2010: A Mandelbrot Odyssey

January 17, 2012

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While the title of this post is slightly beyond my understanding… Which is mainly because I had nothing to do with creating it… I find myself, nonetheless, publishing this rather off-kilter post, which is about nothing in particular, other than for the sake of sharing a magnificently orchestrated zoom into (and out of) the Mandelbrot Set. Truth be known, the title came from a YouTube video, which I Stumbled upon only the other day. And as it was so beautifully done… Not to mention that I so enjoy Johann Strauss’ classical masterpiece, “The Blue Danube”… And with its analogy to Kubrick and Clarke’s potent cinematic magnum opus, which I also adore so… I just had to post it here for your viewing pleasure (it’s best to view it in 720p HD on a full screen with the volume cranked UP for maximum effect).

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Comic… !?

July 5, 2010

What can I say… Too much XAOS and a funny five minutes gave rise to these two little oddities.

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AND/OR

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OR if you’d like to learn more about nonlinear dynamical systems, please read James Gleick’s “Chaos: The Making Of A New Science.” The first three people to E-mail me by clicking here will receive a free copy of the book! Just remember to enter the address you’d like the book to be delivered to…

Analogy As The Core Of Cognition

May 2, 2010

Language… Meaning… Understanding… It’s all about higher concepts ‘globing’ together to create the aspects of cognition, via modes of analogy, that we all use in our everyday lives to impart meaning to, and thus construct, the world that we ‘know’ around us.

Here in this talk, Hofstadter pertinently demonstrates his encyclopaedic knowledge about the development of human language and how the meaning behind words manifest, as concepts develop from their base level ideas, for example “ball,” into higher levels of complexity, as new words are developed to further complement and describe these basic parental precepts – perhaps stemming from new inventions which are driven by new discoveries, from both the empirical and non empirical fields of cognition – into aspects of “roll,” “wheel,” “car,” “moving,” “hovercraft,” “floating,” “flying,” etc… In this endless game of semantic development and lexical evolution, we begin to glimpse at the inherent emptiness behind the words we all so readily use in our lives, and, thus, see how they are really nothing more than eloquent “grunts” that contain mental images of meaning that allow us Human Ape men/women to understand one another and get our ‘points’ across to each other.

In many ways this is a good point of entry for us to see and to begin to understand how words and their meanings come about… From there we can then see that mental categories begin to shape the axiomatic world in which we live. Once we can understand this, we might have a better chance to really grasp the fundamental aspects that lie behind both the Buddhist idea of “Emptiness” and Kurt Gödel’s “Incompleteness Theorems.” *

Interestingly enough… As Hofstadter discusses “word blends” and “phrase blends,” I think we can begin to see a pertinent analogy between feedback loops of all sorts i.e. language here randomly mutates into simple amalgamations of “originality,” much like “video feedback loops” create modified patterns of slightly iterated imprints of what was only just on the visual screen.

Regarding language… This evolution and development of lexical complexity stems from seemingly random conjoining of phonemes, subtle phonetic variances and little – or even crass – syntactical errors, which mostly stem from when two ideas intermingle within the mind of the speaker and are thus ejected into the conversation stream without too much consideration. This process allows a rich source of new etymological meanings to flourish and develop within – initially – small social circles of friends/colleges… Or if particularly ‘catchy,’ they might then spread across wider groups as successful “memes” via infectious minds through any form/medium of social communicability.

In many ways it is this type of evolutionary self-similarity that keeps language alive and “thinking” on its toes – always forcing it to move forwards into new realms of functionality to suite the current semantic landscapes of our ever-changing, ever-evolving social networks and scientific/technological habits. Within these linguistic and memetic structures we may even find multidimensional configurations where complex patterns modulate old ideas into modern modes of ever more complex types of understanding… Old frameworks of thinking are slowly battered into new designs, and then the two co-exist in a sort of symbiosis with one another. And ever iterating forms of lingual cunningness flow from out lips… Just as the Mandelbrot set increases in complexity the further we zoom into its boundary, so language seems to increase in complexity the further We – as human beings – progress in developing ideas, our knowledge database, our inventions and with new observations… In many way these two particular types of pattern i.e. aural language and visual fractals, run parallel to one another in similar veins of iterative novelty. One is mimicking the other… Except it is not an “exact” replica… It is merely uses the same iterative mathematical ideas to regenerate and reorganise itself with.

Here Hofstadter describes a process where two ideas are torn apart and allowed to intermingle in the speaker’s own fractal mind, thus recombining/assimilating into an endless array of sometimes intentional, but probably mostly unintentional, hip analogies. Analogies are all about self-similarity. Hence, in my mind I become some strange feedback loop, reiterating what I have already heard and seen in my life thus far, remixing, recombining and experimenting with neoteric verbal adage to describe new modicums and meanderings concerning my experiential existence… And thus, through my will alone, my “I” become a Möbius strip of etymological and memetic reform, playing with all of you – my iterated equals – in evolution and natural selection’s ancient game.

* This exposé about the illusion of self and the delusion of most types of knowledge – in the light of the Buddhist precept of “Emptiness” and Kurt Gödel’s “Incompleteness Theorems” – is coming in a future blog…

Fractals in Science, Engineering and Finance (Roughness and Beauty)

January 27, 2010

Just the other day a friend dropped over this great little lecture by Mandelbrot which discusses some of the fractal aspects of the world around us… Thanks Martin!

Fractals in Science, Engineering and Finance (Roughness and Beauty)

Roughness is ubiquitous and a major sensory input of Man. The first step to measure and simulate it was provided by fractal geometry. Illustrative examples will be drawn from the sciences, engineering (the internet) and (more extensively) the variation of financial prices. The beauty of fractals, an unanticipated “premium,” helps in teaching and bridges some chasms between different aspects of knowing and feeling.

Self Similarity ~ Fractals, Fractals Everywhere…

September 20, 2009

In this essay I propose nothing more than an idea for nature’s own design… And, by doing so, perhaps one may also find a suggestion for mankind’s ultimate drive within this design. While I’m not saying that there is an absolute answer to “Life, The Universe and Everything,” I am offering up an obvious pattern that seems to recur with common-place regularity throughout every “Universal” system that I have ever studied and/or viewed so far… It is a pattern that nature has followed for eons already; whether through the natural laws of physics, via the designs of evolution, or in the memories we all use each day to build our world around us and develop the meanings that we place on it… This idea can even be found at the heart of all poetry, exuding beauty, simplicity and finesse simply by adding metaphorical adage to the rough-hewn strata of every day activities. Simply put… It is the idea behind all patterns of universal discourse… A discource which has been accursed as being “the thumb print of god.” Self-similarity is what we do. It is what we build our minds with, where our notions come from, and even allows art to fulfill its very purpose of attaining mystical appeal. It allows us to copy others and express ideas of our own in a way that others can relate to. It even allows us to successfully copy and use age old techniques for survival, giving the “user” (or, memetically speaking, should that be the “used”?) that cutting edge over other life forms… (???) And no doubt it is a key process that will shine light into our very Being.

So… Firstly, let’s look at the definition of “self-similarity.”

self-similarity (sělf’sĭm’ə-lār’ĭ-tē)

The property of having a substructure analagous or identical to an overall structure. For example, a part of a line segment is itself a line segment, and thus a line segment exhibits self-similarity. By contrast, no part of a circle is a circle, and thus a circle does not exhibit self-similarity. Fractals such the Sierpinski triangle are self-similar to an arbitrary level of magnification; many natural phenomena, such as clouds and plants, are self-similar to some degree. See more at fractal.

Next… Let’s look at the definition of an attractor.

attractor (əˈtræktər)

–noun

1. a person or thing that attracts.

2. Physics. a state or behavior toward which a dynamic system tends to evolve, represented as a point or orbit in the system’s phase space.

