 18, n) as do a wide variety of sea creatures (Ghyka 1946/1977, page 18; see also Hargittai 1992).
Another connection between § and the natural world is mediated by the logarithmic spiral, which can be easily constructed from a golden rectangle or triangle. The shell of the nautilus, the abalone, and the triton, for instance, show the same sort of geometric growth pattern characteristic of the logarithmic spiral (see Ghyka 1946/ 1977, pages 93-97). The logarithmic spiral is also characteristic of the patterns of growth found on pine cones, pineapples, and sunflowers. A third link between the natural world and is the frequency with which the Fibonacci series is seen in nature.
| Fibonacci (Leonardo of Pisa, ca 1175-1250) himself showed that the growth both of rabbit and of beehive populations can be modeled by a Fibonacci series, even though the reproductive principles of the two are quite different. The relation between Fibonacci numbers and phyllotaxis (the study of the arrangements of leaves on plants) is also widely cited (see Berlyne 1971, page 224; Ghyka 1946/1977, page 16; Huntley 1970, pages 161 - 1 6 3 ; Mitchison 1977 for competing accounts).
Last, there have been claimed to be a wide array of examples of the golden section itself in mammalian anatomy. These claims are more controversial than those previously described because they involve the dubious notion of 'ideal' proportions, and because the actual research is quite old and its details are difficult to establish.
Such claims should not be viewed without a degree of skepticism in the absence of full and replicated scientific reports, but neither is there reason to dismiss them out of hand.
1.3 The history of the golden section The Egyptians are often credited with all manner of arcane mathematical knowledge (see eg Ghyka 1946/1977, pages 60-68). Regarding the golden section specifically, it seems clear that they had an estimate of accurate within 0.5%, and that some | religious buildings might have had explicitly incorporated into their designs. None | of this seems to have been achieved with the assistance of
mathematics, however. Egyptian mathematicians seem to have been akin to Levi-Strauss's (1962/ 1966, pages 16-36) bricoleurs—concrete scientists who develop the applied knowledge they require without raising it to the level of a full theoretical discipline.
Some of this knowledge was transmitted from Egypt to Greece, and there can be no doubt that the Classical Greeks possessed knowledge of, and a fascination with, the golden section (contrary to Gardiner 1994). Euclid discussed it extensively, calling it the "division into mean and extreme ratio" (Berlyne 1971, page 222). Proclus (ca 410-485) reported that the Greek geometers of the Platonic schools called it simply he tome, 'the section' (Ghyka 1946/1977, page 4). The aesthetic valuation of the golden section was transmitted to the Romans as well, and its beauty extolled by Vitruvius himself (Watts and Watts 1986).
942 C D Green The golden section also had a great influence on medieval and Renaissance architects and artists at least partly because of its ease of construction (Coldstream 1991, page 33). Arches and apses of 'golden' proportions can be inscribed by people who have minimal knowledge of engineering or geometry. A tool as simple as a rope, fixed at one end, can be used to inscribe the arc to be built upon (for a hint as to how, see figure 2b). By the 16th century, interest in the golden section was so widespread that Luca Pacioli di Borgo published a treatise on it entitled De divina proportione in 1509, illustrated by Leonardo da Vinci. By the end of the century, a
young mathematics instructor named Johannes Kepler would write: