PREFERENCE JUDGMENT FOR DYNAMIC SYMMETRY
by
Theresa Stroup Ferg
B.F.A., University of Colorado at Denver, 1974
M.A., University of Northern Colorado, 1976
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Integrated Sciences
2005
2005 by Theresa Stroup Ferg
All rights reserved.
This thesis for the Master of Integrated Sciences
degree by
Theresa Stroup Ferg
has been approved
by
IPOS'
Date
Ferg, Theresa Stroup (M.I.S., Master of Integrated Sciences)
Preference Judgment for Dynamic Symmetry
Thesis directed by Professor Peter Kaplan
ABSTRACT
Over significant periods of time and across cultural traditions, dynamic symmetry
images have been consistently preferred by humans in aesthetic design and tiling
patterns. In 1865, Gustav Fechner, the pioneer of psychophysics, presented test
results demonstrating a preference for dynamic symmetry (also known as the golden
ratio or the golden section). In the manner of Fechner, this study investigates
whether, when shown images based upon dynamic symmetry, adult participants
report a consistent (pleasing) aesthetic appeal. Participants were asked to rank, in
order of preference, nine sets of three images (twenty seven total images), and each
set of three was simultaneouslypresented. The comparison set, Group A, consisted
of nine images derived from a novel geometric form based on dynamic symmetry
using Euclidean construction. Groups B and C were a horizontal or vertical
distortion of the Group A images and present a range of ratios from 1.06:1 to 2.5:1.
The Group B images were those from Group A, but distorted 20% or 40%
horizontally (there were six BH20% distortions and there were three BH40%
distortions). The Group C images were those from Group A, but distorted 20% or
IV
40% vertically (there were six CV20% distortions and there were three CV40%
distortions). Participants chose for first choice the dynamic symmetry Group A
images at 48.9%. Participants chose for first choice the BH20% distortions at
19.3% and the CV20% distortions at 15.2%. Participants chose for first choice the
BH40% distortions at 8.2% and the CV40% distortions at 7.6%. A Chi square
goodnessoffit test for a difference between the observed and expected distributions
demonstrates a significant result: X2 (17, n = 139) = 70.91 p < .0001. The evidence
of a preference for the dynamic symmetry images strongly suggests that there is a
correlation between the participants perception of the dynamic symmetry images
and their preference for the mathematical ratio.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Signed
Peter Kaplan
DEDICATION
This is the degree that everyone has earned.
ACKNOWLEDGMENT
There are many, many people who have helped with this study in some way
for the past five years. However, without any one of the people listed here, it would
not have been accomplished. Special thanks go to:
Zenas Hartvigson
Diana Tomback
Peter Kaplan
Mary CoussonsRead
Bill Briggs
Craig Johns
Doris Kimbrough
Ellen Stevens
Erica Ferg
Bruce Urmacher
Jean Ethridge
Connie Buffalo
Jeanne CollopyBach
Rosemary Wormington
Susan Koenig
John Plessinger
Shannon Durkee
Cary Miller
Liz Lhotka
Dave Aker
Ellie Miller
Marc Donsky
Joy Berrenberg
Joan Bihunan
Omar Swartz
Kathy Thomas
Holly Saru
Tali Rosencranz
Phylis Porterfield
Phil Kim
And my incredible and brave children Erica, Stephanie and John.
CONTENTS
Figures...............................................ix
Tables................................................xi
CHAPTER
1. INTRODUCTION........................................1
Dynamic Symmetry in Art and Design....................2
Background of Prior Research..........................3
2. METHOD.............................................11
Visual Stimuli.......................................11
Research Participants................................12
Procedure............................................14
Results..............................................17
3. DISCUSSION.........................................22
NOTES
1. Mathematical Definitions of Dynamic Symmetry.......34
2. Expanded Dynamic Symmetry Definition...............34
3. Creativity Survey..................................35
APPENDIX
A. DESIGN IN NATURE AND ART...........................37
B. DYNAMIC SYMMETRY CONSTRUCTION......................41
C. STUDY PROCEDURES...................................47
GLOSSARY....................................................60
REFERENCES..................................................62
VUl
FIGURES
Figure
3.1 Dynamic symmetry ratio comparison total first choice...........28
3.2 Comparison of ratios pentagram image first choice............. 29
3.3 Comparison of ratios spiral image first choice.................29
3.4 Comparison of ratios mtn image first choice....................29
3.5 Comparison of ratios kite image first choice...................29
3.6 Comparison of ratios portals image first choice................29
3.7 Comparison of ratios thick rhombus image first choice..........29
3.8 Comparison of ratios dart image first choice...................30
3.9 Comparison of ratios thin rhombus image first choice...........30
3.10 Comparison of ratios anvil image first choice................30
3.11 Experienced visual art participants dynamic symmetry ratio
comparison total first choice..................................31
A.1 Islamic design...................................................39
A2 Sunflower.........................................................39
A3 Religious icon....................................................39
A. 4 Le Modulator...................................................39
A. 5 Fivesided tilings.............................................40
A. 6 DNA molecule
40
B.l Dynamic symmetry geometric image.....................................41
B.2a Construction of dynamic symmetry images.............................45
B .2b Construction of dynamic symmetry images............................45
B. 2c Construction of dynamic symmetry images............................45
C. 1 Analysis of the dynamic symmetry preference per image
and quadrant........................................................55
x
TABLES
Table
3.1 Prior research studies....................................................32
3.2 An analysis of variance for each of the nine distributions................33
C. 1 Rankings of the preference for the images and symmetry analysis..........53
C.2 Dynamic symmetry preference matrix by image position.......................53
C.3 Quadrants of the participants by gender and preference.....................57
C.4 Quadrants of the participants by age in years and preference...............57
C.5 Quadrants of the participants by the percent of the number of creative
activities and preference..................................................57
C.6 Quadrants of participants living with children and preference..............57
C .7 Quadrants of the participants living with children by the percent of the
number of creative activities and preference...............................58
C.8 Percentages of the number of creative activities per quadrant..............59
CHAPTER 1
INTRODUCTION
In an effort to develop a cognitive model for sensory aesthetics, researchers
have been investigating a preference judgment for a mathematical ratio called
dynamic symmetry (also known as the golden ratio or the golden section). The
Golden Section Hypothesis states that humans have a preference for objects or
figures whose dimensions fit the mathematical ratio of 1.618:1 (Fechner, 1865,
pp. 115130). At the center of this scientific controversy are sensory aesthetics, and
how an aesthetic response may be related to the perception of that mathematical
ratio. In the debate about a preference for dynamic symmetry, there are two main
theories. The first theory supports an innate aesthetic preference for dynamic
symmetry, and the second suggests that the preference for dynamic symmetry is
learned, and therefore variable.
The mathematical ratio has many different names. Among those is the
golden section, golden mean, divine proportion, dynamic symmetry, 0 (Phi), or the
division of extreme and mean ratio DEMR (HerzFischler, p. 164, appendix I,
1998). Multiple mathematical methods describe this ratio. A dynamic symmetry
rectangle has sides with the approximate ratio 1.618:1. The ratio also exists in an
isosceles triangle (a triangle with two equal sides and 72, 72, 36 angles), in which
1
the sidetobase ratio is 1.618:1. Algebraically, dynamic symmetry is the positive
root of x2 x 1 = 0, the quadratic equation1, which is a seconddegree polynomial
equation, 0 = (1+J5 )/2 = 1.618_In this study, that mathematical ratio will be
referred to as dynamic symmetry2.
Dynamic Symmetry in Art and Design
Historically, art images and patterns follow a rough estimate of dynamic
symmetry in twodimensional form. Lynch and Hathaway (1994, pp. 211220)
conducted a study of the dimensions of450 ancient objects called projectile
points, and statistical analysis of these objects suggests that dynamic symmetry
may have had an aesthetic significance in the creation of objects for some of the
ancient inhabitants in what is now Georgia, USA, during the Middle to late Archaic
periods, 5,500 to 2,000 BCE. The findings show consistent historical use, in both
art and weaponmaking, of a mathematical ratio similar to dynamic symmetry.
Shortess, G.K., Clarke, J.C. & Shannon, K. (1997, pp. 165176) calculated
the ratios of height and width used in both traditional and popular art paintings. The
study found there was excellent agreement with the side ratio of approximately 3:1.
Ratios of twodimensional art supplies and other objects were analyzed; the
majority of the representative ratios for many of these objects were found to be
2
close to a ratio of 3.0:1 and to the side ratio of 1.33:1. Shortess, et al., reports that
these findings did not support a preference to dynamic symmetry.
Recent research on the theory of human preference for dynamic symmetry
by Mario Livio (2002) reveals inconclusive findings. He investigated the role of
dynamic symmetry as a standard for aesthetics (beauty). Livio reviewed numerous
artistic and architecture objects. He found measurable evidence that dynamic
symmetry had been used in some of the cases; however, he concluded that from the
historical record the dynamic symmetry ratio can not be considered a standard for
beauty (Livio, 2002, p. 200).