The basic idea behind an attractor is that a dynamic system will tend toward certain states as time goes on. The simplest form of an attractor is the point attractor. Consider a normal pendulum, it doesn’t matter where you release it from, it will always come to rest in the same position, perpindicular to the ground. This state is the attractor for the system.

Next… Let’s look at the definition of a strange attractor.

Strange Attractor

–noun

The truth of the matter is this, there is not yet a formally accepted definition of a strange attractor. Strange attractors tend to arise in dissipative dynamical systems, such as the first pendulum example given above. Dissipative simply means that the system loses energy as time goes on. The Collins Reference Dictionary of mathematics states the definition of a strange attractor as …such that its Hausdorf Dimension is non-integral, or else dependant on initial conditions… Obviously this is not a complete definition, but it does give us a sort of intuitive way of thinking about a strange attractor.

And lastly let’s understand something about measuring dimensions

Hausdorf Dimension

The Hausdorf Dimension is a means of measuring the dimension of a mathematical object. For instance, the dimension of a point is 0, a line resides in 1 dimensional space, a plane 2, and of course our friend 3-space in which we live. So by definition a strange attractor is an object which is neither a point, a line, or a plane!. For example, the dimension of the Rossler Attractor is estimated at between 2.01 and 2.02. To understand what this means, think about this: the equations which describe the Rossler attractor will describe a curve or line in 3 dimensional space for periodic solutions. But when you have a chaotic solution, which is never periodic, that is, it never visits a point which it has previously visited, then the path of the Rossler attractor as a whole (time to infinity) becomes more than a collection of lines, and just slightly more than a collection of planes. The key point is that for a non-periodic solution, the attractor never retraces a previously traveled path. This non-integer dimension is also what qualifies strange attractors as a fractal. A fractal is defined to be any object with a non-integer Hausdorf Dimension.

And with those four definitions out of the way… I would now like to postulate what my aim within this essay will be, and how I hope to achieve it.

Since the discovery of the Mandelbrot set, many images of its “serpentine” flow have captivated the imaginations of those who have laid eyes on it. The literally “never ending” sequence of divine majesty has held court particularly well with the mystical elements of modern day society i.e. the psychedelic voyagers, healers, pyschics, etc… Not to mention it has even brought an element of the spiritual into math. Something is stired deep inside when people view this marvel of vanilla complexity. But what exactly is “it” that is arroused?

Well… For years Religious and Mystical doctrines have all adorned their sacred texts/ideals with marvelous tapestries of cochlear geometric art and calligraphy. This instinctual linking of the divine with these geometric forms seems to adequately explain how we feel about our lives in the complexity of creation. The images seem to proclaim a deep and beatific connection to the world around us as it unfolds according to God’s will.

Below are a few examples of these decoration:

Fractals vs Religion. The idea of the infinite/God. Bottom mural patterns photographed in Alhambra, Spain.

Imagery from the Book Of Kells vs. The M Set

Ishmael vs. The M Set... More Self-Similarity

Self similarity occurring across many levels. Buddha and the Buddha Brot. Possibly the Buddhist theory of Interdependent Origination should be more closely examined?

But why did We adorn these icons of our saviours with such tortuous and elaborate imagery? Surely they were just as human as we were, albeit slightly more in tune with the divine aspect of reality… And if they knew the divine better than us, then why did they not speak of such intricate complexity? Well… In many ways they did. Buddhists speak of the theory of Interdependent Origination… And even Jesus and Mohammed were known to have talked of the complexity of life and its boundless beauty.

So… Are we all simply seeing the same thing, just from slightly different perspectives… ? Much in the same way Bertrand Russell describes the way various people might perceive a table from different angles in “The Problems Of Philosophy?” I say, “YES, WE ARE!!!”

In Russia, people have used an idea that embodies the essence of fractals and self-similarity to describe the very notion of family life. Babushka dolls relay the obvious similarities that occur between offspring and their parents. A type of affine process that relates to self-similarity…

A matryoshka doll, also known as a Russian nested doll or a babushka doll, is a set of dolls of decreasing sizes placed one inside the other. Matryoshka (Матрёшка) is derived from the Russian female first name Matryona, and babushka is the Russian word for grandmother.

To further the Babushka doll idea… As you may have read just recently in the blog entitled “Human Mutation Rate Revealed” each subsequent generation accumulates new mutations in their genome… Mutations that allow subtle differences to arise within the parameters of the predefined DNA blue-prints that come from both mother and father… DNA which must unravel itself and intermingled to mix into a new generation of human being. Certain traits will be more pronounced than others, but all in all the offspring will resemble both their maternal and paternal side in some way. This game of “tag,” has been played for eons… Ever since life took a hold here on Earth… And here, as seen in the potent image of the “babushka doll,” there is an excellent metaphor that elludes to this deep understanding of the “divine” process of self-similarity and the complexity that comes from it. It’s almost as thought we are aware that all natural processes carry within themselves a part of the whole from whence they originated: denoting a recognizable relationship of “similar object-within-similar object…” It is this abstraction that appears in the design of many other natural AND man-made objects that we see all around us daily…

Even in the “cradle of civilization” i.e. Africa, this notion of self-similarity is well understood. As Ron Eglash has shown here, and in his book entitled “African Fractals,” fractal patterns abound and play a very important roll in the way mankind understood (and still understands) his place within the cycles of nature… Africa is not the only place that this ancient process is celebrated.

Most Indian and Southeast Asian temples and monuments exhibit a fractal structure.

As seen in the above two modest pictures, most Indian and Southeast Asian temples and monuments exhibit a fractal structure: a tower surrounded by smaller towers, surrounded by still smaller towers… The particular examples shown above are Hindu temples. The ideal form gracefully artificed suggests the infinite rising levels of existence and consciousness, expanding sizes rising toward transcendence above, and at the same time housing the sacred deep within. This universe is like a ripe fruit appearing from the activity of the cit [consciousness]. There is a branch of a tree bearing innumerable such fruit. There is a tree having thousands of such branches. There is a forest with thousands of such trees. There is a mountainous territory having thousands of such forests. There is a territory containing thousands of such territories. There is a solar system containing thousands of such territories. There is a universe containing thousands of such solar systems. And there are many such universes contained within what is like an atom within an atom. This is what is known as cit, or the subtle sun, which illumines everything in the world. All the things of the world take their rise in it. Amidst all this incessant activity, the cit is ever in undisturbed repose.

Diagram clearly demonstrating this fractal ideal.

As I might seem like a rambling blogger… I’d like to mention that I am not the only person implying this idea about Indian temple architecture and the cit… Please click here for more details.

No doubt, there is something within us that somehow links us to an idea of the “divine…” Of the infinite… We see the unfolding daily complexity of life… Each action causing its own reaction in a chain of events that unravel from the big bang itself. It is unbounded in nature… And yet bound to what has already been. This is where the infinite truly resides! Perhaps the reason why We, as human beings, express this idea in Self-Similar ways i.e. through religious decree and complex geometrical art, is because deep inside each one of us we use the same structures that nature uses to create the universe around us… Perhaps We are all truly aware of this deep connection that We all share with each other and the cosmos… Even if only subconsciously… And so perhaps We can’t help but intuitively feel this “divine” cosmological phenomenon unraveling around us, and feel the need to express it. Whether We choose to acknowledge it or not, it still courses through our very being each day. We can’t help it. This is what we are… !

HOWEVER… Standing alone, these propositions that I make are nothing more than bold insinuations… Insinuations that might arise from a mind that has become swamped in its own aspirations… Aspirations for demonstrating this “truth” in a hope for peace and unity between all mankind, thereby clouding my reason with foolish dreams and hopes… Prerhaps this is my fate… To suffer from a misguided scoptoma. No doubt…. Without any scientific backing, I wouldn’t expect anyone to take these insinuations any further than the face value with which I have presented them thus far.

