Judgments of Time, Aesthetics, and Complexity as a Function of The

Fractal Dimension of Images formed by Chaotic Attractors

^{1}Depatment of Psychology, Silliman University,
Dumaguete City, Philippines 6200. and Blueberry Brain Institute, Waterbury
Center VT USA 05677

^{2}Correspondence should be sent to: abraham@sover.net

^{3}Department of Physics, University of Wisconsin,
Madison, WI 53706

^{4}Department of Psychology, Moscow State
University, Moscow, Russia

Abstract

* *

In this experiment, we obtained judgments of the duration of
presentation of 3D images of chaotic attractors, and judgments of their
aesthetic value and complexity as a function of their fractal dimension (D_{2}).
We used four levels of fractal dimension (four stimuli at each level, mean D_{2}s
= .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
D_{2} = 1.54, and falling off at D_{2} = 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 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. Academic level, gender, or
cultural differences were not significant sources of variation in judgments,
although there were some interesting individual and cultural findings.

KEY WORDS: psychophysics, time, complexity, aesthetics, chaos, fractal dimension

1. Introduction

1.1 Narrative Background

It was originally my intent to study the perception of time as a function of the fractal nature of stimuli, using auditory or musical stimuli, when Elliot Middleton informed me of a program by Julien Clinton Sprott that generated visual strange attractors, which I immediately downloaded from Sprott’s website, and was immersed in a world of beautiful images. We realized that by switching to these visual stimuli we could have a set of images whose fractal dimension was already computed by his program. I had been using his earlier book (Sprott, 1993) with the newly established Chaos Society of Silliman University, and so was familiar with his approach to dynamics, and with his research on aesthetics as a function of fractal dimension (Aks & Sprott, 1996). I later visited him while he adapted his programs to more general psychophysical explorations. While mainly interested in time perception, I had decided to use aesthetic judgments as well, mainly to attenuate the importance on time for my participants. Then I had the idea to add complexity judgments, thinking at the time that there were no precedents for that, which was true for such images, but in fact, I learned later, has a large history in psychophysical research. At any rate, he adapted his programs for me to be flexible to meet the changing needs of future experimental designs. When some undergraduate students at Silliman asked me for an idea for a research project using chaos for an experimental psychology course they were taking, the opportunity for them to do this exploratory study presented itself. About the same time, while giving lectures at a conference in Moscow in 2000 (Sulis & Tromifova, 2001), I shared Sprott’s computer program with Olga Mitina who subsequently did a nice study with it adding the investigation of personality factors (Mitina & Abraham, 2003). Thus began our collaboration.

1.2 Research Background

Nonlinear dynamical systems have recently been explored in cognitive and perceptual systems by the foundational programs of Freeman, (2006; Skarda & Freeman, 1987), Gregson (1996, 2006), Heath, (2000), Kelso and Engstrøm (2006), Turvey (2005), and Ward (2001), among many others. For a brief earlier history, see Abraham (1997a).

Perceptual/neural organizational features of attending
complex stimuli may affect both the estimation of complexity and aesthetics,
but also, there is the possibility that they could also affect the perception
of time. There is likely a nonlinear interaction between the complexity of
stimuli and experiencing time and aesthetics. Studies of both the perception of
time and of aesthetics have centered on cognitive and biological factors
(Anderson & Mandell, 1996; Eisler, A., 2003; Eisler, H.; Eisler, Eisler,
& Montgomery, 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 could
likely be said for aesthetic judgments (Aks & Sprott, 1996, 2003; Mureika,
Cupchik, & Dyer, 2004; Sprott, 1993, 2003; Taylor, Spehar, Wise, *et al*.,
2005).

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). Also, they are 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 spatial dimensions with the third dimension being represented by color coding, which adds to their aesthetic potential. To check on the assumption that mathematical complexity has a relationship to perceived complexity, it was decided to add an estimate of complexity to the ratings obtained from the participants.

While complexity of stimuli was the principal independent variable, demographic factors as well, namely age and culture (Eisler, Eisler, & Montgomery, 1994; H. Eisler, 1996) were also varied. Therefore, both children and adults, and both urbanized students of Silliman University and rural cultural minority visitors (Ati and Sulod) to a specialized education program to our campus served as participants.

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.

2. Methods

2.1 Participants

(1) Six students (1 female, 5 males, from grades 4-5) at Silliman University Elementary School were recruited from a special computer learning project under the direction of the senior author. Their participation in this experiment had direct relevance to their goals of the integration of art, science, and mathematics.

(2) Six graduate Students (4 females, 2 males), Department of Psychology, Silliman University were recruited from a graduate seminar.

(3) Six Ati and Sulod adults (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 (Negros Oriental Province, where the University is also located).

**2.1.2 Permissions**

In accordance with standard practice for undergraduate research, the locus of informed consent was formally placed in the hands of administrators of the educational units involved. Therefore permission to solicit participants was obtained from Ms. Iyoyo, Principal of the Silliman University Elementary School, Dr. Margaret Udarbe-Alvarez, Chair of the Psychology Department, and Dr. Betsy Joy Tan, Dean of the School of Education and administrator for the Ati Educational Project. Individual participants participated by verbal agreement, but that contained the usual elements of informed consent.

