Judgments of Time, Aesthetics, and Complexity as a
Function of The
Fractal Dimension of Images formed by Chaotic
Attractors
Frederick David Abraham[1],[2],
Julien Clinton Sprott[3],
Olga Mitina[4]
Maureen Osorio1, Elvie Ann Dequito1,
and Jeanne Marie Pinili1
In this experiment, we
obtained judgments of aesthetic value, complexity, and duration of presentation
of images of chaotic attractors as a function of their fractal dimension (F).
We used four levels of fractal dimension (four stimuli at each level, mean Fs =
.59, 1.07, 1.54, 2.27), with small samples of six students from each of three
populations (elementary school students, graduate students, and special
education students from ethnic minority groups in residence on our campus). In
addition to replicating earlier studies of aesthetic judgments, we additionally
asked for judgments of complexity to see if they were also a nonlinear function
of the fractal dimensionality of stimuli (they were, increasing to a maximum at
F = 1.54, and falling off at F = 2.27), suggesting the possibility that
aesthetics and complexity judgments were at least in part mediated by perceived
complexity. Perhaps this non-monotonic result was due to a loss of detail of
contrast and detail within the attractors at the highest dimensional
complexity. Judgments of duration did not yield a significant F-ratio between
groups, but t-tests showed the lowest dimensionality yielded shorter time
estimates than those for the higher dimensionalities. If complexity is a determinant
of subjective duration of these stimuli, it is saturated at fairly low levels
of the fractal dimensionality of the stimuli. Three-way ANOVAs within
participants, showed only the fractal dimension as a significant source of
variation in aesthetic and complexity judgments, and not academic level,
gender, or cultural differences, although there were interesting individual and
some cultural differences.
KEY WORDS: psychophysics, time, complexity, aesthetics,
chaos, fractal dimension
INTRODUCTION
Studies of both the perception of
aesthetics and of time have centered on cognitive and biological factors (Eisler, Eisler, & Montgomery,
1996; Anderson & Mandell, 1996). Many factors of
stimuli, cognition, and biology have been investigated. One of the features of
stimuli which make both biological and cognitive demands for time estimation is
the amount of perceptual/cognitive effort involved, such as can vary with
complexity of stimuli (Cupchik & Gebotys, 1988; Stoyanova & Yakimoff et al., 1987). The same can be said for
aesthetic judgments (Aks & Sprott, 1996; Sprott, 1993, 2003).
Perceptual/neural organizational
features of attending complex stimuli may affect both the estimation of time and
aesthetics, but also, there is the possibility that aesthetics could affect the
perception of time. There is likely a nonlinear interaction between the
complexity of stimuli and experiencing time and aesthetics. Such nonlinearities
have been at the foundation of perceptual psychophysical theories and research
of Gregson (1995).
To investigate the possibility of a
relationship between complexity and judgments of time and aesthetics it was
decided to use abstract stimuli. Chaotic attractors were used for this purpose
as they are easy to generate along with objective measurements of their
complexity (Aks & Sprott, 1996; Sprott, 1993a,b, 2003) and relatively free
of prior associations. Chaotic attractors as stimuli can be created by
integration of three coupled differential equations that produce abstract
computer images in two dimensions, the third being color coded which adds to
their aesthetic potential. To check and see if mathematical complexity has any
relationship to perceived complexity, it was decided to add an estimate of
complexity to the ratings obtained from the participants.
While complexity of stimuli is the
principal independent variable, demographic factors as well, namely age and
culture (Eisler & Eisler,
1994; H. Eisler, , were also varied. Therefore, both
children and adults, and both urbanized students of
And to enrich the dependent variable
side of the experiment, in addition to scales of aesthetics and complexity, and
estimates of stimulus duration, some narrative explorations of the features
upon which the participants might have been basing their judgments were also
attempted.
