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Psychology of Aesthetics, Creativity, and the Arts 2010, Vol. 4, No. 2, 113–126 © 2010 American Psychological Association 1931-3896/10/$12.00 DOI: 10.1037/a0017316 Beyond the Golden Section and Normative Aesthetics: Why Do Individuals Differ so Much in Their Aesthetic Preferences for Rectangles? I. C. McManus, Richard Cook, and Amy Hunt University College London Interest in the experimental aesthetics of rectangles originates in the studies of Fechner (1876), which investigated Zeising’s suggestion that Golden Section ratios determine the aesthetic appeal of great works of art. Although Fechner’s studies are often cited to support the centrality of the Golden Section, a century of subsequent experimental work suggests it has little normative role in rectangle preferences. However, rectangles are still of interest to experimental aesthetics, and McManus (1980) used a paired comparison method to show that although population preferences are weak, there are strong, stable, statistically robust and very varied individual preferences. The present study measured rectangle preferences in 79 participants, particularly assessing their relationship to a wide range of background measures of individual differences. Once again weak population preferences but strong and varied individual rectangle preferences were found, and computer presentation of stimuli, with detailed analyses of response times, confirmed the coherent nature of aesthetic preferences for rectangles. Q-mode factor analysis found two main factors, labeled “square” and “rectangle,” with participants showing different combinations of positive and negative loadings on these factors. However, the individual difference measures, including Big Five personality traits, Need for Cognition, Tolerance of Ambiguity, Schizotypy, Vocational Types, and Aesthetic Activities, showed no correlation at all with rectangle preferences. Individual differences in rectangle preferences are a robust phenomenon that clearly requires explanation, but at present their variability is entirely unexplained. Keywords: Experimental aesthetics, rectangles, Golden Section, individual differences, preference functions lary is that participants find it meaningful and sensible to say which of several rectangles they like the most or like the least and are willing to make such decisions, despite at some surface level their apparent absurdity (see McCurdy, 1954), for why in any immediately rational sense should humans have preferences for one rectangle over another? Explaining such choices, which must surely be regarded as preferences—and aesthetic preferences at that— has remained a challenge to psychology. Humans do, of course, often express preferences in their daily life (e.g., when shopping and choosing one product over another), and such preferences are widely studied by economists, for whom preference is related specifically to cost or more generally to value. The essence of an aesthetic preference is, however, that it precisely does not relate to any objective value, and economists are forced at that point to refer to “hedonic value” when people pay more for objects they regard as more beautiful or attractive than they do for those they find less attractive. Such aesthetic preferences are what Immanuel Kant referred to as disinterested choice. The mathematics, the history, and the application of the Golden Section to aesthetics and other areas could fill several articles, and here only a brief summary needs to be given. More detailed reviews can be found elsewhere (Benjafield, 1985; Boselie, 1992; Green, 1995; Höge, 1995; McManus, 1980; McWhinnie, 1986). Mathematically, the idea of the Golden Section dates back to Euclid’s problem of division in the “extreme and mean ratio”— dividing a line so that the ratio of the larger part to the smaller part is the same as the division of the whole by the larger part (HerzFischler, 1998; Livio, 2002). Those conditions are satisfied when The history of experimental aesthetics, and hence the experimental psychology of art, effectively begins in 1876 with the publication of Gustav Theodor Fechner’s experiments on the aesthetics of simple rectangular figures (Fechner, 1876), which have only partly been translated into English (Fechner, 1997). In a simple but effective experimental design, Fechner laid out 10 white rectangles of different height:width ratios on a black table and asked his 347 participants to say which they liked the most. In a minor variant of the procedure, 245 of the participants were also asked which rectangle they liked least. Fechner’s experiment was in part driven by an interest in the claims of Zeising (1854) that the beauty of many works of art resulted from their components being in the ratio known as the Golden Section, a ratio of 1 to 1.618 (Fechner, 1865). Somewhat to Fechner’s surprise, he did find a population preference at the Golden Section, with almost no participants disliking the Golden Section. In some ways, Fechner’s greatest conceptual leap was in realizing that simply asking individuals which of a range of possible stimuli they preferred— his “method of choice”—allowed individuals’ aesthetic preferences to be assessed. The remarkable corol- I. C. McManus, Richard Cook, and Amy Hunt, Division of Psychology and Language Sciences, University College London. Correspondence concerning this article should be addressed to I. C. McManus, Division of Psychology and Language Sciences, University College London, Gower Street, London WC1E 6BT, United Kingdom. E-mail: i.mcmanus@ucl.ac.uk 113 114 MCMANUS, COOK, AND HUNT the parts are in the ratio ␸:1, the Greek letter ␸ being a common symbol for the proportion, where ␸ has the irrational value (√5 ⫹ 1)/2, which is approximately 1.618803 . . . The number ␸ is similar to ␲, and Euler’s number e, in having a range of intriguing mathematical properties, such as 1/␸ ⫽ ␸ ⫺ 1 and ␸2 ⫽ ␸ ⫹ 1, and it is the limiting proportion of successive numbers in the Fibonacci sequence. A Golden Section rectangle, whose sides are in the ratio 1:␸, has the property that, if a square is removed from one end, the remaining rectangle still has sides in the ratio 1:␸. Suffice it to say that such properties have enchanted not only mathematicians but also many who would like aesthetics to be based in mathematical calculation—see, for instance, Livio (2002). For many such authors, Fechner’s original experiment provides empirical support for what often are hypertrophied theoretical structures derived from mathematics. Without denying any of the beautiful and intriguing mathematics of ␸, and accepting that there is also a beauty in numbers such as e and ␲, shown especially well in that gnomically elegant formula, ei.␲ ⫽ ⫺1, there still remain many open empirical questions about actual aesthetic preferences for the sorts of simple rectangle that Fechner used in his experiments. The history of the golden section in experimental aesthetics since 1876 has, at best, been checkered. Most studies of rectangle aesthetics that followed Fechner and cited him have looked only at the question of whether there is a population preference and, if so, whether it is at the Golden Section. However, implicit in Fechner’s results is a very different finding—that there are individual differences in rectangle preferences. The conventional representation of Fechner’s results emphasizes that the population mode is at the Golden Section. The mode is indeed at the Golden Section, although only 35% of Fechner’s participants actually chose the Golden Section from the 10 rectangles presented to them. Even including in that total the 41% of participants who instead chose either of the rectangles adjacent to the Golden Section rectangle, with ratios 1.50 or 1.77, there still remained 24% of participants who chose one of the seven rectangles far removed from the Golden Section (ⱕ1.45 or ⱖ2.00). Without further evidence as to the consistency of these preferences, little more can be concluded, but it seems likely that there are individual differences. Fechner would have expected individual differences in his experiment and others, as elsewhere he talks of the old Latin tag, De gustibus non est disputandum: “It is an old saying that there is no accounting for tastes, nevertheless people argue about it, about nothing more than taste”; hence, to use Fechner’s words, “es muss sich also doch darüber streiten lassen”—“it must thus be possible to argue about taste” (English translations from Jacobsen, 2004). It is therefore possible to discuss tastes and argue about them because people genuinely differ in their tastes, in their aesthetic preferences, and, hence, in what they regard as beautiful. Nevertheless, differences between individuals have mostly been entirely lost in over a century’s worth of experimental aesthetic studies of the Golden Section that have followed Fechner. Few experiments ask how individuals differ in their preferences and instead concentrate on the similarities of individuals and, hence, the normative question of whether there is a population mean that is precisely at or near to the Golden Section. A problem in identifying individual differences using Fechner’s methodology is that only a single preference judgment (and sometimes one “dislike” judgment) is made by each participant. However one or perhaps two numbers cannot adequately describe what one can call an individual’s preference function—the relative preference of each rectangle relative to all others (and the same objection applies to using Fechner’s Method of Production, with participants producing the single rectangle that they feel looks best; Russell, 2000). For characterization of a complex curvilinear function of unknown shape, multiple judgments must be made across the entire range of stimuli. Rather than simply choosing the best rectangle, it would be better to have participants choose first the most preferred rectangle, then the second most preferred and so on, ranking each of the stimuli until a rank order has been established for all the rectangles. Ranking, however, still has several practical and theoretical problems. With large numbers of stimuli, ranking can be difficult. Participants have to see all stimuli simultaneously, and searching large numbers of stimuli within the visual field requires a large loading on working memory so that participants find it difficult to manage the cognitive complexity of the task. A theoretical problem for ranking is that, although it assumes that all stimuli can indeed be placed within a single preference metric, it may well actually be the case that the preference space is multidimensional. For large numbers of stimuli, rating is sometimes used to establish an aesthetic value for each stimulus. Here, the problem is that absolute judgments are difficult to make on 5-, 7-, or 10-point scales, as at any one time a participant is, to a large extent, judging the current stimulus relative to stimuli that previously have been seen, and they are also anticipating possible future stimuli which the scale needs to accommodate. Although the suggestion is often made that rating measures are absolute, in reality they can rarely be that, for having, say, just given a rating of 9/10 to their most preferred rectangle, what value would a participant give were they to find the next stimulus was the Mona Lisa? Many methodological problems in experimental aesthetics can be solved with the method of paired comparisons, in which each judgment is a relative preference for one of two stimuli that are simultaneously seen side by side. Although not used by Fechner for his rectangle experiment, the