**2.1.3 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.

2.2 Stimuli

Sprott (1993) developed a program for the generation of computer images of chaotic attractors from systems of three nonlinear coupled difference equations (quadratic maps) which were adapted for use in this and similar experiments (Mitina and Abraham, 2003). The equations are:

x_{n+1} = a + bx_{n} + cx_{n}^{2}
+ dx_{n}y_{n} +ex_{n}z_{n} +fy_{n} + gy_{n}^{2}
+ hy_{n}z_{n} + iz_{n} + jz_{n}^{2}

y_{n+1} = x_{n}

z_{n+1} = y_{n }where
_{ }x,y,z are variables and a–j
are parameters.

Solutions (trajectories) are displayed on computer monitors
in two dimensions (x,y, for the horizontal and vertical axes of the plane of
the screen) 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 points, not as a
continuous line) by iteration of the equations, although they evolve so fast as
to be essentially perceived as filling space rapidly over time, rather than 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 or density in filling the 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, divergence
from nearby starting points). The program was used to generate 168 of these
stimuli. A statistics program (STATISTICA) was used to analyze these two
measures of the 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 D_{2}
(introduced to dynamics by Grassberger & Procaccia, 1983; and reviewed in
many books and articles, including Sprott, 2003, p. 307-311, and Abraham,
1997b, pp. 17-18; the fast algorithm of Sprott, ibid p. 317, was used for the
computations here). Stimuli were chosen with D_{2}s in four ranges,
.5–.85, .86–1.4–1.6, and 2.2–2.4 (mean D_{2}s, 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://www.blueberry-brain.org/silliman/jemstim.htm

2.3 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 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:

Once the experimenter finished setting up the computer to run the experiment, the participant pressed any key on the computer keyboard 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)]?**

Then the experimenter explained the rating scale with the first stimulus.

The participant responded on the keyboard, with a number and followed with an entering key-press.

The second question then replaced the first question:

**How complex was the image [1 (least) - 9 (most)]?**

To insure that the participant used a wide range of ratings of complexity, the experimenter provided guidance on numerical values for this and the following three warm-up stimuli.

After that, the participants self-paced themselves for the remainder of the run through the remaining 16 of the 20 images.

The experimenter then reset the computer for the second run that was used to obtain the ratings of perceived duration (time). This second run was completed as follows:

For the estimation of duration of presentation of the images,
the same images were used, with the 5^{th}-20^{th} being in a
new random order. The participant initiated the run with a key press. Each
stimulus was presented for 10.05 seconds, turned off, and then the following
question was presented on-screen:

**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 were completed, the participants were 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.

3. Results

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 (D_{2}) 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 D_{2} (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.

** **

3.1 Judgments of Complexit*y*

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 (mean D_{2} = .59, range: .52–.65), low-mid
(mean D_{2} = 1.07, range:1.05–1.09), high-mid (mean D_{2} =
1.54, range: 1.46–1.55), and high (mean D_{2} = 2.27, range:
2.21–2.36).

This figure shows judgments of complexity to be a
non-monotonic increasing function for most of the range of D_{2} as
might be expected, but decreasing at the highest values. That is increasing
complexity is perceived as such, until the images at the highest D_{2}
values which are reported as less complex. Even within the increasing portion
of the curve, the result is decidedly non-Fechnerian: increases in D_{2}
are not met with proportionately increased magnitude of the judgments of
complexity. The ANOVA showed significant results for D_{2}, 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.

3.2 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 (D_{2}), 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, were
not significant, while the within-participant main independent variable (levels
of DAlan

I will try to find that tape. It will be a blast to see it again. Forgot that I
even taped it.

fred_{2}), was 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 D_{2} also were similar to those
for the complexity judgments (see again the table at http://blueberry-brain.org/silliman/jemanovaaesth.htm).

3.3 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 (D_{2})
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 ratings for the lo-mid complexity stimuli were significantly greater than the lo level stimuli of the mathematical complexity variable.

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 (except for the elementary students which included but one female), and for the Ati/Sulod group compared to the elementary and graduate students.

3.4 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, D_{2}) 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.

3.5 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).

3.6 Correlations: Individual Differences

Half the participants showed a pattern of higher correlations among the fractal dimension and aesthetic and complexity ratings and low correlations involving time judgments (participants 1, 4, 5, 6, 7, 8, 16, 17).

One participant (9), who was also a instructor in art at
Silliman University, expressed a preference for visual art that had a lot of
open space. However, his aesthetic ratings revealed a zero correlation (r =
-.097) rather than a negative relationship, possibly due to the nonlinear
relationship between area covered and D_{2}. The zero correlations
indicate a constant estimation of time, reflecting his accuracy on this factor.
While his claim for a preference for art with open space is not born out, he
appears to be a person with strong independent aesthetic convictions, which are
certainly not related to mathematical complexity.

Two participants (2, 16) showed very strong correlations
among D_{2} and aesthetic and complexity ratings and some fairly strong
positive correlations involving the time estimates.

Another participant (17) also showed very high correlations
of D_{2} to aesthetic and complexity ratings, but his correlation
between the two ratings was somewhat lower (0.346) than the mean (0.638)
showing independence of aesthetic and complexity judgments. This participant
showed the unusual result of moderately strong negative correlation of time
estimation to D_{2} and complexity ratings. A few other participants
showed some correlations to time estimation, both positive and negative.