Background of Prior Research
In 1865, Gustav Fechner, a pioneer in the field of psychophysics, conducted
experiments testing his hypothesis that a correspondence exists between aesthetic
appeal and the mathematical ratio known as dynamic symmetry. Fechner asked
participants to rank their preferences for ten rectangular shapes that ranged from a
square, with a 1:1 side ratio, to a rectangle with a 2.5:1 side ratio. The 1:1 ratio had
equal vertical and horizontal lengths and the ratio 2.5:1 had a 2.5 times greater
horizontal length if horizontally presented or a 2.5 times greater vertical length if
presented vertically. All of the rectangles had an equal area. In the order of long
toshort ratios for the ten rectangles, dynamic symmetry was seventh from the
3
square. These shapes were presented on white cards with a black background.
Fechner used a randomized procedure to present the rectangles in a different order
for each participant. If unable to choose a mostpreferred rectangular shape, the
participants were allowed to select more than one rectangular shape. Fechner also
had the participants chose the leastpreferred rectangular shape. Of the total
responses, 35% of the participants showed a preference for dynamic symmetry,
20.6% of the participants showed a preference for the sixth rectangle with a side
ratio of 1.5:1, and 20.0% of the participants chose the eighth rectangle with a side
ratio of 1.77:1. The results showed a preference for the dynamic symmetry
rectangle that had a 1.618:1 side ratio, as well as the two ratios nearest the dynamic
symmetry rectangle dimensions. The participants least preferred the ratios 1:1 and
2.5:1, which were the furthest from the dynamic symmetry ratio (Fechner, 1865,
pp. 100112) and (Green, 1995, pp. 937968).
Dynamic symmetry has become a controversial topic of research over the
past century because the results of selected studies of varying experimental designs
have indicated inconsistent results Table 3.1. This inconsistency is possibly due to
an absence of the understanding of a broader context of cognitive functioning. In
particular, researchers have used methods of experimentation that produce
confounding factors by combining multiple sensory aesthetics and psychophysics
procedures. The following descriptions of previous dynamic symmetry testing
4
show that when the experimental method was limited to a preference judgment of
just the comparison of ratios, or of line division (a procedure dividing a line into two
smaller lines of varying proportion), the results showed support for the original
Fechner hypothesis.
John Benjafield (1985, pp. 117134) reviewed research on the golden section
hypothesis; specifically, the psychophysics procedure of line bisection. He reported
a preference for the dynamic symmetry ratio in the procedure of dividing a line into
two smaller lines of varying proportions. He concluded that dynamic symmetry was
possibly more pleasing than other ratios. Benjafield proposed that dynamic
symmetry might be useful in understanding other forms of sensory experience.
Benjafield and McFarlane (1997, pp. 143151) investigated the role of
context in determining which rectangular ratios participants preferred. Participants
were shown three different ratios of rectangles in three different orders. The order
of presentation of the rectangles most influenced aesthetic preferences when the first
set of ratios presented contained rectangles with a longer vertical length
measurement. When the first set of ratios contained rectangles with a longer width
measurement and were arranged with the dynamic symmetry ratio at the middle
position, the mostpreferred ratio was dynamic symmetry.
McManus and Weatherby (1997, pp. 209232) used a new design method of
randomized paired comparisons and object placement. The researchers conducted
5
an experiment to have the participants place an object inside a premade picture
frame to find the participants preferred optimal placement. The results showed that
the overall participant preference was to place the objects at the two dynamic
symmetry positions horizontally and the two dynamic symmetry positions
vertically. These findings were supportive of a preference for dynamic symmetry.
Additional empirical studies show that inconsistent data on a preference for
dynamic symmetry might be partly influenced by differences in the verbal
instructions given to the participants. Hekkert (Hekkert, Peper, and van Wieringen,
1994, pp. 185203) showed that when participants, both naive and experienced
(those without and those with art school training, respectively), were given
subjective test instructions, e g., Choose the most pleasing, the naive participants
consistently preferred dynamic symmetry. When the same naive and experienced
participants were given objective test instructions, e g., Choose the most correct,
neither naive nor experienced participant group consistently preferred dynamic
symmetry. Additionally, the researchers concluded that the naive participants
group was significantly more consistent in its judgments than was the experienced
participant group.
In an effort to establish a predictable parameter for the preference for
dynamic symmetry, Macrosson and Strachan (1997, pp. 153163) selected a group
of experienced artists, who were product designers, as participants to do simple line
6
partitioning exercises. The experimental psychophysics procedure was the line
bisection method. The experienced designers mainly preferred to divide a line with
the lengths of the line, 1.0:1 or 2.0:1. The results of preference judgment by the
experienced designers were not supportive for a preference to dynamic symmetry.
The authors stated that the results preferring the side ratios of 1.0:1 or 2.0 indicated
an aesthetic maturity due to their professional backgrounds in the use of learned
design elements.
Researchers have attempted to replicate the traditional Fechner experiments
with novel experimental methods to further explore the dynamic symmetry
preference question. The experimental findings have not been supportive of a
preference for dynamic symmetry. Hoge (1995, pp. 131148) varied the method of
instruction and experimental arrangement of shapes to test for a dynamic symmetry
preference. The participants were given a variety of experimental procedures to
complete. The first procedure was similar to Fechners method of choosing a
rectangle; for the second, the participants were asked to draw rectangles, and for the
third, the participants were asked to sort rectangles. Hoges findings indicated that
preference judgments seem to be the result of information processing using two
cognitive methods: The physical arrangement of the object (tangible) and a
representation of the concept of the object (intangible). He did not find a preference
for dynamic symmetry from the test methods he used.
7
Similar studies that compare varying sidelength rectangles are inconclusive.
Boselie (1992, pp. 118) systematically compared the aesthetic appeal of the 5.0:1
rectangle with the dynamic symmetry rectangle. Boselie added a diagonal line
inside the dynamic symmetry and the rectangles with a side ratio of 5.0:1. This
alteration of dynamic symmetry by an interior diagonal line within the dynamic
symmetry rectangle was a confounding factor because the diagonal line altered the
presentation of the dynamic symmetry. The results did not show that dynamic
symmetry was preferred, and Boselie concluded that other characteristics of visual
patterns were more important to the participants.
Konecni (1997, pp. 177207) focused his experimental methods on
broadening his investigation of a preference for dynamic symmetry to other sensory
aesthetics factors. Konecni investigated the dynamic symmetry preference using
Fechners methods, with new stimuli (contours and cutouts of vases constructed
with dynamic symmetry and nondynamic symmetry). The participants completed
five tasks related to the placement of vases on a laboratorybuilt mantle and an
imaginary mantle. There was no evidence of a preference for the dynamic
symmetry placement position of vases on the mantle. Konecni reported a search for
balance seemed more of a motivation in the participants choice of mantle
placement.
8
The design of the current experiment incorporated three methods taken from
previous research, and employed a different way of generating nondynamic
symmetry stimuli that maintained the basic shape of the target image, but distorted
the stimuli to alter the dynamic symmetry. In the current study, the tests variable
was limited solely to the mathematical construct of dynamic symmetry.
The design of the current study was a forced choice preference method,
which required the participants to rank the images. It also used a new way of
generating the control stimuli. The design was new because it uses the dynamic
symmetry ratio in a variety of different images, rather than in just the image of a
rectangle. The images used can be found in nature and in historical design
(Appendix A: Figures A.1 A.6). Nine different dynamic symmetry images were
randomly placed alongside two distorted ratios of the same image, for a total of 27
images, displayed threepercore board. The distorted images had varying side
ratios with a range from 1.06:1 to 2.5:1 (Appendix B: Dynamic symmetry
construction). In the current study, participants were given subjective instructions
to Choose the most pleasing image. The instructions prepared the participants to
make an aesthetic judgment based specifically on their personal like or dislike of
dynamic symmetry when comparing the various ratios.
This method was consistent with prior studies that showed when the naive
participants were given the subjective instruction of choosing a pleasing image,
9
their preference for dynamic symmetry was predictable (Hekkert, et al., 1994, pp.
185203).
In this experiment, the primary hypothesis is that people would consistently
select the dynamic symmetry version of each shape. A second hypothesis is that
this preference would be consistent across the shapes. A third hypothesis states
there would be a relationship or correlation between the means of the distributions
in the category factors of age, gender, creative activities. In prior research, naive
participants preferred dynamic symmetry and experienced visualartist participants
did not. Therefore, in this study, participant demographics were requested in order
to investigate the third hypothesis.
10
CHAPTER 2
METHOD
Visual Stimuli
For the current study, a geometric construction was developed to create and
verify the comparison set of visual stimuli, Group A, to be used in a test for a
preference judgment for dynamic symmetry ( Appendix B: Figure B. 1 Dynamic
symmetry geometric image). Dynamic symmetry is a ratio, the division of extreme
and mean ratio (DEMR), that begins with various shapes that can be divided into
composite parts via a specific ratio, 0 = (1+V5 )/2 = 1.618...= 1.618:1.
Mathematically the dynamic symmetry implies a spiral, perceived as a cone, coiled
about an axis (Huntley, p. 168, 1970). The spiral is without a terminal point, it
may grow outwards (or inwards) indefinitely, but its shape remains unchanged
(Huntley, p. 102, 1970). The comparison set of stimuli selected for this study were
constructed from a dynamic symmetry geometric image. The dynamic symmetry
geometric image was created on the software computer program Geometers
Sketchpad 4.05s. When animated, the dynamic symmetry geometric form maintains
the dynamic symmetry ratio relationships whether expanding or contracting. This
information provides future empirical investigators a more rigorous mathematical
11
analysis of dynamic symmetry to further explore the description of the mathematical
construct.
