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
Thirtyfour 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
twentyfour 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 foursided figure with all 90degree 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 goodlooking 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, 115130. (Original work published 1876).
Hoge, H.
(1996). The golden section hypothesis – A funeral, but not the
last one… Visual Arts Research, 22, 7987.
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