Fractal Nature

ABSTRACT: Transformations can be effected within the autonomous stream of imagery, through imagery processing via experiential therapy.  The essence of this transformative process is revealed in the fractal nature of imagery and symbols--i.e. their ability to encode, enfold, or compress the informational content of the whole.  Strange attractors condition and govern the transformative process through the complexity of information in dynamic flow.  Emergent consciousness is not an epiphenomenon of the brain.  Rather it is the transformational process of non-manifest, undifferentiated consciousness emerging into manifestation.

Keywords: Fractal Therapy, image processing, information theory, consciousness, chaos theory, image compression, feedback

Our graphics [fractals] did more than inform.  They made people dream.  Colleagues flocked to tell us that we had made them see their own work in a different light, and had helped them by unveiling previously unnoticed analogies.  For the first time, they felt that what they saw directly affected what they did next.

                                                 --Benoit Mandelbrot, Foreword AN EYE FOR FRACTALS

There is a random iteration method for rendering images that is more efficient than direct transformation replacement.  Start anywhere on or near the image.  Choose one of the transformations at random.  Transform the point you are on to a new point and mark it, and choose again.  Shortly you will be on the attractor, and will stay on it as you transform.  The marks you make become the image.  The "dice" for the random choice are loaded proportional to areas of the transformations relative to the total area.

                                                   --Michael McGuire, AN EYE FOR FRACTALS

 FRACTAL DIMENSIONS

Fractal geometry, by revealing the underlying, living, mathematical beauty of nature, allows us to see the same old world of phenomena differently.  It models and helps us comprehend the irregular, broken, fractured, and fractional, yet patterned aspects of the real physical world.  Fractals harmonize complexity and simplicity.

The fractal world reveals wrinkles and crinkles, sometimes infinite detail, and more and more information the deeper we go.  In the fractal world, dimensions are tangled up like a ball of twine, and objects are neither two dimensions nor three but somewhere in between.  Fractals reveal the geometry between dimensions.

Nonlinear systems change radically through their feedback.  Their chaotic behavior arises from the iteration (recycling, feedback, repetition, self-reference) of simple rules, simple processes.  To repeat or rerun processes is analogous to experiential psychotherapy which reruns images.  Iteration has been called the "heartbeat of chaos."  In therapy, iteration equates with memory.  Imagery processing equates with memory processing.  The key is the self referential repetition of the process, over and over, for example, by iterating dream images or using culminating symbols as resources for a period of time.

Chaosology has shown that positive feedback can cause complex, even chaotic behavior concealed inside orderly systems to unfold, and that negative feedback can grow inside an otherwise chaotic system, suddenly organizing it and making it stable.

Since human behavior is essentially chaotic, the extension of the metaphor and analogy yields intuitive insight.  Nonlinear equations are a potent and revolutionary mathematical metaphor for nature, and chaos is one for our nature.  Chaos reflects our ability to exercise freedom in a new context.

Nonlinear systems behave nonlinearly because they are so webbed with positive feedback that the slightest twitch anywhere may become amplified into an unexpected convulsion or transformation.  Until something sets them off they behave in a regular, orderly, cyclical way.  When the critical point is passed, they suddenly go chaotic.  Dynamic systems are ultrasensitive to the "information" continuously circling inside them.

At any moment, the feedback in a dynamical system may amplify some unsuspected "external" or "internal" influence, displaying this holistic interconnection.  So paradoxically, the study of chaos is also the study of wholeness.  If consciousness (rather than matter) is the fundamental basis of reality, chaos initiates a recycling of consciousness.  Dynamic recycling leads to stability, periodicity, or chaos.

Order and chaos have been traditional opponents--antagonists.  But deterministic chaos, rather than complete randomness, is not a common denominator of the two, but a harmonizing third pole.  It describes the complexity of information as it varies during a process of self-referencing iteration.  Information in this sense is an accounting of the number of possibilities, states, or conditions that a system could be in on a continuum ranging from total order to total randomness.