The test for a judgment preference was designed as the set of comparison
visual stimuli developed from the dynamic symmetry geometric construction the
images (designated Group A images, Figures B.2aB.2c) were presented in a
prepared environment (Appendix B: Dynamic symmetry construction). The nine
Group A images were constructed precisely from the geometric model and were
images that have followed natural patterns and artistic design (Appendix A: Figures
A. 1 A. 6). All of the images were presented without the construction lines
(Appendix B: Dynamic symmetry construction). One of the Group A images was
used in each test. There were three different sets of visual stimuli; each set
presented one image from the comparison set of Group A images along with two of
the distortions from groups B and/or C the distorted dynamic symmetry images.
The same nine tests were administered to all the participants in the same order.
Research Participants
The research participants for this study were student volunteers from the
University of Colorado Denver Health Sciences Center (UCDHSC) at the
downtown campus. The location of the study was a variety of classrooms in the
12
North Classroom and Plaza buildings on the downtown Auraria campus during the
months from August through October 2003.
The comparison trials were administered to 139 students at the university.
The student population at the UCDHSC downtown campus is demographically
diverse, which helped to achieve a representative sample of test participants. The
students at the Auraria campus are the most ethnicallydiverse campus in the CU
system and the students range broadly in age. The campus, located in the downtown
area of the city of Denver, attracts nontraditional students. The sample population
for the study was 73 females and 66 males. Forty seven students were between 18
and 20 years of age, 70 students were 21 to 30, 13 students were 31 to 40, 5 students
were between 41 and 50, and 4 students were older than 51 years. The request for
volunteers was accomplished by petitioning the Deans of the Colleges of Liberal
Arts and Sciences, Arts and Media, Engineering, Business and Education. Each
Dean was sent a letter asking permission to approach the faculty of the introductory
program courses. A second letter to the faculty solicited permission to speak to the
students in the classes to request their volunteer participation in the experimental
testing.
Each of the participants received a consent form. The consent form was a
brief overview of the time commitment and stated that the participation was
13
voluntary and confidential. Each participant was listed under a threedigit sequence
number to keep the experimental data confidential. The visual testing lasted about
10 minutes per participant. General demographics information was obtained along
with present family circumstance, creative interests and activities. The creativity
survey3 and demographics questionnaire collected information from the participants
to identify their levels of artistic and creative activity. This was done to determine if
there was a relationship between the participants dynamic symmetry scores and
their creative interests (Appendix C: Study procedures).
Procedure
The participants were tested in an average size classroom. The chairs were
arranged to one side of the room so the participants were able to stand and
simultaneously view the sets of three visual stimuli. The visual stimuli images were
printed on 8.5 x 11 white papers and evenly spaced on the core boards. The visual
stimuli images were made in black ink with lines of equal width. There were nine
white core boards, 20 inches by 36 inches. One core board at a time was placed on
a podium set against the wall. The images were seen at a height of five and a half
feet. The subjects viewed the images from a distance of 12 ft.; the distance was
measured and marked on the floor. The lights in the room were off and any
14
windows were covered. There was a single light source (40 watt bulb) placed
directly above the images. When the participants entered the room one at a time
they were given the instructions for the visual testing and told that a research
assistant would record their responses during the testing. It was explained to the
participants that after the visual testing they would fill out a questionnaire and
survey at a desk outside the testing room. The instructions to the participants were
to choose the most pleasing image and rank them one, two or three, with one being
the most pleasing. (Appendix C: Study procedures) The nine tests were
administered in a continuous sequence; the participants made the observations
without interruption.
The participants were asked to visually discriminate, by forced choice
preference, between Group A images with the dynamic symmetry ratio and Groups
B and C of distorted images with alternate ratios. The geometric construction
mathematically verified the comparison set, Group A, of visual stimuli, which were
used to measure the observations of a preference judgment by the participants using
a ranking method. The names of the Group A images were pentagram, anvil, dart,
spiral, portals, mountain (mtn), kite, thick rhombus and thin rhombus. The other
two groups, B and C, of images, were varying percentages of horizontal and vertical
distortions of the dynamic symmetry visual stimuli. BH20% images were Group A
15
images that were distorted horizontally to 80% and remained unchanged vertically,
and CV20% images were Group A images distorted vertically to 80% and remained
unchanged horizontally. Three of the sets of nine images had one of each of the
distorted images a BH20% and CV20%. BH40% images were Group A images that
were distorted horizontally to 60% and remained unchanged vertically, and CV40%
images were Group A images distorted vertically to 60% and remained unchanged
horizontally. Three of the sets of nine images had one of each of the distorted
images of a BH20% and a BH40%. Three of the sets of nine images had one of
each of the distorted images of a CV20% and a CV40%. The distorted images had
varying side ratios with a range from 1.06:1 to 2.5:1. The BH20% distortions are
ratios between 1.73:1 and 2:1. The CV 20% distortions are ratios between 1.41:1
and 1.25:1. The BH40% distortions are ratios from 1.85:1 to 2.5:1 and the CV40%
are ratios from 1.22:1 to 1.06:1. The orders of the presentation of each set of three
visual stimuli groups A, B and/or C and the placements of the 27 images on the core
boards were randomly determined by computer. Each Group A, B and/or C image
(27 total images) was assigned a number. Then each of the nine test sets was
assigned a number. The order of the nine sets was randomized. The order of the 27
images within each set was randomized.
16
Results
The independent variable for the primary hypothesis of this experiment was the
treatment given to the visual stimuli groups A, B and/or C, i.e., each image of
dynamic symmetry and each dynamic symmetry images variations. Using the
psychophysics method of forced choice preference, the tests measured which image
was chosen first (i.e., preferred), and the number of choices (i.e., recognition and
choice of dynamic symmetry) made by the participants. The preference
observations by participants for the three visual stimuli from each dynamic
symmetry test were judged with a ranking of 1, 2 or 3, with 1 the mostpreferred
rank, and 3 the leastpreferred. Each participant was given a total score, the ranking
of each first choice multiplied by the nine image tests. The average score was the
average of all the trials for all image tests. Each participant was also given a first
choice score, which was the number of times dynamic symmetry was chosen first
for the nine image tests. These scores are the dependent variable the observed
preference for dynamic symmetry.
Participants strongly preferred the dynamic symmetry images at a
probability greater than that expected. Figure 3.1 shows that the dynamic symmetry
stimuli were selected as 49.8 % of the first choice images, across subjects and
repeated trials. Looking at the BH20% distortions and the CV20% distortions, the
17
BH 20% distortions were chosen as 19.3% of the first choice, and the CV20%
distortions were chosen as 15.2% of the first choice. Looking at the distorted
images BH40% and CV40%, the BH 40% distortions were chosen as 8.2% of the
first choice and the CV40% distortions were chosen as 7.6% of the first choice.
While the dynamic symmetry images with side ratios of 1.618:1 had a probability of
three chances out of nine to be chosen, dynamic symmetry received a 50% greater
than expected preference. For the distorted images BH20% and CV20%, there was
a probability of four chances out of nine to be chosen; the images received a 22%
less them expected preference. For the distorted images BH40% and CV40%, there
was a probability of two chances out of nine to be chosen; the images received a
28% less than expected preference. These results indicate a clear preference for the
dynamic symmetry side ratio of 1.618:1.
The rationale for the primary null hypothesis was that there would not be a
difference in the sample distribution of the observed scores vs. the expected scores.
The observed sample would be equally as often selected from the three types of
images from groups A, B and/or C. An appropriate statistical analysis for the null
hypothesis was a Chisquare goodness of fit test. The data (observations of a
preference for dynamic symmetry) scores from the test results were not similar to
the hypothesized expected (chance) distribution. A Chi square analysis showed
18
support for the alternate hypothesis that there was a significant difference between
the observed and expected sample distributions, X2 (17, n = 139) = 70.91 p < .0001.
Looking at first choice alone and all possible orders of ranking, a Chi square
(goodness of fit) test also shows strong evidence that dynamic symmetry was
preferred X2 (9, n = 139) = 52.15 p < .0001.
In looking at the testing method, there was independence in the random
population of student volunteers. Also, the orders in which the images were shown
and the placement of the visual stimuli within the nine sets was randomized.
However, the statistical assumption for independence in this study does not apply to
the analysis of the differences of the means for the nine visual stimuli. The
participants observations of a preference judgment, the dependent values, were
predetermined and the nine test scores are not considered independent events
because the observation score given to each of the participants is based on the
probability of the predetermined ranking. Using a nonparametric statistical
analysis test for the second hypothesis, the dependent variable was the overall
average score of the participants preference for dynamic symmetry and the
independent variable was the observations per participant per each image
distribution, the ranking. Figures 3.2 3.10 show the dynamic symmetry preference
for each of the nine images.
