The Edge of Chaos
We began eons ago in a vast nothingness, yet this seemingly empty void contained the substance for everything. It was an abyss lacking form, yet full of endless potential. Out of this primordial soup forms emerged and life sprang forth in an explosion. Stability and chaos were woven together in paradox, constantly influencing each other, separating, fusing, shifting and eventually becoming polar positions on a spectrum of activity. Early in my artistic career I tangled with the question of why I have the sense that destruction, chaos and insanity seemed to reveal insight and possess utility. I discovered through my research and experience that it is not only part of cyclical system of renewal but is the fuel which initiates the churning of elements that allows for a new form to emerge. It is the spark of life which is so mysterious because it is not rationally perceivable. These personal discoveries led me to the significance of dualities and the dynamic energy in the transmutation from stability to chaos, and back again.
Order in chaos is a scientific cliché according to Glieck. The reverse of this is also true, there is chaos within order. The emergence of structure was born from a chaotic mess. Chaos is integrated within all levels of order. Although polarity is scientifically understood at a magnetic level, and is an axiom of many Eastern philosophies; the Western world still lacks an identification of the chaotic counterpoint to order and an acceptance of its place in our lives. This thesis does not intend to deny the benefits of order and stability, rather it is to exemplify and promote the benefits of chaos. Through this translation of “The Edge of Chaos” I intend to argue that the chaotic; and subsequently disorder, irrationality, adversity & mystery; is necessary not only in the formation and continuation of life but also for the creative process, and social advancement, as well as absolutely essential to the evolution of the mind and spirit. This body of work will investigate the dialectics of substance and absence, logic and irrationality, stability and chaos.
Defining Chaos
Merriam Webster defines chaos as a chasm or abyss; obsolete; a confused mass or mixture; and a state of utter confusion. Moreover, chaos is listed as "a state of things in which chance is supreme; especially : the confused unorganized state of primordial matter before the creation of distinct forms " also "the inherent unpredictability in the behavior of a complex natural system." James Glieck in his book Chaos: Making a New Science collected the following variety of descriptions for the term;
Phillip Holmes, a white-bearded mathematician and poet from Cornell by way of Oxford: The complicated, aperiodic, attracting orbits of certain (usually low-dimensional) dynamical systems.
Hao Bai-Lin, a physicist in China who assembled many of the historical papers of chaos into a single reference volume: A kind of order without periodicity. And: A rapidly expanding field of research to which mathematicians, physicists, hydrodynamicists, ecologists and many others have all made important contributions. And: A newly recognized and ubiquitous class of natural phenomena.
H. Bruce Stuart, an applied mathematician at Brookhaven National Laboratory on Long Island: Apparently random recurrent behavior of deterministic, nonlinear dynamical systems.
James Crutchfield of the Santa Cruz collective: Dynamics with positive, but finite, metric entropy. The translation from mathese is : behavior that produces information (amplifies small uncertainties), but is not utterly unpredictable.
And Ford, self-proclaimed evangelist of chaos: Dynamics free at last from the shackles of order and predictability….Systems liberated to randomly explore their every dynamical possibility….Exciting variety, richness of choice, a cornucopia of opportunity.
The term “Edge of Chaos,” was coined by Christopher Langton who developed the lambda parameter. Lambda[i] allows for the results of Cellular Automata to be tracked across a continuum. Christopher Langton was first inspired toward his discovery by an interest in artificial life. In the 1970s he experienced a subconscious jolt which came to him late one night as he debugged code with pencil and paper while John Conway’s Game of Life ran on the computer screen. “it was like a drug experience, when you run with a crazy fantasy,” Chris explained. “You take down the usual mental barriers to crazy thoughts, and just let them develop freely. It was like a hurricane of ideas sweeping across the mental landscape, and I was just a spectator"(qtd in Lewin 98). James P. Crutchfield was another researcher who ran into similar events in phase transitions and termed it the "Onset of Chaos." Today this phenomenon is the basis of self-organized criticality in Complexity Theory and Chaos Theory.
