A wide variety of fractal images, including the Mandelbrot set, Lorenz attractor and others mentioned in this article, may be found in Sprott’s online fractal gallery, http://sprott.physics.wisc.edu/fractals.htm
The Universe is built on a plan the profound symmetry of which is somehow present in the inner structure of our intellect.
-- Paul Valery
If you like fractals, it is because you are made of them. If you can’t stand fractals, it’s because you can’t stand yourself. It happens.
-- Homer Smith, Computer Engineer, Art Matrix
The rise of cybernetics, the science of information, following World War II, brought a new metaphor to psychology – the notion of mind as mechanism. This metaphor inspired the cognitivist revolumartion, in which psychological activity was likened to information processing in machines. In the decades to follow, a wellspring of new knowledge and empirical methods followed and even continues today.
While invaluable for its early insights, I believe the notion of mind as mechanism has run its course. In this paper, I introduce a different guiding metaphor in order to conceptualize the psyche, one with particular significance to clinicians immersed in the complexity of human affairs. This new metaphor represents the pendulum swung full circle, from machine back to nature, where psychology started when it first diverged from philosophy during Renaissance times. Ironically a return to organic models occurs just as the computer plus related technology ascends takes an ever more central role in most our lives.
More than ever, the computer affords us rich tools for simulating nature’s complexity. Among the most powerful of these is fractal geometry. Because fractals provide a lexicon for nature’s outer complexity, it makes sense that this new geometry is equally as effective for describing the complex terrain characteristic of inner processes.
This paper introduces the significance of fractal geometry to the psyche. In the first section, I describe this new branch of mathematics plus how to render a fractal by computer. I then articulate the significance of fractals to the development of psychological identity. Next, I claim that self-similarity, the hallmark of fractals, is a useful lens for viewing personality organization and especially repetitive patterns of behavior. I also argue that related concepts of dimensionality and scaling help lend breadth to intraspychic analysis. I use the notion of fractal boundaries to illuminate paradoxes of subjectivity and interpersonal relationships. Finally, I assert that fractal boundaries are not just a source of endless confusion and deep psychopathology, but also a fount of novelty, creativity and endless mystery in us all.
Fractal geometry is a branch of mathematics discovered during the 1970s, by a jack-of-all-trades mathematician working for IBM, named Benoit Mandelbrot. The word “fractal” was invented after Mandelbrot thumbed through his son’s Latin textbook and came across the adjective fractus, derived from the verb, frangere, to break. Fractals connote fraction, fracture and fragment. They tap into a central quality of nature – the fractured pieces which yet make her whole.
Technical fractals are rendered, often quite artistically, on the palette of the computer. They consist of very simple formulas, such as the classic Mandelbrot set, X ← X2 + c, iterated on the complex number plane. To render a mathematical fractal, the same equation is computed over and over for every point on the complex plane, as endless cycles of reentry. Each time, the final result of the equation is fed back in again as the new starting point. In theory, this continues indefinitely, as the calculation of fractal dimensionality presumes the presence of infinite feedback loops. In practice, iteration continues until either there is a stable endpoint or an artificial cut-off that gives clear indication of where the equation is headed.
When gazing at a mathematical fractal, the territory outside the fractal is out of control. It gallops unpredictably towards infinity at one speed or another, indicated by gradations of color. By contrast, the territory inside the fractal is ordered. It is relatively stable and settles down to one or more fixed points. The edge between these two realms is what constitutes the actual fractal. Here, in the delicate interface between unbounded and bounded areas, the fractal neither flies out of control nor comes to rest. Instead it self-organizes into an infinitely deep border zone that moves dynamically along with the perspective of the observer.
When the computer is used like a microscope to zoom closer and closer in on this edge, ultimately there is no resolution to be found. Instead there is paradox of the nonlinear kind. That is, the tinier the area under investigation, the more appears to be seen. This may sound like Merlin’s bag of magical tricks, but doesn’t it also sound just like the psyche, where the more we gaze inward, the more there is to see? In both fractal inspection and the act of self-reflection, observed and observer merge seamlessly, as the very act of looking helps to articulate the details to be found.
Fractals are highly complex, dynamic shapes. In fact, the Mandelbrot set is the most complex mathematical object known to humankind. Mathematical fractals contain an ordinary, Euclidean dimension plus a fractional or fractal dimension that indicates its complexity. For example, the fractal dimension of a squiggly one-dimensional line, such as a child’s scribble, might range from 1.256 to 1.894, depending upon how little or much of a two-dimensional plane of paper it occupies. The fractal dimension of a scrunched up sheet of paper tossed away might range from 2.364 to 2.943, depending upon how loosely or tightly we wad it up into a space-filling ball.