19
Looking at an analysis of variance for each of the nine distributions
separately for each Group A image, the second null hypothesis states there is no
difference in the means of the observation rankings of the dynamic symmetry
images and the means of the observation preference per image distribution. A
KruskalWallis nonparametric test for the pentagram, anvil, dart, spiral, portals,
mtn, kite, and thick rhombus statistical analysis Table 3.2 shows there were
significant differences in the means of the rankings per image and the observed
average preference score of the participants. The results of the data analysis for the
thin rhombus distribution showed that there was a marginally significant effect for
dynamic symmetry.
All of the study participants filled out the creativity survey and
demographics questionnaire. Of the 139 individuals, eleven participants answered
that they were experienced visual artists preparing for or already involved in a
professional art career. Four of the eleven did graphic design, their first choice
scores were 4, 3, 5, and 2 out of a possible 9. Four of the participants were painters,
their first choice scores were 4,3, 3, and 1 out of a possible 9. Two of the
participants were art teachers, their first choice scores were 3 and 8 out of 9, and one
was an architect, his score was 5 out of 9. Comparing the visual art participants
dynamic symmetry first choice scores and the scores of the sample population,
Figure 3.11 reveals a similarity Figure 3.1. A paired samples ttest shows a
20
significant result, pvalue = .0001, for a correlation between the total first choice
score sample population of participants and the total first choice score of the visual
artists (Appendix C: Study procedures).
21
CHAPTER 3
DISCUSSION
The results of this study strongly indicate that the participants were able to
discriminate, on first choice, between the dynamic symmetry images and the
BH20%, BH40%, CV20% and CV40% distortions. In general, the participants
preferred dynamic symmetry. Each of the nine dynamic symmetry visual stimuli
images was preferred and the thin rhombus was marginally preferred. The findings
from this current study show a greater level of preference for dynamic symmetry
across nine different images by naive and art experienced participants than has been
previously reported.
The participants were able to discriminate (except for the thin rhombus)
from the visual stimuli that had dynamic symmetry and to prefer the dynamic
symmetry ratio consistently in nine different stimuli image tests. In prior research,
the use of stimuli images has been limited primarily to varying ratios of the
rectangle and line length division. Those images and the use of the nine novel
dynamic symmetry test images focus attention on two important factors in the
understanding of the mathematical concept of dynamic symmetry. First, dynamic
symmetry is an approximation. This approximation is an irrational number, which
is a number that does not have a definitive end. Dynamic symmetry is the most
irrational number. This mathematical concept is known as incommensurability
22
(having no common measure) (HerzFischler, p. 49, 1998). When calculating
dynamic symmetry this incommensurable aspect of the concept invites confusion.
In order to precisely evaluate the proportions of a geometric shape, this
contradiction of exactness of measurement must be understood. It is possible to
designate dynamic symmetry mathematically and yet it is paramount to understand
that the designation is only an approximation. To represent dynamic symmetry as a
definitive number and also use dynamic symmetry as an exact measurement is an
oxymoron. This contradiction has stymied scientific investigations of the dynamic
symmetry concept because the very nature of the scientific method of quantitative
investigation requires complete confidence in the measurements for the treatments
given to stimuli. However, it is possible to empirically test for the aesthetic appeal
of the mathematical ratio, as in the Fechner study. By using a representation of the
dynamic symmetry mathematical ratio (whether a rectangle or line division or an
shape derived from the dynamic symmetry geometric image) and presenting the
representation of dynamic symmetry within a mathematical range of side ratios, for
example 1:1 to 2.5:1. The participant can visually discriminate between the ratio
relationships. Once the participant can determine the differences in the array of
images presented and if the presentation is limited to just dynamic symmetry and
similar stimuli shapes without the dynamic symmetry; a preference for dynamic
symmetry may be revealed.
23
The second factor in the debate about a preference for dynamic symmetry is
whether or not the preference is an innate or a learned preference. This question
addresses the possible origin of the aesthetic appeal of the mathematical ratio. It
can be shown that dynamic symmetry exists in the design of numerous examples in
nature, such as the logarithmic spiral growth of a sunflower and the in design of
DNA. Since the natural world around us and within us has examples of dynamic
symmetry, a preference for dynamic symmetry may exist because humans are of
dynamic symmetry and are predisposed to the design. Or possibly, our preference
for the mathematical ratio exists because humans have been exposed to the design in
nature and humans have therefore learned to use it in art and architecture.
Additional dynamic symmetry preference testing is important in order to find out if
this aesthetic appeal exists in other cultures and societies. Crosscultural testing
using the current dynamic symmetry test design would allow a comparison of the
results from the many individuals that have been exposed to different geographical
and cultural settings around the world. For now there is no agreement in the
scientific community as to the existence of a preference for dynamic symmetry.
Once there is acceptance of the preference for dynamic symmetry (i.e. the Golden
Section Hypothesis), then the number of research investigations may increase to
find an answer as to whether the preference is innate or learned. Prior research
24
shows that this question has not been definitively answered and research on the
possible origin of the preference for dynamic symmetry needs further investigation.
The participants of the current study also preferred the BH20% and CV20%
distortions over the BH40% and CV40% distortions, which was similar to prior
studies. These results are consistent with the original Fechner study that showed the
participants preferred the ratios closest in ratio measurement to the dynamic
symmetry rectangle. Additionally, following the Fechner study, the participants of
the current study least preferred the BH40% and C V40% distortions, which were
furthest in ratio measurement from dynamic symmetry.
The results of a preference for dynamic symmetry of this current study are
consistent with prior studies that have found a positive preference to dynamic
symmetry from naive participants when given subjective instructions. The results of
a preference from this study show that the experienced art participants preferred
dynamic symmetry with a pattern of total first choice preference scores similar to
the scores of the sample population of participants. The results of a preference from
the subset group of experienced visual art participants show greater consistency than
prior studies for total first choice scores of the preference for dynamic symmetry by
experienced artists when given subjective instructions. In prior studies, the
experienced visual art participants chose the ratios least preferred by participants of
prior studies and least preferred by the participants of the current study. This test
25
result for the current study, which contradicts prior research, may indicate that under
specific conditions a preference for dynamic symmetry may be revealed, which
would indicate support for an innate preference judgment for dynamic symmetry.
In conclusion this study advances the knowledge base in the fields of
psychophysics and geometry on the subject of dynamic symmetry. In the manner of
Fechner, quantitative investigation of the functional interrelations of body and mind
such as the aesthetic appeal for the mathematical construct of dynamic symmetry
may help in the continuing scientific efforts to develop a cognitive model that
addresses the integrative design of sensory awareness. Further study of the novel
geometric dynamic symmetry image gives novel mathematical support to the
importance and meaning of dynamic symmetry ratio relationships and aesthetic
appeal.
Investigation of the possible aesthetic significance and origin of dynamic
symmetry (i.e., its constant preference) is important because of the evidence of
dynamic symmetry in nature and mathematics, and because of its continuous
presence in human art and design. By continuing to research ancient historical
records for the use of dynamic symmetry, mathematical incommensurability, it
would be possible to explore the question of a predisposition of humans to the
preference for dynamic symmetry.
26
Increased awareness of the aesthetic appeal to dynamic symmetry may lend
greater support for emotional creative processes especially in education
curriculum. In 1910, John Dewey presented a traditional concept that has been used
in pedagogy to analyze the sequence of events in creative problem solving. The
sequence is: A difficulty is felt, the difficulty is located and defined, possible
solutions are suggested, consequences are considered, and, finally, a solution is
accepted (Gowan, Demos and Torrance, p. 164, 1967). Education programs could
be devised to support and develop integrated art, music, dance, math, science and
literacy curriculum projects, and sensoryawareness systems also designed for
alternative learning.