And yet, I myself, having scrubbed these realizations from my mind’s eye time and again in order to try to see afresh the patterns of universal flow around me, have always been led back to this idea of self-similarity. And I keep wondering why? Why is it that when I forget about “self-similarity” and look for some other process, some other explanation, I am always called back to see the chain of events that brought me here… Why do I see my own originality fade into a sea of past, present and future efforts? Are we doomed to copy ourselves over and over again for an eternity, providing little modifications along the way, so that the present becomes almost stagnant, while the whole becomes flowing? Whatever the answer is… It’s almost as if some innate force within me is pushing me to describe a deep Knowing, like an almost forgotten connection, that we all share with the universe around us. A connection that is based on the idea of coming together, which in turn is founded on the very “laws of attraction” and beauty that guide us all through the experience of our daily lives.

I am aware that for the layman, this is still nothing more than one BIG postulation… All of it hanging only on mere conjecture. So… In order to rectify this loose canon, I proclaim… Within the confines of this essay, I will provide some whole facts that might suggest why we all, somewhere deep, deep inside, have always felt a strong urge to connect with this process, whether throught Religious ideals or through mystical rites, in order to express this obvious self-similar and unknowable complexity as it unfolds into new modes of understanding. And in doing so, perhaps I might also suggest (with regards to understanding the flow within our minds i.e. something that is chaotic) a reason why my experiences have always demanded my return to similar flows of thought – and therefore perhaps even suggest why Religious doctrines, ones that are seemingly unrelated to one another, are infact so similar

Afterwards I will provide some visual pictures of the patterns at work in the chaos of the universe surrounding you, and allow you deduce your own conclusions (something that I always ask all my readers to do, no matter how trivial and obvious a statement I make). Good old empirical thought has been clearly demonstrating of late that the universe is one large dynamical system… And these systems are built not upon perfect ideals, like those used by Euclid to describe his perfectly flat plane (a rather romantic notion of man’s own making)… But rather they are fractured deeply inside, and so sometimes move with a wild abandon that no mere mortal could ever grasp fully or hope to predict… And yet… By providing you with a series of analogies to suggest these complex patterns inherent in many of these natural everyday dynamical systems I hope that the idea of self-similarity occuring across many areas that are, at first glance, seemingly unrelated to one another, will hopefully posit of its own accord. Then one may See that self-similarity resides not only in nature around us, but within the very machinery that allows us to perceive the universe around us i.e. the human body and brain, in a type of all embracing monism.

This is no doubt a very delicate matter, and will demand your utmost attention, as well as some diligent homework on your part too… For even my “facts” are somewhat vague in my own mind, as I allow intuition to guide my reason and dictate the obvious. Thus I ask the following of each reader…

Firstly, to remain open minded while reading this essay. I am not asking you to believe anything writen (or typed) down here straight off… All I am doing, is establishing an idea, along with evidence for it, so you can then do your own ground work to see if it is a valid “ideal” or not… As Socrates once wrote, “…it is the mark of an educated mind to be able to entertain a thought without accepting it…” Ultimately it is up to you whether or not you decide to agree or disagree with this. And secondly, it IS a pre-requisite that you either have some basic understanding about science (biology, chemistry and physics)… As well as being up to date on current science facts and theories… AND/OR read some of the previous blogs that I have posted here… As without these, it like introducing a Mexican national to a Japanese national in Africa, and expecting them to understand eachother straight off. You dig?

Right… Here I would like to introduce several words and terminologies, most of with which you’re no doubt familiar with, to christen the launch of this epic…

A)

empiricism (em-pir-uh-siz-uh m)

- noun

1. empirical method or practice.

2. Philosophy – the doctrine that all knowledge is derived from sense experience. Compare to rationalism.

3. undue reliance upon experience, as in medicine; quackery.

4. an empirical conclusion.

B)

rationalism – (rash-uh-nl-iz-uhm)

– noun

1. the principle or habit of accepting reason as the supreme authority in matters of opinion, belief, or conduct.

2. Philosophy.
a. the doctrine that reason alone is a source of knowledge and is independent of experience.
b. (in the philosophies of Descartes, Spinoza, etc.) the doctrine that all knowledge is expressible in self-evident propositions or their consequences.

3. Theology. the doctrine that human reason, unaided by divine revelation, is an adequate or the sole guide to all attainable religious truth.

4. Architecture. (often initial capital letter)
a. a design movement principally of the mid-19th century that emphasized the development of modern ornament integrated with structure and the decorative use of materials and textures rather than as added adornment.
b. the doctrines and practices of this movement. Compare functionalism.

C)

Egocentric bias

This occurs when people claim more responsibility for themselves for the results of a joint action than an outside observer would credit them.

Besides simply claiming credit for positive outcomes, which might simply be self-serving bias, people exhibiting egocentric bias also cite themselves as overly responsible for negative outcomes of group behavior as well (however this last attribute would seem to be lacking in megalomania).

This may be because our own actions are more “available” to us than the actions of others. See “availability heuristic.”

Michael Ross and Fiore Sicoly first identified this cognitive bias.

The reason why I mention these terms is to demonstrate that I am aware of them. They will no doubt also come up latter on in this essay too. But… Because I am aware of them, I am also aware that what I am about to postulate may be nothing more than a self biased view centered around my own imposition, which is based more on rationalism than empiricism. But non the less, as I’ve had quite a few E-mails asking for further reading on fractals and where they occur, I am writing this lengthy article to demonstrate how close to home they actually are i.e. they are everywhere!!!

So, without further ado… I will begin by demonstrating that our mind uses fractals patterns with which to think.

In Earl R. Mac Cormac and Maksim Stamenov’s book, entitled “Fractals of brain, fractals of mind: in search of a symmetry bond,” they mention that…

We shall present a case for the use of non-linear dynamical systems like fractals that can give an explanatory account which describes both the behavior that we have traditionally called involuntary (reflex) and that which we have traditionally called voluntary (will).

Considerable evidence has been presented showing that both the firing of individual neurons and the activation of patterns of neurons are nonlinear dynamical systems. Walter Freeman’s investigations of the olfactory bulb of the rabbit concluded not only that olfaction was nonlinear but also that this process could serve as a model for cognition (Freeman 1990). Skarda and Freeman state the following:

The observations that brains employ not only self-organization but chaotic dynamics to produce behavior places yet another nail into the coffin of reductionism. Chaotic phenomena preclude long-term predictions. It may seem paradoxical that a deterministic phenomenon is inherently unpredictable, but in systems that exhibit chaotic behavior, small uncertainties are amplified over time by the nonlinear interaction of a few elements. The upshot is that behavior that is predictable in the short run becomes intrinsically unpredictable in the long term. As a result, physiologists cannot make strict causal inferences from the level of individual neurons to that of neural mass actions, nor from the level of receptor activity to internal dynamics. The causal connection between past and future is cut. (Skarda & Freeman 1990:282)

Gregor Schoner and J. A. S. Kelso agree with these conclusions but cast their nonlinear net to include all neural behavior claiming “that it is possible to understand behavioral pattern generation on several levels of description (kinematic, electromyographic, neuronal) by means of the concepts and tools stochastic nonlinear dynamics.” (Schoner & Kelso 1989:311). This comprehensive claim is correct and we will attempt in this paper to demonstrate that fractals, a subset of nonlinear dynamical systems, can present a scientific explanation of neuronal behavior. Further, positron emission tomographic (PET) studies of cognitive behavior hopefully will confirm the fractal nature of specific neuronal processes…”

“Okay… And so what?” I hear some of you saying… Well… Perhaps to understand how chaos can be fractal, it might be beneficial to understand a bit about nonlinear dynamical systems. Then we can propose why they are fractal, without asking you to simply take my word that they are. So what is a nonlinear dynamical system? They’re a bit like linear dynamical system, in that they have dynamics… But apart from that, the similarity ends there. Here’s the compare and contrast…

Linear dynamical systems (LDS) can be solved exactly… In a LDS, the variation of a state vector (an N-dimensional vector denoted ) equals a constant matrix (denoted ) multiplied by . This variation can take two forms: either as a flow, in which  varies continuously with time.

or as a mapping, in which  varies in discrete steps

These equations are linear in the following sense: if  and  are two valid solutions, then so is any linear combination of the two solutions, e.g.  where α and β are any two scalars. The matrix  need not be symmetric.