In general, fractals carve out system dynamics in one of two ways. Either, they are lower dimensional objects that strive towards higher dimensions by recursively adding structure. An example of this is the Koch snowflake, where each side of a triangle is replaced by tinier triangles.
Or, fractals are higher dimensional objects that retreat into lower dimensions by recursively removing structure. An example of this is Cantor dust, where the middle third of a line is successively removed, over and over, until only a sprinkling of points remain.
In psychology, the distinction between progressing towards higher versus regressing towards lower dimensionality may help us model whether a person’s psyche is better characterized by structure building evolution of consciousness or structure eroding involution. Note that high or low levels of complexity are equally as likely at any dimension. When it comes to consciousness, this could help clinicians make important distinctions between issues of complexity versus those of dimensionality.
For example, a person with borderline personality disorder often displays high levels of complexity at low levels of involuted consciousness. A description of borderline complexity, which results in self and other getting sucked into irresolvable boundary confusion, follows in a later section. By contrast, a spiritually enlightened person characteristically displays low levels of complexity at high levels of consciousness. Many religious leaders utter simple, yet profound statements, like “God is love.” While a Marxist might argue that such simplicity appeals to the lowest common intellectual denominator, a mystic might counter that such simplicity reflects deep truth that flows from the invisible interconnectedness of all things. In support of the mystic, fractal geometry actually provides evidence for such hidden, invisible connection, especially beneath the surface of chaos in nature, despite all its surface uncertainty and unpredictability.
Self-Similarity, Weather Storms and Brainstorms
To recognize the significance of fractals is to understand its hallmark – self-similarity. Self-similarity is a newly discovered symmetry in nature by which parts of fractal objects relate to their wholes. That is, the overall pattern of a fractal is repeated at multiple size or time scales, from small to large scale. Sometimes this repetition is exact, as with a linear fractal. Most often, especially in natural fractals, self-similarity is approximate or statistical. This nonlinear property allows fractals as they appear in nature to embody irregularity, discontinuity, evolution and change.
Some natural fractals are detectable only via mathematical abstraction, one reason why computers are sometimes necessary for their detection. Computer modeling reveals hidden, fractal order beneath apparently random, surface behavior typical of many chaotic systems.
An example of this is the Lorenz attractor, which models the unpredictable flow of weather patterns. A slice of this strange attractor, known as a Poincaré section, reveals fractal form not unlike Cantor dust.
One profound insight to be derived from contemporary nonlinear science stems from the fact that human nature is embedded within nature at large, whose essence is chaotic and fundamentally unpredictable. Fundamental unpredictability means that the local, or minute-to-minute details of specific instances can never be precisely anticipated. Yet, beneath the surface of even the most turbulent chaos, usually lurks invisible, exquisite order in the form of fractal attractors. While self-similar patterns can be reliably detected at the global level, their local details remain uncertain and fundamentally unpredictable.
I believe the presence of nonlinearity accounts for the abysmal performance of predictive experiments historically in psychology. Because statistics capture global attractors of highly nonlinear, complex phenomena, this means that even in theory, we can never hope to predict specific behavior in specific individuals. Yet with this new perspective, there is a wealth of new information to be found in old experiments, when statistics are re-examined as evidence for global attractor patterns.
Brownian motion of particles, the ups and downs of the stock market, the spread of epidemics as well as the destructive paths of forest fires are all examples of chaos in nature, with invisible, fractal order lurking underneath. When time series data is plotted for each, a strange attractor is revealed, such as the well-known Lorenz attractor. Lower dimensional cross-sections of these attractors usually display order in the form of fractal structure, including self-similarity at multiple scales of observation.
In highly nonlinear systems, which are characterized by output extremely disproportionate to their input, fractal structure becomes significant in that it serves as the mechanism for sensitive dependence on initial conditions. Sensitive dependence, the hallmark of chaos, means that tiny changes in any starting condition can send a chaotic system careening off into new, entirely unpredictable directions. Underlying fractal structure aids rapid escalation of change by facilitating propagation of pattern from tiny to large scales.