27
 Dynamic Symmetry (DS) Ratio Comparison
Percentage of First Choice per Ratio
o
DSRATIO BH20 BH40 CV20 CV40
BH = Group B Horizontal distortion
CV = Group C Vertical distortion
Figure 3.1 Dynamic symmetry ratio comparison total first choice
DS ratio (48.9%) BH20 (19.3%) BH40 (8.2%) CV20 (15.2%) CV40 (7.6%)
I
i
28
I
Â§ Pentagram Ratio Comparison
I Percentage of First Choice per Ratio
OSRXnO CV2G CS*0
BH Qvup B Hotanta! dMrton
CV* Group CVbScddhtorflon
Figure 3.2 Comparison of ratios pentagram image first
DSratio (70.50%) CV20(12.94%) CV40(16.55%)
Â§ Spiral Ratio Comparison
J Percentage of First Choice per Ratio
DSRATIO CV2Q CV*0
BH Qtmp B HorterW dwtoffeon
CV*QoupCtericrt distort**
Figure 3.3 Comparison of ratios spiral image first choice
DSratio (61.15%) CV20(25.89%) CV40(12.95%)
JMtn Ratio Comparison
Percentage of Fins* Choice per Ratio
0SRATIO BH20 CV20
BH Group Bllminroi tfclortcn
CV* Group C Vfcrtcai Morton
Figure 3.4 Comparison of ratios mtn image first choice
DSratio (53.96%) BH20(28.06%) CV20(20.14%)
I* Portals Ratio Comparison
E
> Percentage fo First Choice per Ratio
D68AJ10 8H20 CY20
BH * B HortanM dMorlsn
CV * C MU dstorton
Figure 3.6 Comparison of ratios portals image first choice
DSratio (46.04%) BH20(25.90%) CV20(28.06%)
 Kite Ratio Comparison
J Percentage of First Choice per Ratio
DSRATIO T 20 BMC
BH Gmro BHorupnTOt dhtorton
CV* Group C VMcai Askxtat
Figure 3.5 Comparison of ratios kite image first choice
DSratio (53.24%) BH20(30.22%) BH40(16.55%)
f Thick Rhombus Ratio Comparison
J' Percentage of First Choice per Ratio
DSRXnO BH20 BH40
BH bGinp BHariaontol distortion
CV* Qoup C tortical drstorfon
Figurc3.7 Comparison of ratios thick rhombus image first choice
DSratio (42.45%) BH20(36.69%) BH40(20.86%)
29
Dart Ratio Corrparison
f Percentage of First Choice per Ratio
U 5Q>
DSftATD BH20 CV20
BH Onip B Hatanhi tistofton
CV C VMkal divMon
Figure 3.8 Comparison of ratios dart image first choice
DSratio (42.45%) BH20(27.34%) CV20(30.22%)
f Thin Rhonrfcus Ratio Comparison
Â£ Percentage of First Choice per Ratio
DSRATIO CVQO OMC
BH = On*) B HanonW thfcrton
CV Qvup C VMctf dstMfon
Figure 3.9 Comparison of ratios thin rhombus image first choice
DSratio (39.57%) CV20(21.58%) CV40(38.85%)
I" Anvil Ratio Corrparison
j Percentage of First Choice per Ratio
DSRATIO 3H20 DH40
BH Group 6 Horton* dotation
CV* Group C Vwlcal dtstorfon
Figure 3.10 Comparison of ratios anvil image first choice
DSratio (38.85%) BH20(25.18%) BH40(35.97%)
30
I* Experienced Visual Art Participants
I* Ratio Comparison
ARTISTDS /JRTBH20 ARTBH40 ARTCV20 4RTCV40
BH = Group B Horizontal distortion
CV= Group C Vortical distortion
Figure 3.11 Experienced visual art participants dynamic symmetry ratio comparison
 total first choice
DS ratio (41%) BH20 (19%) BH40 (10%) CV20 (17%) CV40 (12%)
31
Table 3.1 Prior research studies
Order Scientist Method Result
1 Fechner, 1865 10 rectangles shown simultaneously ratio 1:1 to 2.5:1 DS preference
2 Bcnjafield, 1985 line division bisection of various line lengths DS preference
3 Boselie, 1992 DS rectangle compared to the 5:1 ratio with diagonals No DS preference
4 Hekkert, et aL, 1994 Fechner method: subjective and objective instructions Subjective: (artists & naive) DS preference objective: (artists & naive) no DS preference
5 Lynch & Hathaway, 1994 Measured ancient artifacts 450 ct. Shows DS preference
6 Hoge, 1995 Fechner method with additional tests draw, sort; objective and subjective instructions No DS preference
7 Benjafield & McFarlane, 1997 Rectangles three different ratios and three different orders, thick and thin lines Some preferences for DS for specific orders
8 Konccni, 1997 Line bisection, Fechner method, make rectangle, novel stimuli; cutout vases and placement No DS preference
9 Me Manus & Weatherby, 1997 Objects (vase) placed on a picture frame Horizontal and vertical DS points preferred
10 Macrosson & Strachan, 1997 Line bisection by experienced artists No DS preference, 1:1 or 2: lpreferred
11 Shortess, et aL, 1997 Measured existing rectangles in design Preferred 3:1 and 1.33:1 NoDS preference
32
Table 3.2 An analysis of variance for each of the nine distributions
Image KruskalWallis test
Pentagram X2(l,n = 139) = 26.105 p< 0001
Spiral X*(l, n = 139) = 25.660 p<0001
Mtn X^l, n = 139) = 25.274 p< 0001
Portals X2(l,n= 139) =15.039 p<0001
Kite X^l, n = 139) = 32.635 p<.0001
Dart X^l, n = 139) = 15.254 p<0001
Thick Rhombus X^l, n = 139) = 33.697 p<0001
Anvil X2(l,n= 139)= 11.803 p<001
Thin Rhombus X2(l,n= 139) = 3.534 p<060
33
NOTES
1. Mathematical Definitions of Dynamic Symmetry:
a. The division of line AC at point B. Point B divides the line in extreme and mean
ratio (EMR), which creates a constancy of the ratios; AC, the whole line: AB, larger
segment = AB, larger segment :CB, shorter segment.
b. To cut a given straight line so that the area of the rectangle contained by the
whole line and one of the segments is equal to the area of the square on the
remaining segment. (Euclid, Theorem II, 11)
c. The numeric value of Phi can be calculated by a quadratic equation:
AC = x, CB 1, so that AC/CB = x = 0
x+l/x = x/l, i.e., x2x1 = 0 the positive solution is (l+>/5)/2 = 1.61803, the
negative solution is (1V5 )/2 = 0.61803. If, AC = 1 and CB = x', then
x' + 1/ 1 = 1/x', i.e., x'2 x' 1 = 0 the positive solution is (JE1)/2 = .61803
(Huntley, p. 26 1970)
2. Expanded Dynamic Symmetry Definition:
Dynamic symmetry is a mathematical process that begins with various shapes that
can be divided into composite parts via a specific ratio 21:13 = 1.618, 0 =
(1+^5 )/2. The subdivisions of dynamic symmetry reduce to fractions (ratios) from
which emerges a spiral. The spiral is without a terminal point: it may grow
34
outwards (or inwards) indefinitely, but its shape remains unchanged (Huntley, p.
102, 1970). The dynamic symmetry geometric image retains these dynamic
symmetry ratios whether it is reduced or enlarged. The perception of this three
dimensional geometric patterning creates an aesthetic appeal.
3. Creativity Survey:
From the Creativity in Arts and Sciences: A Survey by A. Heck published in
Organizations and Strategies in Astronomy II, 2001.
1. Do you have a creative interest?
[painting, writing, music, science,...]
2. If so, what are your motivations for creating?
[none (spontaneous process), making a living, conveying messages, knowledge
advancement, career progress,...]
If several motivations, please rank them by decreasing order of importance.
3. Is the result of your creativity expected (you know in advance what you will
achieve) or (even partially) unexpected?
4. Would you say that creating is giving birth to something?
[feel free to elaborate]
5. Would you say that your creativity is produced by another 'person' inside you?
[feel free to elaborate]
6. What is the time of the day/week/month/year when the creative process is
working best?
7. Is weather influencing your creativity?
8. Is there any influence helping your creativity?
35
9. Please describe your creative process in a few words.
Are there several phases?
[preparation, concentration, depression...]
10. Did you notice an evolution/changes with age?
11. Would you say your creativity is a family gift?
[other creative people in your family?]
12. What do you think of the claimed parallel between artistic and scientific
creativity?
13. Additional comments?
36
APPENDIX A: DESIGN IN NATURE AND ART
Nine images were used in this experiment and they have dynamic symmetry.
Evidence of these dynamic symmetry images can be found throughout the ages.
They have been used as archetype designs and in tiling patterns. These designs and
patterns have persisted from early civilization to the present day; their continued use
suggests an innate preference. The iconic names are the spiral or shell, architecture
portals, and the pentagram, or fivesided star. The underlying construction of the
icons follows Euclidean geometry and the nine experimental images were derived
from that geometry (EuclidHeath, 1926), (Appendix B: Dynamic symmetry
construction).
Some of the geometric patterns of Islamic art are based on dynamic
symmetry. The dynamic symmetry images pentagram, kite, thick rhombus, dart,
anvil and thin rhombus are represented in (Figure A. 1 Islamic design). It has been
appreciated since antiquity that beauty arises if and only if the constituent parts of a
structure are harmoniously proportioned in relation to each other and in relation to
the whole. Beautiful patterns have to be based on some form of inner logic of
proportions, (Abas, p. 18, 1995). Natural patterns of logarithmic spiral dynamic
symmetry image can be found in numerous examples of botanical organisms, such
as the sunflower (Figure A.2 Sunflower), (Huntley, p. 164, 1970).
37
The iconography of religious figures is precisely drawn according to the
design of the dynamic symmetry geometric image (Figure A. 3 Religious icon)
(Sendler, pp. 107118, 1988).
French architect Le Corbusier proposed a universal standard of architectural
portal design. Le Modulor, Figure A.4 is the theory of structural design based on
the proportions of the human body (Hurlburt, p. 12, 1978).
Roger Penrose in 1974 (Livio, p. 208, 2002) discovered two basic sets of
fivesided tilings Figure A. 5 that exhibit pentagonal dynamic symmetry. The
patterns are nonperiodic and have matching rules that are applied to the shapes that
make up the patterns. This patterning has led to the discovery in 1984 of the
crystals for an aluminum manganese alloy.
The representational picture of DNA shows overlapping decagons (DNA
molecule) Figure A.6, which have dynamic symmetry. The DNA molecule is a
decagon geometric shape when viewed from above (Brooks, pp. 117, 1988). And
the DNA molecule is a double helix structure with an underlying design of dynamic
symmetry. The measurements of DNA are dynamically symmetric, 34 angstroms in
height and 21 angstroms in width (Watson and Crick, p.737, 1953).