Self-Organized Criticality
Dynamic systems include a variety of system types[ii]. They were developed to discover a more accurate understanding of how the world we live in works. Rather than identify an isolated phenomena, we can see the dynamics of interactivity and how change occurs. This critical state of complexity within them is the Edge of Chaos identified by Langton and others. Cellular Automata are squares of a grid in which each part has a limited number of states (i.e. on or off, moving forward or backward, or down etc. (see Figure 1). These elements follow rules that are dictated by their neighboring squares. The results of Cellular Automata programs provide information on how local interactions create global behavior which is discussed further under Emergence. Wolfram separated CAs into four broad categories:
Class 1: CA which evolve to a homogeneous state;
Class 2: displaying simple separated periodic structures;
Class 3: which exhibit chaotic or pseudo-random behavior;
Figure 1.Avnet, Jeremy. Theory of Cellular Automata. Graph
Class 4: which yield complex patterns of localized structures and are capable of universal computation.
Figure 2. Eric D. Beinhocker. Robust Adaptive Strategies. 1999
The significance of this concept is that between chaos and order there is an area of maximum complexity within which intricate functions generate endlessly. The unending manifestation of new information and the combination of stable forms and periodic chaos generate adaptability; it creates a space in which change can occur. The phenomena of this phase transition in automata, the Game of Life, and other programs creates a visual image that is recognizably lifelike. Early on large dynamical systems were hypothesized by Der Bak to naturally evolve toward this critical state. Chaos mathematics, theory and science have become their own realm of study due to the boundaries it crosses through scientific disciplines. The Edge of Chaos or critical state is a pattern found over and over in the results of research and simulation. Some dispute the concept’s validity as an evolutionary model, such as Melanie Mitchell, James P. Crutchfield, and Peter T. Hraber who wrote Dynamics, Computation, and the Edge of Chaos: A Re-Examination. In this paper they concluded that the interpretation of Packard and Langton’s experiments were incorrect. These authors feel all computational systems do not automatically enter into this particular phase transition, and that the Edge of Chaos requires redetermination and further defining. Others such as Stuart Kauffman look for similar phenomena in their own work and the work of others. Kauffman concluded that evolution is maximized in this critical state during his research of biodiversity and fitness adaptations of organisms to their environment. (See figure 2)
Whether or not all systems are attracted to the Edge of Chaos, we still see it’s presence over and over again in our history and in our current world. Most geometry people learn has existed for two millennia, originating from the days when the irrational was feared. This math is Euclidian and exists in an abstract imaginary state of Platonic harmony. These new geometric mappings of phase space mirror “a universe that is rough, not rounded, scabrous, not smooth. It is a geometry of the pitted, pocked, and broken up, the twisted, tangled, and intertwined" (Glieck 94).
Evolution and Emergence
The origins of life began in a wild explosion of variation and growth. "The sudden increase and richness of life forms in the Cambrian has been attributed to the occupation of a 'vacant ecology,' an environment which was available for and receptive to evolutionary experimentation" (Lewin 18).
Such life can only evolve, become more, develop and elaborate itself to the extent that there is something fundamentally unstable about both its milieu and its organic constitution. The evolution of life can be seen not only in the increasing specialization, elaboration and bifurcation or differentiation of life forms, but above all, in their becoming-artistic, in their self-transformations which exceed the bare requirements of existence. Sexual selection, the consequence of sexual difference or morphological bifurcation - one of the earliest evolutionary upheavals in the evolution of life on earth, and undoubtedly the most momentous invention that life has brought forth, the very machinery for guaranteeing the endless generation of morphological and genetic difference - is the opening up of life to the indeterminacy of taste, pleasure and sensation (Grosz 16).
Complex adaptive systems include weather, fluid motion, turbulence, biodiversity, swarm activity and more. The systems which are considered adaptive exhibit an ability to form emergent properties which improve functionality. "The movement from low-level rules to higher-level sophistication is what we call emergence" (Johnson 18). These emergent properties can be seen in natural and dynamic systems as well. "Keller and Segel saw it in the slime mold assemblages; Jane Jacobs saw it in the formation of city neighborhoods; Marvin Minsky in the distributed networks of the human brain” (Johnson 18). Emergence can be described as the formation of a pattern that was not inherent in the local rules of interacting elements. Goldstien describes emergence as "the arising of novel and coherent structures, patterns and properties during the process of self-organization in complex systems" (Corning 7).