Within the human psyche, chaos is ubiquitous in the brain, where it forms a background for more ordered, perceptual states (Freeman, 1991). Because of high sensitivity and easy mobility, a background of chaos enables us to mobilize attention and change perceptual states rapidly in response to unpredictable environmental shifts. Chaos also appears to form the background of our ordinary mood shifts. Perhaps this sheds light on the use of weather-related metaphors to express human nature, e.g., “feel in a fog,” or “have a brainstorm.” The ups and downs of normal emotion usually display more chaos than the excessive periodicity of certain psychopathological states, like manic-depressive disorder (see Hannah, 1990).
Motor development involves the presence of technical chaos in the unpredictable, jerky movements of the infant, which become constrained and channeled over time (e.g., Thelen & Smith, 1994). Finally, impulsive behavior in adults may represent the immature condition of original chaos improperly constrained (Marks-Tarlow, 1993). With the psyche as with other parts of nature, where chaos is present, fractal structure tends to lurk invisibly beneath in fractal form.
Along with their unseen presence under chaos, many fractals manifest concretely and quite visibly. Fractal geometry offers a cornucopia of shapes more befitting of nature’s uniqueness and complexity than Euclidean geometry ever did. This is so even though mathematicians and artists have been madly cutting and pasting simple Euclidean shapes to approximate natural complexity for centuries.
Why is [Euclidean] geometry often described as ‘cold’ and ‘dry’? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line (Mandelbrot, 1977, p. 1).
As is common with any new field, beginning stages are dominated by defining and classifying phenomena under study. This has been the case for fractal geometry as well. While the precise definition of a fractal is still the subject of hot debate, since Mandelbrot first discovered this branch of geometry, thousands of articles in a multitude of scientific disciplines have detailed the presence of fractals, sometimes in the strangest of places. Physical fractals occupy all niches in nature. They appear at the microscopic level, e.g., diffusion dynamics of chemical leaching, as well as at the cosmic scale, e.g., self-similar clustering of galaxies. They also pervade the macroscopic world in which we live, where self-similar branching patterns are just as likely to appear outside as inside our bodies.
To date, many scientists have dismissed fractals as mere pretty pictures. Perhaps this is because most scientific papers primarily involve fractal identification and classification. I foresee a new era in which fractals can be understood more deeply. I believe this is inevitable, because I perceive fractal form to relate intimately to its function – as record keepers of history as well as boundary keepers between various strata of existence.
Fractals are a means by which time, or system dynamics, gets etched into form via self-similar, recursive loops that exist on multiple size scales. Fractals exist in the paradoxical space between dimensions, levels and forces of existence. They arise at the interface between processes, at boundary zones where they serve both to connect and separate multiple levels.
Before turning in greater depth to the psychological level, here I’ll sketch fractal boundaries and recursive, self-similar dynamics that exist at multiple levels within the human body. My list is not meant to be exhaustive, only indicative. At each level, we can see how fractals are involved with the communication, transportation or transformation of energy, matter and information in and out of the mind/body or between its various subsystems. This is the hallmark of open, complex, self-organizing systems existing in far-from-equilibrium conditions (Prigogine & Stengers, 1984).
At the biological level, our skin is pocked with fractal pores that negotiate the transportation of oxygen inside and of water and toxins outside. Wrinkles, physical evidence of our unique histories becoming etched into our faces, are fractal as well. So too are the pattern of animal markings, such as spots and stripes on leopards and zebras, lending each animal a unique fractal signature.
Many of our internal organs display fractal structure. These include the lungs, which bring air into the body; branching patterns of our arteries and veins, which circulate blood and nutrients throughout the body; the intestines, which transport waste outside; and the brain, our executive center for communication, transportation, navigation and broadly modulating relations between internal and external worlds.
In the field of perception, many of our sensory systems, such as sight and hearing, follow psychophysical power laws. Power laws involve nonlinear, exponential relationships between variables, in this case between how energy is transduced from the material level of signals outside the body to the spiritual level of conscious perception. For example, the formula that relates the internal quality of subjective loudness ( L ) to the external quantity of physical sound intensity
( I ) is L ~ I 0.3. The fractional exponent means that in order to double the loudness of a string quintet, we must increase the number of players tenfold – to fifty, with all musicians maintaining equal power output.
Power laws are self-similar, because the same relationship holds between their variables no matter how they are scaled or rescaled. Newton’s universal law of gravitational attraction, F~ r -2, is another example of a power law. The same relationship holds between mass and gravitational attraction in Newton’s formula, whether manifest at the tiny scale of the wavelength of light or the cosmic scale of light-years.
