Fractals and Chaos in Psychology, Biology, History, and Sociology

Link to Fractal Science Documentaries post
Link to Potential Geometry in Quantum Physics
Link to Fractal Evolution

Link to Complexity and Chaos


What is the proper place of infinity in physics?



In mathematics and theoretical physics, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system containing a finite number (two or more) of incommensurable frequencies.

That is, if we imagine that the phase space is modelled by a torus T (that is, the variables are periodic like angles), the trajectory of the system is modelled by a curve on T that wraps around the torus without ever exactly coming back on itself.

As for

same, but allowing T to be a torus with an infinite number of dimensions.   (QUOTE ALL MENTIONS OF INFINITY)



“All things come out of the one, and the one out of all things.” Heraclitus ~ 500BC

“All have similar form yet none are the same as the other, and thus the choir shows a secret law, a sacred mystery.” – Goethe

“The worst of all possible misunderstandings would be that psychology be influenced to model itself after a physics which is not there anymore, which has been outdated.” – Robert Oppenheimer

“I believe that scientific knowledge has fractal properties, that no
matter how much we learn, whatever is left, however small it may
seem, is just as infinitely complex as the whole was to start with.
That, I think, is the secret of the Universe.” —Isaac Asimov

“Reality cannot be found except in One single source, because of the interconnection of all things with one another. … I maintain also that substances, whether material or immaterial, cannot be conceived in their bare essence without any activity, activity being of the essence of substance in general. … In conclusion, nothing should be taken as certain without foundations; it is therefore those who manufacture entities and substances without genuine unity to prove that there is more to reality than I have just said; and I am waiting for the notion of a substance, or of an entity, which successfully comprehends all these things; after which parts and perhaps even dreams will be able one day to lay claim to reality.” -Leibniz

“The contradiction, the paradoxical evaluation of humanity by man himself is in truth a matter for wonder … in other words …’man is an enigma’.” – Carl Jung, The Undiscovered Self

“In all chaos there is a cosmos, in all disorder a secret order.”
– Carl Jung

“Chaos often breeds life, when order breeds habit.”
– Henry Adams

…to suppress the pictures is to suppress a powerful source of suggestion. … Pictorial representation is essential for discovery and rapid understanding…
J.L. Synge in The Hypercircle in Mathematical Physics (1957), p. 12
The Double Spiral Arrangement of the Human Musculature

The Alexander Technique
and the String Pedagogy of Paul Rolland

chaos, complexity, and entropy

The first question that I want to examine, then, is: why is it that, among
all the practitioners of science, applied science, engineering disciplines, and human sciences,
physicists were practically the last ones to be interested in chaos and to use it in their work?
There were exceptions, of course. The people who built the large particle accelerators knew
about chaos; in fact they discovered much of it. There are many other exceptions. But the
majority of physicists did not know about chaos, and still does not. Engineers of many sorts,
meteorologists, some types of chemists, population biologists, cardiologists, economists, and
even psychologists, seized upon chaos long before the physicists did.
One-dimensional cyclic cellular automata can be interpreted as systems of interacting particles, while cyclic cellular automata in higher dimensions exhibit complex spiraling behavior.

Animation of two dimensional cyclical cellular automaton growing to repeating patterns from a random beginning.

Could the classical relativistic electron be a strange attractor

quote parts of introduction — universality..
Society for Chaos Theory Blog

Infinite Space and Self-Similar Form

in Alchemy and Fractal Geometry

By Laura Strudwick

add indra’s pearls fractal metaphor
spiral of dialectics and philosophy
Quote book, Chaos Theory in Psychology

Rupert Sheldrake – Science and Wonder

The Secret Life of Chaos BBC

Ilya Prigogine: Science and Religion

Ilya Prigogine – Chaos is Order

spiral fractal aloe
Spiral aloe again.