method of paired comparisons seems first to have been described by him in what has been described as a “surprisingly little known account” (David, 1963), in the Elemente der Psychophysik of 1860. Each judgment in a paired comparison design requires no memory of previous stimuli or anticipation of future stimuli, nor is there any cognitive complexity to be managed. The method of paired comparison does require large numbers of judgments to be collected from each single participant (and that may be the reason why Fechner did not use the method), so that for n stimuli, a complete paired comparison requires n ⫻ (n ⫺ 1)/2 comparisons, the number of pairs being proportional to the square of n. Paired comparison also has the important advantage that a significance test for the presence or absence of preference can be applied to individual participants’ results. The significance of preferences (and, implicitly, the dimensionality of preference space) can also be assessed by examining what are called circular triads, illogical triads, or inconsistent triads (David, 1988). If there is indeed a single underlying dimension that if A p B (read as, A is preferred to B), and B p C, then it should also be the case that A p C. However, if preferences are occurring for different reasons (A has a nicer color than B, and B has better composition than C, it may then be reasonable that C p A). In principle, paired comparison allows such multidimensionality to be assessed, although it needs to be distinguished from random variation or noise. The finding of McManus (1980) with rectangles, that circular triads were associated overall with weaker judgments, indeed BEYOND THE GOLDEN SECTION suggests that triads mostly result from participant error or measurement error. The present study is an extension and a development of the paired comparison study of rectangle and triangle preferences by McManus (1980). That study suggested that participants find the paired comparison method straightforward to use, that most participants do have strong preferences that are statistically significant, that participants appear to have very different preference functions, and that those different preference functions are stable over several years. A Q-mode factor analysis also suggested that there were three underlying factors behind participants’ different preference functions. Important also for the question of the Golden Section was that, although population preferences were small in comparison with the size of individual preferences, there was a hint of a population preference broadly around the Golden Section, and in addition, there was a clear separate mode visible at the square. The aims of the present study were severalfold. Some of the questions could not be asked in 1980, for a host of technical and practical reasons, but can now be asked with computerized stimulus presentation and better statistical analysis of results. In particular, we wanted to develop a more efficient incomplete paired comparison design that allowed a wider and better range of rectangles to be assessed in all of the participants, without the study becoming impracticably large. The fitting of an incomplete paired comparison design requires the estimation of what is, in effect, a Bradley–Terry model (Bradley & Terry, 1952), which can be carried out by conventional regression models (Critchlow & Fligner, 1991). Regression models also have the advantage over the methods of McManus (1980), in which standard errors can be fitted to preference functions. Computer presentation of stimuli and responses also allows collection of response times, and they in turn can be used to assess the details of the process by which aesthetic preferences are made. Finally, and it was the primary purpose of the study, we wanted to collect a wide range of individual difference measures of personality and behavior to assess whether any of them were related to the large individual differences in rectangle preference functions. Method The data presented here were collected in two separate studies and carried out in successive years; therefore, there are minor differences between them. For many purposes, the data can be combined, and in general we do so, indicating where that is not the case. Study 1 was carried out from October 2006 to June 2007 and was primarily exploratory. Study 2 was carried out from October 2007 to February 2008, with a number of minor differences from Study 1, as other hypotheses were also being tested in the principal part of the study, which was concerned with rectangle classification, and the classification and preference of quadrilaterals. However, Study 2 required the collection of rectangle preferences in a manner similar to that of Study 1; therefore, as far as possible, the two studies used the same stimuli, conditions, and background questionnaire-based data. The Description of Rectangles A rectangle’s shape is readily described by the aspect ratio (hereinafter, “the ratio”), which is the width divided by the height. 