Due to their nonlinear, exponential increases, power law dynamics in psychophysics ensure that at the lower end of the signal spectrum, tiny amounts of a signal can be detected, as little as a single photon for the eye or single decibel for the ear. Meanwhile at higher ends, because distinctions are made with far less precision, we become capable of perceiving the widest possible range of signals. There is a paradoxical element to power laws – that the same relationship holds between variables at every scale means that they possess no characteristic scale. This hallmark of fractal dynamics is critical not just to psychophysics, where we enjoy the widest range possible of perception, but also to the psyche in general.
We take for granted that our minds generate patterns across broad categories of space, time and person – large and small, short and long, personal and universal – wandering freely within various scales, without being restricted to any characteristic one. Yet this fact is actually quite remarkable. I believe the issue of scale will prove highly fruitful for harvesting regarding how fractal dynamics affect psychotherapy. For example, as a clinician, if I persist in paying attention to tiny, microlevel, process details that seem trivial or irrelevant to my patients, because they are busy with macrolevel, survival level concerns, this can be conceptualized as a misattunement of scale. The degree to which scale matching is critical to feeling understood is an empirical issue in need of further investigation.
Power laws appear not just in psychophysics and Newton’s law, but also all over nature (see Schroeder, 1991). At times their appearance takes on a magical feel, because they connect seemingly unrelated things. For example, over a hundred years ago, the Italian economist Vilfredo Pareto recognized that a simple power law models the number of people whose personal incomes exceed particular values. More recently George Zipf recognized that a power law connects word rank and word frequency for many natural languages. Suppose we take any ordinary book and count all the words it contains. If we list the words first by rank order of word popularity and next by the actual number of many times each word appears, we find a power law relationship connecting the two. Power laws seem magical when they relate the apparently unconnected, e.g., qualities like rank order, to quantities like frequency.
Continuing with this survey of fractal dynamics related to human boundaries, along with Zipf’s law, self-similar dynamics are evident in language even more broadly, whose symbolic arena is one of the cornerstones of our humanness. Fractal dynamics afford language its remarkable flexibility – the ability for a limited number of words and grammatical rules to enjoy unlimited combinations. At a purely formal level, language clearly consists of self-similar structures: words are embedded within words, phrases within phrases, thoughts within thoughts, etc. That our number system is fractal is so obvious as to seem almost trite. Yet, interestingly, only at the point when it became so, did the concept of infinity arise. That is, only when numbers served as place-holders, could they be recycled infinitely. This made way for continued novelty in the form of calculus and other mathematical advances.
While dozens of books cover the role of fractal geometry in human physiology (e.g., Iannaccone & Khokha, 1995; West, 1990), fractal geometry and the fluid, relational, nonlinear, dynamic framework it demands, is just now beginning to enter the field of psychology. One reason fractals may have been slow in coming is because of the measurement problem. It takes a tremendous number of data points to a plot strange attractor or measure fractal dimensionality. Because fractals in the psyche exist at symbolic, invisible realms, psychologists as of yet can only describe them qualitatively. Yet fractal expert Manfred Schroeder (1991) suggests definitively that fractals exist symbolically, which at least opens the door for psychologists.
Cyberneticist Ron Eglash (1999; Eglash & Broadwell, 1989) examines fractal processes cross-culturally, observing both physical and symbolic fractals as they appear in art and architecture. Eglash and Broadwell make the interesting suggestion that the two levels of concrete versus symbolic fractals correspond to analog versus digital information processing. They illustrate this by examaming the Dogon culture of Mali, where the human body is the primary organizer of meaning.
Reoccurring on multiple size scales among the Dogon, self-similarity manifests partly concretely on smaller size scales and partly symbolically on larger ones. At a small scale, the human form is carved into doorposts, pots and other parts of houses. At a larger scale, the Dogon house is arranged in the form of a human, with various spaces and rooms serving as head, arms and torso. At a larger scale still, the village is laid out in the form of a person, with the smithy symbolizing the head, etc. Finally at the cosmic level, heaven is conceptualized yet again in the shape of a human being. But here the realm is entirely symbolic. In this example, as is typical of fractals broadly, each level is recursively embedded within the next. Meanwhile self-similar structure progresses from concrete to abstract levels, as if from analog to digital processing.
