THE USE OF
FRACTALS FOR THE STUDY OF THE PSYCHOLOGY OF PERCEPTION:
Olga V. Mitina
&
Frederick David Abraham
The
present paper deals with perception of time (subjective assessment of temporal
intervals), complexity and aesthetic attractiveness of visual objects. Fractal
pictures were chosen as stimulus material. This choice was determined by the
fact that fractals can be described with the help of mathematically rigorous
information measures (fractal dimension, the Lyapunov exponent, etc.). These
information parameters vary in a rather broad and continuous range of real
numbers, so the experimenter can easily manipulate them. Besides, fractals
appear on computer monitor not as static images, but dynamically in the
experimentally preset time range, and that allows assessing the subjective
perception of time during which the image is being displayed. Finally, fractals
are nonstandard, abstract visual objects, which permit elimination of the
effect of recognition and related sideeffects (hidden associations, variable
or confounded conditions that are difficult or impossible to control in the
course of an experiment, etc.).
Fractals
enable visual representation of the mathematical idea of determined chaos
[Schroeder 2001; Bozhokin & Parshin, 2001]. This word combination itself
contains a semantic opposition and, therefore, a certain ambiguity. A fractal
curve can remain irregular and chaotic even in an indefinitely small area. In contrast
to the objects of traditional Euclidean geometry like point, line,
circumference, sphere, etc., which are complete forms, it is impossible to
finish the drawing of a fractal. One can determine the fractal algebraically,
for instance, but any finite realization is necessarily incomplete, emphasizing
that a fractal is a dynamical process, more than a static one that we may view
at any point in its evolution.
The
semantic opposition of dynamic and static, may raise fractals from mere
geometrical objects to the level of Art, in that any perceptual process is also
dynamical, which may relate to the dynamics of the creation of the stimulus
process being viewed, which is not surprising as biological evolution interacts
with both the external and internal aspects of perception [Yevin, 2001]. It is
not by chance that fractals appeared in the pictures of Dutch painter Maurice
Escher even before those of mathematician Benoit Mandelbrot who gave fractals
their rigorous mathematical description, which showed the static product of the
fractal process as a structure consisting of parts that in a certain sense are
similar to the whole [Mandelbrot, 1977 ; Peitgen & Richter, 1986].
Hence the main principle of fractal is selfaffinity.
The
incompleteness of any iterative fractal process implies that imagination is
necessary to complete the fractal object in human the human perceptual process.
This suggests, in a certain sense, a third semantic opposition, that between a
logical/mathematical process (algorithmic) and an intuitive one. Studying
subjective perception Sprott [1993] suggests the interpretation of fractal
dimension as the measure of fractal’s “strangeness”, and the Lyapunov exponent
as the measure of its chaoticity. Studying subjective perception Sprott [1993]
suggests the interpretation of fractal dimension, its degree of complexity in
space, as the measure of fractal’s “strangeness”, and the Lyapunov exponent,
its degree of complexity in time, as the measure of its chaotic nature.
Art
and other objects (all of which are, of course, produced by dynamical
processes), like fractals, can vary in their mathematical complexity. Fractal
dimension and the Lyapunov exponent are the most widespread mathematical
characteristics of fractals [Sprott, 1993; Abarbanel 1996]. Thus, doing
research on the perception of fractal objects may reveal insights into
perceptual process generally. Bearing in mind the fact that fractal dimension
is the measure of information, we can say that it is a measure of the object’s
“actual” complexity, this parameter providing a convenient independent variable
for perceptual experiments.
One
more semantic (and dynamical) opposition represented by fractal processes is
between constriction and expansion existing simultaneously in the same process.
Trajectories constrict infinitely on the dimensions that correspond to negative
Lyapunov exponents and infinitely diverge from each other on the dimensions
that correspond to positive Lyapunov exponents [Abarbanel 1996; Ott, Sauer,
& Yorke 1994]. This opposition, the struggle between convergence and
divergence, is responsible for the development of chaotic trajectories in
dynamical systems [Abraham, Abraham, & Shaw, 1990].
Since
William James [1890] many prominent psychologists [Fraisse, 1967; S.L.
Rubinstein ; B.G. Ananjev; and many others] were interested in the
problems of psychological time. Psychological time is essentially different from
objective chronological time. People have individual peculiarities of time
perception, for example, as smooth or uneven, compressed or stretched out,
empty or rich.
The
studies of the subjective length of intervals for stimuli representation and
their complexity are rather controversial. Schiffman and Bobko [1974]