Methods
Participants
Participants were from three
populations:
(1)
Students
(six, 1 female, 5 males, from grades 4-5)
(2)
Graduate
Students (six, 4 females, 2 males), Department of Psychology,
(3)
Ati
and Sulod (six, 4 females, 2 males) were recruited from a residential special
learning program for the current school year hosted by the University’s
Department of Education. They were visiting from several Ati and Sulod
communities (
Permissions
In accordance with standard practice for undergraduate
research, the locus of informed consent was formally placed in the hands of
administers of the educational units involved. Therefore permission to solicit
participants was obtained from Ms. Iyoyo, Principal
of the
Selection
The elementary school students were
not randomly conscripted from the school, and could be considered as among the
high achieving students, being the whole of the members of the experimental
enrichment program. The same could be said of the graduate students, who
represent a high achieving profile. This achieving profile could work against
our finding an age difference, due to the sophistication of both populations.
The Ati, also are not a random sample of Ati, being leaders in their
communities interested in bringing their Silliman education back to their
communities, and from among those, the more adventuresome were among those who
volunteered to participate. While these sophistications could have minimized
differences in the results, their differences in age, education, life styles,
and environment could possibly have influenced the way they responded to the
abstract visual stimuli.
Stimuli
Sprott (1993) developed a program for
the generation of computer images of chaotic attractors from systems of three
nonlinear coupled difference equations which were adapted for use in this and
similar experiments (Mintina and Abraham, in press).
The equations are:
xn+1 = a + bxn + cxn2 + dxnyn +exnzn
+fyn + gyn2 + hynzn + izn
+ jzn2
yn+1 = xn
zn+1 = yn where x,y,z are
variables and a–j are parameters.
Solutions (trajectories) are displayed
in two dimensions (x,y, the plane of the screen of
the computer) with the third dimension (z) represented by color coding. These
stimuli vary in complexity depending on the parameters used in each
integration, and are selected by the program at each integration. These
programs develop the trajectory (attractors, plotted by point, not as a
continuous solid line)) by iteration of the equations, although they evolve so
fast as to be essentially perceived as filling space rapidly over time, but not
as a trajectory growing over time. Thus of the measures of complexity produced
for each image by the program, the fractal dimension of the attractor (which
measures complexity in space) is a more realistic measure of the complexity of
the trajectory than the largest Liapunov exponent (which measures aspects of
the evolution of the trajectory in time). The program was used to generate 168
of these stimuli. A statistics program (STATISTICA) was used to analyze these
two measures of these images. Aks and Sprott (1986) used both measures and
found aesthetic responses to vary with both measures for some participants.
However we chose our stimuli on the basis of having the Liapunov exponents well
correlated with the fractal dimension, so it would be irrelevant to try to
separate the two aspects of complexity. The fractal dimension, F, employed here
is the correlation dimension, usually designated as D2 (introduced
to dynamics by Grassberger & Procaccia,
1983; and reviewed in many books and articles, including Sprott, 2003, p.
307-311, and Abraham, 1997, pp. 17-18). Stimuli were chosen with Fs in four
ranges, .5–.85, .86–1.4–1.6, and 2.2–2.4 (mean Fs, 0.59, 1.07, 1.54, 2.27).
Twenty stimuli, five from each of these four ranges were selected. Four were to
be used at the beginning of each run in a fixed order to give a warm-up for the
participant, and the remaining 16 were presented in a randomized sequence for
the remainder of the run, as described below. These 16 stimuli were used in
analyzing the results. These images can be viewed at http://blueberry-brain.org/silliman/jemstim.htm.
Computer Implementation and Instructions
A session on the computer started
with a set-up phase by the experimenter in which the instruction set and
presentation method were selected, and the participant’s identification and
data file name were entered. Next, the participant sat at the computer, and the
experimenter gave the participant verbal instructions on how the experiment was
to be run. Then the stimuli were presented in succession, going through the
sequence twice. The participant responded to each stimulus with three judgments
as follows.
The following comprised the two runs
through the stimuli, the first run to obtain the participant’s ratings of
aesthetics and complexity, and the second run to obtain estimation of the
duration of presentation of the images. The first run was completed as follows:
(1) When the experimenter finished setting up the computer to run the experiment, the participant pressed any key to begin. Then the first stimulus appeared on the screen. For 1.04 seconds, the stimulus remained on, and then the first question appeared at the top of the screen:
How beautiful was the image [1 least
- 9 (most)]?