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? For example, participant 15 had identical scores for the two
ratings aesthetic and complexity to D_{2 } (0.559). Likewise, the correlations of aesthetics and complexity
to time estimates were also identical (.604) for participant 15. Thus correlations of aesthetic ratings were
higher to complexity judgments than to actual mathematical complexity. These
could reflect response bias or generalization, possibly due to a sense of
uncertainty. Two other participants (3, 10) 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 (11) was similar to these two participants,
except that the aesthetic to complexity correlation was smaller.

Of the remaining three correlations (all for the time judgments), a few were modestly stronger, 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 more strongly negative (Participants 7 & 17), and four (Participants 2, 14, 15, & 16) strongly positive. Three were strongest for the correlations to the mathematical 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 largest being r(A,T) = –0.434 for participant 13. The remaining correlations of aesthetics ratings to time estimates were negligible except for three of the Ati/Sulod group (participants 13, 15, 16) who had correlations of .-0.434, 0.604 and .434 respectively.

The only hint of cultural and gender differences that can be found in the data are swamped by individual differences, and the relationships to mathematical complexity are much stronger and therefore survive individual differences.

4. Discussion

4.1 Complexity

Since the judged complexity displayed the classic
∩-shaped function (also known as the inverted U-shaped function; Walker,
1971) of aesthetics to the mathematical complexity of the images as measured by
D_{2}, the question arises as to why that should be? Visual inspection
of the images gives one possible answer. For stimuli where the mean D_{2}
= .59, the space is quite empty; the attractors fill little of the available 3D
Cartesian space for the image. When going to the next higher D_{2}
(mean D_{2} = 1.07), the attractors fill the state space (the 2D
computer screen) more completely, and the space within the attractor is also
more completely filled. Going to the next higher (mean D_{2} = .154),
there is a further increase in the filling of space and more visual boundaries
and features within the attractor (the third, colored dimension helps a lot
with that). At the highest level (mean D_{2} = 2.27), the attractors again
occupy slightly more of the viewing space, but the most apparent difference is
the increase in density of points within the attractor, filling the attractor
more completely, but with less interior details. The attractors appear more
cloud-like, more homogeneous, with less interior definition and boundaries,
which could be the basis for their being judged less complex.

Why does the measure, D_{2} not see all that the eye
sees? D_{2} sees mainly the filling of space and does not enjoy the
advantages of the eye, brain, and cognition in giving weight to the boundary
effects, and other subtle features, especially those arising from the coloring.
And unlike other measures of complexity, such as D_{0,} (also known as
the capacity or box-counting dimension; an upper bound on the
Hausdorff-Besicovich dimension), which weight the hypercubes of the
computational algorithm equally no matter the amount of the occupancy of each
hypercube, D_{2} weights the hypercubes by the extent to which they are
each occupied, and thus the more dense images produce higher estimates of
dimension.

What else might one investigate concerning judging complexity? Our instructions to participants were quite nonspecific, giving little guidance to participants as to what meaning they might wish to construct or assume for it. Very likely there 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 participants’ narratives, but many participants corroborated the kinds of details mentioned above. The similarity of the three groups suggests that all participants were rather sophisticated in observing these details. Garner (1962, 1974) speaks of quantifying aspects of spatial dependencies of images within an informational theoretic framework as distributional redundancy and correlational redundancy. Symmetry, a feature studied in many psychophysical studies that tends to reduce perceptual complexity (Atteneve, 1955; Day, 1967, 1968; Perkins, 1932) could not be a factor here, as our images are devoid of this feature along with many other features used in earlier studies. While our images do not contain subsymmetries (Alexander & Carey, 1968) either, their study could be relevant in terms of the nonhomogeneities present in our fractal images.

Randomly constructed polygons were frequently used
for studies of complexity during the 1950s to the 1970s. Some studies found
that the number of sides of random polygons were a major determinant of
perceived complexity (Arnoult, 1960; Attneave, 1957, Attneave & Arnoult,
1956; Day, 1967; Munsinger & Kessen, 1964; Stenson, 1966). Generally, these
studies showed complexity judgments as an increasing monotonic function of the
number of sides, considered a measure of complexity, of the judged images (Day,
1967). Attneave (1957) also showed that
P^{2}/A (P is perimeter/ A area
of a polygon) and angular variability also contributed to judgments of
complexity.

Another popular basis for image formation for the
study of complexity were checkerboard patterns formed of black and white
squares (Berlyne, 1958; Chipman, 1977, Chipman & Mendelson, 1979; Dorfman
& McKenna, 1966; Houston, Garskof, & Silber, 1965; Karmel, 9166, 1969;
Nicki, 1972; Smets, 1973). Some of these showed perceived complexity as an
increasing function of the number of elements and as an inverse function of
redundancy (information theoretic term for orderliness; Houston *et al*.,
1965; Karmel, 1966, 1969). Karmel (1966, 1969) showed that the amount of
contour of such figures was more important than other aspects of pattern
complexity. This feature could be a candidate considered analogous to both
external and internal contours of our attractors, and to similar boundary-like
features within our attractors.