Complexity & Chaos – Part 15 (A New Biology)

Complexity & Chaos – Part 7 (Time, Entropy & The 2nd Law)

CBL – Machinery

“The hierarchical structure of many social systems suggests a possible fractality of political systems. Federal is to states, as state is to counties, as county is to cities, as city is to neighborhoods, as neighborhood is to households, as household is to families, as family is to individual. Are these just loose metaphors, or do they represent functional relationships across several scales?”

The Landscape of History (As Fractal)

“John Gaddis’ The Landscape of History: How Historians Map the Past advances the interesting point that historians and biographers have been aware of scaling aspects of history for a very long time.

Drawing distinctions between history and the social sciences, Gaddis notes

in the social sciences, narrative is embedded within the model, whereas

in history the model is embedded within the narrative.

This is one way scaling is used in history: the illustrative anecdote as narrative device. Historians use particular events to make make general points, implying a similar structure across many scales. That is,

“how phenomena so small that they totally escaped notice at the time could shape phenomena so large that we’ve always wondered why they occurred.” (pg 25)

Another example of scaling is revealed in the work of Michel Foucault, who shows similar patterns of authority from cultures to nations to states to cities to families.

Biography also seeks similarity across scales. “Biographers seek pattens that persist as one moves from micro- to macro-levels of analysis.”

Gaddis notes similarities between history and the non-laboratory sciences (astronomy, paleontology, ecology, for example): process is deduced from structure. History is much more ecological than reductionist.

Another interesting point is that Gaddis says historians deny the distinction between independent and dependent variables. “Historians have a web like sense of reality, in that we see everything as connected in some way to everything else.” This is a ripe setting for noticing the presence of chaos. Indeed, historians do: “particular actions had larger consequences than might otherwise have been expected.” By following the work of Poincare, Henry Adams seems to have noted this aspect of history with remarkable clarity. In fact, now chaos is providing historians with a collection of tools for making sense of the problem that small events can have large consequences.”
“The generative sciences originate from the monadistic philosophy of Leibniz.”
Of circles and spirals: Bridging the gap between the leading circle and spiral wave concepts of cardiac reentry

Human Development VII: A Spiral Fractal
Model of Fine Structure of Physical Energy
Could Explain Central Aspects of
Biological Information, Biological
Organization and Biological Creativity

Published November 14, 2006

Ventegodt, S., Hermansen, T.D., Flensborg-Madsen, T., Rald, E., Nielsen, M.L., and Merrick, J. (2006) Human development
VII: a spiral fractal model of fine structure of physical energy could explain central aspects of biological information, biological
organization and biological creativity. TheScientificWorldJOURNAL 6, 1434–1440. DOI 10.1100/tsw.2006.256.

add quotes

Geological History Time Spiral

Full Size Image

Ilya Prigogine – Misconception of Science

Ilya Prigogine – Society concept

Ilya Prigogine – Mix of laws and events

High Anxieties The Mathematics of Chaos 3/7

Quote Sally Goerner

Terence Mckenna – The Importance of Human Beings
In recent times, reaction–diffusion systems have attracted much interest as a prototype model for pattern formation. The

above-mentioned patterns (fronts, spirals, targets, hexagons, stripes and dissipative solitons) can be found in various types of

reaction-diffusion systems in spite of large discrepancies e.g. in the local reaction terms. It has also been argued that

reaction-diffusion processes are an essential basis for processes connected to morphogenesis in biology[16] and may even be related

to animal coats and skin pigmentation.[17][18]
See also: The chemical basis of morphogenesis

Another reason for the interest in reaction-diffusion systems is that although they represent nonlinear partial differential

equation, there are often possibilities for an analytical treatment.[7][8][19][20][21]