115 On that basis, a square has a ratio of 1. A problem with ratios is that horizontal rectangles, which have ratios between 1 and infinity, when rotated through 90° produce vertical rectangles (ratio ⬍1) with ratios that are compressed into the range from 0 to 1, meaning that vertical and horizontal rectangles are not symmetric around the square. Following McManus (1980), we therefore describe rectangles in terms of the log ratio (LR), calculated as LR ⫽ 100.log10(ratio), for which vertical and horizontal rectangles of the same shape differ only in their sign, and the scale is more likely to be psychologically equi-interval (although see Schonemann, 1990). Rectangle Preference Task A set of 21 rectangles was chosen with several constraints: They should sample a wide range of rectangles, from tall, thin, vertical rectangles, through the square, to wide, flat, horizontal rectangles; they should be at approximately equal intervals on a logarithmic scale; and they should include the Golden Section and the square. An important feature was also that the range should be somewhat wider than that in McManus (1980). The rectangles chosen had ratios of 0.205, 0.259, 0.320, 0.387, 0.460, 0.537, 0.618 (Golden Section), 0.704, 0.795, 0.893, 1.000 (Square), 1.121, 1.258, 1.421, 1.618 (Golden Section), 1.863, 2.175, 2.582, 3.125, 3.866, and 4.903. The LRs were therefore ⫺69.1, ⫺58.7, ⫺49.5, ⫺41.2, ⫺33.8, ⫺27.0, ⫺20.9 (Golden Section), ⫺15.3, ⫺9.97, ⫺4.93, 0 (square), 4.93, 9.97, 15.3, 20.9 (Golden Section), 27.0, 33.8, 41.2, 49.5, 58.7, and 69.1. Design A complete paired comparison experiment with 21 rectangles would have 210 pairs (ignoring side of presentation), which would have been impractically long. The basic rectangle preference experiment therefore used an incomplete paired comparison design in which participants saw a sample of 84 pairs of rectangles. The design (which is described fully at www.ucl.ac.uk/medical-education/ other-studies/aesthetics/resources/rectangle-aesthetics) has the advantages of sampling the entire stimulus domain while allowing more detailed attention to be paid to pairs that are adjacent in stimulus space. The Web site also contains complete sets of stimuli that can be downloaded, as well as a detailed description of the analysis of the incomplete paired comparison design. Pairs of rectangles were presented on a computer screen in a darkened room with a specially written program written in Matlab and Psychtoolbox (Brainard, 1997; Pelli, 1997). Stimuli were at a medium gray level (128 on an eight-bit scale), and all rectangles had an area of 20,000 pixels so that luminous flux was held constant. At a typical viewing distance with a 15” (38.1-cm) VGA monitor, the square subtended a viewing angle of about 4.3°. The rectangles in each pair were centered vertically and spaced to either side of the midline, with the side of presentation randomized. Participants indicated their preferences by using the keys Z, X, C, N, M, and the comma key, which were indicated with colored labels and corresponded to a strong, medium, or weak preference for the stimulus on the left and a weak, medium, or strong preference for the stimulus on the right. Response times were measured from the time of presentation until a response key was hit. After each response, there was a brief pause of approxi- 116 MCMANUS, COOK, AND HUNT mately 0.5 s before the next pair was presented, and stimuli were arranged in blocks so that participants could take rests. Participants conducted the experiment at their own pace. Test–retest stability. Stability of preferences was assessed at three time periods: immediate, short term, and medium term. For the immediate period, the participants in Study 1 repeated the basic rectangle preference task immediately after completing the main set of 84 stimuli, without any pause and without being told that the set of stimuli was being repeated, so that they made judgments of a total of 168 pairs of rectangles. For the short-term period, after carrying out the 84-item basic rectangle preference task, the participants in Study 2 carried out a range of other aesthetic tasks, lasting about 30 min, and then repeated the 84-item basic preference task. The second presentation to these participants can therefore be regarded as providing an estimate of short-term stability. For the medium-term period, 9 participants in Study 1 were traced about 5 months after the main experiment and repeated the rectangle preference experiment; as in their first testing, they gave preferences for two successive sets of 84 paired comparisons. These data allow an assessment of medium-term stability. Questionnaire measures. The questionnaires given to participants asked about a broad range of individual difference measures that might be expected to relate to differences in rectangle preference. These were as follows: • An abbreviated (30-item) measure of items from the Big Five personality traits (Costa & McCrae, 1992), which contained one item from each of the six facets of the five traits, with half of the measures on each trait being negatively scored. participants to circle any of 24 adjectives that described the rectangle preference task. A single yes/no question was also asked about whether the participants had ever heard of the Golden Section, Golden Ratio, or Divine Proportion. Procedure Participants were informed that they were taking part in an experiment relating to aesthetics and were led to a small cubicle. The cubicle was lit by a single spotlight facing an adjacent wall, and participants were seated about 57 cm away from the computer display. Participants received a written instruction sheet, which was purposely minimal, mainly concentrating on the practicalities of using the computer and responding. Concerning the task itself, the sheet said only, “The task is very simple. You will be presented with pairs of gray shapes and asked to identify which one you prefer (i.e., which looks nicer or more attractive).” The instructions purposely referred to “shapes” rather than “rectangles.” Statistical Analysis Conventional statistical analyses were carried out using SPSS 13.0. The use of multiple regression for analyzing an incomplete paired comparison design is described formally on the Web site at www.ucl.ac.uk/medical-education/other-studies/aesthetics/ resources/rectangle-aesthetics, where the syntax for carrying out the analysis in SPSS can also be found. Analyses of paired comparisons and circular triads used methods based on the approach of David (1988) and were programmed in Matlab. Ethical Permission • An abbreviated (9-item) measure of the need for cognition (NfC; Cacioppo & Petty, 1982) using the modified items of Thorne and Furnham (in preparation), with three items from each subscale. This study was approved by the Ethics Committee of the Department of Psychology at University College London. • The Budner Tolerance of Ambiguity Scale (ToA; Budner, 1962), which has 16 items. Participants • The (22-item) short form of the Schizotypal Personality Questionnaire (SPQ-B; Raine & Benishay, 1995). Three factor scores can also be derived (see http://www-rcf.usc.edu/ ⬃raine/). • A 14-item measure of aesthetic activities (AA), described by McManus and Furnham (2006), except that reading nonfiction and reading poetry were omitted. • A brief measure of the six vocational types (RIASEC) described by Holland (Holland, 1997). Participants rankordered six brief pen portraits of one-word labels of the RIASEC groups: doer (R), thinker (I), creator (A), helper (S), persuader (E), and organizer (C). Subscales were available for several of the measures, but to avoid alpha inflation, we used them only in statistical analyses if there were clear indications that the overall factor was significant. The questionnaire also asked about basic demographics (gender and age), and it then finished with a checklist of adjectives asking Results Forty participants took part in Study 1 (study numbers S101 to S140), most of whom were undergraduates (mean age ⫽ 22.3, SD ⫽ 4.8, range ⫽ 18 – 42; 25% male, 75% female). Thirty-nine participants took part in Study 2 (study numbers S1 to S39), most of whom were undergraduates (mean age ⫽ 21.1, SD ⫽ 1.4, range ⫽ 18 –26; 39% male, 61% female), of whom 7 were pilot participants for the principal study, and 4 were excluded from the principal study because of technical problems (although all 39 participants had complete rectangle preference data and questionnaire data and hence are included here). Individual Rectangle Preference Functions For each pair of rectangles, participants made a response on a 6-point scale, which was converted to scores of ⫺1.0, ⫺0.6, ⫺0.2, 0.2, 0.6, and 1.0 for preference of the right-hand rectangle relative to the left-hand rectangle, positive numbers indicating a preference for the right-hand stimulus. In the basic rectangle experiment, each participant made 84 preferences for the same subset of all of the possible 210 comparisons between the 21 rectangles. Statistical analysis used multiple regression. A series of dummy variables was created, one for BEYOND THE GOLDEN SECTION each of the 21 stimuli. For any particular pair, all but two of the dummies were set at zero, with the left-hand stimulus having a dummy variable with a value of ⫺1 and the right-hand stimulus having a dummy variable of 1 (see Appendix 2). Arbitrarily, the dummy variable for the first stimulus was set at zero to prevent singularity, and preference values for the 21 rectangles were then scaled so that the mean preference was zero. In general, preferences ranged from ⫺1 to 1, although occasionally, because the preferences are fitted values, they can sometimes be slightly outside that range. Rectangle preference functions were calculated for each of the 79 participants on each occasion that they were tested. Using the regression across all 20 dummy variables and considering just the first time the basic rectangles were presented, we found that, overall, 69/79 (87%) participants showed significant preferences, with p ⬍ .05; 60/79 (76%) showed significant preferences, with p ⬍ .001; and 10/79 (13%) had preference functions that were not significant overall. Figure 1 shows examples, selected to emphasize their diversity, with the constraint that none of them subsequently appear in Figures 3 or 5. 1.0 S2 Because a principal interest of this study was in describing individual differences, we analyzed the structure of the differences using a Q-mode factor analysis (as was carried out by McManus, 1980). Q-mode analysis differs from conventional factor analysis in that the data are transposed so that the correlations are not between the stimuli but instead are between the participants. For this analysis, the correlations were between the 84 judgments made by one participant with another participant. On a technical note, this means that the factor analysis does not “know” that the 84 judgments correspond to preferences between pairs of 21 stimuli that are organized on a line but merely recognizes that some pairs of participants are very similar in their judgments (they are positively correlated), some are the opposite of one another (they are negatively correlated), and others seem unrelated to one another (no correlation in preference judgments). The Q-mode factor analysis of the judgments from the 79 participants, with principal components followed by a Varimax rotation, suggested two main 1.0 S3 0.8 0.6 0.4 0.2 0.2 0.0 Preference 0.4 0.2 -0.2 0.0 -0.2 0.0 -0.2 -0.4 -0.4 -0.4 -0.6 -0.6 -0.6 -0.8 -0.8 -0.8 -1.0 -1.0 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 -1.0 -70 -60 -50 -40 -30 100.log10(aspect ratio) -20 -10 0 10 20 30 40 50 60 -70 70 1.0 1.0 0.8 0.8 S23 S26 06 0.6 0.4 0.4 -0.2 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -0.8 -0.8 -1.0 -1.0 -10 0 10 20 30 40 50 60 70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 -70 1.0 1.0 0.8 0.8 0.8 S31 S36 0.6 0.2 0.0 -0.2 02 0.0 -0.2 02 -0.4 -0.6 -0.8 -0.8 -0.8 -1.0 -1.0 -1.0 -20 -10 0 10 20 100.log10(aspect ratio) 30 40 50 60 70 60 70 -40 -30 -20 -10 0 10 20 30 40 50 60 70 S115 -0.2 02 -0.6 -30 50 0.2 -0.6 -40 40 0.0 -0.4 -50 30 0.4 -0.4 -60 -50 0.6 Preference P 0.2 Preference P 0.4 20 100.log10(aspect ratio) 1.0 -70 -60 100.log10(aspect ratio) 0.4 10 -1.0 -70 100.log10(aspect ratio) 0.6 0 0.0 -0.6 -20 -10 -0.2 -0.4 -30 -20 0.2 -0.6 -40 -30 0.4 -0.4 -50 -40 S27 06 0.6 Preference Preference 0.2 0.0 -60 -50 100.log10(aspect ratio) 0.8 -70 -60 100.log10(aspect ratio) 1.0 06 0.6 S17 0.8 0.6 0.4 Preference Preference 0.6 Preference Q-Mode Factor Analysis 1.0 0.8 Preference P 117 -70 -60 -50 -40 -30 -20 -10 0 10 20 100.log10(aspect ratio) 30 40 50 60 70 -70 -60 -50 -40 -30 -20 -10 0 10 20 100.log10(aspect ratio) Figure 1. Examples of diverse preference functions from 9 different participants who have been chosen so that they are not among the example participants in Figure 3 or the medium-term follow-up participants in Figure 5. In particular, asymmetric functions are emphasized because they are underrepresented elsewhere. 