ascertained that the more complex is the stimulus within a given interval of
time, the longer seemed the time interval to the subject. F. Macar [1996] obtained the results that
the increasing of stimuli complexity lead to the perception of time intervals
as shorter.
Nevertheless,
his work contains references to the results of numerous empirical studies over
several years indicating that the increased complexity of stimuli leads to the
perception of time interval as longer. This uncertainty agrees with the
hypothesis about the existence of the Ushaped (quadratic) dependence between
the complexity of stimulus and the evaluation of time. According to this
hypothesis simple and very complex stimuli are perceived as more lengthy in
comparison with the stimuli of moderate complexity [Lomranz, 1983].
The
search for mathematical correlates of the attractiveness of art, sculpture, and
architecture has very old traditions—e.g., the golden mean (the limit of the
Fibonacci ratio, 1.61818…—probably the most studied number in
mathematics : why should the eye like it so much ?), the formulas of
Leonardo da Vinci, etc. It is possible to say that Fechner laid the foundations
of experimental aesthetics studying which forms and sizes are most pleasant to
the eye [Schroeder, 2001; Crozie, 2002]. The study of the interdependence
between the magnitude of the objective stimulus and the evoked subjective
sensation is a classic field of psychophysical research. The relations between
physically or mathematically measured intensiveness of auditory, visual,
temporal and other signals and loudness, brightness, length perceived by man
were studied since the middle of the 19th century by E. Weber, G. Fechner, S.S.
Stevens [1975] and many others. For instance, the wellknown Weber–Fechner Law
states that within a certain range the dependence between the magnitude of
physical stimulus and the evoked subjective sensation measured in some
conditional units obeys the logarithmic law [Torgerson, 1958]. Early studies
were mainly on acoustic, visual, haptic, and tactile stimuli. The study of more
complex aspects of time perception appeared much later, including
multidimensional interactions [Gregson, 1994; Stevens, 1975]. As for the
study of the perception of complexity and aesthetical attractiveness of
objects, these endeavors, despite their multiplicity, still remain topical.
Even the meaning of the “golden section” is no clearer at present than in the
times of Fechner [Crozie, 2002].
According
to I. Yevin [2001] there is the optimum of arousal potential when the pleasure
from aesthetical object reaches its maximum. It is quite obvious that the
possibility to achieve the optimal value ФС is connected with many
factors—individually and situationally.
The consideration of individual and situational factors has led to the
exploration of individuality factors in such studies, such as age, gender, and
personality (e.g, Eisler & Eisler, 1994, for time scaling). Thus our
studies of the psychophysics of time, complexity, and aesthetics here include
several measures of personality factors in order to identify if any are key to
the utilization of fractal properties perceptually.
3.1. Apparatus and
Stimuli
As
stimulus material we used a program for generating strange attractors by J.
Sprott [1993]. He adapted the basic program to one for presenting the stimuli
and recording results from participants via a computer, in a flexible format
for us. The basic algorithms for the generation of these attractors is as
follows:
The
iteration procedure of are difference equations of the general form:
X_{n+1}=
a_{1}+a_{2}X_{n}+a_{3}X_{n}^{2}+a_{4}X_{n}Y_{n}+a_{5}X_{n}Z_{n}+a_{6}Y_{n}+a_{7}Y_{n}^{2}+a_{8}Y_{n}Z_{n}+
a_{9}Z_{n}+a_{10}Z_{n}^{2}
Y_{n+1}=
a_{11}+a_{12}X_{n}+a_{13}X_{n}^{2}+a_{14}X_{n}Y_{n}+a_{15}X_{n}Z_{n}+a_{16}Y_{n}+a_{17}Y_{n}^{2}+a_{18}Y_{n}Z_{n}+
a_{19}Z_{n}+a_{20}Z_{n}^{2}
Z_{n+1}= a_{21}+a_{22}X_{n}+a_{23}X_{n}^{2}+a_{24}X_{n}Y_{n}+a_{25}X_{n}Z_{n}+a_{26}Y_{n}+a_{27}Y_{n}^{2}+a_{28}Y_{n}Z_{n}+ a_{29}Z_{n}+a_{30}Z_{n}^{2}^{}
We fixed the quotients a_{i}_{ }within the
interval from 4.5 to 5 and initial values X_{0}
Y_{0} Z_{0}. If after a large number of iterations (from 500000
to 10 million) the function values remain bounded in a small domain of
threedimensional state space and the largest positive Lyapunov exponent
exceeds 0.005, the obtained graphic trajectory can be regarded as a usable
fractal. Two dimensions are represented in the plane, and the third with a
colorpalette.
The Lyapunov exponent was calculated in
the usual manor. For two arbitrarily close initial conditions (X_{0}
Y_{0} Z_{0}) and (X^{/}_{0} Y^{/}_{0}
Z^{/}_{0}) the distances d_{n} between the images of those points
obtained at the nth step and the
distances d_{n+1} between the images of the same points at
the (n+1)th step (generated by
Sprott’s program as the trajectories evolve). Then the Lyapunov exponent L equals
L
= S_{n=1}^{N } log_{2} (d_{n+1}/d_{n})/N
After the end of iteration process we
obtained the fractal dimension (also generated by the program as the trajectory
evolves; the trajectory evolves too rapidly to follow visually; it appears
complete early during it’s the course of its presentation; and is not a