In previous papers (Marks-Tarlow, 1995; 1999), I examine fractal processes related to the purely abstract level of identity formation in the psyche. Perhaps partly because it is so highly symbolic, the self is a vague concept that has never been precisely defined. Some treat the self from a phenomenological perspective, completely as an interior experience. Others emphasize social and interactional components of the self. Still others broaden the lens to include cultural determinants along with their physiological underpinnings. From the widest perspective of all, the self has been viewed in universal, transpersonal terms. Here, pan-cultural themes are woven into tapestries of local, personal and cultural variation.
Historically these perspectives have vied for legitimacy. They compete for the prize of univalent, objective truth. They fight to the death, as if only one correct definition of the self exists. The difficulty in precisely defining the self may stem primarily from the lack of a fractal vantage point where identity is conceptualized multivalently, in terms of self-similar processes repeating on multiple size and event scales.
From the intrapsychic to the universal, I believe each level of analysis to be equally as valid and useful. Each folds into the next, displaying recursively embedded dynamics that recur on multiple levels of observation. Using fractal lenses, I conceptualize the self as an open, multileveled system coupled to other dynamical systems of broadening scope, from biological, physiological levels through intrapsychic, social, cultural, and even transpersonal ones.
A personal identity exists in the interior phenomenological space of our heads; a family identity supplies the uniqueness of each person’s relational dynamics; a regional identity dons the local garb of particular geographical areas; a national identity forms the butt of international jokes and stereotypes. A global identity may even struggle to emerge and bring geographical differences into harmony. In general, such multivalent existence is highly suggestive of fractal dynamics.
Fractals also come into play within the internal attractor structure of personality, which organizes self-similar patterns of behavior at various scales of observations. A timid middle-aged housewife tries to enter the professional world once her children have grown and her nest is empty. One day, early in therapy, she forgets to turn on the red light in the waiting room. By so doing, she keeps her presence invisible. Coincidentally, similar themes of disconnection and invisibility have also appeared in a dream from the night before. In it, the woman frantically tries but fails to get the attention of her boss in order to alert him to a critical flaw she has found in his business machinery. The flaw is miniscule, but so serious it could shut down the whole enterprise.
Together, patient and therapist interpret the dream. Deeply rooted fears are uncovered of tiny flaws under the surface, which endanger the very continuation of this woman’s enterprise, both personal and professional. With her children gone and caretaker role all but eliminated, her primary identity is threatened. This woman feels invisible and disconnected at multiple, recursively embedded levels. She wonders if she will be seen as vital to anyone, including her boss, therapist, and most importantly herself.
When the dream is examined self-referentially, as all dreams can be, this patient’s concerns about visibility and vitality recursively implicate her relationship to her own inner world. In self-fulfilling, self-similar fashion, as this housewife struggles to transform and broaden her role in society, she unconsciously enacts the very conflict she fears most. Ironically, by “forgetting” to turn on the light, she isolates herself at the very moment that consciously she is most eager to discuss and share her predicament.
Because of the capacity for the tiniest fragment of a dream to reflect the whole of the psyche, it is easy to see how beneath the surface, every dream carries fractal structure. Every dream carries full potential for an infinitely deep and wide nexus of interpreted meaning.
The fractal display of personality at multiple scales in everyday life is something most of us pick up intuitively. It comes as no surprise that someone who interrupts us frequently during conversation might simultaneously brag about road rage. This person, who seems to take pleasure in running over us verbally, also enjoys trying to run a fellow traveler off the road. Meanwhile we hear rumor of this man backstabbing a colleague or undercutting his business competition unfairly.
From the micro level of speech patterns, through a medium-scale event of a chance encounter on the highway, to the large-scale level of ongoing business relations, people generally demonstrate self-similar behavior across multiple scales of observation. When this gets rigidly stereotyped, we might think in terms of Freud’s notion of repetition compulsion. But a certain degree of self-similar repetition of behavior is natural, perhaps representing the characteristic “signature” of personality by which others recognize us and we recognize ourselves. When it comes to behavior emitting from the depths of personality, the same fractal dynamics crosscut every scale. This is because when it comes to the psyche, there is no characteristic scale of operation.
Selves in the Paradoxical Space Between
The perspective of the self I offer is of an open, dynamical system that is fractally constellated. My view dovetails with Francisco Varela’s framework of autonomy in biological systems(e.g., Varela, 1979). Varela’s model involves endless feedback loops, which allow biological systems to re-enter themselves continuously. This results in paradoxical dynamics when biological systems are characterized in opposite terms, as being functionally closed yet structurally open. Selves follow the same pattern. They too are “closed” in that, when we are healthy, we retain a cohesive, coherent, ongoing sense of identity. Yet, selves are clearly open via interaction with the others, which constitutes the social mechanics of their negotiation.