38
These illustrations depict examples of the historical dynamic symmetry
design similar to the geometric forms used for the nine test stimuli.
Figure A. 1 Islamic design Figure A.2 Sunflower
Figure A. 3 Religious icon
Figure A. 4 Le Modulor
39
Figure 108
Figure A. 5 Fivesided tilings
..: . .>;!*?*. smss jjl *<<< VvCfeiV/a#'/;;:
i>: >wJwv
Figure A.6 DNA molecule
40
APPENDIX B: DYNAMIC SYMMETRY CONSTRUCTION
mZEDH = 10a 00000
m^DHG= 108.00000
Radius *E = 2.59292 cm
EM= 259292 cm
kH = 259292 cm
PM*5 259292 cm
Radius CE = 1 60251 cm
CD= 1.60251 cm
QCi = 1.60251 cm
CC= 1.60251 cm
Radius CA = 259292 cm
m^XCTiDi = 90.00000
m^YEiEt = 90.0000CP
mZS'ST = 90.0000CP
UV= 1 60901 cm
Stf = 258185 cm
ST= 1.60901 cm
SU = 258185 cm
D1E1 =259292 cm
DiDt = 4.19543 cm
DiDi
D1E1
1.61803
X'R= 259292 cm
X'X = 4 19543 cm
X*
X'R*
1.61803
Red = points
Green = circles
Blue = lines, pentagons, triangles, rectangles
Figure B. 1 Dynamic symmetry geometric image
RT = 259292 cm
YY1 = 4.19543 cm
YY'
RT = 1 61803
WB1 = 518563 cm
BW= 259292 cm
BW
WB'
= 0.50000
ZX' = 240377 cm
WD= 259292 cm
R3 = 3.04816 cm
DF= 4.93202 cm
DF
FG
1.61803
ETf= 4.93202 cm
E*N = 7.98018 cm
EN
EH
1.61803
ST= 1.60901 cm
St) = 2.58185 cm
St)
ST'
1.60462
F1F1 = 1.60251 cm
FiGi = 259292 cm
FiGi
F1F1
1.61803
XT'= 518583 cm
41
Dynamic Symmetry Image Construction and Proof
Geometers Sketchpad 4.05s
1. Construct point A toward the left side of the image.
2. Construct point B toward the right side of the image.
3. Construct line segment AB.
4. Construct point C at the midpoint of line segment AB.
5. Construct circle CA by using center C and point A.
6. Construct point D by rotating point A 270 degrees about point C.
7. Construct point E by rotating point D 72 degrees about point C.
8. Construct point F by rotating point E 72 degrees about point C.
9. Construct point G by rotating point F 72 degrees about point C.
10. Construct point H by rotating point G 72 degrees about point C.
11. Construct line segments DE, EF, FG, GH and HD to inscribe a pentagram in
circle CA
12. Construct point I by rotating point A 90 degrees about point C.
13. Construct circle IC by using Center I and point C.
14. Construct point J at the left intersection of circles CA and IC.
15. Construct point K at the right intersection of circles CA and IC.
16. Construct line segment JK. Note that line segment JK is used as a
line for later construction.
17. Construct points A, B, and D through H by reflecting points A, B, and D
through H about line segment JK.
18. Construct line segments AB\ DE\ EF\ FG\ GH, and HD by
reflecting line segments AB, DE, EF, FG, GH, and HD about line segment
JK.
19. Construct line segments DF and DG.
20. Construct line segments DF and DG by reflecting line segments DF and
DG about line segment JK.
21. Construct line segment EF. Note that line segment EF is for construction
purposes only.
22. Construct point L at the midpoint of line segment EF. Note that point L is
for construction purposes only.
23. Construct a line through point L and perpendicular to line segment EF.
This line is for construction purposes only.
24. Construct a line through point C and perpendicular to line segment AB. This
line is for construction purposes only.
25. Construct point M at the intersection of lines created in steps 23 and 24.
26. Construct circle ME by using center M and point E.
42
27. Construct point N at the upper intersection of circle ME and the line created
in step 24.
28. Construct point O at the lower intersection of circle ME and the line created
in step 24.
29. Construct line segments NF and NG\
30. Construct line segment EH. Line segment EH is for construction purposes
only.
31. Construct point P at the midpoint of line segment DE. This point is for
construction purposes only.
32. Construct a line through point P and perpendicular to line segment DE. This
line is for construction purposes only. Construct point Q at the intersection
of this line and line segment NF.
33. Construct circle QE by using center point Q and point E.
34. Construct points M and N by reflecting points M and N about line segment
JK.
35. Construct circle ME by reflecting circle ME about line segment JK.
36. Construct line segments NF and NG by reflecting line segments NF and
NG about line segment JK.
37. Construct line segment EH by reflecting line segment EH about line
segment JK.
38. Construct point R at the midpoint of line segment EH.
39. Construct point R by reflecting point R about line segment JK.
40. Construct line segments RF and RG.
41. Construct line segments RF and RG by reflecting line segments RF and
RG about line segment JK.
42. Construct point S at the intersection of line segments RF and DF.
43. Construct point T at the intersection of line segments RG and DG.
44. Construct line segment ST.
45. Construct points S and T by reflecting points S and T about line segment
JK.
46. Construct line segment ST by reflecting line segment ST about line
segment JK.
47. Construct line segments SS and TT.
48. Construct point U at the intersection of line segment SS and circle IC.
49. Construct point V at the intersection of line segment TT and circle IC.
50 Construct line segment UV.
51. Construct points U and V by reflecting points U and V about line segment
JK.
43
52. Construct line segment UV by reflecting line segment UV about line
segment JK
53. Construct line 1 through point A perpendicular to line segment AB.
Construct line 2 through point B perpendicular to line segment AB.
54. Construct a line through point D perpendicular to line 1. This line is for
construction purposes only.
55. Construct a line through point R perpendicular to line 1. This line is for
construction purposes only.
56. Construct a line through point 0 perpendicular to line 1. This line is for
construction purposes only.
57. Construct point W at the intersection of line 2 and the line created in step 54.
58. Construct point X at the intersection of line 1 and the line created in step 55.
59. Construct point Y at the intersection of line 2 and the line created in step 55.
60. Construct point Z at the intersection of line 1 and the line created in step 56.
61. Construct point Ai at the intersection of line 2 and the line created in step 56.
62. Hide the construction entities created in steps 21, 22, 23, 24, 31, 32, 54, 55,
and 56.
63. Construct line segment DW.
64. Construct point Bi at the midpoint of line segment DW.
65. Construct line segment ZAi.
66. Construct points X and Y by reflecting points X and Y about line segment
JK.
67. Construct line segment DR.
68. Construct point Ci at the intersection of line segments EH and DG.
69. Hide line segment EH.
70. Construct line segments RX and RY.
71. Construct point Di at the midpoint of line segment RX.
72. Construct point Ei at the midpoint of line segment RY.
73. Construct points Di and Ei by reflecting points Di and Ei about line
segment JK.
74. Construct line segments EiEi and DiDi.
75. Construct point Fi at the intersection of line segments FG and DiDi.
76. Construct point Gi at the intersection of line segments FG and EiEi.
77. Construct points Fi and Gi by reflecting points Fi and Gi about line
segment JK.
44
Construction of the dynamic symmetry Group A images
from the geometric image with ratio examples:
Figure B.2a Construction of dynamic symmetry images
/ / \.N '
V X
i" / A 4 / ,
1 '
/ V" ' kJ
Figure B.2b Construction of dynamic symmetry images
Figure B.2c Construction of dynamic symmetry images
45
Comparison of Ratios
Image sets comparison of ratios measured in centimeters.
BH20% BH40% CV20%
image set DS ratio ratio ratio ratio
Thick rhombus 1.618:1 1.73:1 1.85:1
Kite 1.618:1 1.86:1 2.17:1
Anvil 1.618:1 1.89:1 2.50:1
Pentagram 1.618:1 1.41:1
Spiral 1.618:1 1.25:1
Thin rhombus 1.618:1 1.33:1
Mtn 1.618:1 1.86:1 1.43:1
Portals 1.618:1 2.00:1 1.30:1
Dart 1.618:1 1.75:1 1.50:1
Dynamic symmetry (DS)
Group B horizontal distortions BH20%, BH40%
Group C vertical distortions CV20%, CV40%
Order of presentation and placement of the comparison of ratios:
Pentagram CV20% DSratio CV40%
Anvil DSratio BH20% BH40%
Dart CV20% BH20% DSratio
Spiral CV20% DSratio CV40%
Portals CV20% BH20% DSratio
Mtn DSratio BH20% CV20%
Kite DSratio BH20% BH40%
Thick rhombus DSratio BH20% BH40%
Thin rhombus CV40% DSratio CV20%
Total scores for first choice
DSratio BH20% CV20% BH40% CV40%
54 35 18 50 23
75 38 42 23 18
74 36 36 29 54
59 36 39
98 42 28
55 51 30
85
59
64
CV40%
ratio
1.22:1
1.06:1
1.06:1
46
APPENDIX C: STUDY PROCEDURES
Participants Information Form
Sequence number
Thank you for your willingness to participate voluntarily in this study. Please
answer each of the following items by circling the answer or writing the answer that
most accurately represents you. Please print. Your responses will be strictly
confidential.