The studies by Chipman (Chipman, 1977; Chipman &
Mendelson, 1979) are strongly suggestive of our contention that non-homogeneous
features of our attractors are relevant for the subjective complexity of our
images. She used 6x6 checker-board-like matrices, 12 of the 36 squares being
black, scattered within the matrix producing differing degrees of complexity
that was quantified as to number of turns (corners, which is correlated with
number of sides), P^{2}/A—a measure of contour complexity; think of
Koch or Peano curves, or better, the Koch Island which builds with squares
rather than the Koch snowflakes built algorithmically using a Cantorian
replacement process from triangular components (Mandelbrot, 1977, Chap. 2). She
also used various measures of symmetry (partial and whole) around vertical,
horizontal, and diagonal axes, and repetition (missed by symmetry measures).
She performed stepwise regression analysis of magnitude estimates to these
variables. Separate sections of her study progressively orthoganalized
confoundings among these independent variables. Thus she was attempting to
determine that **“a wide variety of pattern structures may be effective in
reducing perceived complexity.” (p. 271.)**. She points out that for stimuli
constructed from particular elements, that there could be arbitrarily different
ways of measuring dimensions of complexity, as Brown and Owen (1967) had also
shown. In discussing the work of Alexander & Carey (1968) on subsymmetries,
she points out that “The particular form in which the pattern variables were
computed was determined both by the desire to detect partial organization and
by the stipulation that they should be readily extended to patterns of
arbitrary size or into a continuous function.” The concept of fractal dimension
introduced by Mandelbrot (1977) accomplishes those objectives of a continuous
measure independent of size of image, although it fails to determine some the
features of internal structure within our images.

Chipman found that complexity judgments increased
with increasing number of corners, and that almost all aspects of patterned
organization, most especially symmetry features, reduced complexity judgments
from those limits. This may seem at odds with our result—the ∩-shaped
function—unless we identify our internal organizational features with her
corners, i.e., her complexity, rather than with her patterned features like
symmetry, which are lacking in our images. Thus when you get to the homogeneous
figures of our highest fractal dimension, there is a lack of much internal
structure. It might be noted that, while most of her experiments showed
complexity judgments as an increasing function of image complexity, in one
experiment where range of the number of corners was very high, there is also
the hint of the ∩-shaped function, resulting from a drop in complexity
judgments at the highest value of the number of corners in her images. That is,
when her image complexity gets great enough, perhaps it loses perceptual
organization like ours, and appears more homogeneous or its organizational
features are too small or too overwhelmed to detect. She did show, with lesser
complexity, that partial areas of organization did tend to reduce perceptual
complexity, which is akin to Wertheimer’s *Prägnanzstuƒen* which means
“regions of figural stability” (Wertheimer, 1923).

Her findings also show that the reduction in
perceived complexity becomes effective above a threshold of image complexity,
when it assumes a power function whose constant quantifies the magnitude of the
reduction. Also there was a interaction between the number of corners and
degree of symmetry suggesting that symmetry becomes more effective when closer
to an “ideal of perfect organization”, *Prägnanz* (Köhler, 1920), i.e., a
perceptual attractor, a result similar to that of Zusne (1971) who found
symmetrical forms were perceived as more symmetrical than they actually were, a
classic Gestalt result.

Chipman also showed that if different types of organization are combined in an image, they compete rather than summate to produce the reduction in perceived complexity. That suggests there is a self-organizational process going with bifurcations allowing differing basins of attraction to be attempted in the perceptual process of assessing pattern and pattern complexity. As is well known, systems are unstable the closer control parameters are to bifurcation points (Abraham, Abraham, & Shaw, 1990; Aks & Sprott, 2003). It might require complex nonlinear psychophysical models, such as Gregson’s Γ recursion models (Gregson, 1995a,b), to adequately model such a dynamical perceptual process, which we will not attempt here. Aks and Sprott (2003), point out that another modeling process, that of self-organized criticality, may also be employed for bifurcations “from the interaction of the system’s component parts” which could apply to the attentional components within our images, similar to the dynamics within Bak’s sandpiles (Bak, Tang, & Wiesenfeld, 1987).

We might mention some other historical precedents for studying stimulus complexity. Many studies have used complexity as a property of stimulus pattern in a variety of investigations of curiosity, arousal, emotion, and aesthetics, etc, as one of a set of factors which Berlyne calls ‘collative’ which also included ‘novelty, surprisingness, ambiguity, and puzzlingness’ (Berlyne, 1971, p. 69) as well as ‘structural’ and ‘formal ones (Berlyne, 1974, p. 5); Cupchik & Berlyne, 1979). Here we concentrate on those studies simply for their concern with ratings of complexity, and take up those involved in aesthetics and time estimation later. Day (1965) used rank order to relate subjective complexity to objective categories of complexity of stimuli, but not to quantified properties of the physical complexity of stimuli. Most of his stimuli were geometric and curvilinear shapes, but some were not unlike our attractors in some respects. Berlyne, Ogilvie, & Parham (1968) used the same (Day’s) stimuli (Figure 4) and the Shepard-Kruskal multivariate-scaling procedure with principal-axis rotation obtaining results that indicate “that subjective complexity depends primarily on the two principal determinants of information content.” (Berlyne, 1971, p. 201.)