(2) Then the experimenter explained the rating scale with the first stimulus.
(3) The participant responded on the keyboard, with a number and followed with an entering key-press.
(4) The
second question then replaced the first question:
How complex was the image [1 (least) - 9 (most)]?
(5) To insure that the participant used a wide range of ratings of complexity, the experimenter suggested numerical values for this and the following three warm-up stimuli.
(6) After that, the participants self-paced themselves for the remainder of the run through the remaining 16 of the 20 images.
(7) The experimenter then reset the experiment for the second run that was used to get the time estimates.
The second run to obtain the ratings of perceived duration
(time) was completed as follows:
(1) For the estimation of duration of
presentation of the images, the same images were used, with the 5th-20th
being in a new random order.
(2) The second run was started with the
experimenter preparing the run as in the first run, choosing the new
presentation format. Then the participant, as before, sat in front of the
computer and initiated the run.
(3) The stimulus was presented for 10.05
seconds, turned off, and the following question presented:
How many seconds was the image on?
(The participant had to hide the face of a watch if worn, and asked not to
count.)
Finally, after the runs are
completed, the participants were then asked to describe which features of the
images determined their aesthetic and complexity judgments.
Each run took about 5-8 minutes, the
whole session took 20-30 minutes. Some participants were run singly, and some
were run simultaneously, using up to the 6 computers in the psychology computer
laboratory where the experiment was run. When six participants were run
simultaneously, it was found that each of the three experimenters could
supervise two participants at a time without difficulty.
Results
Data Analyses
The computer program used to run the
experiment recorded the participant’s responses automatically. These were then
transferred to an Excel spreadsheet program as follows. The responses to the
test stimuli for each of the three measures at each of the stimulus complexity
levels were averaged for each participant. Also the Pearson product moment
correlations were computed for all six pair-wise combinations of the three
response measures and the fractal dimension for each participant. Thus a
case-by-variables data table was assembled with 24 variables per case: group,
gender, the fractal dimension (F) and the judgments of aesthetics, complexity,
and duration at each of the four complexity levels, and the six correlation
coefficients.
This table was imported by the
STATISTICA data analysis program, where descriptive statistics and tables were
completed (box & whisker plots, Figures 1, 2, & 3,). Inferential
statistics were also computed, namely 3-way ANOVAs for a within-participant
design for each of the three measures (with the complexity level being the
repeated variable, and group and gender being the independent between-group
variables. T-tests for paired measurements were also used to determine the
differences for each of the main measures (complexity, aesthetic, and duration
judgments) between each pair of adjacent levels of F (low vs
low mid, low-mid vs high mid, and high-mid vs high). An alpha level of .05 was used for all
statistical tests.
Statistical Results
Judgments of Complexity
The box & whisker plots (Figure
1) show the means, standard error of the mean, and the 95% confidence limits of
the complexity judgments. The y-axis represents the means of the judgments of
the complexity for each level of complexity of the images. The x-axis
represents the four levels of the complexity of the images. The levels of that
variable, going from left to right are: low (F = .59, .52–.65), low-mid (F =
1.07, 1.05–1.09), high-mid (F = 1.54, 1.46–1.55), and high (F = 2.27,
2.21–2.36). F is the mean of the fractal dimension for the four stimuli in that
level of the fractal dimension; the range is given after the mean.
Insert Figure 1 about here (click)
This figure shows judgments of
complexity to be a non-monotonic increasing function for most of the range of F
as might be expected, but decreasing at the highest values. That is increasing
complexity is perceived as such, until the images at the highest F values which
are reported as less complex. Even within the increasing portion of the curve,
the result is decidedly non-Fechnerian: increases in
F are not met with proportionately increased magnitude of the judgments of
complexity. The ANOVA showed significant results for F, but not for gender or
group, nor for any interactions. The t-tests showed all adjacent means to
differ significantly from each other. All probabilities were way below the
alpha level. Tables can be found at http://blueberry-brain.org/silliman/jemanovacmplx.htm.