Cantorian Fractal Space- Time Fluctuations in Turbulent Fluid
Microscopic spiral waves reveal positive feedback in subcellular calcium signaling
Inward Rotating Spiral Waves in Glycolysis
Spiral wave structure in pre-biotic evolution: Hypercycles stable against parasites
“The Soliton model in neuroscience is a recently developed model that attempts to explain how signals are conducted within neurons.”
Although spiral waves are ubiquitous features of nature and have been observed in many biological systems, their existence and

potential function in mammalian cerebral cortex remain uncertain. Using voltage-sensitive dye imaging, we found that spiral waves

occur frequently in the neocortex invivo, both during pharmacologically induced oscillations and during sleep-like states. While

their life span is limited, spiral waves can modify ongoing cortical activity by influencing oscillation frequencies and spatial

coherence and by reducing amplitude in the area surrounding the spiral phase singularity. During sleep-like states, the rate of

occurrence of spiral waves varies greatly depending on brain states. These results support the hypothesis that spiral waves, as an

emergent activity pattern, can organize and modulate cortical population activity on the mesoscopic scale and may contribute to

both normal cortical processing and to pathological patterns of activity such as those found in epilepsy.

Spiral dynamics are frequently observed in cortical population neuronal activity . Rhythms created by spirals influence and entrain

ongoing cortical activity . The occurrence rate of cortical spirals depends heavily on brain states . Cortical spirals are short

lived and fast drifting, neither robust nor persistent

There are many spiral forms in nature, both on Earth and in space. Spirals occur in physical forms such as DNA and the shell

formation of mollusks such as the conch and chambered nautilus. They also occur in wind patterns, including hurricanes and

tornadoes. They are present in air and flame forms known as vortexes and whorls. And they occur in the way things fall in the

atmosphere, from leaves to aircraft. In the human body, the spiral pattern of the heart’s bioelectric impulses causes the chambers

to beat with a spiral pulsing rhythm. Brain waves, comprised of neuron impulses, seem to flow along the neurons and down the spinal

cord in a spiral pattern. Some evidence shows bioelectrical spiraling in the labor impulses during birth. Finally, we see spiral

forms omnipresent throughout the visible and invisible universe, in galaxies, accretion disks around black holes, coalescing

interstellar clouds and many other forms of matter and energy.

Cardiac Beat-to-Beat Alternations Driven by Unusual Spiral Waves
Some key features of these phenomena can be well reproduced in computer simulations of a non linear reaction diffusion model.
There are spiral waves in other systems, for example also in the brain. In the neural
tissue there are also spiral waves. Spiral waves are produced by a chemical reaction.
Cells in the brain have less calcium, but a rather similar system.
About SIEO

Bohm 1980, p. 11 said: “The new form of insight can perhaps best be called Undivided Wholeness in Flowing Movement. This view

implies that flow is, in some sense, prior to that of the ‘things’ that can be seen to form and dissolve in this flow”. According

to Bohm, a vivid image of this sense of analysis of the whole is afforded by vortex structures in a flowing stream.


“[In] evolutionary processes, causation is iterative; effects are also causes. And this is equally true of the synergistic effects

produced by emergent systems. In other words, emergence itself… has been the underlying cause of the evolution of emergent

phenomena in biological evolution; it is the synergies produced by organized systems that are the key. (Peter Corning 2002)
Many aspects of nature are essentially unpredictable over the long term, even when quantum effects are completely absent. A

frequent cause is chaotic dynamics, which can arise when the rates of change of important dynamical variables depend in a nonlinear

fashion on the variables themselves.
Among other applications, the circle map has been used to study the dynamical behaviour of a beating heart.

Chaos and the Imagination

Form and Void
“Macrocosm and microcosm is an ancient Greek Neo-Platonic schema of seeing the same patterns reproduced in all levels of the cosmos, from the largest scale (macrocosm or universe-level) all the way down to the smallest scale (microcosm or sub-sub-atomic or even metaphysical-level). In the system the mid-point is Man, who summarizes the cosmos.

Many ancient peoples other than the Greeks (Incas, Aborigines, Vikings and Celts, amongst others), observed the golden ratio in many parts of the ordered universe both large and small. The Greeks were philosophically concerned with a rational explanation of everything and saw the repetition of the golden mean throughout the world and all levels of reality as a step towards this unifying theory. In short, it is the recognition that the same traits appear in entities of many different sizes, from one man to the entire human population.