continuous trajectory, but a series of points, such that the trajectory is lost
to view in the cloud of points as the stimulus becomes an image). To do so we
calculated N_{2}, the number of points in the course of the whole
iteration process that lay at the distance of 0.2 from each other, and N_{1}, the number of points that lay at the
distance of 0.1 from each other. The fractal dimension F was then computed as
F =
log_{2}(N_{2}/N_{1})
For
the present study we used 20 threedimensional fractals {0.52 < F <
2.36, 0.01 < L < 0.22, r(F,L) = 0.91} that
were displayed on the monitor of the computer in random order. Presentation
times varied from 2 to 30 seconds. The participants had to assess the
subjective complexity and aesthetic attractiveness of each fractal on the
ninepoint scale (from 1 to 9) and to estimate the duration of the presentation
of each stimulus. The stimulus disappeared from view before the set of three
questions for each was presented.
When
fractals with high fractal dimension were displayed there always was a certain
moment when there appeared so many points on the screen that the respondent
could no longer notice the appearance of new points and it seemed that the
object had already completed its shape. Therefore answering the question about
the time of fractal formation the respondents tended to underestimate the time,
though the tendency was not very marked.
3.2. Participants and
procedure
The
participants were 140 students and postgraduates of the Department of
Psychology of Moscow State University. We carried out two experiments. These
differed only in the durations of the presentation of the stimuli. In the first
(93 Ss), the stimuli varied randomly in duration from 2 to 30 sec. In the
second (47 Ss), three fixed intervals were used of 5, 15, and 25 seconds.
Besides assessing the fractals,participantsanswered the questions of the follow
personality inventories:
Internality
inventory for determining Locus of control (Internal control orientation « external control orientation) (Rotter 1982), Eysenck
Personality Questionnaire Revised (Eysenck, 1990) for determining temperamental features, 16PF Sixteen
Personality Factor Questionnaire (Cattell, 1946; Cattell et al, 2002) and
Spielberger's Test Anxiety Inventory (Spielberger et al., 1983, 1995) (See
Table 1AD). The participants that took part in the second series were divided
into three subgroups of 15 or 16 people. This was a betweensubjects design
where each subgroup received just one of the three presentation times (5, 15
and 25 sec.) for all 20 fractals.
We
would note that subjective complexity and aesthetic attractiveness rather
highly correlate (Table 2). For both experiments, we constructed the
clustertrees of fractals’ similarity on their aesthetic attractiveness and
complexity. As one might expect both trees were very similar in their
structure. Conventionally we can describe the fractals of low, middle, and high
dimensionality as the following types: (1) “points and thin lines” (F
< 1), (2) “lace” (1 < F < 2) and (3) “cushions” (F
> 2). Figure 1 illustrates typical
fractals of each type.
As
the result of correlation and determination analysis (Chesnokov, 1982) we
specified significant determinations of predispositions to assess presented
fractals as more complex or simple, aesthetically attractive or unattractive by
certain personality traits. The most robust personality determinations are
follows (see Tables 1A,B,C for identification of personality factors):
Ss
disposed to selfaccusation, vulnerable, and diffident (O+) overtly tended to
assess fractals of type 1 (separate points and lines) as very beautiful and
complex. One of the possible interpretations of this fact is that those people,
being uncertain, were afraid that they had missed something on the screen,
failed to grasp the essence and were not competent enough to discover the
complexity and beauty of that singular points and lines. After the experiment
one of the respondents supposed that the program simply “made an error” on
those pictures and didn’t draw the whole fractal to the end. Evidently we
observe the feeling of guilt – the program worked poorly on her computer.
Nevertheless, she evaluated these fractals from the first group as complex,
i.e. “invented” their complexity and aesthetic attractiveness.
A
tendency to regard the fractals of type 1 as complex was revealed in the
participants for whom approval and support from the others was in many respects
the decisive factor (Q2–). It may be connected with the desire to demonstrate a
nonstandard understanding of the process from the mathematical standpoint and
deserve an approval from the experimenter. After all, their assessments of
aesthetical attractiveness didn’t deviate from the norm. In contrast,
independent people (E+), who are independent in making decisions and actions,
evaluate the fractals of type 1 as the least complex. Such evaluation is based
exclusively on their own perception and the opinions of others are ignored.
As
we noted before, the whole sample revealed the trend to evaluate fractals with
higher fractal dimension as more aesthetically attractive and complex. However,