Judgments of Aesthetics
The box & whisker plots (Figure
2) show similar results for the aesthetics judgments. This plot also shows a
non-monotonic form nearly identical to that for the complexity judgments, with
maximum aesthetics judgments also occurring at the high-mid level of the
fractal dimension of the images. That means that for the three lower levels of
the fractal dimension, the participants scaled the aesthetics as increasing
with increased complexity, but for the highest level of mathematical complexity
(F), they tended to judge the aesthetics as less, quite comparable to the
low-middle level. Again, that result is
non-Fechnerian even within the monotonic increasing
phase of the curve.
The ANOVA results yielded results
almost identical to those for the complexity variable (a table is at http://blueberry-brain.org/silliman/jemanovaaesth.htm).
That is, the two between-group independent variables, group and gender, where
not significant, while the within-participant main independent variable (levels
of F), was also significant. The one difference compared to the results for the
complexity judgments were that the 3-way interaction was also significant, a
result difficult to interpret (see the figure at
http://blueberry-brain.org/dynamics/jemint3xaesth.jpg). The t-tests for
adjacent values of F also were similar to those for the complexity judgments
(see again the table at http://blueberry-brain.org/silliman/jemanovaaesth.htm).
Judgments of Duration
The results of the judgments of the
duration of the presentation of stimuli were not significant. The box &
whisker plot (Figure 3) shows (1) no simple relationship to the mathematical
complexity (F) of the images, (2) remarkable accuracy of the means of the time
estimates to the actual time of 10 sec. despite (3) great variability in the
responses (range: 2-16.25 sec). This accuracy is reflected in the means of the
four levels, which were 9.31, 10.06, 9.69, and 10.31 sec. (left to right on the
box & whiskers plot), with a grand mean of 9.84 sec.
The ANOVA (the table is at http://blueberry-brain.org/silliman/jemanovatime.htm)
shows no significant main effects or interactions, which is not surprising
considering the lack of trends and the great variability seen in the box &
whisker plot. Despite that, the t-test for one of the comparisons, the lo-mid
was significantly greater than the lo level of the mathematical complexity
variability.
While the interaction terms were not
significant, the plots of the means for the 3-way interaction (see table at http://blueberry-brain.org/silliman/jemint3xtime.jpg)
showed consistently lower time estimates from females compared to males, and
for the Ati/Sulod group compared to the elementary and graduate students.
Correlations: Overall
Pearson product-moment correlations
(see the Table) mirrored the results of the box & whisker plots and the
ANOVAs. The mathematical complexity (fractal dimension, F) was positively
correlated to both the complexity ratings (r = .37) and the aesthetic ratings
(r = .46), but not the time estimations (r = .03). The highest correlation was
between the aesthetic and complexity ratings (r = .64). The remaining two
correlations were remarkably unremarkable, the one between time estimation and
complexity ratings (r = .07), and the one between time and aesthetics judgments
(r = .05). Thus, the three correlations
involving relationships among fractal dimension, complexity judgments, and aesthetic
judgments were strong, while the three involving time estimates were
essentially zero.
Correlations: Group Comparisons
The students from the elementary school
were similar to the overall means of the correlations, with the exception that
their correlation between aesthetic and complexity ratings were higher, and in
fact were the highest among the groups (r = .84). The graduate students were similar to the
overall means of the correlations except for being lower (lowest compared to
the other groups) on the correlation between fractal dimension and the
aesthetic ratings (r = .24). The students from the Ati and Sulod had the
highest correlation compared to the other two groups for the relationship
between the fractal dimension and aesthetic ratings (r = .67), and the lowest
correlation compared to the other two groups for the relationship between the
aesthetic and complexity ratings (r = .44).
Thus the highest correlations were
those between the complexity and aesthetic judgments (r = .64), with the
elementary students being the highest (r = .84), the Ati/Sulod-adult education
students being lowest (r = .44), and the graduate students being in between,
right at the mean (r = .63). Another interesting group difference was for the
correlation of the fractal dimension to aesthetic ratings, which was highest
for the Ati/Sulod students (r = .67), lowest for the graduate students (r =
.24), and intermediate, right at the mean for the elementary students (r =
.47).