Macrocosm/microcosm is a Greek compound of μακρο- “Macro-” and μικρο- “Micro-“, which are Greek respectively for “large” and “small”, and the word κόσμος kósmos which means “order” as well as “world” or “ordered world.”

Today, the concept of microcosm has been dominated by sociology to mean a small group of individuals whose behavior is typical of a larger social body encompassing it. A microcosm can be seen as a special kind of epitome. Conversely, a macrocosm is a social body made of smaller compounds.”

Chaos is Order Order is Chaos – Nithyananda

Li: The Patterns of Nature

“Li” is a Chinese word that refers to the underlying intelligence and order of nature as reflected in its organic forms. This film explores in poetic, non-narrative terms, the myriad patterns of nature that are spontaneously generated in the physical world and blurs the distinction between living and inanimate phenomena. -By Echomedia

“In comparing alchemy and fractal geometry, four subtopics reveal themselves, similar to the alchemical stages: chaos with a focus on infinity and nothingness; process with a focus on looping spirals, unpacking and repacking secrets; hidden order with a focus on self-similar form; and, finally, a glimpse of the big picture, the alchemical Philosopher’s Stone. Chaos is the space for beginnings. Chaos precedes and pervades creation, in both the creative process and creation as a noun. Disorganized drama that leads to emotional stress is a popular connotation of the word chaos, but the original mythological meaning was different: instead of a sense of overcrowded, swirling, patternless particles, mythological Chaos meant a dark emptiness, an abyss filled with infinite possibilities.”

Stuff on Self Organization , Autopoiesis,Quantum-Mechanics-and-origins_13927.html

“Jung aligned alchemy’s stages with the individuation process: a process of discovery, of digging for inner treasure, of Making, of Art. The snake or dragon symbolizes the process in its various guises. The uroboros snake biting its tail is only the beginning of its metamorphosis. It changes throughout the alchemical opus: it forms the vessel as the uroboros, then “burrows into the alchemical egg that begins the process,” vanishes, returns, then wraps around the egg that is now the philosopher’s stone (Robertson 63). As it disappears and returns, it oscillates, tantalizes, and tricks like Mercurius, exposing bits of secrets, then hides again. Jung visualized the individuation process as a spiral that “seems chaotic and interminable at first. . . . Owing to the diversity of the symbolical material it is difficult at first to perceive any kind of order at all. . . . But, as I say, the process of development proves on closer inspection to be cyclic or spiral” (277-78). In any process of discovery and creation, be it alchemy, psychological individuation, writing, or scientific experimentation, there is a sense of snaking one’s way through a labyrinth, looping back through spiral patterns (recursion) to find familiar images and ideas as the spiral crosses itself (scaling, self-similarity).

The reflections of the micro, macro, and mesocosms come into play through process. Complex systems do not form causal chains but instead “nature forms feedback loops in which information is constantly fed back into the system. . . . We each exist in an eternal moment that in some way contains both everything that came before and everything that will come, linked together in a complex feedback loop” (Robertson 65). When one is inside the process, the pattern mostly remains hidden because perspective is limited when one’s position is a point within the system.” – Laura Strudwick (Link at top)

“I’ve always felt that biology is a strategy, a chemical strategy, for amplifying quantum-mechanical indeterminacy into macrophysical systems called living organisms, and that living organisms somehow work their magic by opening a doorway to the quantum realm through which indeterminacy can come. And I imagine that all nature works like this”

“What is happening to our world is ingression of novelty toward what Whitehead called “concrescence,” a tightening gyre. Everything is flowing together. The “autopoetic lapis,” the alchemical stone at the end of time, coalesces when everything flows together. When the laws of physics are obviated, the universe disappears, and what is left is the tightly bound plenum, the monad, able to express itself for itself, rather than only able to cast a shadow into physis as its reflection. I come very close here to classical millenarian and apocalyptic thought in my view of the rate at which change is accelerating. From the way the gyre is tightening, I predict that the concrescence will occur soon—around 2012 AD. It will be the entry of our species into hyperspace, but it will appear to be the end of physical laws accompanied by the release of the mind into the imagination.”