respondents with prevailing modesty, tactfulness and pliability (E–) were among those who displayed this trend most of all. It is
their E+ counterparts — selfassured, independent participants, inclined rather
to leadership than subordination or submission, uncompromising and disregarding
authority and existing norms — that markedly differed from the whole sample and
didn’t admit the aesthetic attractiveness of these fractals. One may make many
hypotheses about this tendency, but more definitive answers will require more
research.
Another
fact, still remaining unclear, is connected with the disposition of
temperamentally and emotionally stable people (NE–) to underestimate the
complexity of fractals of type 2.
It
is interesting to mention that although the ratings of aesthetic attractiveness
of fractals of type 3 with the highest fractal dimension, given by respondents
with high selforganization and selfdiscipline (Q3+) do not differ from average in the sample, their assessment of
fractals’ complexity tends to be higher. It may be possible, that here they
latently showed their ability to answer directly the question that was asked
(fractals of this group actually have the highest objective complexity) and
leave the emotional component “behind the brackets”. (The question of aesthetic
attractiveness unequivocally touches upon the emotional sphere).
The
participants with moderate emotional openness and need for communication (A±)
assessed fractals of type 1 (points and lines) as most simple and mostly disliked
them. These participants generally gave lower average ratings to all pictures
on both attractiveness and complexity.
We
revealed no personality determinant in the assessment of time intervals.
To
discover the influence of the duration of presentation and fractal dimension of
the stimuli on the assessment of their subjective complexity and aesthetic
attractiveness we used twofactor analysis of variance. It enabled us to make
the following conclusions: the time of fractal generation on the screen does
not affect the assessment of its aesthetic attractiveness and complexity.
Fractal dimension is the main correlate of subjective evaluation of fractal’s
attractiveness and complexity (p<0.0001). The more is the fractal dimension
the more attractive and complex is the fractal perceived. (Table 3).
J.
C. Sprott was the first to conduct experiments on assessing fractals’ aesthetic
attractiveness of. Although the number of pictures being assessed was very
impressive — 7500 fractals, the number of respondents was small (7 people).
Most attractive appeared fractals with fractal dimension from 1.1 to 1.5 and
the Lyapunov exponent from 0 to 0.3 [Sprott 1993; Aks & Sprott 1996].
Osorio et. al [2000] used four levels of fractal dimension (and the same
stimuli as used in this study) obtained a nonmonotonic function of aesthetic
and complexity judgments peaking in the similar fractal range as that of
Sprott, namely, 1.41.6. The Osorio study also failed to show any simple
relationship between time estimation and fractal dimension.
Mandelbrot
[1977] classic establishment of fractals as mathematical discipline showed that
many natural objects could have dimension. He and others have noted the beauty
in fractals, perhaps because of their imitation of nature [Peitgen & Richter,
1986]. In Sprott’s opinion, the preference for fractals with relatively small
Lyapunov exponent is connected with the fact that excessive chaotization of the
object does not favor its harmonic perception. We obtained different results.
And here we have the possibility of culture factor influence. Russians like
more “dense” things. And our sample may be more representative. The Osorio
study looked at age and cultural factors, but found no such clear results. We
have hopes of evaluating such cultural influences in future studies.
In
conclusion we would note that our hypothesis on the applicability and
fulfillment of WeberFechner law for the perception of time, complexity and
subjective attractiveness was confirmed. Namely, the subjective estimate of
time, τ, obeys the logarithmic dependence from the actual time of
fractal’s formation (presentation) on the screen, t:
t = 7.15LN(t) – 6.53 (R^{2 }= 0.47).
The
subjective assessment of fractal’s complexity S and aesthetic
attractiveness σ obeys the logarithmic dependence upon fractal dimension F:
S =
2.56Ln(F) + 4.88 (R^{2 }= 0.61)
s = 2.06Ln(F)
+4.89 (R^{2 }= 0.59)
and upon the Lyapunov exponent L:
S =
1.02Ln(L) + 8.07 (R^{2 }= 0.34)
s = 0.85Ln(L) +7.53
(R^{2 }= 0.35)
In
all three cases the significance level p<0.0001.
This work was supported by a grant
from the Russian Foundation for Basic Research. The authors wish to thank
S.Fomichev for assistance in data collecting, I.Timofeev for translation this
text in English, M.Gambarian for help in preparation the manuscript for
publication. Clint (J.C.) Sprott kindly spent considerable time patiently
adapting his strangeattractor engine for use in our series of psychophysics
experiments. Deborah Aks gave us important advice on the conduct of
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Tables below ¯
http://www.blueberrybrain.org/dynamics/mitinafractalperceptionfigure.htm.
Lable 
Factor Name 
Low index of factor 
High index of
factor 