Gender 1. Female 2. Male
Age 1. younger than 20
2. 2130
3. 3140
4. 4150
5. 5160
6. older than 60
Education background
High School Bachelors Degree Masters Degree PhD other________________________
Occupation or career goals:
Administrative support Executive Professional Restaurant Sales Service
Teaching Technical Transportation
47
Present family circumstance (circle as many as are appropriate)
Living alone Living with another adult Living with more than one adult
Living with children/child under 3 living with children/child over 3
Living with another adult/s and children /child under 3
Living with another adult/s and children/child over 3
Living with an elderly person (over 65) Living with more than one elderly person
(over 65)
Interests you have and activities you may do someday. (Circle as many as you want)
reading writing skiing scuba diving hiking drawing knitting sewing cooking
jogging biking art shows museums concerts movies parties travel
teaching acting religion shopping bowling cultural events cars trains
boats airplanes water sports professional sports TV computers family plays
holidays inventions sculpture electronics machines games construction
photography design furniture gardens making friends parades radio
newspapers comedy health dancing exercise music pets learning
48
Interests you have and activities you do occasionally, (circle as many as you want)
reading writing skiing scuba diving hiking drawing knitting sewing cooking
jogging biking art shows museums concerts movies parties travel
teaching acting religion shopping bowling cultural events cars trains
boats airplanes water sports professional sports TV computers family plays
holidays inventions sculpture electronics machines games construction
photography design furniture gardens making friends parades radio
newspapers comedy health dancing exercise music pets learning
Interests you have and activities you do frequently, (circle as many as you want)
reading writing skiing scuba diving hiking drawing knitting sewing cooking
jogging biking art shows museums concerts movies parties travel
teaching acting religion shopping bowling cultural events cars trains
boats airplanes water sports professional sports TV computers family plays
holidays inventions sculpture electronics machines games construction
photography design furniture gardens making friends parades radio
newspapers comedy health dancing exercise music pets learning
49
Test Score Sheet
Sequence No. _________
Dynamic Symmetry Test Response Sheet
most pleasing = 1, second most pleasing = 2, third most pleasing
1. L M R
2. L M R
3. L M R
4. L M R
5. L M R
6. L M R
7. L M R
8. L M R
9. L M R
50
Demographics Analysis
The demographics of the participants of the experimental testing were:
Total number of participants N= 139
Gender of participants Age of participants
Female = 73 52.5% of N 1. 18 to 20... ...47 33.81%
Male = 66 47.4% of N 2. 21 to 30... ...70 50.35%
3. 31 to40... ...13 9.35%
4.41 to 50... 5 3.5%
5. 51 to 4 2.87%
Analysis of statisticalfindings, Questions of interest:
Did the subjects prefer dynamic symmetry?
The results of the testing showed significant support for the alternate hypothesis that
there was a difference in the observed and expected population distributions,
X*(17,n = 139) = 70.914 p< 0001. Looking at the total score per image per
participant from a two sample t.test (two sided pvalue<0001). The grand mean of
the difference of scores for the nine tests was 15.75 on a 9 to 27 point scale. The
observationpreferencescoreperparticipantfrequency statistics are a mean of
1.7435 and a median of 1.77. A line graph of the mean score of the preference
judgment observations for dynamic symmetry reveals two modes, one at 1.77 and
the other at 2.22. A 95% confidence interval for the preference to dynamic symmetry
scores is from 15.16 to 16.34. A Chi square test for goodness of fit also shows
strong evidence, X^n = 139) = 52.151p<000). A Chi square goodness of fit, Chi
square: Â£(obsexp)A2/exp, on the Binominal distribution, null hypothesis: score
(responses) is Binominal (9, 1/3) looking at first choice alone shows strong evidence
(p value< 0001) that dynamic symmetry was preferred. A Welch two sample t.test
shows a significant difference of the means between the first choice (mean 69.33)
and second choice (mean 35.44) (pvalue=.00015, twosided) with a 95%
confidence interval of 19.45 to 48.55. And a significant difference in the first choice
(mean 69.33) and third choice (mean 34.33) (pvalue = 0005 with a 95% confidence
level of 18.13 to 51.87. The t.test of the scores between second choice (mean 35.33)
and third choice (mean 34.33) (pvalue=.8975) does not show a significant
difference, the 95% confidence interval is from 15.28 to 17.28.
Was there a difference in the choice of images?
Overall, the ranking of preference for the images by the total study
population was for first choice: pentagram, spiral, mtn, kite, portals, there was a tie
between the dart/thick rhombus, thin rhombus and anvil. For second choice it was
51
the dart, mtn, portals, thick rhombus, spiral, kite, thin rhombus and a tie between the
pentagram/anvil. The third choice ranking was the anvil, thin rhombus, thick
rhombus, kite, portals, spiral, dart, pentagram and mtn. A onesample t.test
analyzing the differences in the first, second and third choices for each image shows
a pvalue of 0.215 for the pentagram, 0.065 for the anvil, 0.063 for the dart, 0.140
for the spiral, 0.052 for the portals, 0.128 for the mtn., 0.084 for the kite, 0.023 for
the thick rhombus, and 0.050 for the thin rhombus. The images thin rhombus and
anvil were the least preferred.
Looking at the participants choices of radial, bilateral, nonradial or non
bilateral shows for a first choice in ranking the pentagram (1st), spiral (2nd) and mtn
(3rd), for a second choice the ranking was the dart (Is1) mtn(2nd) and portals (3rd),
for a third choice the ranking was the anvil (Is*), thin rhombus (2nd) and thick
rhombus (3rd ). For first and second choices combined the ranking was the mtn (1st),
pentagram(2nd) and dart (3rd). A Friedman nonparametric test shows a mean rank
for the pentagram 4.02, spiral 4.37, mtn 4.46, portals 4.93, kite 5.00, dart 5.06, thick
R 5.45, thin R 5.80, and anvil 5.91.
Did the position of the images make a difference?
Looking at the differences in the scores for the left, middle and right
positions of the images a Welch two sample t.test shows a pvalue 0.7823 of the
difference of left first choice to middle first choice, a pvalue of0.0991 for the
middle first choice to the right first choice, and a pvalue of 0.1112 for the right first
choice to the left first choice. The results do not indicate that the position of the
images left, middle, or right was significant in the choice of images by the
participants. Looking at the column totals on the dynamic symmetry matrix shows a
relationship of the t.test results to the position of the dynamic symmetry images.
There were four dynamic symmetry images in the left column, three in the middle
and two on the right side.
There was no conclusive evidence of a position effect or preference to
symmetry other than dynamic symmetry found for the presentation or design of the
images. The pentagram was the only image with radial and bilateral symmetry and it
was ranked Is* in the first choice, 9 in the second choice, and 8th for the third
choice. The spiral did not have radial or bilateral symmetry and was ranked 2nd the
second for first choice, 5 for the second choice, and 6 for the third Looking at the
combined first and second choices shows the mtn (bilateral symmetry) as ranked the
1st, the pentagram 2nd and the dart 3rd. The dart (bilateral symmetry) was ranked 1st
in the second choice.
52
Table C. 1 Rankings of the preference for the images and symmetry analysis
First choice Second choice Third choice First and Second
Choices
rank image scores rank image scores rank image scores rank image scores
1 pentagra m 98 1 dart 58 1 anvil 63 1 mtn 127
2 spiral 85 2 nun 52 2 thin R 59 2 pentagra m 120
3 nun 75 3 portals 49 3 thick R 46 3 dart 117
4 kite 74 4 thick R 34 4 kite 39 4 spiral 116
5 portals 64 5 spiral 31 5 portals 26 5 portals 113
6 kite 26 6 spiral 23 6 kite 100
6,7 dart, thick 59 7 thin R 25 7 dart 22 7 thick R 93
8 thin R 55 8 pentagra m 19 8 thin R 80
9 anvil 54 8,9 pen tag ram, anvil 22 9 mtn 12 9 anvil 76
total 623 319 309
mean 69.22 35.44 34.33
median 64 31 26
range 44 36 51
Green = radial and bilateral symmetry, Blue = bilateral symmetry, Violet = no radial or bilateral symmetry
Total
451
441
359
Image Left 1 Left 2 Left 3 Midi Mid 2 Mid 3 Right 1 Right 2 Right 3
pentagram 18 81 40 98 22 19 23 36 80
anvil 54 21 64 35 80 24 50 38 51
dart 42 53 44 38 28 73 59 58 22
spiral 36 90 13 85 31 23 18 18 103
portals 39 47 53 36 43 60 64 49 26
mtn 75 52 12 36 39 64 28 48 63
kite 74 26 39 42 88 9 23 25 91
thick R 59 34 46 51 66 22 29 39 71
thin R 54 37 48 55 25 59 30 77 32
476
422
353
324
388
539
Blue = Group A (dynamic symmetry)
Red = Group B 20% (horizontal distortion)
Green = Group B 40% (horizontal distortion)
Black = Group C 20% (vertical distortion)
Violet = Group C 40% (vertical distortion)
53
Contingency tables of count bv the factor of dynamic symmetry preference on
gender, age, creative activities, and the participants living with children.