My colleagues and I (Marks-Tarlow, Robertson & Combs, under review) have modeled the emergence of identity through endless cycles of reentry. Here, consciousness arcs away from the self, in order to enter into the perspective of another, and then circles back round again. (I see you seeing me; as a result, I see myself ever more clearly.) This model conforms nicely to social mirror theory (e.g., Baldwin, 1902; Cooley, 1968; Mead, 1934; Whitehead, 2001), which posits the development of self and the perception of others to arise hand in hand.
We can readily perceive such feedback cycles through observing young children. A two-year runs carefree, yet inevitably falls down. Not terribly hurt and mostly startled, she immediately looks towards mommy for a reaction. If her mother becomes scared or concerned and protectively rushes over to her child, this toddler will probably become upset, start to cry and take refuge in the very comfort being offered. Alternatively, if mommy smiles and nods approvingly, treating the fall as a necessary part of living and learning, chances are good this toddler will comply with mother’s call for independence by getting up, brushing herself off and taking the event literally into stride.
Of course, such feedback loops work in both directions. If the toddler really is hurt and mother merely nods and smiles, this parent is failing to take any cues from her own child. Mother is in danger of missing the emotional mark. If similar events, however minor, occur frequently over time, the toddler may become confused by mother’s misattunement. The child might begin to mistrust either her own internal signals or mommy herself, who increasingly appears as disconnected and invalidating.
Contrary to popular lore, which maintains that it takes huge, traumatic events to shape basic personality, it is increasingly evident to most clinicians that the tiny falls, tweaks and mishaps are equally, if not more, significant in the formation of basic personality. Like endless waves on a shoreline, ever-similar yet ever-changing at a minute-to-minute level, day in and day out, mommy or other caretakers plus their children are embedded in paradoxical, feedback cycles of subtle nuance. Tiny events, like the toddler’s stumble, form endless feedback loops in both directions, from self other, other to self. Over time, these cycles shape both people by building a repertoire of memory and experience. Out of this foundation, at the next level of complexity, self-image emerges to form self-referential loops in consciousness.
By requiring the ongoing presence of others to become present to our selves, this model of development emphasizes the paradoxical dimensions by which self and other, observer and observed are inseparable. Selves retain a paradoxical quality because the truth of a fixed identity, i.e., its functional closedness, rests precisely on its underlying falsity, i.e., its structural openness.
The idea of selves arising in the paradoxical space between people was articulated elegantly by British object relations psychoanalyst, D. W. Winnicott (e.g.,1971). Winnicott’s most important contribution was the notion of the transitional object. This consists of baby's first possession, such as a blanket or teddy bear, which occupies the fertile space between mother and baby. The transitional object is the first symbolic object that serves both to connect and separate baby and mother. This object is partly discovered and partly created, neither wholly of the one nor of the other, yet it partakes of both. Out of the nebulous space of the transitional object, Winnicott envisions the creative emergence not only of symbol and play, but also most broadly of culture at large.
Winnicott came to his idea of transitional objects after returning again and again to a Tagore poem, "On the seashore of endless worlds, children play." Like a barnacle, this fragment lodged in his psyche upon first encounter. Over the years, wave after wave of meaning washed over him. At first the poem represented endless intercourse between man and woman, with the child emerging from their union. Then the sea represented the mother's body and the land her ego, with the baby spewed upon the land like Jonah from the whale.
Finally out of a long, chaotic state of not-knowing, the notion of "transitional object" crystallized in Winnicott's mind. Both psychoanalyst Stuart Pizer (1998), a pioneer who conceptualizes psychotherapy in nonlinear dynamical terms, and myself believe it no coincidence that Winnicott’s creativity emerged through contemplation of a fractal image. Because fractals inhabit the nebulous territory of the “space between,” their borders provide endlessly fertile, endlessly deep frontiers.
I suggest that the self arises in the paradoxical space between people and events as an ongoing, co-creative, interactive and iterative process. Just as with any fractal, internal structure gets added or removed through the ongoing negotiation of boundaries. This complex border area, where inner and outer, self and other are melded, can be conceptualized in terms of fractal separatrices.