Name 
Characterization 
Psychomedical term 
Name 
Characterization 
Psychomedical term 

A 
Warmth 
Cool 
Reserved, impersonal, detatched, formal, aloof 
Sizothymia 
Warm 
Outgoing, kindly, easygoing, participating, likes people 
Affectothymia 
B 
Intelligence 
Concrete Thinking 
Less intelligent 
Lower scholastic mental capacity 
Abstract thinking 
More intelligent, bright 
Higher scholastic mental capacity 
C 
Emotional Stability 
Affected by feelings 
Emotionally less stable, easily annoyed, upset 
Lower ego strength 
Emotionally stable 
Mature, faces reality, calm 
Higher ego strength 
E 
Dominance 
Submissive 
Humble, mild, easily led, accommodating, not assertive 
Submissiveness 
Dominant 
Assertive, aggressive, stubborn, competitive, bossy 
Dominance 
F 
Impulsive ness 
Sober 
Restrained, prudent taciturn, serious 
Desurgency 
Enthusiastic 
Spontaneous, heedless, expressive, cheerful 
Surgency 
G 
Conformity 
Expedient 
Disregards rules, selfindulgent 
Weaker superego strength 
Conscien tious 
Conforming, moralistic, staid, rulebound 
Stronger superego strength 
H 
Boldness 
Shy 
Threat sensitive, timid, hesitant, intimidated 
Threctia 
Bold 
Venturesome, uninhibited, can take stress 
Parmia 
I 
Sensitivity 
Toughminded 
Selfreliant, nononsense, rough, realistic 
Harria 
Tenderminded 
Sensitive, overprotected, intuitive, refined 
Premsia 
L 
Suspicious ness 
Trusting 
Accepting conditions, easy to get on with 
Alaxia 
Suspicious 
Hard to fool, distrustful, skeptical 
Protension 
M 
Imagination 
Practical 
Concerned with 'down to earth' issues, steady 
Praxernia 
Imaginative 
Absentminded, absorbed in thought, impractical 
Autia 
N 
Shrewdness 
Forthright 
Unpretentious, open, genuine, artless 
Artlessness 
Shrewd 
Polished, socially aware, diplomatic, calculating 
Shrewdness 
O 
Insecurity 
Selfassured 
Secure, feels free of guilt, untroubled, selfsatisfied 
Untroubled adequacy 
Apprehen sive 
Selfblaming, guiltprone, insecure, worrying 
Guilt proneness 
Q1 
Radicalism 
Conservative 
Respecting traditional ideas 
Conservatism of temperament 
Experimen ting 
Liberal, critical, open to change 
Radicalism 
Q2 
SelfSufficiency 
Group Oriented 
A 'joiner' and sound follower, listens to others 
Group adherence 
Self sufficient 
Resourceful, prefers own decisions 
Self sufficiency 
Q3 
SelfDiscipline 
Un Disciplined, selfconflict 
Lax, careless of social rules 
Low integration 
Following selfimage 
Socially precise, compulsive 
High selfconcept control 
Q4 
Tension 
Relaxed 
Tranquil, composed, has low drive, unfrustrated 
Low ergic tension 
Tense 
Frustrated, overwrought, has high drive 
High ergic tension 
Table 1B. Internality inventory
Lable 
Low index of scale 