Basically, without the mathematical talk, this simply means that a variable changes in some kind of linear manner with relation to its output. This might be a complex linear relationship, so do be aware that, when plotted, it might not look linear. However, when a system qualifies as a linear system, it is possible to use the responses to a small set of inputs to predict the response to any possible input. This can save the scientist enormous amounts of work, and makes it possible to characterize the system completely i.e. it is predictable.

In contrast to LDSs, most nonlinear ones cannot be solved exactly. Occasionally, a nonlinear system can be solved exactly by a change of variables to a linear system. Moreover, the solutions of (almost) any nonlinear system can be well-approximated by an equivalent linear system near its fixed points. Hence, you can see why understanding LDSs and their solutions is a crucial first step to understanding the more complex nonlinear dynamical systems.

So… What is a nonlinear dynamical system (NDS)?

In mathematics, a NDS is a system which is not linear. That is, it is a system which does not satisfy the “superposition principle” i.e. that, for any linear system, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually… To put it another way, a NDS is a system whose output is not proportional to its input. Less technically, a NDS is any problem where the variable(s) to be solved for cannot be written as a linear combination of independent components.

The equation X = rx, which is a linear equation, has a solution. However X = rx(1-x), shown above, demonstrates that as r is increased, chaotic behavior can occur. Therefore it is a NDS and has no definitive solution.

The above logistics map shows how chaos can come about in a NDS. The reason why I chose to use a logistics map to demonstrate a NDS is because it is a relatively simple way of demonstrating how a seemingly obvious system can generate totally random outputs i.e. become chaotic. A rough description of chaos is that chaotic systems exhibit a great sensitivity to initial conditions… And the human mind is no exception.

Why are NDSs of interest to scientists? Nonlinear problems are of interest to physicists and mathematicians because most real-world physical systems are inherently nonlinear in nature. Even in neuroscience, nonlinear dynamics has been shown to play a very big hand in brain dynamcis, understanding spatiotemporal inputs to the brain, etc…

However… Nonlinear equations are unsolvable because they give rise to a phenomena called chaos. The weather is famously nonlinear i.e. simple changes in one part of the system produce complex effects throughout, something that we have seen in a previous post, entitled “A Strange Attraction – A Brief Study Of Strange Attractors And Their Implications… In Tribute To Edward Lorenz.”

But it should be noted that just because a system is nonlinear i.e. it is not predicatable, does not mean that it will be in a state of chaos all the time… Even nonlinear systems have stable patterns that can occur within their temporal unfolding – see the diagram below.

Chaos and stability in a nonlinear dynamical system.

As we have already seen earlier in “The Ultimate Hack: Reverse Engineering The Human Brain,” Henry Markram is developing a three dimensional model of the human brain, specifically of the circuitry in the neocortex. Researchers at IBM have teamed up with Markram and his team at L’Ecole Polytechnique Fédérale de Lausanne (EPFL) to build the model using IBM’s Blue Gene supercomputer. The goal is to gradually expand the model until it encompasses the entire brain. How can they do this?

Well… In order to understand how one can model a computer simulation on something so arbitrary like the human brain, one really needs to understand what the brain is, what it’s made up of, and how it all interlinks to form a complete unit. So firstly, let’s look at the neuron, the basic building block of the brain’s structure.

A neuron (also known as a neurone or nerve cell) is an excitable cell in the nervous system that processes and transmits information by electrochemical signaling.

The real work of the brain goes on in these individual cells. An adult brain contains about 100 billion neurons with branches that connect at more than 100 trillion points. I know… A trillion is a big number… And it might not mean a lot to one who does not use it daily. So let’s put it into context. A trillion seconds is 31,688 years. Yeap… That’s 31,688 years… Pretty big, eh? So imagine a hundred trillion seconds… That’s three million, one hunred and sixty eight thousand, eight hundred years… Or numerically, 3,168,800 years! So… Is it any wonder scientists call this dense, branching network a “neuron forest?” Signals traveling through the neuron forest form the basis of memories, thoughts, and feelings… But we’re jumping the gun somewhat.

Neurons exist in a number of different shapes and sizes and can be classified by their morphology and function. The anatomist Camillo Golgi grouped neurons into two types; type I with long axons used to move signals over long distances and type II without axons. Type I cells can be further divided by where the cell body or soma is located. The basic morphology of type I neurons, represented by spinal motor neurons, consists of a cell body called the soma and a long thin axon which is covered by the myelin sheath. Around the cell body is a branching dendritic tree (made up of dentrites, as shown in the diagram above) that receives signals from other neurons. The end of the axon has branching terminals (axon terminal) that release neurotransmitters into a gap called the synaptic cleft between the terminals and the dendrites of the next neuron. The anatomy and the properties of the surface membrane determine the behavior of a neuron. The surface membrane is not uniform over the entire length of a neuron, but is modified in specific areas: some regions secrete transmitter substances while other areas respond to the transmitter. Other areas of the neuron membrane have passive electrical properties that affect capacitance and resistance. Within the neuron membrane there are gated ion channels that vary in type, including fast response sodium channels that are voltage-gated and are used to send rapid signals.

Neurons communicate by chemical and electrical synapses in a process known as synaptic transmission. The fundamental process that triggers synaptic transmission is the action potential, a propagating electrical signal that is generated by exploiting the electrically excitable membrane of the neuron. This is also known as a wave of depolarization.

Also, it should be noted that neurons communicate with one another via synapses, where the axon terminal or en passant boutons (terminals located along the length of the axon) of one cell impinges upon another neuron’s dendrite, soma or, less commonly, axon. Neurons such as Purkinje cells in the cerebellum can have over 1000 dendritic branches, making connections with tens of thousands of other cells; other neurons, such as the magnocellular neurons of the supraoptic nucleus, have only one or two dendrites, each of which receives thousands of synapses… Thus we can begin to see the complexity that is already beginning to manifest from just the variance in cell types and the way they are interconnected…

A computer simulated diagram of the connections between neurons in a rats brain. And bearing in mind a human brain is 40 times larger, the complexity of these interconnections would be vast.

So while the traditional “computational metaphor” really is an oversimplified model of the way in which the brain works i.e. where the brain is viewed in a succinct analogy of a computational device similar to that of a computer – the mind serves as a software program controlling the operations of the brain – I feel it very much leaves out the beauty and essence from which its originality stems. You see, human thinking and neuronal processes combine to produce a computational process. And it is this very computational process that is interactive i.e. the mind affects neuronal processes as demonstrated by phenomena like biofeedback and the way memories are stored… AND the brain also affects the mind, as shown by the effects of hallucinogenic drugs upon thinking, leaving the computer analogy way behind.