Total order has negligible information content; there is only one state available.  Total chaos (complete randomness) is a maximum of possibilities, thus maximum information content.  The chaotic pole lies between, distinguished by complexity in the sense of the amount of computation, the amount of effort it takes to describe what is there.

Total order is obviously null complexity.  Total randomness is also really null complexity.  At phase transitions, such as the "liquification" of consciousness, there is filigreed entanglement and diversity flourishes at boundaries.  In consciousness terms, this translates into undifferentiated chaotic consciousness and cosmic consciousness of the web of life, a truly fluid identity.

Transition areas--the points at which the system moves from simplicity to complexity, from bright, stable order to black, impenetrable gyrations of total chaos--are the most interesting places.  Inside these transition zones and boundary regions, systems degenerate and emerge in patterns. Though unpredictable in detail, one can predict the patterns and ranges of a system's movement.  There are certain repeatable, rough patterns that systems seem attracted to as they break down into or emerge from chaos.

These fractal graphics not only inform, but according to their discoverer Benoit Mandelbrot, they make people dream.  Fractals are map-like geometrical descriptions of dynamic processes--the dimensional terrain of abstract thought and imagination.

Though computer generated, they seem natural or organic in their unfolding emergent form, rather than artificially contrived.  With an eye for fractals, we find them in all kinds of natural settings: lines of force in nature such as swirls, growth patterns, flow marks, erosion, etc.  They describe the infinitely complex surfaces of clouds, mountains and trees.  On the computer screen, they indicate the degree of detail or crinkliness in the object, how much it occupies the space between the Euclidean dimensions.

The word fractal comes from the Latin fractus, which means broken, hinting at their self-similar reiteration at all levels or scales.  Fractals generate a sense of perceptual wholeness.  To iterate means to repeat an operation, usually using the last result of that operation as the input.

In experiential therapy, this might means feeding back and mirroring the last comment of the client back into the co-consciousness stream, flowing into the morphing image...becoming it.  In chaos theory and experiential therapy, iterations often quickly lead to spirals toward infinity.  Experientially there are sensations of a dizzying magnetic pull into the primal vortex of information.

Self-similarity means a similar, though not identical appearance at all scales.  Fractals pack an infinity of iterations into the smallest space.  It is possible to bring out or unfold finer and finer detail, by "zooming in" on the set's boundary or "magnifying" it.

In terms of image processing, they facilitate an "image compression" whose redundancy insures the continued accurate transmission of the informational content.  The analogy in the fractal essence of imagery is that symbols and imagery compress or economically encode condensed sequences or cycles of experiences, called COEX systems by Stan Grof.

Fractal self-similarity exists at the edge of waves.  If this chaos is influencial at the subatomic level, it may influence the geometry of quantum waves.  This nonlocal phenomena is part of our inherent quantum essence.  Furthermore, quantum states and chaos influence the brain/mind.  Fractal rhythms and distinct fractal signatures have been found in dopamine and serotonin receptors in the brain, and in enzymes.

Fractal resolution falls between classical dimensions.  They describe a fragmented, broken, bumpy or otherwise irregular topology of states in phase space.  According to McGuire (1991):

Fractals pack an infinity into "a grain of sand."  This infinity appears when one tries to measure them.  The resolution lies in regarding them as falling between dimensions [non-integer dimensions].  The dimension of a fractal in general is not a whole number, not an integer.  So a fractalcurve, a nominally one-dimensional object in a plane which has two dimensions, has a fractal dimension that lies between 1 and 2.  Likewise, a fractal surface has a dimension between 2 and 3.  The value depends on how the fractal is constructed.  The closer the dimension of a fractal is to its possible upper limit which is the dimension of the space in which it is embedded, the rougher, the more filling of that space it is.