Cupchik’s work on experimental aesthetics goes back to his
work with Berlyne and their colleagues at Toronto, with some of his work
related the dimensional analysis (factor analytic rather than fractal) of art
(Cupchik, 1974). This first paper of his (1974) upon which we wish to focus
starts with Wölfflin’s (1915) characterization of five dimensions or stylistic
factors of art as linear versus painterly, plane versus recession, closed
versus open form, multiplicity versus unity, and absolute versus relative
clarity (p. 236). We would view these as in dynamic interplay, creating various
basins of perceptual attraction, for both artist and viewer, which would depend
upon the interaction of features of the art, personal/cognitive features of the
artist/viewer, and cultural/contextual factors. For our study, some of these
dimensions are more heavily loaded than others, for example, dimensions of
linear-painterly (objective or *systematic nomalization* versus subjective
or *systematic distortion* in perceptions of art, pp. 237-239) and
multiplicity-unity would be more important. Cupchik mentions Zimmermann (1858),
Riegl (1893), and Worringer (1908) for the roots of Wölfflin’s distinctions,
but when he mentions that “Neither style depicts external reality with complete
fidelity.” (p. 237) it suggests the distinction between Parmenides’ world of
fixed, eternal, or static ideals and his *doxa*, the world of the
illusory, and his efforts to explain the dynamics between the two. Thus for any
given dimension, such as linear-painterly, or multiplicity-unity, there could
be two basins of attraction, for linear-painterly, one attractor for external
ideals, and one for perceptual creative involvement. The coupling constants could
make one predominate with resulting bifurcations, say eventually to a single
attractor, perhaps obeying a simple cusp-catastrophe. Or the parameters could
allow a continued perceptual trajectory in a multi-basin portrait, or could
create bifurcations to cyclic or complex chaotic attractors, all of which
requires self-organization and navigation in parameter space.

It might seem something of an excursus to continue a bit
more on Wölfflin, but we might point out that Wölfflin was influenced by
Dilthey’s historical/scientific hermeneutics, which may be reflected especially
in his linear-painterly distinction, which in some ways is like the
logocentric-deconstructive distinctions made in post-modern literature (e.g.,
Derrida), much of which was centered on not only social progress, but also on
linguistics, art and architecture (Benjamin, 1936/1979; Baudrillard,
1981/1983). Kristeva (1980), for example makes a linguistic distinction between
symbolic (Oedipal, paternal) and semiotic (pre-Oedipal, maternal), which is
paralleled by a distinction in personal space as being more metric (Aristotle’s
*topos*) or more amorphous (Plato’s *chora*). And all of these
distinctions are paralleled by the long-running difference of approaches in
philosophy and history between the Parmenidean-vs-Heracletian ontological
perspectives. Can short-term perception and aesthetics be a fractal zoom on
different time scales of self-similar processes that are themselves
interdependent? Cupchik notes these parallels also in characterizing the “external
factors of the model: democracy-autocracy and consistency-inconsistency of
communication and reinforcement.” (p. 244.)

Cupchik performed two experiments using 16 paintings from the Art History Library at the University of Wisconsin that varied on four of Wolffin’s dimensions, as selected by expert judges. In the first experiment, student participants rated the dissimilarity of each of the 120 pairings of the paintings. Nonmetric scaling showed “The dominant dimension distinguished paintings which share an emphasis on outline, and contrasts them with others that appear to emphasize qualities, and whose forms are blurred or indeterminate.” (p. 348.) He compared this to the linear-painterly dimension, thus giving it further definition. Similarly, the scaling procedure showed a dimension close to the tonal-versus-colorful discrimination, one close to the abstract-versus-representational distinction, and the weakest dimension turbulent, active versus simpler, calmer, which seemed to correspond to the multiplicity-versus-unity dimension. (p. 349.) A second experiment was used in which participants rated each of the same paintings on 16 scales, and was analyzed with factor analysis. The first factor emphasized line, clarity, ideas, and coldness, thus being like the linear characteristic, which he called “Classicism”. The second factor de-emphasized accurate reproduction and emphasized color composition and shapes, and thus called “Subjectivism”. (p. 253.) A third was labeled “Complexity”, and the fourth, emphasized emotion, which he called “Expressionism”.

The images in our study may have confounded the
complexity dimension, as defined by D_{2}, with the linear-painterly
dimension, since not only does the space get filled more completely, but the
attractors have more linear features, going to greater homogeneity as D_{2}
increases. It is easy to see how this could affect judgments not only of
complexity, but of aesthetics and time as well.

Information theoretic measures of aesthetic patterns
have been employed in music (Pinkerton, 1956) and images (Atteneave, 1957,
1959). Some studies related subjective complexity to informational measures of
polygons randomly constructed with increasing number of sides (Arnoult, 1960;
Atteneave & Arnoult, 1956; Day, 1967; Munsinger & Kessen, 1964), while
other used checkerboard patterns (Houston, Garskof, & Silber, 1965; Karmel,
1966, 1969). For some of these studies, subjective complexity was proportional
to objective complexity, and inversely proportional to redundancy (Day, 1965;
Houston *et al.*, 1965; Karmel, 1966, 1969) as have been several studies
of complexity judgments for music (Crozier, 1981). We will consider that when
the function is increasing monotonic instead of ∩-shaped in those
studies, that the parameter space (independent variable) was not sufficient to
show the complete function, that is, the upper limb of the ∩-shaped
function.