But due to much diligent research over the years, scientists have no doubt begun to grasp the very idea and essence of what the human brain actually is. You see, much work has been done in understanding the various types of neurons present within the brain, where exactly they are located, the difference between their morphologies i.e. variences in their respective anatomy/structure is, as well as the properties of their surface membranes, all of which determine the behavior of the specific neuron and speed at which they fire and recharge at, their near exact electro-potentials, etc… AND THUS how they interact with one another. This highly sensitive organic molecular “computer” is the driving process for the human body… In fact it is the driving process for most complex multicellular organisms here on Earth. In many ways, if we study the mechanisms within these brain structures, we will begin to peer into how we function… And thus see what we truly are. From all this data that is being amassed on how our brains and bodies work, some people down at IBM are building a near on exact computer simulation out of all this statistical accumulation of scientific data on brains, where by the complex dynamical system of the various neurons is mapped with great precision into mathematical analogies, which are then structure together using highly accurate brain maps, so that a model of the human brain can be brought to life as a virtual simulation. Viewing the brain as a highly complex interacting dynamical system should help researchers understand processes like thought, perception and memory a lot better than if they were to simply chop up the parts and observe them interacting via fMRI. And ultimately these simulations could also shed light on how and why particular parts of the brain malfunction, leading to a better understanding of autism, depression and schizophrenia.

Markram’s team with the IBM researcher are focusing firstly on the neocortex, which is unique to mammals, and accounts for around 85% of the mass of the human brain. It is thought to be responsible for our higher cognitive functions, such as language, learning, memory and complex thought.

According to Markram, the EPFL professor heading up the project, the collaboration is one of the most ambitious initiatives undertaken in neuroscience. “Modelling the brain at the cellular level is a massive undertaking because of the hundreds of thousands of parameters that need to be taken into account.”

Markram expects the simulation to accelerate the pace of brain research, whereby the computer simulation will help show holes with our current understanding of the brain, and thus will be able show researchers where to look and what studies to perform next. This guiding “presearch” could therefore direct current neural research done “in vitro” with a lot of the pre-testing and planning done “in silico”, saving a lot of time and money in physical research. “With certain simulations we anticipate that a full day’s worth of wet lab research could be done in a matter of seconds on Blue Gene,” Markam added.

The model of the brain will be based on 10 years of the wet lab experiments and research that Markram has been a part of at EPFL. The IBM researchers will use this data to build the model of the electrochemical interactions of the human brain on four racks of Blue Gene. The machine will have a very respectable peak processing capacity of 22.8 Teraflops, and will take up about the same space a four fridges.

The model of the brain is just one of the projects allocated time on the machine. Other teams will use Blue Gene to investigate how plasmas might be used in energy production, and how the folding of proteins plays a role in diseases like Creutzfeldt-Jakob Disease, the human variant of BSE, or mad-cow disease.

by Lucy Sherriff

It is from this unpredicatable interaction between neurons firing across the brain that consciousness and perception comes about. The chaos inherent in the system allows a great versatility to ensure survival in the light of any threatening possibility. Without this chaos, it would be totally deterministic i.e. completely predictable… And this predictability could only serve to ensure stagnation. I’m sure you could imagine how this would effect one’s mental development! To have the same response to all situations clicking over and over again in our brains would turn us into little more than cyclical morons who continue walking into closed doors… Never thinking to find the handle.

For one to catch up with the essence of Chaos, I would like to take a moment to present a pertinent book, entitled “Chaos: Making A New Science,” written by James Gleick, that explains how chaos works in the universe at large… And, in many ways, what a good thing it is. And I highly recommend it as a great starting point to delve further into understanding nonlinear dynamcial systems and Chaos Theory… As a taster, the following quote comes from the back of the book:

Over the last decade, physicists, biologists, astronomers and economists have created a new way of understanding the growth of complexity in nature. This new science, called chaos, offers a way of seeing order and pattern where formerly only the random, the erratic, the unpredictable–in short, the chaotic–had been observed. In the words of Douglas Hofstadter, “It turns out that an eerie type of chaos can lurk just behind a façade of order–and yet, deep inside the chaos lurks an even eerier type of order. “

The science of chaos cuts across traditional scientific disciplines, tying together unrelated kinds of wildness and irregularity: from the turbulence of weather to the complicated rhythms of the human heart, from the design of snowflakes to the whorls of windswept desert sands. Highly mathematical in its origin, chaos nonetheless is a science of the everyday world, addressing questions that every child has wondered about: how clouds form, how smoke rises, how water eddies in a stream.

In Chaos, James Gleick tells the remarkable story of an idea–an idea that at once frightened and fascinated the scientists who began to explore it. Gleick describes the stunning and unexpected insights of these scientists: Edward Lorenz’s discovery of the Butterfly Effect that underlies weather’s unpredictability and constancy; Mitchell Feigenbaum’s calculation, prompted by his meditations on nature and art, of a universal constant; Benoit Mandelbrot’s concept of fractals, which has created a new geometry of nature.

Chaos is a history of scientific discovery. It chronicles, in the participants’ own words, their conflicts and frustrations, their emotions and moments of revelation. It is a record of a revolution, the birth of a new science. After reading Chaos, you will never look at the world in quite the same way again.

While neurons are remarkably slow relative to components in mordern computers, they still are able to assimilate a complex array of data into the experience we know as “Life”, which we are all (hopefully) having right now. Each cell in the body produces its own highly specific biochemical response/signal, which is produced from an array of factors that affect the cell’s internal and external workings (such as other external chemical messengers triggering internal cellular responses, OR internal cellular concerntrations of molecules rising or falling to initiate an external cellular change), as discussed in the blog “What Is This “Experience” We Call Being Alive… And How Does It Come About?.” This feeds-back into the brain, which in turn feeds-back into the body, and so a cyclical system of experience is born.

The computer program that Markram will be developing with IBM, will be a mathematical representation of the human brain. And, as is the case with NDS (nonlinear dynamical systems), under certain values the computer simulated NDS algorithm will produce chaos, while under other conditions it will produce a stable pattern.

In seeking this mathematical description of neuronal process, we are construction a rational reconstruction rather than a simulation. In looking at brain functions, one can investigate various hierachial levels of explanation beginning with the operation of individual neurons, how their excitatory and inhibitory actions take place through ionic exchanges across membranes that produce changes in electrical potential, as we’ve already discussed…

In Skarda and Freeman’s 1990 paper, entitled “Chaos And The New Science Of
The Brain
,” they postulate the following:

The idea that perception can be explained in terms of feed-forward processing, that it is caused by the stimulus or can be explained as the sum of responses to stumuli, is no longer acceptable. Our model tells us that perceptual processing is not a passive process of reaction, like a reflex, in which whatever hits the receptors is registered inside the brain. Perception does not begin with causal impact on receptors; it begins with the organism with internally generated (self-organized) neural activity that, by re-afference lays the ground for processing of future receptor input. In the absence of such activity, receptor stimulation does not lead to any observable changes in the cortex. Percpetion is a self-organized dynamic process of interchange inaugurated by the brain in which the brain fails to respond to irrelevant input, opens itself to the input it accepts, reorganizes itself, and then reaches out to change its input.

In many ways this is a very logical observation… For example, we need to be able to decern what type of particular object, rushing towards us at speed, might cause our body irreparable damage. For example, a leaf blown towards us in a gust of wind will not do our body any harm if it were to “bash” into us. We recognize (or rather choose NOT to recognize) the leaf as it is blow towards us. However, we have learnt that a spear, which has been hurled through the air at us, will cause our body great harm if the two were to meet… So our brain brings into focus the need to distinguish between the kind of action to be taken in order to either avoid or ignor one of these types of object flying towards us. Either way, the leaf and the spear are still both objects in the physical world… But our brains are able to discern between the two almost instantly, and therefore efficate a reaction that is appropriate i.e. ignor or avoid, and thereby ensure self-preservation.

Skarda and Freeman go on to say:

We suggest that the self-organzing process that replaces environmental input with an internally generated, chaotic activity is one that gives “biological meaning” to the stimulus.