Fractal dimensions are characterized by the amount of space the fractal structure fills.  For example, the spectrum or spread in spatial frequency of a landscape is closely related to its fractal dimension.  Impossibly jagged fractal mountains fill more space in their fractal dimension than gentler mountains as observed in nature.

A fractal figure lying in a plane has a dimension between 1 and 2.  The closer it comes to the upper limit of 2, the more "spacefilling" it is.  The notion extends readily to surfaces in three-dimensional space...using a sequence of random numbers, but varying a constraint to produce a different value of fractal dimension.

Analogous to the perpetual flow of internal imagery, there is a replacement rule in moving from one stage of construction of fractals to the next.  Each graphical object [read image] is replaced with another which is usually more complex, but which fits into the place of the original.  In graphics it can be appropriately scaled, turned, or flipped over, creating symmetries from logarithmic iterations of the seed form.

The length of a fractal boundary is virtually infinite in one dimension.  And a set of line segment can paradoxically occupy zero area.  There are fractals that fall between 0 and 1 in their dimension.  At the limit of infinite iteration one value becomes infinite while the other becomes infinitesimal and we get a measure of the fractal in its dimension which is neither zero nor infinite.  In dreamhealing, people often speak of a boundary place that is the edge of infinity, where nothing and infinity meet.

Fractals are visually complex mathematical objects that are similar in shape and structure over an infinite range of scale.  They arise from the iterated application of a simple rule.  Nature is visually complex and has material shapes and structures that are similar over a finite range of scale.

Over this finite range of scale with self-similar structure, we might analogously expect a simplicity in the rule or law of formation or growth that produced the structure.  This could be a constancy of the underlying geology of a mountain, or the genetic identity of DNA.  Both erosion and growth are in some sense iterative.

Self-similarity is statistical, not absolute.  Thus a constrained randomizing occurs which yields the natural variety and unexpected rhythms in nature and our nature.  But randomness is not required for all fractal iteration, such as fractal branching like in a tree.  Infinite variety relieves the automatic boredom which comes from a predictable, regular structure.

Perceptual psychologists allege that sensory processing is "efficient" in the way it encodes and handles information presented to it.  And fractals are nothing if not efficient in generating visual complexity from the least amount of information.  So a brain shaped by evolution in a natural world might be expected to have structures complementary to, and adept at apprehending fractal structure in nature.

See illustration for a fractal structure based on hexagons which has been proposed as a possible way the receptive fields of the brain's visual cortex are organized.  The smallest hexagons correspond to the close-packed structure of the cells of the retina and the perception of finest details.  The layers of larger hexagons organize the underlying layers to respond to progressively coarser detail.  And further, this organization seems to fit the spectra of natural images.

FRACTAL TRANSFORMATIONS

Fractals of different types can also be generated through linear and nonlinear transformations.  Linear transforms include moving, rotating, scaling, shearing, and inverting.  Any primary starting shape--a circle, asterisk, single point--on which these transformations are made leads to the emergence of fractal forms.  Fineness of detail is a matter of persistence in iteration.

Fractal images are a collage of transformed copies of the whole.  The transformation is encoded in a sequence of values which store the information.  In this sense they are an analogy of image compression.  Just as a therapeutic process can be "stored" in a culminating symbol, a run of pixels with the same value can be encoded with two numbers symbolizing the value common to them, rather than recording all the values in the run.

All transformative methods somehow find and remove redundancies in the image.  Fractals are in a way the ultimate in redundancy, so this method of coding or symbolization is potentially the most effective compression of information.  Cycles of interaction and iteration reveal the underlying emergent order with no loss of resolution as the image is more clearly defined.

Deterministic chaos arises from the iteration of simple nonlinear rules.  In terms of human experience, effects are no longer proportional to causes when things go nonlinear.  They can only be described geometrically in complex phase space which consists of a plane of interfacing real and imaginary numbers.  Solutions are written as the sum of a real number and an imaginary number, a mixture of reality and imagination.