Walker (1981) offers other explanations for both a rising function of subjective complexity as a function of objective complexity as well as for the ∩-shaped function, depending on the circumstances. For the rising function,

“There can be situations in which the range of complexity values of the physical stimulus can be quite large while the range of psychological complexity values can be considerably restricted. This can occur when the subject fails to process all of the material present in the physical stimulus, especially of the more complex stimuli. A result could be an inverted U when all of the information is processed, but an essentially linear rising pattern when it is not.” (Walker, 1981, p. 47.)

This explanation is consonant with an experiment by Olson (1977) who obtained the monotonic rising result, and with our present experiment with its ∩-shaped function, since our participants may not have been able to perceive or appreciate the mathematical complexity of the images of highest complexity.

We conclude that the ∩-shaped function of subjective
complexity as a function of the fractal dimension could be due to a combination
of the more amorphous nature of the images with the highest D_{2}, and
the lack of time or ability to appreciate the complexity within those images.
The obvious nature of the increased complexity of the images with the lowest
three ranges of D_{2} is obvious and responsible for the increasing
limb of the ∩-shaped function.

4.2 Aesthetics

While aesthetics did not provide the principal motivation
for the present experiment, it deserves some attention because aesthetics has
long been known to be, in part, a function of complexity. The early history of
ideas of unity and diversity (chaos depends on the interaction of these as
tendencies of convergence and divergence) in aesthetics is summarized by
Berlyne(1960) and Gilbert & Kuhn (1953). “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 (Berlyne, 1960, p. 237).” Fechner mentioned the
veining in marble as an example.

Birkhoff, an early developer of mathematical dynamics
(Birkhoff, 1927, 1932) formulated a mathematical theory of aesthetics in which
complexity was a factor (Birhoff, 1932). 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 should be proportional to order, and inversely
proportional to complexity. Very shortly, there were attempts to test this theory
(Davis, 1936; Eysenck, 1941) which found, similar to our result, that there was
a maximum of aesthetic judgment at intermediate values of Birkhoff’s M. Our
measure of complexity, D_{2}, 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 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).

In the 1950’s, there was a dramatic development in ‘intrinsic’ motivation, moving beyond the ‘extrinsic’, interoceptively driven drives such as hunger and thirst, touched off primarily by the work of Harry Harlow and his students at Wisconsin (Harlow, 1950). These included studies of manipulatory and exploratory behavior, arousal, and curiosity. Some studies of exploratory behavior had already appeared (Dashiell, 1925; Mote & Finger, 1942; Nissen, 1930). Daniel Berlyne also developed a major program of research at Toronto (1960). Visual stimuli are important in such studies, and Berlyne (1958, 1969) was thereby apparently drawn to art, beginning as evident in a note on the golden mean, which Fechner (1876) had also studied. One of the extraordinary features of his program in addition to the innovative experimental methods was his development of a coherent theoretical framework throughout all these directions of exploration, curiosity, play, humor, and aesthetics which brought together elements of neurophysiological arousal theory and behavior conflict theory (Berlyne, 1971). His two-factor approach began with Wundt’s arousal theory (Wundt, 1874), which showed a ∩-shaped arousal curve resulting from the summation of positive and aversive hedonic tendencies (Figure 5; from Berlyne, 1971).

Wundt proposed the relevance of this conflict-arousal theory to aesthetics: “. . . the simplest cases of the pleasing and displeasing have been almost entirely lost sight of, although they constitute a necessary basis for psychological theory including the explanation of the most complicated aesthetic effects.” (Wundt, 1874, p. 222; quoted in Berlyne, 1971, p. 28).

Berlyne also used the linear dynamical approach-avoidance
conflict model of Miller (Miller, 1937, 1959) to explain ∩-shaped
aesthetic curves as a function of positive and negative affect, including the
influence of novelty and complexity on response strength (Berlyne, 1971, p. 166).
Lewin, known for his dynamical field theories, also had a model of
approach-avoidance conflict (Lewin, 1951). There have been some simple
nonlinearizations of Miller’s model (Figure 6; Abraham, 1995; Abraham *et al*.,
1990; Townsend & Bussmeyer, 1989) and of Lewin’s model (Abraham, 1997a).

Most of the stimuli used in earlier experiments were not particularly aesthetic (Figure 4), being mostly simple black white line drawings varying in complexity, informational content, dissonance, novelty, etc, they thus did not cover a broad range of aesthetic possibilities. It is easy to imagine the stronger aversive affect toward them dominating the weaker pleasing affect toward them, thus generating the ∩-shaped function of aesthetics to complexity, but that involves some assumptions about the homogeneous figures in our highest fractal images being either aversive compared to the expectation of the more aesthetic figures included, or at least being more boring. To further conjecture on such possibilities, it may first be worthwhile to look at some of the more recent work based on dynamics and more aesthetic images.

There are two principal lineages of contemporary studies, those based on aesthetic judgments of fractal images, inaugurated by Sprott and his colleagues (the Sprott Squad); and those using images based on art, such as those of Taylor and his colleagues (Team Taylor).