If we allow emergence to assume a metaphoric substance the overall aesthetic is creative formation. Out of endless possibilities a pattern forms, when and if it fails the process starts over, when it succeeds it builds upon itself. In dynamic systems the results are somewhat unpredictable yet rely heavily on their initial conditions. This I liken to a choice, every choice made by the artist will change the course of the work, and every choice we make in life might not lead to a predictable outcome, but will greatly influence the outcome that does result. The artist practices again and again until they find something real. Then using those initial conditions pushes to its limits discovering the extent of possibilities.
To generate images of the Edge of Chaos requires the modern advancement of computers. They are a necessary tool to process this information "A computer can address the problem by simulating it, rapidly calculating each cycle. But simulation brings its own problem: the tiny imprecision built into each calculation rapidly takes over, because this is a system with sensitive dependence on initial conditions. Before long, the signal disappears and all that remains is noise" (Glieck 44). In sciences this noise used to be ignored for the preference of clean simplified data. Due to this a lot of information was overlooked. With the advancement of processing capabilities in computers new programs were developed to fine tune these extensive mathematics into something more precise. Such advanced mappings of data are only made possible by modern computers. The visual output of this information allows scientist to see their data from an omnipotent view. Our experience of these images speaks deeply within us, resonating of something familiar yet mysterious.
Strange Attractors
In complex system dynamics the pull toward or around a particular area creates an attractor. These specifically follow certain base rules[iii]. They also exist is phase space[iv]. "Phase space gives a way of turning numbers into pictures, abstracting every bit of essential information from a system of moving parts, mechanical or fluid, and making a flexible roadmap to all it's possibilities… In phase space the complete state of knowledge about a dynamical system at a single instant in time collapses to a point… The history of the system time can be charted by the moving point, tracing it's orbit through phase space with the passage of time" (Glieck 134). When the attractor is chaotic it is known as a Strange Attractor.
Figure 3. Lorenz, Edward N. Lorenz Attractor
Visualize the empty space within the strange attractor as a void, as zero; it is the unknown, nothingness, empty. It represents that which is not; it is the abyss that we can only see the edges of, we cannot peer into it from outside of it. In the Mandelbrot set, as in others, this edge of the void can be magnified infinitely to show similar yet always changing patterns. Recursion is not only a known phenomenon of nature, but it is a strong visual element in art. Recursion contains both repetition and variation which we will come back to later. Attractors bring the motion of the system toward stability, even if perturbed the system will gravitate to its replicating tendencies. Often they are mapped with more variables then than a more easily conceived three-dimensional form.
The Lorenz Attractor and the Attractor of Henon were of the originating images that influenced further examination of attractors. What was so alluring about these graphs? In the Lorenz Attractor where the orbits around the attractors visually seem to merge there are actually infinitely complex surfaces that come very close to one another yet do not actually intersect. In the Henon Attractor where it appears to be a solid line is actually parallel lines very close to each other. The orbits of strange attractors appear to mimic the feedback in an emergent system. Positive feedback propels, while negative feedback "keeps the system in check" (Johnson 130). It is not too farfetched to see the metaphoric capabilities of strange attractors, Cellular Automata , and self organized dynamic systems to represent life. "Pattern-matching negative feedback, ordered randomness, distributed intelligence. The only difference is the materials they are made of: swarm cells, sidewalks, zeros and ones" (Johnson 220). We can extend Johnson’s list to include concepts, artists, paint, beliefs… and well almost anything.