This study can be categorized as a single population with four categories of
response designed by a 2x2 table, and the independence hypothesis is appropriate.
For the participants living with children, Retrospective Product Binomial sampling
was used, as the subpopulation was randomly chosen, and the placement of scores
in the cells was also random. The four levels of the row factors correspond to the
crossclassification of gender (male, female), and score (preferred, not preferred).
The sampling scheme is Poisson because none of the marginal totals were known in
advance. The design of the contingency tables and factors is:
1. The overall preference for dynamic symmetry response (preferred, not preferred),
corresponding to the factor (male, female), in a 2x2 table.
2. The overall preference for dynamic symmetry response (preferred, not preferred),
corresponding to the factor (male, female), accounting for five levels of age
categorization, in five 2x2 tables.
3. The overall preference for dynamic symmetry response (preferred, not preferred),
corresponding to the factor (male, female), accounting for three levels of creative
activity categorization, in three 2x2 tables.
4. The subpopulation of participants, living with children, dynamic symmetry
responses corresponding to the factor (male, female), in a 2x2 table.
54
court
1. Pentagram
2. Anvil
3. Dart
4. Spiral
5. Portals
6. Mountain
7. Kite
8. Thick rhombus
9. Thin rhombus
Dynamic Symmetry preference
per image and quadrant
Choice 1
Dynamic Symmetry preference
per image and quadrant
Choice 2
2 4 6 8
Image number
Quad I = females who preferred dynamic symmetry = *
Quad II = females who did not preferred dynamic symmetry = :
Quad HI = males who preferred dynamic symmetry = #
Quad IV= males who did not preferred dynamic symmetry = o
Figure C.l Analysis of the dynamic symmetry preference per image and quadrant
55
count
Dynamic Symmetry preference
per image and quadrant
Choice 3
Quad I = females who preferred dynamic symmetry = *
Quad II = females who did not preferred dynamic symmetry= :
Quad HI = males who preferred dynamic symmetry = #
Quad IV= males who did not preferred dynamic symmetry = o
Figure C.l Analysis of the dynamic symmetry preference per image and quadrant (cont.).
56
Table C.3 Quadrants of the participants by gender and preference
Quad I Pref = 52....37.41% Quad IQ Pref 41 29.49%
Quad n Non = 21...15.10% Quad IV Non 25... 17.98%
Table C.4 Quadrants of the participants bv age in years and preference
Age 1820 2130 3140 4150 51
Quad I 13 30 3 3 3
Quad II 7 12 1 1 0
Quad in 15 19 7 0 1
Quad IV 12 9 2 1 0
Percent of participants in age categories (1820), (2130) and (3140) is 93.53% of the total
population
Table C.5 Quadrants of the participants bv the percent of the number of creative
activities and preference
Activities Frequent Occasional Future Total Activities
Quad I 12.8 24.86 26.76 64.42
Quad H 8.31 6.83 9.63 24.77
Quad m 8.93 12.58 17.91 39.42
Quad IV 6.93 8.04 9.38 24.35
Quad I + Quad in = 103.84
Quad II + Quad IV = 49.12
103.84 is 67% of 152.96
Table C.6 Quadrants of the participants living with children and preference
n = 40 28,00% ofN
Quad I 15= 37.5% Quad HI 13 = 32.5%
Quad n 7= 17.5% Quad IV 5 = 12.5 %
Quad I + Quad III = 28 70% of n
Quad I + Quad III = 12 30% of n
57
Table C.7 Quadrants of the participants living with children bv percent of the
number of creative activities and preference
Activities Frequent Occasional Future Total
Quad I 8.22 5.39 3.43 17.04
Quad II 2.89 1.94 1.22 6.05
Quad in 5.98 3.74 2.73 12.45
Quad IV 1.70 1.12 1.00 3.82
Quad I + Quad HI = 29.49 29.49 is 75% of the total 39.36
Quad II + Quad IV = 9.87
Did those participants with an artistic background prefer dynamic symmetry?
All of the study participants filled out the creativity survey and
demographics questionnaire. Of the 139 individuals, eleven participants answered
that they were experienced visual artists preparing for or already involved in a
professional artist career. Four of the eleven did graphic design, their first choice
scores were 4, 3, 5, and 2. Four of the participants were painters, their first choice
scores were 4, 3, 3, and 1. Two of the participants were art teachers, their first
choice scores were 3 and 8, and one was an architect, his score was 5. Comparing
the visual art participants dynamic symmetry first choice and distortion first choice
scores with the total participants dynamic symmetry first choice and distortion first
choice scores Figure 3.11 shows a close correlation to Figure 3.1.
Did the participants who were living with children prefer dynamic symmetry?
There is strong evidence from the ANOVA and Welch two sample test
showing a correlation of the subpopulation of subjects with children to a positive
dynamic symmetry score. An analysis of variance test (ANOVA) with a tstatistic of
4.989 (pvalue<0001) and a predicted mean score of 4.53 with predicted 95%
confidence intervals of min = 4.33 and max = 4.68 supports the hypothesis that
participants living with children preferred dynamic symmetry. Of the 139
participants 40 were living with children. The category, participants living with
children, was females who preferred dynamic symmetry were 37.5%, males who
preferred dynamic symmetry were 32.5%, females who did not prefer dynamic
symmetry were 17.5%, and males who did not prefer dynamic symmetry were
12.5% of the distribution. Overall, 70% of the subpopulation preferred dynamic
symmetry while 30% did not.
58
Scope of Itrference
The contingency tables of count show a similarity in the percentages of the
four quadrant groups showing the preference or non preference for dynamic
symmetry for the categorizations of age, activities and the participants living with
children. The consistent similarity in the responses of the participants shows that for
each category of activity the percentage of females who preferred dynamic
symmetry was the more active group 42 % (table 5) 37.41% (percent of Quad I,
table 3) = 4.59% increase. The males who preferred dynamic symmetry were 25.8%
creatively active, which was less active 25.8% (table 5) 29.49% (percent of Quad
m, table 3) = 3.69% decrease. The females who did not prefer dynamic symmetry
were slightly more active 16.2% (table 5) 15.10% (percent of Quad II, (table 3) =
1.1% increase and the males who did not prefer dynamic symmetry were slightly
less active 15.9% (table 5) 17.98% (percent of Quad IV, (table 3) = 2.08%. The
differences in the categories were not significant but the differences show a
similarity to the prediction of a positive relationship of dynamic symmetry to the
number of creative activities of the participants. Overall 67% of the participants
were creatively active following the overall population preference to dynamic
symmetry.
Table C.8 Percentages of the number of creative activities per quadrant
Activities Frequent Occasional Future Total
Quad I 34.6% 47.5% 42.0% 42.0%
Quad n 22.5% 13.0% 15.0% 16.2%
Quad III 24.2% 24.0% 28.0% 25.8%
Quad IV 18.7% 15.4% 14.7% 15.9%
59
GLOSSARY
Aesthetic is a sensitivity to that which is beautiful.
Design arrangement of objects in harmonious relationship to each other and to the
space they occupy. The linkage of mathematical systems and design can be traced to
the earliest culture, science and art find a common denominator in the search for
perfect form.
Emotion is a complex, usually strong subjective response, such as love or fear. It is
the part of the consciousness that involves feeling or sensibility.
Fivesided tiling a surface can be completely tiled in an asymmetrical non
repeating manner in a fivefold symmetry with just two shapes based on phi.
Golden Section (GS) Hypothesis is an empirical investigation following Gustav
Fechner of a preference judgment for the proportion dynamic symmetry.
Icon is an image, a symbol, a representation or picture of a scared Christian
personage, traditional to the Eastern Churches.
Incommensurability the discovery that the ratio of diagonal and side of the square
is not equal to the ratio of two integers produced the necessity to extend the number
system to irrational numbers. The length of the diagonal in the unit square, square
root of two, is irrational.
Islamic Art is a reflection of that which is truth, and therefore, of God. Geometry is
at the base of all Islamic art forms. Islamic decorative patterns are geometric shapes,
Islamic architecture is entirely geometrical, calligraphy is based upon geometric
shapes, and the music is entirely mathematical and geometrical.
Isosceles triangle is a triangle having two equal sides with two 72 and one 36
angles.
Le Modulor a system of architectural proportion and the logarithmic spiral. The
design system links dynamic symmetry to the scale and proportion of the human
anatomy.
60
0, Phi is the Golden Mean or Golden Section ratio, it is the most irrational number,
1.618033988749 This is the ratio found in nature to define growth patterns for
spiral shells, leaf growth on plants, and humans.
Psychophysics is the division of psychology that studies the physiological aspects
of mental phenomena and in particular the quantitative relations between stimuli
and the resultant sensations.
Ratio is the quantitative relationship between two comparable magnitudes.
Religious iconography a set of traditional or specified symbolic forms associated
with the subject or theme of a stylized work of art
Sensory aesthetics is a branch of philosophy dealing with the nature of beauty. The
word aesthetics was first used by German philosopher Alexander Gottlieb
Baumgarten, who helped to establish the study of aesthetics as a separate
philosophical field of study. The word aesthetic can be used as a noun meaning "that
which appeals to the senses." Someone's aesthetic has a lot to do with their artistic
judgement.
Spiral coiled in a plane or as if around a cylinder or cone.
61
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