30 40 50 60 70 MCMANUS, COOK, AND HUNT 118 factors, and a scree-slope analysis showed three factors above the general “scree” (first 10 eigenvalues ⫽ 27.777, 6.708, 2.769, 2.224, 2.167, 1.977, 1.905, 1.836, 1.671, and 1.625), the first two factors together accounting for 44% of the total variance. At first, it was thought that the third factor might be significant, but exploration suggested that it did not seem to show any meaningful structure and therefore was ignored. We conducted reification of the factors by summing the individual preference functions of all the participants, weighted by their loading on each of the factors. Figure 2 suggests that Factor 1 is essentially a preference for squares, although the peak is very slightly displaced from a pure square toward a slightly horizontal rectangle with a ratio of 1:1.12. Factor 2 is essentially bimodal, with peaks that are at somewhat more extreme rectangles than the golden section, at ratios of 1:1.863 and 1:0.537, as well as a minimum that (like that of the square factor) is slightly to the right of the square, at a ratio of 1:1.12. We call these factors the square factor and the rectangle factor, respectively. Figure 3 shows the loadings of individual participants on the two factors. Most participants loaded on the first factor, the second factor, or both, with few participants loading on neither of the factors (shown in the center of the plot). The Population Preference Function Given the range of individual differences in preference functions, the overall preference function for the whole group of participants is necessarily going to be fairly flat. Nevertheless, it is presented in Figure 4, primarily to emphasize both the small size of the preferences in absolute terms, the solid black circles being on the same ordinate as the data in Figures 1 and 3. The open circles show a magnified version of the same data and emphasize that although the function is small, compared with the individual preference functions, it is still significantly different from random, F(20, 6616) ⫽ 12.121, p ⬍ .001; with an overall preference for squares and little evidence of any population preference around the golden sections. Asymmetry of the Preference Function A striking feature of both the square factor and the rectangle factor is their symmetry, yet some participants seemed to show very asymmetric preference functions, as seen in Figure 1. An asymmetry score was therefore calculated by subtracting the mean preference score for vertical rectangles from the mean preference score for horizontal rectangles. A positive score therefore indicates an overall preference for horizontal rectangles, and a negative score indicates an overall preference for vertical rectangles. The overall mean asymmetry score was .0076, indicating that, on average, horizontal and vertical rectangles are equivalent, but the standard deviation was .245, with the minimum and maximum scores being ⫺.61 and .71, indicating large differences in a few participants. The presence of large asymmetries in a few participants, coupled with the essential symmetry of the two extracted factors shown in Figure 2, suggests that the two factors are not explaining all of the explainable variance, perhaps because of idiosyncratic factors corresponding to only a few participants. A low communality was therefore also used as an indicator of the possible presence of other systematic factors (although it may also correspond to random, nonsignificant preferences). Correlations With Personality and Demographic Factors The demographic factors consisted of gender and age (2 measures), the personality factors consisted of the Big Five, ToA, SPQ-B, NfC, AA, and Holland types (15 measures) and knowledge of the Golden Section was also included, for a total of 18 measures. The primary interest was in assessing how these related Figure 2. Summary preference functions for (a) the square factor and (b) the rectangle factor. The functions were calculated from the preference functions of all 79 participants, weighted by their loadings on the Q-mode factors, and then arbitrarily scaled around zero so that the maximum absolute value of function was 1. BEYOND THE GOLDEN SECTION 119 Figure 3. The graph in the center shows the loadings of the 79 individual participants on the square factor (horizontal) and the rectangle factor (vertical), with participants indicated by their participant numbers. Participants in boxes are statistically significant ( p ⬍ .05) on both the multiple regression analysis and the analysis of circular triads (n ⫽ 66), whereas those in italics are significant only on the regression analysis (n ⫽ 3), those underlined are significant only on the triad analysis (n ⫽ 6), and those in normal font are not significant on either criterion (n ⫽ 4). The eight preference functions around the central scatterplot show examples of participants with high or low positive or negative loadings on the two factors. Participants have been chosen who have not been included in other figures (and note that Participants 120 and 140, who have both positive rectangle factors and negative square factors are both shown in Figure 5). to the square factor and the rectangle factor, with an additional interest in the asymmetry measure. A total of 18 ⫻ 3 ⫽ 54 correlations were therefore calculated. A strict Bonferroni correction for multiple testing would set a nominal alpha level of about 0.001, although that is likely to be overly conservative, given that not all personality and other measures are strictly independent; in addition, the study was to some extent exploratory. A compromise significance level of .01 was therefore chosen. The correlation matrix is shown in Table 1. Of the 54 correlations, only one was significant, with p ⬍ .01, and can be regarded as possibly significant at the compromise alpha level. Total AA correlated ⫺.294 ( p ⫽ .0089) with the rectangle factor. Total AA is composed of 14 subitems, and when these were correlated with the rectangle factor, only two showed significant correlations at the .01 level: “going to classical music concerts/opera” (r ⫽ ⫺.357, p ⫽ .0014) and “going to theater (plays/musicals, etc.)” (r ⫽ ⫺.342, p ⫽ .00228). Perceptions of the Rectangle Preference Experiment On average, participants used about four adjectives to describe the rectangle preference experiment, (M ⫽ 3.97, SD ⫽ 1.82, range ⫽ 0 –11), with abstract, boring, hard, restrictive, theoretical, MCMANUS, COOK, AND HUNT 120 Figure 4. The preference function for all 79 participants. For comparative purposes, the solid black circles with a solid line indicate the value of the preference function on the same scale as the participants in Figures 1, 3, and 5. For better visibility, the open circles with dashed line show the same data rescaled so that the absolute maximum value is 1. easy, scientific, and cold being used as descriptive terms by 20% or more of the participants. At the .01 significance level, the only correlations with the square factor, the rectangle factor, and the asymmetry measure, were that “creative” correlated positively with asymmetry (r ⫽ .318, p ⫽ .0043; see Table 2). Factor analysis of the 24 adjectives suggested that there were three underlying factors, which can be labeled using the highest loading adjectives as creative/artistic (and not boring), practical/ sensible (and not abstract), and scientific/academic (and not profound). Scores for these three factors showed no significant correlations with the square factor, rectangle factor, or asymmetry measure. The only significant correlations ( p ⬍ .01) with demographic and Table 1 Correlations of Three Factor Scores With Demographic Measures and Measures of Personality and Interests Square factor Rectangle factor Asymmetry score Demographic Loading p Loading p Loading p Age Gender (1 ⫽ male, 2 ⫽ female) Knowledge of Golden Section (1 ⫽ yes, 0 ⫽ no) Big Five: Openness to Experience Big Five: Conscientiousness Big Five: Extraversion Big Five: Agreeableness Big Five: Neuroticism Tolerance of Ambiguity Need for Cognition Aesthetic Activities Schizotypal Personality Questionnaire Holland type Doer Thinker Creator Helper Persuader Organizer ⫺0.264 0.215 0.003 ⫺0.112 0.074 ⫺0.055 ⫺0.074 0.083 0.012 0.018 ⫺0.194 0.044 .019 .057 .983 .328 .514 .633 .517 .466 .915 .872 .086 .701 ⫺0.119 0.145 ⫺0.075 ⫺0.044 0.108 ⫺0.156 0.008 0.023 0.079 0.072 ⴚ0.303 ⫺0.044 .295 .201 .513 .702 .346 .170 .942 .843 .489 .529 .007 .697 ⫺0.066 ⫺0.106 0.270 0.072 ⫺0.059 0.045 ⫺0.170 ⫺0.194 ⫺0.032 0.212 0.001 0.087 .564 .353 .016 .526 .608 .694 .135 .086 .784 .061 .990 .445 0.081 0.153 ⫺0.202 ⫺0.239 ⫺0.011 0.240 .477 .178 .074 .034 .921 .033 0.167 0.163 ⫺0.028 ⫺0.254 ⫺0.141 0.078 .141 .151 .804 .024 .214 .497 ⫺0.078 0.075 ⫺0.046 ⫺0.155 0.117 0.094 .494 .512 .685 .172 .305 .409 Note. N ⫽ 79 in all cases. The one correlation that is significant with p ⬍ .01 is shown in bold type. BEYOND THE GOLDEN SECTION 121 Table 2 Correlations Between the Adjectives Used by Participants to Describe Their Perception of the Experiment (Left Side) With Speed of Responding, and Scores on the Square Factor, Rectangle Factor, and the Measure of Asymmetry Correlations with: Mean response time Square factor Rectangle factor Asymmetry Adjective % Respondents Loading p Loading p Loading p Loading p Abstract Boring Hard Restrictive Theoretical Easy Scientific Cold Practical Interesting Artistic Creative Sensible Applied Superficial Sensual Academic Profound Ugly Realistic Irrational Intellectual Beautiful Emotional 62.0 34.2 30.4 27.8 26.6 24.1 22.8 20.3 19.0 19.0 17.7 15.2 15.2 11.4 10.1 8.9 6.3 6.3 5.1 5.1 5.1 3.8 3.8 2.5 ⫺0.028 ⫺0.081 0.062 0.147 0.117 ⫺0.114 0.178 ⫺0.041 0.114 0.180 0.252 0.122 ⫺0.052 ⫺0.038 ⫺0.096 0.071 ⫺0.011 ⫺0.205 ⫺0.001 0.061 ⫺0.086 0.144 ⫺0.115 ⫺0.171 .805 .477 .588 .196 .305 .319 .117 .721 .316 .113 .025 .284 .646 .737 .398 .536 .920 .070 .994 .592 .452 .206 .315 .131 ⫺0.035 0.230 ⫺0.066 ⫺0.020 0.020 0.160 ⫺0.111 ⫺0.032 0.144 ⫺0.122 0.041 ⫺0.047 ⫺0.278 0.140 ⫺0.021 ⫺0.118 0.137 ⫺0.040 0.047 ⫺0.033 ⫺0.002 ⫺0.212 0.017 ⫺0.199 .761 .042 .562 .859 .864 .158 .330 .777 .207 .286 .717 .681 .013 .218 .855 .299 .229 .724 .681 .773 .986 .061 .881 .078 0.103 0.022 ⫺0.134 0.150 0.061 0.164 ⫺0.097 0.081 0.112 0.142 ⫺0.022 ⫺0.065 ⫺0.261 0.121 ⫺0.097 0.048 0.091 ⫺0.100 ⫺0.029 ⫺0.001 ⫺0.029 0.005 ⫺0.080 ⫺0.012 .365 .844 .240 .188 .592 .149 .397 .480 .327 .211 .849 .569 .020 .287 .394 .672 .424 .382 .797 .996 .797 .962 .482 .914 ⫺0.098 0.064 ⫺0.127 ⫺0.118 0.068 0.022 0.032 ⫺0.030 0.035 ⫺0.072 0.198 0.318 ⫺0.125 0.074 0.073 0.050 0.037 0.010 ⫺0.010 ⫺0.102 ⫺0.077 0.126 0.112 ⫺0.129 .391 .576 .265 .298 .550 .845 .782 .791 .760 .530 .080 .004 .272 .519 .520 .663 .745 .932 .930 .369 .499 .270 .326 .258 Note. N ⫽ 79 in all cases. The sole correlation that is significant with p ⬍ .01 is shown in bold type. personality measures were that older participants and those with a greater NfC saw the study as more scientific/academic (r ⫽ .344, p ⫽ .0019 and r ⫽ .383, p ⫽ .00049, respectively). Stability of Preferences Immediate test–retest reliability. A Q-mode factor analysis was carried out