Taking a sliced 2D of one of these 3D images generates a return map or Poincare map. The continuous line becomes points as they intersect the plane. This sampling at particular intervals or planes looses one type of data while clarifying another. Dissection provides a view of the interior of attractors, a hidden structure within the chaotic system. Many of these strange attractors and strange loops whether viewed as a Poincare map, limit cycle, or 3d rendering (see Fig.4) felt reminiscent of many automatic drawings. They also made me think of children’s drawings (see Fig. 5) with their many instances of circling back across their own marks. There is something greater being provided by the exploration of chaos and complexity then attempts to replicate the natural world. The results of the data are not strings of numbers nor a direct answer, it is not linguistic or rational in any sense. Through our perspective these results are not even mathematical, they are images. Visual language operates as an experience; we perceive these graphs and states of
Figure 5. Claira Tenny. Age 4. Untitled. April 2010
systems as an optical input through which understanding begins to unfold in a way closer to art then to science. The good news is that a little uncertainty is always present and usually ends with beautiful visual results. I have embraced the idea that venturing into the unknown and chaotic leads to wisdom of experience.
The Human Condition
While Western science draws closer to acceptance of chaos and unpredictability, our philosophy, religions, and cultural perspectives have not. Perhaps this is due to the origins of discourse in Western schools of thought. Just as the most minor change in the early stages of a complex system will result in drastically different results; a minor difference in the establishment of a philosophy can make all the difference in the development of belief systems between the west and the east. “Terrified by the titanic and primordial forces disrupting the mind, Platonism, Pythagoreanism, Stoicism, and similar schools of philosophy exposed the irrational as a danger and disgrace which reason or the soul must combat” (Porter 36). This fear of the unknown can be seen in the history of insane asylums[v]. Religions and governments reinforce that enjoyment of disorderly conduct is inappropriate and intolerable. Still despite regulation on social and political levels “children seek disequilibrium and sensations of vertigo as source of pleasure” (Virilio 27). Even into adulthood we embrace excitement in amusement parks or films of intense nature. Music is filled with layered orders and empty spaces that move us in a chaotic manner. Such enjoyment is a driving force in the lives of many. Chaos operates within the voids structure leaves like the intermingling of many free willed peoples within a structure both physical and conceptual. Chaos also operates externally of structure as a shaping agent. War, economic flux, the decisions of someone’s mind which operates on a creative level and more all help shape what form a physical or conceptual structure will take. In contemporary times the prediction of how "the entertainment world will self-organize into clusters of shared interest" (Johnson 219) has come to pass. Netflix, Amazon, Facebook, YouTube, any networking or media site has created a new space within which clusters of people begin to operate like a swarm.
Self-Organization rather than external control lends itself to a system that is dynamic and capable of expansion or advancement. It is true that adversity can create discomfort rather than excitement; some may be frightened rather than in awe of the vast chaotic and bewildering truths of the universe. The binary actions of polar forces such as stability and chaos generate a whole that is larger than its parts. I liken this to the dialectic method of discourse. Two disagreeing views utilizing a common language or set of terms to convince one another as structures and voids do each casting light and shadow on each other to combine two perspectives. More precisely they inform each other. Chaos as a generator of creative potential allows our culture and society to advance and adapt. “What philosophy and art share in common - their rootedness in chaos, their capacity to ride the waves of a vibratory universe without direction or purpose, in short, their capacity to enlarge the universe by enabling its potential to be otherwise, to be framed through concepts and affects. They are among the most forceful ways in which culture generates a small space of chaos within chaos where chaos can be elaborated, felt, thought” (Grosz 29).