Mac Cormac and Stamenov, having quoted the above in their book “Fractals of brain, fractals of mind: in search of a symmetry bond,” then mention…

Our search, therefore, must be for nonlinear dynamical systems that actively organize themselves into neuronal patterns that result in recognizable cognitive behavior. Rather than beginning with a simple simulation of the firing of a neuron (a sigmoid function) and then building up from that simulation to hierarchial layers as in artificial neural networks, we will attempt to employ nonlinear systems, especially fractals, as rational reconstructions of neural processes.

Mac Cormac and Stamenov next look at the analogies between neuronal processes and glycolysis in order to demonstrate a direct physical relationship between mind and body.

Our search for nonlinear systems to rationally reconstruct neuronal processes begins with the analogy between nonlinear systems that explain the chemical process of glycolysis and the activation of neurons. That analogy arises from the consumption of glucose by neurons when they are activated. But why would one seek an analogy from an underlying physical process to explain cognitive behavior? This seems like trying to explain the meaning of telephone messages by giving an explanation of the physics of how sounds are transformed into electronic signals, transmitted by photons in fiber optics and then reconverted into sounds at the receiving end. Why then can one have any confidence that an underlying physical process can have any integral relationship to a mental process?

Few would deny that physical processes like the activation of neurons and mental processes like recognizing words are related. The question, however, is whether the explaination of how neurons operate is like the explanation of how cognitive acts take place. Hope for such an analogous correlation was provided by a series of ingenious experiments conducted by Georgopoulos and his research group (Georgopoulos et al. 1989:234-236). They trained a rhesus monkey to move a handle in the direction of a light when it came on dim ad perpendicular to and counterclockwise from the direction of the light when it came on bright. During these tasks, they measured the activation of neurons in the motor cortex and constructed a population vector of these neurons. With the bright light, the monkey had to rotate the image 180 degrees and counterclockwise; here was a cognitive task that was measured in a motor cortex, the basis of physical movement. Georgopoulos et al. state:

“The rotation of the neuronal population vector is of particular interest because there was no a priori reason for it to rotate at all. It is also interesting that the population vector rotated consistently in the counterclockwise direction: this suggests that the spatial-motor transformation imposed by the task was solved by a rotation through the shortest angular distance. Given that the mental rotation is time consuming, this solution was behaviorally meaningful, for it minimzied both the time for the animal to get the reward and the computational effort which would have been longer if the rotation had been through 270 clockwise. (Georgopoulos et al. 1989:234-236)

In their abstract for the aritcle, the authors make an even stronger claim: “These results provide direct, neural evidence for the mental rotation hypothesis and indicate that the neuronal population vector is a useful tool for ‘reading out’ and identifying cognitive operations of neuronal ensembles.” (Georgopoulos et al. 1989:234-236) this direct evidence shows that the explanation of a physical neuronal process describes the structure of a cognitive process.

Armed with this insight that a genuine analogy exists in at least one case between a physical and cognitive process, glycolysis offers another possible candidate as a real analogy between physical and cognitive. Benno Hess, Stefan Muller, Theo Plesser and Mario Markus of the Max-Plank Institute fur Ernahrungsphysiologie in Dortmund have investigated the nonlinearity of glycolysis through an ingenious method of capturing the dynamics of the reaction by employing a video camera to record spectrophotometry (Muller et al. 1989, Markus et al. 1988, Hess et al. 1988). Pictures of the spectrophotometry of these reactions reveal not only a scientific beauty but also a suggestive analogy to computer graphics representations of other nonlinear systems like fractals.

They then go on make a very interesting statement about the dynamics of the way in which the brain uses its physical processes to cause mind… The way the mind can effect the physical processes:

Even though cognition and consciousness depend upon neuronal processes for their very existence, each maintains an emergent independence from the brain. Conscious thinking can and does change the physical activity of the brain as in the case of biofeedback where the subject can consciously lower blood pressure, pulse rate and other bodily functions. Mind and body interact as changes in neurotransmitters through drugs direct the contents of thinking. Neuronal processes are a necessary but not sufficient condition for cognition and consciousness. One cannot, however, predict the contents of cognition by fully understanding the organization of neuronal patterns as these activations are nonlinear and self-organizing. The neural chaos described by these nonlinear algorithms is both complex and unpredictable. Calling these systems ‘deterministic’ means that various patterns of neuronal activation can be described by algorithms but not that every future event can be predicted. This is contrary to the more tradtitional assumption that deterministic systems generate precise predictions about future behavior. But these more familiar dterministic systems were composed of linear rather than nonlinear equations. The chaos of nonlinear systems does not imply a formless interdeterminism. Instead, nonlinear dynamical systems can produce a self-organizing organism with rational but unpredictable future behavior.

The brain operates as an organic machine generating self-organized patterns of neuronal activation that yields unpredictable, creative congnitive behavior. Ingredient in this self-organizing machine that produces chaos are both mind and consciousness. Both emerge from neuronal processes in ways that remain mysterious. Each may be a series of subsets of the processes OR they may be a collection of activations of definite geographical areas of the brain.

And there you have it… The brain uses these unpredicatable leaps of chaotic intelligence, something that standard computer programs to date have lacked. It is this inherent chaos that adds intelligence to a dynamical system… If a system, a biological organism for example, has pre-defined set of rules within it that allow it to fuction in a given environment, then it usually only follows a series of commands given to it by the programmer. It’s as basic as that. This system, can be compared to a linear dynamical system i.e. a set input will always provide a given and predicatable output. It follows its programming. However, for something to become intelligent, it needs to have that spark… That “je ne sais quoi” about it. It should be able to vary its own response to things… And it should be able to learn from that new behavior i.e. did it provide a benefit or not to the device, by feeding-back the data to a “situation…” The “situation” being: an input occurs, and rather than following “standard protocol output response”, it varies the output response (called a “varied the output response”) and then performs a reflection upon the “varied the output response” to understand if this provides any benefit over the “standard protocol output response” program. If it does actually benefit i.e. saves time OR saves energy, then this can provide a uselful alternative to the “standard protocol output response”. But futher “tests” will need to be performed by the organism in order to ascertain exactly what situations this new “varied the output response” will be benefical in.

These responses will have to built into a large database of sorts and then cross referenced with eachother. Some of these responses will no doubt provide useful alternatives to “conditioned reflexes” or “standard protocol output responses”. Some will not. But inorder to gauge which are beneficial and which are not, these “self-similar actions” will have to be crossreferenced against a catalogue of “varying situations”. Infact, the organism should be able to crossreference the outcomes with the inputs in many various ways i.e. “self-similar actions” against “varying situations”, “self-similar situations” against “varying actions”, “self-similar situations” against “self-similar actions” in order to find which might be best… And the originality and efficiency with which an organsim mixes these combinations up is determined by nonlinear dynamics…

So perhaps it is with good fervor that the saying, “Chaos often breeds life, while order breeds habit,” arises?

From what we have seen so far, I think we can understand how chaos might actually benefit a living organism, and allow it “evolve” in novel and unpredicatable ways. Mac Cormac and Stamenov go on to state that:

‘Mathematical chaos’ describes the properties of nonlinear dynamical systems, namely, algorithms that are complex and deterministic and yet unpredicatable. For some variables, these algorithms are stable with results clustered around point attractors. For other variables, the results are scattered or chaotic and unpredictable.

Now… This is where it gets interesting. I have no idea whether this has already been documented in some scientific study or not… If it has, then perhaps it is simply that I am not the originator. But… My sensibilities tell me that this idea of a strange attractor can have a parrallel drawn between it and Pavlov’s experiments with behavior in his dogs.

