Golden Section Hypothesis and
Proportions of Computer Drawn Shapes
by Alyssa M. Wallis

© 2006 Alyssa M. Wallis

Originally Submitted for Psychology 304: Experimental Psychology
Professor Craig Clarke

Abstract

The effect of given side length was used to replicate Gustav Fechner’s Golden Section study, which found that shapes with a ratio of 1:1.618 are more pleasing to the eye than shapes with other proportions. Using different lengths of horizontal sides, as designated by two red dots on a computer screen, college students used a computer program to draw a rectangular quadrangle that looked good to them. Solid and open shapes were tested to see if they had any effect on the proportions of the drawn shape. The results did not support the golden section hypothesis, but indicated that the length of the given side had an effect on the length of the drawn side and that shape color had no effect. 

The Golden Section Hypothesis and Proportions of Computer Drawn Shapes

Gustav Fechner (1997) felt that “certain shapes and proportions may reveal some preferences of pleasingness to others” (p. 119). The Golden Section states that people tend to prefer shapes that are 1:1.168 in ratio proportion. There are three ways to test the Golden Section Hypothesis (GSH), the method of production, the method of use, and the method of choice. The method of production is when participants are asked to make or draw shapes that they think look good, and then the ratios of the shapes are measured  to determine if a majority of them fit into the golden section ratio. The method of use is when objects that were already made are measured to see if they fit the golden section ration, such as measuring paintings, mouse pads, or wallets. The method of choice involves giving participants different sized shapes and having them choose which one looks best to them, or rating which ones look better to see if most people choose shapes that fit the golden ratio.

 Fechner was the first to empirically test the golden section hypothesis through an experiment using the method of choice. He had 347 participants choose out of 10 rectangles, which one they preferred. The 10 rectangles had different side ratios ranging from 1:1 (a square) to 5:2. He found that the square with the golden section ratio of 34:21 had the most preference judgements. One problem that arose with Fechner’s study was that he only used people from educated backgrounds, who might have different views of what is beautiful or “could have known Zeising’s law of the golden section” (Hoge, 1996, p. 87). Also, some participants felt they could not decide which shape was better until they knew for what reasons the rectangle would be used, or participants had a hard time choosing from the several rectangles they had narrowed it down. Later problems arose in replicating his study because there were no standardized procedures for reporting methods and results in those times.

                In 1996, Holger Hoge performed two experiments that dealt with the Golden Section Hypothesis. The first experiment used the method of production in which 62 participants were either asked to make four beautiful rectangular quadrangles (experimental group) or to just make four rectangular quadrangles (control group). He avoided using the word rectangle because he thought that this implied a shape with sides of different lengths and wanted the participants to have the choice of making a square if that is what they felt was aesthetically pleasing. Hoge’s  (1996) hypothesis was “that a verbally given positive aesthetic criterion (beautiful) should result in a preference for the golden section whereas this preference should be lower or absent when no criterion is given” (p.80). Each participant was given four horizontal sides of different lengths, therefore a total of 248 shapes were made. The results showed that “the verbal criterion is one of the sources which show an influence on how to make a proportion of a simple figure” and the longer the given horizontal side was, the longer the drawn vertical side was. (Hoge, 1996, p. 81). Also there was no peak at the golden section in either the control group or the experimental group.

                In Hoge’s second experiment, he used the method of choice. Twenty participants sorted 88 figures that were drawn in the first experiment (44 figures from the beauty criterion and 44 from the control group). They were told that certain figures were drawn with the beauty instruction and that some figures were drawn without the beauty instruction. Their job was to sort the 88 figures into two baskets, one assigned to the beauty condition and one assigned to the control condition. The results showed that there was “considerable agreement between both experimental tasks (sorting vs. drawing)” (Hoge, 1996, p.85). The average proportions for the drawn shapes were 0.74 for the beautiful condition and 0.64 for the control condition, while the average proportions for the sorted figures were 0.71 for the beautiful condition and 0.66 for the control condition. Therefore, the experiments showed that “there is no special aesthetic attractivity of the golden section” (Hoge, 1996, p. 86).

The current experiment used the method of production because participants created shapes on a computer screen. The question was whether students would draw a 1: 1.168 rectangular quadrangle (golden section proportion) even if the given sides for each of the four trials were different lengths. The variables manipulated in the study were side length and color. The hypothesis was that even if the given side length varied, students would still draw a rectangular quadrangle close to the golden section that was 1:1.618 in proportion and that as given side length increased, drawn side length would increase. This would be in conjunction with Hoge’s findings in his experiment. The variable of color was two levels. Participants either made a black shape (closed) on a white background or a white shape (open) with a black outline on a white background. Fechner had only used solid (closed) shapes in his experiment and only open shapes could be made in the method of production studies until now since paper and pencil could make only an outline of a shape. Using a computer allowed a test to see if there were differences between the two. There was no hypothesis as to whether solid or open rectangles would create shapes closer to the golden ration. There was also no hypothesis as to whether there would be an interaction between color and given side length.

Method

Participants

Thirty-four student enrolled in an experimental psychology course from Salisbury University participated in the experiment as part of a class activity. There were 28 females and 6 males. The participants were from the three different sections of the class offered during the fall 2005 semester and were all psychology majors. All participants were treated ethically.