Mythos in the History of Art
Figure 6 Pollock, Jackson. Moon Woman Cuts the Circles. 1946
Myths are models of behavior that are influenced by culture and in turn may influence culture. The elements of chaos in myth are not new and trace far back to the origins of art. “Chaos is viewed as a primordial completeness or unity with no division… the idea of the very first beginning, the ‘temps d’origine’, is identified with chaos…the artist should begin like God from chaos and a void” (Kępińska 123). The first movement I examined was the schitzoids shortly after finding the prinzhorn collection. Interestingly enough the schitzoid movement grew out of coping with an oppressive society. The Dada movement brought discordianism. “The veneration of Eris, a.k.a. Discordia; widely popular among hackers. Discordianism was popularized by Robert Shea and Robert Anton Wilson's novel Illuminatus! as a sort of self subverting Dada-Zen for Westerners — it should on no account be taken seriously but is far more serious than most jokes. Consider, for example, the Fifth Commandment of the Pentabarf, from Principia Discordia: ‘ A Discordian is Prohibited of Believing What he Reads.’ Discordianism is usually connected with an elaborate conspiracy theory/joke involving millennia-long warfare between the anarcho-surrealist partisans of Eris and a malevolent authoritarian secret society called the Illuminati.” (Babylon, "Discordianism")
The surrealists and the abstract artists believed that chaos in the form of action painting and automatism is an expression of freedom. In particular Josef Sudek utilized chaos as disorientation, uncertainty and vertigo. “Chaos was a crucial notion for Sudek’s apprehension of reality: in chaos, it is possible to find secrets, because it is an endless source of imaginative investigation” (Lahoda 1). Then abstract expressionism pursued chaos as a mythological element beginning with symbols which extend back to the dawn of man, such as the Mandala. A lot of the themes prior to abstraction were related to cycles and goddess worship. “[Action painters] have a common denominator: the dialectic struggle and union of contradictions in the cosmic model of the cycle of life (birth, growth, and decay). But death is not the end of life but the necessary condition for life to assume a new form” (Kępińska 121). My own artistic background, similar to these artists, includes the influence of philosophical and psychoanalytical literature, a fascination with ancient cultures, an attempt to reconcile the need for myth in a modern world, and the decay of form into something more primal that leads one to experience and embrace chaos. The early work of the action painters of this time period was full of mythical motives, some of the oldest most universal elements of the human condition.
Kępińska links the basic shapes/symbols to their original meaning in the context of ancient myths. All of these symbols are intricate to the process of birth, growth, decay, death, and rebirth. The origin of chaos and the return to chaos are a constant in our existence from the formation of the earth, to the evolution of species, to the changing of the seasons and the experience of being. The basic reoccurring motifs detailed in this article are “the sun, circles, concentric rings, the moon, a serpent, a spiral, a bird, a wolf, a woman, a head(or face), an eye (or a pair of eyes), a hand, abstract signs (squares, crosses, dots, lines), and numbers”(109) (See Figure 6). Mandalas (circles/spheres) as well as spirals, women, eyes and abstract signs frequently occur in my work. For this series of pieces I have chosen the most basic and universal, the Mandala. It is important to note how mythic themes assist in understanding the direct experience of these visual signs. “The symbolism of the circles and the sphere is multivalent, which applies to nearly all mythological symbols since they join order with chaos, life with death, day with night, (each element with its opposite, with the first containing the second respectively)” (Kępińska 111).
Kępińska explains in regards to the action painters that “this change of spiritual perspective, this journey into the ‘vale of the ancestors’ was a sort of death suffered deliberately in order to be ‘born anew’ on another level of consciousness and to reexamine and renew the questions of art within a broad anthropological, philosophical perspective” (122). This is the purpose of art, and the inspiration it provides to the world, one that indulges in peering over the Edge of Chaos into the abyssal chasm from which life springs forth, and to pursue a sublime experience in addition to mere survival. There is universality to basic symbolic elements, just as there is universality to the premise of myths that ran parallel in various traditions. There is also a universality that is produced through complexity theory, and the ratio of convergence, or scaling.
Visual Elements
Figure 7. M. C. Escher, Spiralen, 1953, Holzschnitt
Artists, like dynamic systems, begin with a limited set of variables which indefinitely changes into something unpredictable from those initial conditions. We take the raw creative material vacant of form and begin to cycle through moments of intense chaotic divination and calculated structuring. M.C Escher (see Fig.7) had a strong capability to visually represent the paradox of recursive dialect between stability and chaos. A visual paradox forces our minds to recognize and experience the irrational. Optical Illusions are translated as impossibilities through rational reasoning, yet meditated on they can open our minds to expanses beyond physicality.