As we have already seen, the brain is essentially a complex nonlinear dynamical system. This fact, however, does not predetermine it to being chaotic all the time… While a nonlinear dynamical system is prone to output chaotically under certain stimuli, it does also have the abiltiy to achieve stable states under certain conditions. As you may have seen in one of the previous links provided (click here to see), it has been thought that people who have been diagnosed with behavioral issues i.e. schizophrenia, depression, bi-polar, etc… tend to exhibit more chaotic neuronal patterns/functioning than healthy minds. Bearing this in mind, I am now going to present an extract from William Sargant’s book entitled “Battle For The Mind”… This will be a lengthy interlude, but please do read it, as it beautifully describes the importance of Pavlov’s own work in reflexes, plus clears the path into the connection between mind and body, which will allow for a deep understanding about how we all, dog, monkey and man, operate alike:

In the course of over thirty years of research Pavlov accumulated a mass of observations on various methods of building up behavior patterns in dogs and then breaking them down again. He interpreted his findings in mechanistic terms which have since been frequently disputed by psychologists and psychiatrists. Yet the findings themselves have been confirmed again and again. Horsley Gantt attributed the absence of any important errors in Pavlov’s work to his ‘painstaking methods, his adequate controls, his habit of giving the same problem to several collaborators working in separate laboratories or institutes, with whom he checked results and supervised experiments…’

Pavlov had won the Noble Prize, in 1903, for research on the physiology of digestion before turning to study what he called the ‘higher nervous activity’ in animals. What changed his line of enquiry was a sense that he could learn little more about digestive functions until he had investigated the workings of the brain and nervous system, which often seemed to influence digestion. He then became so deeply absorbed in the implications of this new study that he concentrated on it until his death in 1936, at the age of eighty-six.

Pavlov was one of the Russian scientists of the old regime whose work Lenin thought valuable enough to encourage after the Revolution; and even though extremely critical of the Soviet regime, Pavlov continued to receive generous support from the government. Both inside and outside Russia he was admired for the courageous attitude he adopted, and only at the very end of his life did he become reconciled to living under a dictatorship. Ironically, he is now regarded as a hero of the Revolution, and no mention is made in recent Soviet publications of his persistent opposition to the regime. Horsely Gantt, revisiting him in 1933, asked why his political attitude was now more conciliatory; and Pavlov replied half jokingly that at the age of eighty-three his heart could no longer stand the strain of infuriated outbursts against the authorities who were sponsoring him. About this time, also, the Nazis had begun to threaten Russia, and Pavlov’s great mistrust of Germany inclined him to abate his hostility to the Russian Government. But though he was now relating his discoveries about animals to problems of human behaviour, it is extremely doubtful whether he ever foresaw that his work could be used as an instrument of Soviet policy. Since he always demanded and obtained freedom of thought for himself, it is unlikely that he would have wished to curtail freedom of thought for others. He insisted on travelling abroad to maintain contact with his scientific colleagues, and won a great ovation when lecturing in England just before his death.

Pavlov cannot therefore be considered a typical scientist of the soviet regime; even if much of his finest work had not been done before the Revolution. Yet the Communists must have found his mechanistic approach to the physiological study of behaviour in dogs and men most helpful while persuing their policy of indoctrination. In July, 1950, a medical directive was issued in Russia for re-orientation of all Soviet medicine along Pavlovian lines – probably partly because of the impressive results obtained by applying Pavlov’s research to political ends. Yet outside Russia its implications still sometimes tend to be ignored.

As soon as Pavlov expressed a desire to apply his experimental findings on animal behaviour to problems of morbid psychology in human beings, the Soviet government placed a near-by psychiatric clinic at his disposal. His first public lecture on this topic was dilivered in 1930: he called it ‘The Trial Excursion of a Physiologist in the Field of Psychiatry.’ It may be that these new interests date from an operation for gall-stones which he underwent in 1927; because he then published his significant ‘A Post-operative Cardiac Neurosis Analysed by the Patient: Ivan Petrov Pavlov.’

Pavlov’s work seemed to have influenced the techniques used in Russia and China for the eliciting of confessions, for brainwashing and for inducing sudden political conversions. His findings, applicable to these, should be easily understood even by the non-technical reader, without the need of spending too much time on the details of his actual experiments. Most of these findings are well reported in a series of Pavlov’s later lectures translated by Horsley Gantt, and published in Great Britain and the United States in 1941 under the title ‘Conditioned Reflexes and Psychiatry.’ Professor Y. P. Frolov’s enlightening book on these experiments, ‘Pavlov and his School’ (1938), has also been translated and published in English. Professor Babkin’s more recent ‘Life of Pavlov,’ however, makes little reference to some of his most important findings from the view point of our study. And though Dr. Joseph Wortis in his ‘Soviet Psychiatry,’ published in the United States, emphasizes the importance in modern Russian medicine of Pavlov’s experimental approach to psychiatric problems, few details are given of the last important phase of Pavlov’s work. An official ‘Life of Pavlov,’ published in Moscow in 1949, written by E. A. Asratyan, also contains many details of Pavlov’s earlier experimental work on conditioned reflexes in animals, but very few details of his later work relevant to conversion and brain-washing techniques. At all events, no publication in English has hitherto explained these for the benefit of ordinary readers but recently a good new translation of Pavlov’s selected works has become available in English.

Thirty years of research convinced Pavlov that the four basic temperaments of his dogs approximated closely to those differentiated in man by the ancient Greek physician Hippocrates. Though various blends of basic temperamental patterns appeared in Pavlov’s dogs, they could be distinguished as such, rather than as new temperamental categories.

The first of these four correspond with Hippocrates’ ‘choleric’ type, which Pavlov named the ‘strong excitatory.’ The second correspond with Hippocrates’ ‘sanguine temperament’; Pavlov named it ‘lively,’ the dogs of this type being a more balanced temperament. The normal response to imposed stresses or conflict situations by both these types was increased excitement and more aggressive behaviour. But whereas the ‘choleric,’ or ‘strong excitatory,’ dog would often turn so wild as to be completely out of hand, the ‘sanguine’ or ‘lively’ dog’s reactions to identical stresses were purposeful and controlled.

In the other two main temperamental types of dog imposed stresses and conflict situations were met with more passivity, or ‘inhibition,’ rather than aggressive responses. The more stable of these two inhibitory temperaments was described by Pavlov as the ‘calm imperturbable types, or phlegmatic type of Hippocrates.’ The remaining temperament identified by Pavlov corresponds with Hippocrates’ ‘melancholic’ classification. Pavlov named it the ‘weak-inhibitory’ type. He found that a dog of this type shows a constitutional tendency to meet anxieties and conflicts by passivity and avoidance of tension. Any strong experimental stress imposed on its nervous system reduces it to a state of brain inhibition and ‘fear paralysis.’

Yet Pavlov found that the other three types, too, when subjected to more stress than they could cope with by usual means, responded in the end with states of brain inhibition. He regarded this as a protective mechanism normally employed by the brain as a last resort when pressed beyond endurance. But the ‘weak inhibitory’ type of dog was an exception: protective inhibition occured more rapidly, and in response to lighter stresses – a difference of the utmost significance to this study.

Pavlov fully recognized the great importance of environment, as well as of constitution, in deciding the final behaviour patterns of his dogs. He found that certain fundamental instincts, such as sex or the need for food, were constantly adapted to changes of the environment by the formation of appropriate behaviour patterns. A dog without a brain cortex (which contains some of the more complicated connections between the main brain centres) might still swallow food placed within its mouth; but it needed a brain cortex and means of forming complicated conditioned reflexes, if it were to learn that food would be given only after an electric shock of a certain definite strength, or after a metronome has been heard beating at one particular rate and no other.