Correlations: Individual Differences
Half the participants showed the
typical pattern of high correlations among the fractal dimension and aesthetic
and complexity ratings and low correlations involving time judgments
(participants: at, fd, lh, rn, mr, jq,
jy, ly, am, although one of
these—jy— had a somewhat lower correlation between
aesthetic ratings and fractal dimension). One (ly)
had identical correlations of F to aesthetic and complexity ratings, and a
perfect correlation (+1) between aesthetic and complexity ratings, a clear case
of response bias or generalization. Two other participants (en, ml) showed this
response generalization reflected in high correlations between aesthetic and
complexity ratings while the remaining five correlations were negligible,
including no relationships to the fractal dimension. Participant (jn) was similar to these two participants, except that the
aesthetic to complexity correlation was smaller.
Another participant (am) also showed
very high correlations of F to aesthetic and complexity ratings, but their
correlation between the two rating was somewhat lower than the mean (r = .35
compared to the mean of .64) showing independence of aesthetic and complexity
judgments while both sets of judgments were strongly related to the fractal
dimension. This participant showed the unusual result of fairly strong negative
correlation of time estimation to F and complexity ratings.
Two participants (ao,
ma) showed very strong correlations among F and aesthetic and complexity
ratings and some fairly strong positive correlations involving the time
estimates.
One graduate participant (ms) a
collegiate art instructor as well as a graduate student in psychology,
expressed a preference for open space in art, but his response revealed this
preference in a negative relationship between aesthetic ratings and F, perhaps
small due to the nonlinear relationship between area covered and F. His low
correlations indicate of a strong independence of complexity from F as well,
and the slight positive correlation of complexity to F, and slightly negative
correlation between aesthetic and complexity indicate an independence of these
judgments from each other, while somewhat related to F. The zero correlations
indicate a constant estimation of time. He appears to be a person with strong
independent convictions.
The individual correlations between
complexity judgments and aesthetic judgments were the highest of all, with 12
of them being greater than .8, and one was a perfect 1. This is a key finding.
Why is it not so for all individuals? Can there be different reasons for
different individuals to have high correlations? Can one unravel some of the
individual differences by examining the pattern of responses across all
measures for particular individuals? The reader is invited to try, before we
look at a few in the discussion section later. For example, S15 had identical scores for the two ratings at
each of the level of mathematical complexity. Thus (1) correlations of
aesthetic ratings were higher to complexity judgments than to actual
mathematical complexity (functionally because both were similarly nonlinearly
related to mathematical complexity), (2) These were degraded for a few
individuals into lower correlations, because some people had aesthetic
preferences for lower dimensional stimuli. One of these, S10, a graduate
student and art teacher, stated such a preference in his interview, but this
preference did not make for large negative correlations which his claim would
indicated, but certainly did degrade not only this correlation to near zero
(-.066), but all of his correlations. And (3), the only hint of cultural (and
gender) differences that can be found in the data are swamped by these individual
differences, and the relationships to mathematical complexity are so strong as
to survive the individual differences.
Of the remaining three correlations
(all for the time judgments), only a few were strong, mainly for participants
2, 7, 14, 15, 16, 17. Thus, there was greater individual variability in the
Ati/Sulud group (4 Participants compared to 1 each in
the other two groups). Of these, two were strongly negative (Participants 7
& 17), and four (Participants 2, 14, 15, & 16) strongly positive. Three
were strongest for the correlations to the mathematical (fractal) complexity
(Participants 2, 7, 15), and three were strongest for the correlations to the
complexity judgments (Participants 14, 16, & 17), all three of these in the
Ati/Sulud group.
There were no large negative
correlations of other variables to aesthetic ratings. The exceptions could be
with the time variable, where two of the Ati/Sulod group had correlations of
.604 and .434. The single large positive correlation between time and aesthetic
ratings also came from a participant in this group.