Chaos and stability are polarities in a recursive loop of varying size just as void and matter are. They are layered and embedded in each other like a story inside a story inside a story. In aesthetic terms this type of recursion functions as both repetition and variation in one. Benoit Mandelbrot’s mathematical set brings this repetition and variation to an endless fractal, it has been called the philosophy of art although its origins are mathematical and computational. "Rossler felt that these shapes embodied a self organizing principle in the world. He would imagine something like a wind sock on an airfield, 'an open hose with a hole in the end, and the wind forces its way in.' he said. 'Then the wind is trapped. Against its will, energy is doing something productive, like the devil in medieval history. The principle is that nature does something against its own will and, by self-entanglement, produces beauty'" (Glieck 142).
Figure 8. Pic, Sylvia. Untitled. 1998.
With the overlaying veils of attractors orbits in 3-d space become a "dark scribble" inside of which a structure is imperceptible. Sylvie Pic’s rubber tire sculpture (see figure 8) presents the difficulty in perceiving the space within which has a form of its own different from the outside. My work for this thesis is also layered to a point where the original 2d image cannot be known, yet it's remnants are visible through the veils of the overlaying layers. “Art indeed struggles with chaos, but it does so in order to bring forth a vision that illuminates it for an instant, a Sensation… Art is not chaos but a composition of chaos that yields the vision or sensation, so that it constitutes, as Joyce says, a chaosmos, a composed chaos - neither foreseen nor preconceived…Art” (qtd in Grosz 9).
Aesthetic surprise gives rise to originality in thought, inventiveness, and discovery much more so then any rational or orderly thought. I intended to have a strong visual impact for this work which influenced the color palette. First I experimented with many colors, and narrowed them down based on instinct. Further investigation confirmed my choice of red, black, and some white as the dominant palette. (See figure 9) The complex dynamics of how we process what we see reflects "A perception that sifted the messy multiplicity of experience and found universal qualities. Redness is not necessarily a particular bandwidth of light, as the Newtonians would have it, it is a territory of a chaotic universe, and the boundaries of that territory are not so easy to describe - yet our minds find redness with regular and verifiable consistency” (Glieck 165). Studies found that red closely followed black and white in most cultures as the most recognizable and earliest learned colors.
Figure 9. Taylor, Megan. Critical Mass #3, 4.22.2010
"Overwhelmingly, BLACK, WHITE, and RED are the most salient colors; they account for 86.3 per cent, 72.2 per cent, and 73.1 per cent of all color terms elicited on Tasks I, II, and II respectively"(Bolton 298).[vi] Other studies showed red as the most salient, with black then white following closely behind. Black and red have the highest potency. I also have included silver which is not only a color but an identifying trait of precious metals. The salience of color is integral to survival and learning.
Conclusion
If there ceased to be order all form would collapse into entropy and nothing would exist. Likewise if there ceased to be chaos all would become still, and life would cease to exist. Not only is chaos an essential part of the cosmos and necessary for the emergence of life and evolution, it is inherent in the creative process. It is inherent in all social development, and cultural evolution. “Teetering on the Edge of Chaos unlocks the unconscious and reconstructs the conscious to trigger emergence, or a state of becoming that opens closed structural boundaries” (Sherman & Schultz, 7). If we do not embrace chaos as disorder, adversity or the benefit of many orders then our society will be one which goes into lock down. The warnings were apparent long ago when novelists tried to subtly warn us. Underground movements still push strongly in music, art and more inspiring us to live truly free, on the whim of free will. My studies of complexity science and chaos in mathematics and philosophy have led me to believe in free will’s existence and purpose.
Chaos as disorder is the fluid that lubricates change, allows for order to be rearranged, stacked and morphed. Our adaptability in this world relies heavily on the variety chaos as chance provides. "To understand how the human mind sorts through the chaos of perception, surely one would need to understand how disorder can produce universality"(Glieck 165). Universality means that each person can identify with it, that it is recognizable in varying levels of experience. "If much of nature dances to the tune of complex dynamical systems, then the consequences should be apparent from single organisms through to the way the entire planet works. The phenomena of spontaneous generation of order and of adaptation to the Edge of Chaos would shape what we see, level built upon level, a hierarchy of effects" (Lewin 108).