High
index of scale 
E 
External control orientation 
I 
Internal control orientation 

A belief about that the outcomes of our actions are contingent on the events outside our personal control, are under the control of powerful others or is determined by fate, luck or chance 

A belief about that the outcomes of our actions are contingent on what we do, are directly the result of behavior 
EG ES EF 
General External control orientation External control orientation in successes External control orientation in failures 
IG IS IF 
General Internal control
orientation Internal control
orientation in successes Internal control
orientation in failures 
Table 1C. Eysenck Personality
Questionnaire (EPQ).
Lable 
Factor Name 
Low index of factor 
High index of
factor 

Name 
Characterization 
Name 
Characterization 

EX 
Extraversion 
Intraversion 
In need of peace
and quiet 
Extraversion 
Outgoing, talkative, high on positive affect (feeling good), and in need of external stimulation 
NE 
Neuroticism 
Stable 
High activation thresholds and good emotional control, experience negative affect only in the face of very major stressors, calm and collected under pressure 
Neuroticism 
Unable to inhibit or control their emotional reactions, experience negative affect (fightorflight) in the face of very minor stressors. easily nervous or upset 
PS 
Psychoticism 
Normal 

Psychoticism 
Aggressive, Assertive, Egocentric, Unsympathetic, Manipulative, Achievementoriented, Dogmatic, Masculine, Toughminded 
Table 1D. Spielberger's Test
Anxiety Inventory
Lable 
Factor Name 
Characterization 
SA 
State anxiety 
refers to transient emotional states, consisting of "consciously perceived feelings of tension, apprehension, nervousness, and worry, and associated with activation or arousal of the autonomic nervous system, [which] vary in intensity and fluctuate over time as a function of perceived physical or psychological danger" "(Spielberger et al., 1995, p. 44). 
TA 
Trait anxiety 
"is conceptualized in terms of relatively stable individual differences in anxiety proneness "(Spielberger et al., 1995, p. 44). 
Table 2. Significant
correlations between mathematical and psychological characteristics

Lyapunov exponent 
Aesthetic attractiveness 
Subjective complexity 
Fractal Dimension 
.907 (.000) 
.664 (.002) 
.718 (.001) 
Lyapunov exponent 

.594 (.006) 
.626 (.003) 
Aesthetic attractiveness 


.934 (.000) 
The parentheses contain
the level of significance.
Aesthetic
attractivenes 
F 
Significance 


Fractal
Dimension 
44.92 
.000 

Time 
1.246 
.291 
2Way Interaction 
Fractal
Dimension
+ Time 
0.709 
.600 
Subjective
complexity 
F 
Significance 


Fractal
Dimension 
87.26 
.000 

Time 
0.729 
0.511 
2Way Interaction 
Fractal
Dimension
+ Time 
2.102 
0.083 
Three Levels of Fractal
Dimensions: Three Levels of Time:
<1 5 sec
from 1 to 2 15
sec
>2 25 sec