using the immediate retest data for the 40 participants of Study 1. Calculating the loadings separately for the first 84 paired comparisons and their immediate repetition as the second 84 paired comparisons, there were correlations of .888 and .920 for the loadings on the square and rectangle factors (n ⫽ 40; p ⬍ .001 in each case). Short-term reliability. In Study 2, after an interval of about half an hour during which the participants carried out a range of other tasks, the participants again carried out the basic rectangle preference task. The correlations for the loadings on the square and the rectangle were .911 and .810 ( p ⬍ .001 in each case). Medium-term reliability. Nine participants repeated the rectangle preference task after an interval of about 5 months (average interval ⫽ 159 days, range ⫽ 134 –193 days). Figure 5 shows their preference functions, and it is clear that in general there is a strong similarity across the two occasions, although Participant 102 is an obvious exception. Considering only the preference functions based on the first 84 rectangle pairs, the retest correlations for the square and rectangle loadings were .586 and .648, respectively (n ⫽ 9; ps ⫽ .097 and .059, respectively). However, examination of Figure 5 and of scatterplots suggests that this relatively low correlation was mainly due to Participant 102, whose preference function had changed dramatically over the 5-month period. Removal of Participant 102 resulted in correlations across the 5-month period for the square and rectangle factors of .905 and .761, respectively (n ⫽ 8; ps ⫽ .0020 and .028, respectively). Response times. Participants varied in the speed with which they carried out the task. The mean response time was calculated for each participant and showed an average value of 2.23 s (Mdn ⫽ 2.10, SD ⫽ 1.05; fifth and ninth percentiles ⫽ .87 and 4.30; range ⫽ 0.45–5.30). There was no correlation between response time and loadings on the square factor or rectangle factor, nor with the communality, the significance of the preference function, or with any personality variables. The only correlation with perceptions of the experiment was that the 14 participants describing the study as “artistic” had longer response times, t(77) ⫽ ⫺2.28, p ⫽ .025 (see Table 2). Circular triads. Paired comparison designs can be analyzed with the methods described by David (1988) for looking at triads and assessing the number of circular triads, in which one assesses the number of triads of preferences of the form A p B and B p C but C p A. The incomplete paired comparison design used here has a total of 84 triads. Triads were assessed only considering the direction of preference (right- or left-hand stimulus), ignoring the strength of preference. Considering just the basic rectangle preferences by the 79 participants, the mean number of circular triads MCMANUS, COOK, AND HUNT 122 Figure 5. Study 1. Medium-term stability plots for all of the 9 participants followed up after a 5-month interval in was 15.1 (SD ⫽ 12.52, range ⫽ 0 – 45). For 7 participants, the number of triads was similar to that expected by chance in a random matrix (ⱖ36); 8 participants were significant with .01 ⬍ p ⬍ .05 (30 –35 triads), 3 participants were significant with .001 ⬍ p ⬍ .01 (24 –29 triads), and 61 participants were significant with p ⬍ .001 (ⱕ23 triads). Six participants had no circular triads at all. There was a close correspondence between significance using the method of circular triads and the regression approach described earlier, although there were 3 participants significant at p ⬍ .05 on the regression analysis who were not significant on the circular triads, and 6 participants who were significant on the circular triads and not on the regression analysis. Four participants were nonsignificant on either method, and 66 were significant on both methods. The number of circular triads was lower in participants who had higher loadings on the square factor (r ⫽ ⫺.370, p ⫽ .00078) and the rectangular factor (r ⫽ ⫺.216, p ⫽ .055) but who showed no correlation with overall response time (r ⫽ .092, p ⫽ .420). Response times in circular triads. Circular triads may reflect overly rapid and, hence, careless responding by participants, or alternatively they may reflect genuine uncertainty, and, hence, be associated with longer, more careful deliberation. Mean response times (after log transformation to stabilize variance) were calcu- lated for all response times included in any circular triad and were compared with all response times included in noncircular triads. A positive difference indicates that participants took longer when making judgments that were a part of a circular triad. Figure 6 plots the average difference in log response time in relation to the number of circular triads (excluding the 6 participants with no circular triads). There is a significant negative correlation (r ⫽ ⫺.384, p ⫽ .00078, n ⫽ 73), indicating that, in general, participants take longer when making a circular triad, suggesting that greater deliberation is taking place. However, calculation of approximate significance tests for individual participants (shown in Figure 6) suggests that 6 of the participants were actually faster when making circular triads, suggesting careless or overly rapid responding in these individuals. Response times and circular triads in relation to preference values. Some rectangles are more similar in their preference values than others (as calculated in the preference function). When a comparison is made of two rectangles with a very similar preference then it might be expected that the task is harder and hence will take longer than when two rectangles are very different in their preferences. That was tested by calculating, separately for each participant, the correlation between the log of the response