Atoms are the smallest known structures; the materials that make them up may be yet another level; however this is where we will start. The nucleus is surrounded by an orbit of electrons, as an attractor it pulls them toward it. The exchange of electrons in molecules is a delicate set of rules, these dynamic interchanges make up geography, biology, and allow for the exchange of energy necessary for life. Now let’s pull back to single cell organisms. Bacteria and organisms such as the slime mold self organize themselves into adaptive dynamic habits and processes. Back up a little more and we can use ants as an example of small multi-cellular organisms. A little further and we can see the function of bodily tissues, organs, and the immune system within us. In 2009 the University of Cambridge announced that elaborate testing led them to conclude the brain is a dynamic system operating in a self-organized critical state, this is chaos at the individual level.
Expanding to the social level, networks of sidewalks, streets, passing of information, the functioning of cities shows an adaptive complex system. Open up the view to a global level and we see weather and ocean systems. The orbiting of the planets around the sun may even operate on an attractor which orbits yet never repeats, and Jupiter’s Red Spot is a dynamic system of gases with a central attractor. Looking into the far reaches of the universe we have observed the birth and death of stars, black holes, nebulas and various other phenomena that might have a recognizable process but must be recognized as chaos at its origin, our origin, creation and destruction mixed indistinguishably in a beautiful display.
Pure anarchy is not the goal nor is it effective. Relying on base rules (whether morals or ant pheromones or computer code) will generate emergent macro behavior. Local rules are necessary to building global intelligence. Global rules cannot be enforced on the local level; the system collapses from the inside. There may be some truth to the idea that one who embraces nothingness and chaos becomes greatly disturbed or unstable. “The Edge... There is no honest way to explain it because the only people who really know where it is are the ones who have gone over” (Thompson). The sacrifice in going over the metaphorical Edge of Chaos is an inability to return to a fully rational perspective. Perhaps a little irrationality is just what we need to emerge anew out of the ashes of our failing systems. When we are met with adversity, we are taken out of a one way perspective.
Although the idea of chaos seems disorienting, it is the way in which your brain shuffles through endless possibilities to make a choice, to decide your own future. In his essay on chaos theory, complexity and art Jay Kappraff concludes that we will enter into a new era where these ideas become integral to our myths, philosophies, religions & general culture. Is it a paradigm shift? I hope so; the mere meaning indicates adaptation on a connected mental level. An emergent formation from the collective of us all indicating the mind and the sum of all minds will evolve.
[i] Lambda Parameter- a dimensionless measure of complexity and computation potential in cellular automata, given by a chosen state divided by all the possible states. For a 2-state, 1-r neighborhood, 1D cellular automata the value is close to 0.5. For a 2-state, Moore neighborhood, 2D cellular automata, like Conway's Life, the value is 0.273.
[ii] Other terms indicating similar functioning concepts include: Complex natural systems, complex dynamic systems, and complex adaptive systems. See glossary for full descriptions.
[iii] An attractor is a subset A of the phase space characterized by the following three conditions:
A is forward invariant under f: if a is an element of A then so is f(t,a), for all t > 0.
There exists a neighborhood of A, called the basin of attraction for A and denoted B(A), which consists of all points b that "enter A in the limit t → ∞". More formally, B(A) is the set of all points b in the phase space with the following property:
For any open neighborhood N of A, there is a large positive constant T such that f(t,b) ∈ N for all real t > T.
There is no proper subset of A having the first two properties.
Roughly speaking, an attracting set for a dynamical system is a closed subset of its phase space such that for "many" choices of initial point the system will evolve towards (Milnor)
[iv] For a system with N-variables, a phase space is a 2N dimensional space composed of N-variables and their time derivatives. (Kitano, 2004)
Phase space is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. A plot of position and momentum variables as a function of time is sometimes called a phase plot or a phase diagram. Phase diagram, however, is more usually reserved in the physical sciences for a diagram showing the various regions of stability of the thermodynamic phases of a chemical system, which consists of pressure, temperature, and composition. (Findlay, 1911)
[v] In his book Madness and Civilization Foucault describes what he titles the great confinement of the 17th Century where large numbers of shunned people who may have been poor, strange, small time criminals, vagrants, or disliked by someone of power were all thrown into asylums.