STRANGE ATTRACTORS

The core of a chaotic system is called a strange attractor.  They arise from iterations of transformations in the complex plane.  The shape of a figure which is the boundary between attractor points is no longer a circle, but a fractal, whose shape depends sensitively on one of the attractor values.

Attractors are not just single points or cyclic arrays of points.  They are fractals in the space of the behavioral variables of a chaotic system.  The strangeness is their fractal properties.  The trajectory of a chaotic system never intersects itself, but packs infinite fractal structure into finite space.

Chaos has two faces.  On the microscopic scale there is the butterfly effect instability, while on the macroscopic scale there is a great stability--the strange attractors.  Just where a dynamical system will end up on its attractor is exquisitely sensitive to its initial conditions.  But it will inevitably be pulled onto its attractor, regardless of initial conditions.

Solid regions within a fractal pattern indicate that only the boundaries are described by fractal sets.  When a fractal has an inside (stable areas), there are attractor points other than infinity.  The set is generally most interesting near its boundary.  There is similarity between what we see "zooming in" on a region and the sets in that region.

Nonlinear transformations don't have much direct connection to the natural world we see.  They depict an unending filigreed entanglement burrowing infinitely into the smallest areas.  They exhibit a diversity that flourishes particularly well at boundaries.

WINDOWS ON CHAOS

Fractal structure arises from the iteration of simple rules.  While it need not create a visible fractal structure, a process can leave tracks in the natural world (or imaginal world) which are fractal in essence, for example turbulent flows and their erosion patterns.

Nonlinearity arises when iteration leads beyond exponential growth toward its (natural, existential, or environmental) limit.  The rate can no longer be constant, leading to oscillation which may develop supercritical perturbations.  Damped oscillations indicate an attractor influencing cyclic iterations.  When the number of cycles becomes infinite, it is chaotic and unpredictable yet completely deterministic.  Sensitivity to initial conditions is a basic property of chaos.

In therapy it is the overwriting of the "initial conditions" programming which leads to radical transformation in co-evolving systems of adaptation.  It is like a symbolic, cosmic "do over," referred to as a "change history" in NLP.  It dissolves the old, rigid system into undifferentiated consciousness and substitutes new "initial conditions" with different emergent values and behaviors.  It is a leap to a new context.

In plotting supercritical junctures in far from equilibrium processes, certain cascades or bifurcations have been noted.  The bifurcation diagram is somewhat universal, describing period doubling cascades in ripples and waves of transitions to turbulent flow, nonlinear oscillators, and the gamut of other chaotic systems.  They are probably implicated in the chaotic dynamics of sub-quantum processes such as probability.  The ratio of successive amplitudes at bifurcation is independent of a particular physical system.

In the stable cycle region, the iterations all fall on top of 1,2,4,8...discrete points; while in the chaotic region they don't, yielding a fractal dust.  The bifurcation diagram reveals that in the chaotic region THERE ARE WINDOWS WHERE THE CHAOS DISAPPEARS--interdimensional portals.  In psychological terms they may represent temporarily stable, but nonordinary states of consciousness.  Perhaps they represent states such as healing, creativity, and visionary intuition. The "windows" are self-similar to the whole, reflecting the universal patterning principle.

SELF-MAKING OR AUTOPOESIS

Consciousness, rather than matter, is the fundamental ground of existence. It may also be the ground of non-existence or potential, the virtual domain.  Nothing is outside consciousness.

This primal consciousness has intentionality which is expressed through creative choice, chaotic choice, natural selection following the path of least resistance and natural economy of action.  Evolution is this process by which more and more complex structure evolves.

The conditions which give birth to structure are far from equilibrium.  Therefore, our physiology is a holistic activity laced with fractals and chaos.  The shape of many pathogens is also fractal, making us vulnerable to them.  Pathology is also related to loss of "natural" background chaos (noise) in the body.  For example, the immune system's method for making antibodies may involve chaotic activity.  Even aging may be the result of a decrease in degrees of chaos.