Discussion
Complexity
Since the judged complexity is not a
simple increasing monotonic function of complexity as measure by F (D2),
falling off when F> 2, the question arises as to why that should be? Visual
inspection of the images gives one answer. For stimuli where the mean F = .59,
the space is quite empty. When going to the next higher F (mean F = 1.07),
there are considerably plumper images filling the space more completely (both
the viewing space as a whole and within the attractor itself) and having
greater intuitive complexity. Going to the next higher (mean F = .154), the
increase in the filling of space and the apparent boundaries within the
attractor (the third, colored dimension helps a lot with that) is increased
only a little, thus the judgments likewise are not much increased. At the
highest level (mean F = 2.27), the attractors occupy about the same amount of
the viewing space as a whole, but the increase in density of points within the
attractor fill the attractor more completely and the attractors appear more
like clouds, with less interior definition and boundaries, they are more
homogeneous, leading probably to their being judged less complex.
Why does the F (Grassberger-Procaccia
D2) measure not see all that the eye sees. There could be other measures of the
mathematical complexity of the images, but F sees only the filling of space and
does not enjoy the advantages of the eye, brain, and cognition in giving weight
to the boundary effects, especially those from the coloring. And unlike other
measures of complexity, such as D (the Hausdorff, capactity, box-counting dimension which weights a hypercube
in the space equally no matter the amount of occupancy), D2 weights the
hypercubes by the extent to which they are each occupied, and thus the more
dense images are given greater dimension. We shall return to some of these
alternative properties when considering other judgments as well.
A final note could be made as to what
one might investigate further concerning judging complexity. Our instructions
were quite nonspecific, giving little guidance to what meaning they might wish
to construct or assume for it, and there very likely was a great range in both
what they considered complex, and the extent to which they may have been aware
of features of the images to which they were responding. We have not made a
formal analysis of the narratives, but many corroborated the kinds of details
mentioned above. The similarity of the three groups suggests that all were
rather sophisticated in observing these details, and we shall return to a group
difference later after discussing the other judgments. The possibility of
interest is to see if people can learn to discriminate different types
dimensionality by viewing images and training with them. Can the mathematical
definitions of images be learned through experience judging them?
[will check out Atteneve
(1957) who used polygons to study judgements of
complexity; search for follow up work; derived from his interest in information
theory, which is mathenmatically related to nonlinear
measures of complexity.]
Aesthetics
The history of the psychological
study of aesthetics is long and large. Since this feature was not part of the motivation
for the present experiment, we will review only the part of it that relates to
issues related to complexity and order. While much of the literature deals with
simple properties, e.g., those on the well known Golden Mean, surprisingly the
empirical work on complexity goes back to Gustav Fechner
(1801-1887) and also surprising is that some of the conjectures about
complexity apply to some simple forms as well as complex ones (Berlyne, 1960). Berlyne concludes
that “The work on the experimental aesthetics of simple visual forms that began
with Fechner’s Vorschule der Äesthtik
(1876) tends to confirm the view that some intermediate degree of complexity
produces the most pleasing effect and the extremes of simplicity or complexity
are distasteful (p. 237).” Fechner mentioned the
veining in marble. Also, interestingly enough, two of the early contributors to
this literature included pioneers of dynamics, the mathematician George David Birkhoff (1884-1944), and the mathematical biological
scientist, Nicholas Rashevsky (1899-1972) (see
Abraham & Shaw, 1982, 1992), although their analysis of aesthetics
apparently did not involve dynamics, but did look at complexity.
The early history of ideas of unity
and diversity in aesthetics (summarized in Berlyne,
1960; Gilbert & Kuhn, 1953) was addressed mathematically by Birkhoff (1933) with his formulation that aesthetic value
(M) was a function of complexity C of the image (diversity or numerosity) upon which attention and tension depended, and
order (unity, due to properties such as symmetry) of the image, upon which (as updated by Graves, 1951) resolution of
the tension depended. Birkoff’s formula was M = O/C.