Sprott first announced his study of aesthetic evaluations of
chaotic attractors in his earlier book (Sprott, 1993). He used monochromatic
(black-on-white) two-dimensional quadratic maps displayed on computer screen
and ratings on a five-point scale, and obtained the ∩-shaped function
with preferred D_{2}s in the 1.1 to 1.5 range. Aks and Sprott (1996) replicated this result
using a slightly different psychophysical method, that of choosing the most
preferred image from four varying in D_{2} and the principal Lyapunov
exponent. While they got a ∩-shaped function for both measures, we did
not investigate the Lyapunov exponent as our images were drawn so fast on the
computer screen it did not seem to offer any possibility for visual evaluation
of the divergence of trajectories. That possibly may have been premature, both
on the basis of the positive results of both the Sprott (1993) report and the
Aks & Sprott (1996) study, and it could be that the Lyapunov, λ, may
have contributed to the internal detail at mid-range dimension, and to the
homogeneity of the for the high dimensional images, with D_{2} mainly
measuring the result. That is, D_{2 }may be more a result of the
dynamics in filling the space, but λ may be a better reflection of the
flow of the dynamics that determines the clumping or internal detail, the
tendencies to converge as well as diverge. Aks & Sprott these properties,
that D_{2} measures the filling of space and λs measure the
sequential properties of the divergence and convergence of trajectories.

We think that our study shows a preference for images with
higher D_{2}s not merely because the Cartesian embedding space is 3D,
making three the upper limit of possible D_{2}s_{ }compared to
the upper limit of two for the 2D space of Aks and Sprott’s images, but
possibly the greater aesthetic value that color affords and its emphasis on the
internal heterogeneity of the images. We conclude that the preference for
slightly higher D_{2}s in this study is real and due to the changes in
images, mainly in the use of color and context making our images a bit more
appealing.

Scott Draves developed an ingenious system of obtaining
voting from thousands of internet users for his animated art deployed as
screen-savers which he called electric sheep (Draves, 2005a,b; based on Dick,
1986). These were brief animations using 2D nonlinear iterated function systems
to create fractal attractors of considerable aesthetic value. The population of
deployed screen-savers evolved over time based on voting, and were generated by
(a) random seeding of computational parameters, (b) genetic algorithms, and (c)
users’ contributions. Sprott’s fast algorithm (Sprott, 1994, 2003) was later
applied to the images to determine their D_{2}s (Draves *et al*.,
in press). They obtained a positively skewed ∩-shaped function of voting
to D_{2} with an average D_{2 }= 1.52 +/- 0.23 for the most
highly rated sheep, possibly an underestimate of central tendency due to the
skewing. While the frequency distributions of the images were also extremely
highly skewed, and not evenly distributed as in the Sprott, Mitina, and Abraham
studies, it appears that the voting did not simply follow these underlying
distributions, but represented real preferences for the mid-dimensional
animated images, especially in rejection of the very highest D_{2}s and
the randomly seeded animations. Thus there was not only a preference for the
mid-dimensional images, but an evolution toward those preferences (Draves et
al., in press, figure 3). There was a similar evolution of Pollack’s drip
paintings from D_{0} = 1.1 to 1.7 over the years 1944 to 1954 (Taylor;
2002, 2005; Taylor *et al*., 2003), representing a tendency to increase
toward greater complexity, while the Draves’ evolution seemed to be more
downward, resulting from the extinction of higher dimensional images.

Taylor began by establishing that Pollock’s paintings
satisfied the mathematical criteria as fractal images from resolving the
measurements of D_{0} (“box counting” or “capacity” dimension, whose
values are close to D_{2} although D_{2} weights denser areas
of the attractor more (Taylor et al., 1999). Taylor found that D_{0}
increased for Pollock, both rapidly in the early stages of crating one painting
(1950), from about .5 to 1.9, as well as over the 10 year period mentioned
above; presumably representing Pollock’s own aesthetic preferences for D_{0 }(Taylor,
Micolich, & Jonas, 2002).

Many have speculated that aesthetic appreciation of fractal images and environment could be due to nature (evolution of neural properties) or nurture (experience in natural and social environments). The former would suggest the possibility of universal preferences for an ideal value of fractal dimension, and to test this concept, Taylor, using the method of paired comparisons, and images from nature, art, and computer-mathematical generation (Taylor, 2001, 2005; Spehar, Clifford, Newell, & Taylor, 2003) and found a maximum preference for mid-value D0s = 1.3-1.5 “irrespective of their origin”. They note that our failure to obtain gender and cultural differences supports this finding of universality, it should be obvious that our small N and cultural and educational conditions were not optimal for finding such differences, and more careful work in that direction would still be worth doing despite exponential growth of global cultural homogenation.