“My notion of what’s going on in the informational phase space of contemporary existence is that… – we are under the influence of a kind of attractor. This is this thing I mentioned which seized hold of us as a higher animal and steered us toward language, ritual, religion, the calculus, so forth and so on. This attractor is literally sucking the world of three-dimensional space and time into itself. This is what history is. History is biological time turning into some other kind of time. History is speeding up as we approach the omega point.” – Terence Mckenna

“…the story of the universe is that information, which I call novelty, is struggling to free itself from habit, which I call entropy… and that this process… is accelerating… It seems as if… the whole cosmos wants to change into information… All points want to become connected… The path of complexity to its goals is through connecting things together… You can imagine that there is an ultimate end-state of that process—it’s the moment when every point in the universe is connected to every other point in the universe.” – Terence Mckenna

Terence McKenna: Syntax Of Psychedelic Time – 1/8

Lorenze Attractor - Views from Science

“Here is a Lorenz-type image showing the beauty in this strange attractor. Within this image, one can detect an underlying fractal pattern. The apparently elliptical path is that followed by a particle drawing the trajectory in 3D.”

“While, in theory, the location of any point on a chaotic attractor is determinable, in practice it usually is not, due to computational difficulties and other things. This deterministic chaos is seen in the Lorenz diagram show above.

(…)”More formally, a plotted attractor depicts an equilibrium state of a dynamic system. The system’s state converges to the attractor.” – Views from Science . com

“Not all chaotic attractors exhibit a fractal (or self-similar) nature as clearly as the one seen here. Those attractors that do can be “zoomed-in” to reveal hidden structure, just like Mandelbrot fractals.

There is an obvious similarity between the Lorenz attractor and a butterfly. But the real reason chaos is forever associated with the butterfly effect is a paper Edward Lorenz presented at a session of the AAAS in December 1972. The title of the paper: “Predictability- Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?”

Lorenz admitted that while one butterfly’s flapping wings could trigger off a tornado in the Lone Star state, another butterfly’s flapping wings could prevent it. (What this means is that when you hear about a tornado in Texas, it’s a pretty safe bet that there must be an odd number of butterflies in Brazil—at least for the moment!)”

Spiral wave, scroll wave, and stationary pattern generated from 2- and 3-dimenstional arrays of Chua Circuits.

A caricature of the double spiral fractal geometry of the double scroll attractor.

“The Chua Circuit has been built and used in many laboratories as a physical source of pseudo random signals, and in numerous experiments on synchronization studies, such as secure communication systems and simulations of brain dynamics. It has also been used extensively in many numerical simulations, and exploited in avant-garde music compositions (Bilotta et al, 2005), and in the evolution of natural languages (Bilotta and Pantano, 2006).

Arrays of Chua Circuits have been used to generate 2-dimensional spiral waves, 3-dimensional scroll waves, (Munuzuri et al, 1993) and stationary patterns, such as Turing and other exotic patterns, (Munuzuri and Chua, 1997), (Madan, 1993), as illustrated in Figures 11(a), (b), and (c), respectively. Such high-dimensional attractors have been exploited for applications in image processing, neural networks, dynamic associative memories (Itoh and Chua, 2004), complexity (Chua, 1998), emergence (Arena et al, 2005), etc.”

addquotes and more attractor models

Nicholas Rashevsky in Looking at History Through Mathematics

Terence McKenna — The World Soul Part 1/6

Nature is a communicating system of some sort.
One way of looking at nature is that it is entirely linguistic intent. That DNA is in fact a way of uttering protein syntactical structures into matter. The problem that we have is to transcend cultural languages, historically created languages with very limited applications. And instead fall into phase with the kind of communication systems that nature has placed all around us. – Mckenna
Terence Mckenna on Nature and Language
“There is much of language in nature; in fact, you could argue that all of nature is a linguistic enterprise.”
“…I saw what most of us only see on National Geographic specials; the real fact of the rain forest; the real fact of organic nature. And how nature is communication. Not only are the species that comprise the biota linked by pheromones and acoustical signals and color signals and other various methods by which communication is seeping around.