Chaology measures systems as they breakdown, disintegrate, come apart, or fluctuate unpredictably and transform themselves.  It has been suggested by Ben Goertzel that dreams exist in order to gnaw away at the self-reinforcing behavior of thought systems, such as logic.

During iteration, equations are calculated to an output and then the output becomes an input as the equation is calculated again--amplifying feedback.  Strange attractors can convulse and change their basic shape if the system is perturbed enough.

But short of total metamorphosis, each iteration of feedback "folds" the chaotic system into itself, exploring a new region of space within the tangled outline of its strange attractor.  Strange attractors fill space in a self-similar way, leaving the imprint of the chaotic dynamical system they plot.

Webb (1993) has postulated that "normal" personality might be modeled by the simple spiral form of the Rosseler strange attractor, a single regime attractor.  This notion assumes a normal personality has some sort of underlying chaotic attractors which keeps it moving through the phase space of possible conscious states.

Likewise bipolar disorder could be represented by the two "eyes" of the Lorenz attractor, indicating unpredictable mood swings between lobes analogous to multiple discrete regimes.  Webb questions whether, "a small perturbation at the right point in the trajectory can flip the system over into the other lobe."

Self-generating processes emerge at the fundamental level of oscillation--flux.  Some starting values, when they are plugged into an iterative equation cycle, explode toward infinity, others fluctuate, some don't change very much.  Equations that undergo unpredictable behavior when iterated are nonlinear.

A cycle where one chemical begins to produce more of itself is a feedback process chemists call "autocatalysis."  This positive feedback acts like a pump creating wave fronts of active regions.  Related concepts are self-reinforcing systems, self-stabilizing, recursive loops, standing waves, and self-resonance.

Pumping action leads to the formation of vortexes.  The paradox of turbulence is that in the midst of disorderly motion, vortexes may appear and remain stable while the disorderly current boils on around them.

Behind these fronts are quiescent regions and adjacent to them are receptive regions into which the reaction proceeds.  However, within the wave fronts, the same design repeats itself on smaller and smaller scales--making the evolving pattern of the reaction fractal.

The background in each plot describes the dark domain of total chaos.  In chaosology, the background is the real subject, the nature of the primal grain of experience.  Black internal regions are quiescent, so black holes represent a region of points (starting guesses) that haven't begun converging toward solutions by the time iteration ceases.

Black holes obscure regions of chaos, and the boundary everywhere is covered by black holes.  Black areas mark the best-behaved numbers, the most stable points--points that lie within the Julia set itself, for example.  No wonder, participants are drawn magnetically into the "black holes" during consciousness journeys.

Work at the Santa Fe Institute on the so-called "edge of chaos," models systems that move between periodic behavior and chaotic behavior, riding on the interface between them.  Complex patterns can arise and maintain themselves for relatively long periods.

In primal reactions, classically chaologists have detected spirals of self-organizing order developing out of chaos.  This phenomenon has been found in the propagation of signals along nerves, which may be influencial in terms of the interface of chaos and perception.  Chaos may be present in neural nets, biased by chaotic noise.

According to Webb (1993), "If psychological processes can be linked to the behavior of neurual nets, then introduction of chaotic behavior in such neural nets will obviously impact on such psychobiological processes."  Hopfield proposed a neural net model based on the idea that the system moves toward attractors (memories) with maximum entropy.

The spiral pattern symbolizes activity in the life-giving boundary between order and chaos.  Anthropologists say the spiral is the ancient symbol for the labyrinth, the twisted pathway for a journey to the core of being and perhaps into non-being.

Our "personal equation" is solved creatively, and the result is folded back into the equation and solved again.  In therapy we gain insight about the many dimensions of our personality--all the possible states of consciousness we become locked into--as well as our deeper essential nature.  What chaotic attractor keeps us moving through the phase space of possible conscious states?