In addition to their confirmation of the ∩-shaped
function, they report that the galvanic skin response (skin conductance, a
measure of sympathetic activity related to arousal) exhibits a U-shaped
function, that is, arousal is minimal when aesthetics is high (Taylor, 2005,
2006; Taylor, Spehar, Wise, Clifford, Newell, & Martin, 2005; Wise &
Taylor, 2003). It would be nice to see this work replicated with a wider range
of D_{0} and with GSR measures to the images rather than only arising
from their attenuation to the stress of cognitive tasks. Their finding,
nonetheless, could be important to the Miller-Dollard-Berlyne-Abraham conflict
theory, showing minimal aversion and maximal positive affect at mid-dimensional
values. The generic ∩-shaped function is similar the gradients postulated
by Miller for his rats staying or oscillating near a midway point in a runway
where they have been both shocked and given water at the goal end (Miller,
1937a,b; 1959; see also Abraham, 1995; Abraham *et al*., 1990; Berlyne, ).
Their discussion of evolutionary or ‘neuro-aesthetics’ (see especially Taylor,
Spehar, *et al*., 2005; Zeki, 1999) addresses a broad history of
speculation on that subject (Barrow, 2003; Mandlebrot, 1979, 1989; Zeki, 1999).
While there is often a confounding of phylogenetic and ontogenetic factors
involved in such speculations, the possibility is consistent both with the
universality mentioned previously, and the limbic aspects of conflict theory.

**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 Silliman University Elementary School, Dr. Margaret Udarbe-Alvarez, Chair of the Department of Psychology, and Dr. Betsy Joy Tan, then Dean of the School of Education, and now Vice President of Academic Affairs, for their support, Dr. Christian Schales for setting up our computer laboratory, and Dr. Agustin Pulido, President of Silliman University, for establishing that laboratory for the Department. Thanks also to Robert Gregson, dynamical psychophysicist extraordinaire, for comments on our manuscript, and on the contributions of Berlyne, Birkhoff, Davis, Eysenck, and Graves. Gregson also provided analyses of eigenvalues of matrices from our results giving sophisticated support to the obvious limitations of the ANOVA’s and of the averaging of correlation coefficients due to heterogeneity of variances, large variations in individual differences, nonlinear features of psychophysical processes (his series of books on that subject being leading proponents of that field), and possible correlations in variances. I pass some of these on in an appendix, to show not only his sophistication, but the extent to which he went to assist us in our analyses. A true scholar in our field, and one who much deserved the award given him by the Society for Chaos Theory in Psychology and the Life Sciences.

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Notes for
further development, incorporation, deletion (memory aids):

Updates since February 24, 2006. Main updates since last version are in section 4.1 and references.

, and Cupchik and his colleagues (The Cupchik Cabel). Cupchik was a protégé of Berlyne (Cupchik, 1981; Cupchik & Berlyne; 1979.

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Kluwer Academic Publishers. [abstract]

Notes for further development of the paper:

Imagine prey-pred or other handy-dandy tool for model, bifurcations and portraits, response diagrams. In our strange attractor art, is the viewer tending to find some ideal form in the internal boundaries of our art, and thus finds the attraction in this attractor, or is the viewer made happier from freedom from form found in a mid-dimensional attractor, as a better less static ideal form?

His dissertation on the psychological aspects to
architectural appreciation is a synthesis of the major intellectual influences
in his life: Burckhardt's broad view of what constitutes an historical
document, Wilhelm Dilthey's psychological approach to historical hermeneutics, *Geisteswissenschaften*,
and Giovanni Morelli's (q.v.) technique of visual comparison, to name but three

http://www.blueberry-brain.org/chaosophy/semiotics/kristeva2.htm

Aks, Mitina, Richards

Gregson, Kocic-Stevanovska, others? Taylor Zausner2007bifurcations

The gsr, implication for conflict model, implication for stability/instability

Remaining discussion.

Mitina result similar; Draves result, Berlyne/Yevin dynamics, inverted U, aversion at top end, preference for intermediate complexity with some detail. How do the personality studies (Aks; Mitina; Richards) bear on the Berlyne 2-factor theory; aversion or homogeneity: both probably contribute.

We have noted the high correlation between complexity and aesthetic ratings, which could be attributable to any or all of three factors: (1) response generalization (non independence of response tendency independent of the rating category, (2) both ratings are responding to the same dynamical interaction between stimuli and participant, or (3) subjective complexity mediates aesthetics, linear rather than an inverted U.

Prey-predator model?

2^{nd} lineage: real art

time function: from Mitina,paper of Macar

fragment removed from section 4.1: disposition?

Another approach to the dynamics involved would be to see if people can learn to discriminate dimensionality by viewing images and training on them with feedback or labeling based on their dimensionality. Can the mathematical aspects of images be learned through experience judging them? Besides such learning being an obvious possibility, there was a series of experiments in the 1970’s by Nicki and his associates showing that learning could be influenced by the degree of blurring of images of everyday objects (Nicki, 1970, Nicki & Shea, 1970) and cubist paintings (mainly Braque and Picasso) for which information theoretic uncertainty values (Laffal, 1955; the methodology depending not on objective measurement of stimulus properties but on participants free associations to the paintings) had been established (Lee, 1972; Nicki & Lee, 1977). These studies showed a ∩-shaped function of subjective uncertainty to degree of blurredness (Nicki, 1970; subjective uncertainty of intermediate blurredness at just over .5 bits). They also indicated that learning could depend on a reduction of uncertainty and conflict. Nicki (1981) also showed that increases in ambiguity could increase reward value. These findings raise the issue, that while we often consider complexity as one determinant of aesthetic preference, might it be that that aesthetics can also influence subjective uncertainty (ratings of complexity), that they are involved in a complex system of interactive factors? We take this up later after considering aesthetics.