Materials

 Participants used a Gateway EV700 computer with 17 in (43.18 cm) monitor to produce the rectangular quadrangles. The computer was set on 800 x 600 resolution. The program used by the participants was version 5.1 of the Microsoft Paint Program. Participants used a standard wired mouse to drag from one point to another creating a rectangular quadrangle that looked good to them. The coordinates (the location for each dot) were set at 238,281 and 368,281 for 130 pixels, 238,231 and 418,281 for 180 pixels, 238,281 and 468,281 for 230 pixels and 238,281 and 518,281 for 280 pixels. They sat on a 24 in (60.96 cm) stool at a 36 in (91.44 cm) lab table.

Design and Procedure

            There were four different lengths of given sides, 130, 180, 230, and 280 pixels. With the four different given lengths, 24 different combinations of the lengths could be made. All 24 were used in the study. Participants were assigned to one out of the twenty-four conditions using block randomization and counterbalancing. Each class was told to send in the next student to the research lab after the first one came back based on their seating position in class. As the students came in they were randomly assigned by each block. The order of students that were to draw solid rectangular quadrangles or open ones was counterbalanced

The participants read and signed the informed consent statement before any instruction began. They were then asked to sit in front of the computer and to read the instructions on the screen while the experimenter read them aloud.  The instructions asked the participants to create a rectangular quadrangle that looked good to them, and it was explained that a rectangular quadrangle is a four-sided figure with all 90-degree angles. They were to draw the shape on the screen by using the mouse and clicking on one of the two red dots and then dragging the mouse to the other red dot (which determined the given length).

The participants were given two practice trials to ensure they understood the instructions. During this time they were told that if they made a mistake, such as not holding down on the mouse, stopping it before they wanted to, they could click on the undo button at the top of the screen.  Each participant completed four trials, with a different given side length on each trial. No participants were dropped from the experiment. As the participants completed the experiment they were debriefed and told that the experiment was interested in the proportions of the shapes they drew. They were also told that if they had any questions they could be addressed at that time or by contacting the experimenter later on. The participants were asked not to discuss the experiment with anyone for a few days.

Results

As Figure 1 indicates, there were 21 rectangular quadrangles drawn (out of 136) that were in accordance with the Golden Section hypothesis of an aesthetically pleasing figure with a short side to long side ratio of 0.618. There were also 21 rectangular quadrangles drawn that were about half that size (0.32), creating a bimodal distribution. This means that there was no preference for the golden section proportion.

Figure 2 shows that the means of drawn sides increased as the length of the given side increased. At an alpha level of .05, the effect of given side length was statistically significant, F(3,  96) = 12.617, p < .01. This means that the longer the given side was, the longer the drawn side. As a measure of effect size, partial eta – squared came out to be 0.283, this means that about 28% of the variability in drawn side length can be explained by given side length.  A Tukey’s HSD post hoc test reveals that the mean differences between certain given side lengths (measured in pixels) are significant at the .05 level. Between130 pixels and 230 pixels (p = .000), between 130 pixels and 280 pixels (p = .000) and finally, the mean difference between 180 pixels and 280 pixels is also significant at the .05 level (p = .006).

            The condition of solid rectangular quadrangles versus open rectangular quadrangles was not statistically significant, F(1,  32) = 0.311, p = .581. Drawing a white (open) or a black (solid) shape had no effect on the drawn side length and the difference that did occur between the means ( M = 107.985 for solid , and M = 99.529 for open with a standard error of 3.968 for each) probably occurred due to chance. The interaction between solid versus open shapes and given side length was also found not to be statistically significant F(3,96) = 2.695, p > .05. The analysis also showed that large individual differences had occurred, F(32,  96) = 7.294, p < .01. Partial eta – squared was 0.709, meaning that about 70% of variability in drawn side was due to participant’s individual differences.

Discussion

            The results of the experiment do not support the hypothesis that Fechner’s golden section proportion of 1:1.618 would be drawn more often than any other proportion (therefore, it is not more aesthetically pleasing than any other proportion).  The findings of the present experiment are similar to those in Holger Hoge’s experiment in which he also found no support for the golden section, either with drawing a shape, or sorting shapes into beautiful or control piles. On the other hand, the results did support the hypothesis that as given side length increased, drawn side length would increase as well. This finding also supports Hoge’s findings from his experiment done in 1996. The question of open versus closed figures was not solved in the current experiment.Using different colors may produce different results.

            Since the experiment was very controlled and had high internal validity, the problem of external validity arises. The findings may not be able to be generalized to the general population because psychology students may differ systematically as to what they feel a good-looking rectangular quadrangle looks like. The limited sample of participants makes it hard to say that most people do not have a preference for the golden section ratio. Also, participants may have drawn what they believed to be a beautiful shape on the first try, but may have become bored after six trials (2 practice, 4 test) with such a remedial task, or may have just not wanted to make the same shape more than once. On the other hand, since the experiment did have high internal validity, one can say with confidence that the differences in the dependent variable (drawn side length) are mostly due to the independent variable (given side length). The only other possibility being individual differences, which can always be found because people will differ. For future studies, experiments could be done using more trials with more lengths of given sides, different colors of shapes, or a different sample of participants.  

References

Fechner, G. T. (1997). Various attempts to establish a basic form of beauty: Experimental aesthetics, golden section and square (M. Neumann, J. Quehl, & H. Hoge, Trans.). Empirical Studies of the Arts, 15, 115-130.  (Original work published 1876).

Hoge, H. (1996). The golden section hypothesis – A funeral, but not the last one… Visual Arts Research, 22, 79-87.

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