In fact, nature ultimately resolves itself into a self-reflecting, syntactical metasystem, right down to the DNA. DNA working as it does, with nucleotide sequences that code — that means arbitrarily assign association — code for certain amino acids. It means that organic objects are essentially utterances in three dimensional space and express of some kind of universally distributed linguistic intent. This is what it means when it says, “In the beginning was the word.” Nature is that word. This infinitely self-adumbrating, fractal, syntactical hallucination with an infinite number of facets for potential regarding and self-regarding.

And having said all of this, I might invoke here Godel’s Incompleteness Theorem, which as I’m sure many of you know was Kurt Godel’s brilliant contribution to theoretical mathematics where he showed that the possible set of true formal statements generated by any formal system exceeded the possible set of true formal statements which the rules of that system allowed. He showed this for simple arithmetic. And what this means, friends, is that what was called truth up until the beginning of the twentieth century, is absolutely impossible. That’s what Godel’s Incompleteness Theorem secures. It shows that there is no ultimate closure in an effort to describe a formal system.

And so in a way, my take on nature, and culture, and man, is that human language is a meta-linguistic system, generated out of the necessary formal incompleteness of nature. Nature is a self-describing genetic language and yet out of it arises something which is not formally predicted by its constraints and rules. There’s a symmetry break there, and a so-called emergent property comes into view. This emergent property is our unique ability to provisionally code sound to meaning so that we then can freely command and reconstruct the world. We imagine that we do this for our own purposes of communication. The analysis that I’m suggesting would seem to indicate that actually we do it because we are complicated enzyme systems that are moving linguistic charge around inside some kind of metasystem. A metasystem that is very important for the emergence of new order out of nature.

The fact that it is contrived, provisional, is very interesting. It doesn’t arise out of the gene structure. Rather it is agreed upon by individuals who are living at the time that the linguistic structure, whatever it is, emerges into consciousness. Since individuals are replaced, the language is much more in flux than the genome. The genetic component of an organism is a physical structure stabilized by atomic bonds — possibly stabilized by a phenomenon like room-temperature’s superconductivity. In that way nature works to conserve the genes. Molecular machinery has evolved to do that. But there is no mechanism in nature with the same kind of binding force that conserves meaning. Meaning is some kind of freely-commanded, open-ended, self-evolving system. The rules are that there are no rules.”
The significance of the spiral as a form of the Great Mother inevitably led to its identification with the thunder weapon, like all her other surrogates. I have already referred (Chapter II, p. 98) to the association of the spiral with thunder and lightning in Eastern Asia. But other factors played a significant part in determining this specialization. In Egypt the god Amen was identified with the ram; and this creature’s spirally curved horn became the symbol of the thunder-god throughout the Mediterranean area, 1 and then further afield in Europe, Africa, and Asia, where, for instance, we see Agni’s ram with the characteristic horn. This blending of the influence of the octopus- and the ram’s-horn-motifs made the spiral a conventional representation of thunder. This is displayed in its most definite form in China, Japan, Indonesia, and America, where we find the separate spiral used as a thunder-symbol, and the spiral appendage on the side of the head as a token of the god of thunder.

quotes on self-reference in language art media, macluhan

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One thought on “Fractals and Chaos in Psychology, Biology, History, and Sociology”

  1. dan winter says:

    golden ratio is shown to be the generalized
    solution to constructive wave interference
    therefore compression / phase conjugation.. and gravity

    shown in the geometry of hydrogen
    (new equation- planck length x golden ratio exponenets=hydrogen radii

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