Those insights influence our fantasies, creativity, and even our dream life, as dreamlife influences our insight.  In process-oriented therapy we are drawn irresistably (attracted) to the "region" which reflects our issues and identity, though within that region our path is chaotic.  No one knows where the imagery will go next, but we will be inexorably pulled toward our "strange attractor".

Research has shown that in chaos-based learning, the network can be made "smarter" by increasing the parameters until the network displays chaotic behavior, then decreasing it until the chaos goes away, then increasing it again, and so on...which prevents getting stuck in suboptimal solutions.  Dipping into and out of chaos thus functions as an amplifier or pump in psychotherapy.  This process is fundamentally related to the spiral form in nature and imagery.

Biomathematicians believe that the spiral lens structure may form from self-organizing waves.  The pattern is the trace left by an autocatalytic (feedback) process, which generates order from chaos.  Synchronization or resonance can phase-lock or phase-couple oscillators in many kinds of systems, including neural networks.

As each oscillator fires, its neighbors are influenced by the feedback of repeated signals so that an oscillator close to its firing threshold senses a signal from its neighbor and fires off immediately.  At that point oscillators become locked together.  This process can proceed until all the oscillators become coupled.

A chaotic system constantly mixes things up, creating new directions in which the system can go.  These moments of possibility are called bifucation points.  Here an impulse can amplify through the system's feedback  The phases or frequencies of the feedback become locked together and a structure emerges.

The rapport or resonance that occurs in co-consciousness reflects such phase-linking (perhaps through brain wave patterns) and creates an intentionality through self-referencial feedback.  In the midst of chaos a slight change in frequency can set them oscillating or dancing in an orderly pattern of phase locking.

When the therapeutic process goes "chaotic" there are frequently reports of spiral imagery, various forms of vortices perceptible in one form or another.

Once formed, the self-organized structure stays "alive" by drawing nourishment from the surrounding flux and disorder.  Especially long-lasting forms of phase-locked feedback are called solitons.  They may be analogous to what we have refered to elsewhere as the intentionality of the "Dream Wave," permeating conscious life.

The phases of the soliton elements in a soliton wave are so synchronized that two soliton waves that collide at angles or from opposite directions will pass through each other, emerging on the other side as if no collision whatsoever had taken place.  (Ref. B. Fuller in SYNERGETICS and COSMOGRAPY on Vector Equilibrium Matrix, where waves can pass through waves without interference; and Thomas Beardon's zero vector summation for tapping the vacuum potential).

"Positive" feedback can actually push a system to explode or spiral out of control.  The famous example of this occurs when a TV camera is pointed at its own monitor, producing a "hall of mirrors" effect of infinite reproductions.

Yet there is a structure-making dimension to this positive feedback, creating a place where new forms come into being.  Positive feedback can cause complex, even chaotic behavior concealed inside orderly systems to unfold, while negative feedback can grow inside an otherwise chaotic system, suddenly organizing it and making it stable.

Feedback in nature, weaves order out of disorder and holds order tenuously in place.  Evolution itself is actually an infinitely intricate feedback system.  It also relies on negative feedback to keep mutation changes from spiraling out of control.  It is the checking power of many negative feedback loops which weeds out most mutations, keeping species stable for long periods of time.

Briggs, John; FRACTALS: THE PATTERNS OF CHAOS; Simon & Schuster, New York, 1992.

Goswami, Amit et al; THE SELF-AWARE UNIVERSE; Tarcher/Putnam, New York, 1993.

McGuire, Michael; AN EYE FOR FRACTALS; Addison-Wesley Publishing, New York, 1991.

Swinney, Graywolf Fred and Miller, Iona; DREAMHEALING: CHAOS AND THE CREATIVE CONSCIOUSNESS PROCESS; Aesculapia, Wilderville, 1992.

Webb, Burt; online discussion of chaos in nerual networks on CHAOPSYCH, Oct. 1993.