That is, aesthetic value was proportional to order, and inversely proportional
to complexity. Very shortly, there were attempts to test this theory (Eysenck, 1941; Davis, 1936) which found, similar to our
result, that there was a maximum of aesthetic judgment at intermediate values
of Birkhoff’s M, or our F. Our measure of complexity,
F, treats complexity as a single dimension stretching between order and
complexity, rather than assuming a composite function, although much dynamical
thinking looks at oppositional forces along dimensions of the state space, and
our generative equations utilize a complex combination of three variables. The
curvilinear relationship of all our judgments suggests that these judgments are
not unidimensional.
Rashevsky (1938) suggested a possibility based
on mathematical assumptions of excitation and inhibition in cortical neurons
which gave a “measure of total excitatory effect, which is identified with
aesthetic value. Rashevsky’s measure sounds very much
like a measure of complexity, but it actually bears a curvilinear relation to Birkhoff’s M, reaching a sharply delineated maximum when M
is at an intermediate value (quoted in Berlyne, 1960,
p. 239).
Summary & Conclusions
ACKNOWLEDGMENTS
We thank Debbie Aks for sharing her expertise in conducting
psychophysical experiments and Anna Lourd Villaneuva, a friend and graduate student, for assisting at
making her friends among the Ati and Sulod feel comfortable participating in
our experiment. We also thank Ms. Iyoyo, Principal of
the
REFERENCES
Abraham,
F.D. (1997). Nonlinear coherence in multivariate research: Invariants and the
reconstruction of attractors. Nonlinear Dynamics, Psychology, and Life
Sciences, 1, 7-33.
Abraham,
R.H., & Shaw, C.D. (1982/1992). Dynamics:
The geometry of behavior.
Aks, D.J.,
& Sprott, J.C. (1996). Quantifying aesthetic preference for chaotic
patterns. Empirical Studies of the Arts, 1(1), 1-16.
Atteneve,
F. (1957). Physical determinants of the judged complexity of shapes. Journal of Experimental Psychology, 53, 221-227.
Berlyne,
D.E. (1960). Conflict, arousal, and
curiosity.
Birkhoff,
G.D. (1933). Aesthetic measure.
Cupchik,
G.C., & Gebotys, R.J. (1988). The experience of
time, pleasure, and interest during aesthetic episodes. Empirical Studies of
the Arts, 6(1), 1-12.
Eisler,
A.D., Eisler, H., & Montgomery, H. (1996).
Prospective and retrospective time perception: Cognitive and biological
approaches. In S. Masin (Ed.), Cognitive models of
psychological time (pp. 251-256).
Hillsdale: Erlbaum.
Eisler,
H. (). Time perception from a psychophysicist’s perspective. In H. Helfrich (Ed.), Time and mind. (pp. 65-86).
Eysenck,
H.J. (1941). The empirical determination of an aesthetic formula. Psychological Review, 48, 83-92.
Fechner,
G.T. (1876). Vorschule der Äesthtik.
Gilbert,
K.E., & Kuhn, H. (1953). A history of
aesthetics.
Grassberger,
P., & Procaccia,
Gregson,
R.A.M. (1995). Cascades and Fields in Perceptual Psychophysics.
Gregson,
R.A.M. (1995). N-Dimensional nonlinear psychophysics. In
Mitina,
O.V., & Abraham, F.D. (in press). The use of fractals for the study of the
psychology of perception: Psychophysics and personality factors, a brief
report. International Journal of Modern Physics C.
Rashevsky,
N. (1938). Contribution to the mathematical biophysics of visual perception
with special reference to the theory of aesthetic values of geometrical
patterns. Psychometrika,
3, 253-271.
Sprott,
J.C. (1993a). Strange Attractors: Creating patterns in chaos.
Sprott,
J.C. (1993b). Automatic generation of strange attractors. Computers and Graphics, 17,
325-332.
Sprott,
J.C. (2003). Chaos and Time-Series Analysis. New York/Oxford:
Stoyanova,
Y., & Yakimoff, N., et al., (1987). In search of
painting’s influence on time judgment. Acta
Neurobiologiae Experimentalis,
47(2-3), 103-109.
This view was also reflected by Külpe (1895), “The more complicated the object of affective appreciation, the less possible, of course, is it to decide a priori what aspect of it will be effective, and consequently what will be the nature of the resultant feeling.”