The ordinary conception of a boundary is literally a bounded, or fixed area whose resolution is easily detectable, e.g., the door of our houses or edge of our desks. By contrast, fractal separatrices can never be resolved. Instead they form endless, infinitely complex zones of articulation and negotiation. Here, between any two points, e.g., of self and other, inside and outside, exist infinitely many other points.
The image of fractal separatrices can be viewed in terms of intrapsychic dynamics, such as the dilemma of a person with obsessive-compulsive personality disorder in an ice-cream store. Each color, or basin of attraction, represents a different flavor, and the complex fractal boundary between the four choices illustrates the obsessive nightmare of attempting to use intellect or logic to figure out the “right” choice, when rightness isn’t the issue at all.
In order to understand the complex, nether zone of a fractal separatrix interpersonally, it is useful to examine the psychopathology of the borderline personality. People with this character disorder display chaotically organized psyches, which includes intense, shifting affect and highly unstable relationships. These individuals tend to oscillate between subjective poles of engulfment and abandonment, often harboring central issues of rage and shame. They repeatedly express confusion between self and other. At times, interpersonal confusion reaches a crescendo, to the point denying psychological existence altogether. That is, the borderline is wont to claim that she has no self, to assert in essence, “I don’t exist.” This is the ontological equivalent of the Liar’s paradox, “This statement is false.” In both cases, if it’s true, then it’s false; and if it’s false, then it’s true.
Because of such intense confusion, extreme defensiveness and rigidly closed boundaries, interaction with a borderline personality frequently results in what anthropologist and scientist Gregory Bateson identified as the double-bind. The double-bind, which Bateson postulated as the cause for schizophrenia (e.g., 1972/1956), consists of seemingly impossible, paradoxical demands made on relationships, that involve contradiction at multiple levels of communication. When it comes to borderline personality disorder, paradoxical demands often center upon the issue of blame. For example, “You’re to blame for my hurt. If you don’t think so, you’re wrong, because I know you better than you know yourself. But even if you’re right, you’re still to blame, because you’re always trying to be right at my expense.” In this closed feedback loop that serves to keep contradiction in place, which is what Ben Goertzel (1994) calls “chaotic logic,” we see that engagement with a borderline easily becomes a paradoxical morass, including the potential for endless recursion.
Attempts to ignore multiple realities and ambiguity by always being right while making others wrong results in failure to recognize the fractal quality of boundaries, along with their irresolvable openness. People with borderline personality disorder have often been so hurt or abused by letting others in emotionally, they now feel entirely threatened. Yet the more they fight the open, contradictory nature of psychological boundaries, ironically the more everyone gets sucked into the endless vortex in the space between.
Based on this example alone, we may be tempted to conclude that fractal separatrices characterize only severely pathological states, such as borderline or paranoid personality disorders or psychotic states. Whereas in paranoid states, confusion between inside and outside leads to delusions of trailing FBI Agents or invisible alien attack, in psychosis, such confusion takes on even more profound proportions to invade perception itself, in the form of visual or auditory hallucinations.
Here and previously (Marks-Tarlow, 1999) I propose that fractal separatrices are not just evidence of pychopathology, but characterize all psychological boundaries. Fractal separtrices between inside and outside mean we all carry the potential for confusion between self and world, projections, delusions, hallucinations or self and other, e.g., introjections, projective identification, delusions and borderline double-binds. Yet, except under extreme conditions of stress, most of us resist these vulnerabilities. We can usually let this seam alone, because in ordinary daily functioning, it proves irrelevant. By contrast, psychopathology is characterized either by excessive rigidity or too much chaos that causes us to deny, fight, reject, ignore or repress this potentially scary, disheartening condition.
The major difference between psychological health and psychopathology is not so much the possession of clearer or cleaner boundaries. Rather it is more that in health we possess the wherewithal and flexibility to recognize, tolerate, and if we’re lucky even welcome, the vagueness, uncertainty and ambiguity inevitable with fractal separatrices. We don’t need to lose ourselves in infinitely complex, irresolvable boundaries as long as we understand them.
Along with a source of psychopathology, open boundaries are a fount of aliveness, creativity, and even higher consciousness. They preserve the mystery and wonder of life. We grow through our ability to tolerate ambiguity, to hold opposites without succumbing to the tension of reducing one side to the other, and to understand ambivalence. All these emotional skills relate to embracing rather than rejecting underlying fractal dynamics, along with their paradoxical elements.