In discussing the ‘weak inhibitory’ type, Pavlov pronounced tha though the basic tempramental pattern is inherited, every dog has been condtioned since birth by varied environmental influences which may produce long-lasting inhibitory patterns of behaviour under certain stresses. The final pattern of behaviour in any given dog will therefore reflect both its own constitutional temperament and specific pattern of behaviour induced by environmental stresses.

Pavlov’s experiments led him to pay increasing care to the need for classifying dogs according to their inherited constitutional temperaments before he subjected them to any of his more detailed experiments in conditioning. This was because different responses to the same experimental stress or conflict situation came from dogs of different temperaments. When a dog broke down and exhibited some abnormal patterns of behaviour, its treatment would also depend primarily on its constitutional type. Pavlov confirmed, for instance, that bromides are of great assistance in resotring nervous stability to dogs who have broken down; but that the doses of sedative required by a dog of ‘strong excitary’ type is five to eight times greater than that required by a ‘weak inhibitory’ dog of exactly the same body weight. In World War II the same general rule applied to human subjects who had temporarily broken down under battle and bombing stress and needed ‘front line sedation.’ The required doses varied greatly according to their temperamental types.

Towards the end of his life, when he was experimentally applying his discoveries about dogs to research in humans psychology, Pavlov gave increasing attention to what happened when the higher nervous system of his dogs was strained beyond the limits of normal response; and compared the results with clinical reports on various types of acute and chronic mental breakdown in human beings. He found that severer and more prolonged stresses could be applied to normal dogs of the ‘lively’ or ‘calm unperturbable’ type without causing breakdown, than to those of the ‘strong excitatory’ and ‘weak inhibitory’ types.

Pavlov came to believe that this ‘transmarginal’ (it has also been termed ‘ultraboundary’ or ‘ultramaximal’) inhibition which eventually overcame even the two former types – changing their whole behaviour dramatically – could be essentially protective. When it occurred, the brain might have no other means left of avoiding damage due to fatigue and nervous stress. He found a means of examining the degree of protective transmarginal inhibition in any dog at any given time: by using his salivary gland conditioned reflex technique. Thought the dog’s general behaviour might seem normal, at first sight, the amount of saliva being secreted would tell him what was beginning to happen in it brain.

In these tests, the dog would be given a definite signal, such as the beating of a metronome at a certain rate, or the passing of the a weak electric current into its leg, before being given food. After a time the signal would provoke an anticipatory flow of saliva, without the need of letting the dog see or smell the food. A conditioned reflex having thus been established in the brain between a signal and the expectation of food, the amount of saliva secreted could be precisely measured in drops, and any changes in the response of the brain conditioned reflexes and induced patterns, could be plainly registered.

Here let me digress by emphasizing the relevance of Pavlov’s experiments on condtioned reflexes to the ordinary happenings of everyday human life. Much human behaviour is the result of the conditioned behaviour patterns implanted in the brain, especially during childhood. These may persist almost unmodified, but more often become generally adapted to changes of environment. But the older the person, the less easily can he improvise new conditioned responses to such changes; the tendancy thn is to make the environment fit his, or her, increasingly predictable responses. Much of our human life consists also in the unconscious following of conditioned reflex patterns originally acquired by hard study. A clear example is the way a car-driver builds up numerous and varied conditioned responses before being able to negotiate a crowded city street without paying much conscious attention to the process – this is often called ‘driving automatically.’ If the driver then gets into the open country, he will change to a new pattern of automatic behaviour. The human brain is, in fact, constantly adapting itself reflexly to changes of environment; although, as with car-driving, the first lessons in any given process may demand difficult, and even tedious, efforts of concentration.

Human and canine brains are obliged to build up a series of both positive and negative conditioned responses and behaviour patterns. Most people in business and the Armed Forces learn by experience to behave negatively in the presence of their superiors; and positively, even perhaps aggressively, in that of their juniors. Pavlov showed that the nervous system of dogs developes extraordinary powers of discrimination in building up these positive and negative responses. He showed that a dog can be made to salivate when a tones of 500 vibrations a minute is sounded, if this is a food signal; but not if the rate is only 490, and no food can therefore be expected.

Negative conditoned responses are no less important that the positive ones, since the members of civilized societies must learn how to control normal aggressive responses almost automatically, though sometimes obliged to release them in a split second when a vital emergency arises. Emotional attitudes also become both positively and negatively conditioned: one learns an almost automatic revulsion from certain classes of people, and an automatic attraction to others. Even such words as Catholic and Protestant, Worker and Employer, Socialist and Conservative, Republican and Democrat, evoke very strong conditioned responses.

One of Pavlov’s most important findings was exactly what happens to conditioned behviour patterns when the brain of a dog is ‘transmarginally’ stimulated by stresses and conflict beyond its capicity for habitual response. He could bring about what he called a ‘rupture in higher nervous activity’ by employing four main types of imposed stresses.

The first was, simply, an increased intensity of the signal to which the dog was conditioned; thus he would gradually increase the voltage of the electric current applied to its leg as a food signal. When the electric shock became a little too strong for its system, the dog began to break down.

A second powerful way of achieving the same result was to increase the time between the giving of the signal and the arrival of food. A hungry dog might be conditioned to receive food, say, five seconds after the warning signal. Pavlov would then greatly prolong the period between a signal and the giving of food. Signs of unrest abd abnormal behaviour might become immediately evident in the less stable of his dogs. He found, in fact, that the dogs’ brains revolted against any abnormal prolongation of waiting under stress; breakdown occurred when a dog had to exert very strong or very protracted inhibition. (Human beings, too, often find prolonged period of anxious waiting for an event more trying than when it finally comes).

Pavlov’s third way of producing a breakdown was confuse them by anomalies in the conditioning signals given – continued positive and negative signals being given one after the other. The hungry dog became uncertain what would happen next, and how to face these confused circumstances. This could disrupt its normal nervous stability – just as happens with human beings.

A fourth way of producing a breakdown was to tamper with a dog’s physical condition, fevers, or by disturbing its glandular balance. Though the three other means listed above failed to produce a breakdown in a particular dog, this might be engineered later by using the same sort of stresses immediately after the removal of its sexual glands, or during an intenstinal disorder. The advantage taken of debilitation and other changes of bodily function in human beings for their political and religious conversion will be discussed later. In some cases, Pavlov’s findings may have been exploited; in others, anticipated.

Pavlov found not only that after castration or intestinal disorders a breakdown might occur even in temperamentally stable dogs; but also that the new behaviour pattern occurring afterwards might become a fixed element in the dog’s way of life, though it had long recovered from the debilitating experience.

In the ‘weak inhibitory’ type of dog new neurotic patterns thus implanted could often be readily removed again: doses of bromide might be enough to achieve this – though they did not alter the dog’s fundamental weakness of temperament. But in ‘calm imperturbable’ or ‘lively’ dogs who needed castration, for instance, before they could be nervously disrupted, Pavlov found that the newly implanted pattern was more often ineradicable once the dog had recovered its normal physical health. He suggested that this was die to the natural toughness of the nervous systems in such dogs. The new patterns of behaviour had been difficult to implant without a temporarily induced debilitation; now they might be held with as much tenacity as the old.

The relevance of this last experiment to similar changes of behaviour in humans hardly needs to be emphasized: towards the end of a long period of physical illness, or after a period of severe debilition (sometimes produced by enforcing fasting), people of ‘strong character’ are often known to make a dramatic change in their beliefs and convictions. If they then recover strength, they may remain true to the new orientation for the rest of their lives. Case-histories of people ‘converted’ in times of famine or war, or in prison, or after harrowing adventures at sea, or in the jungle, or when brought to destitution by their own self-will, are frequent. The same phenomenon is often observed in both psychotic and neurotic patients who have suffered from glandular operations, fevers, loss of weight and the like, and only then developed their abnormal patterns of behaviour: if they had strong previous personalities, these new patterns may persist long after physical recovery.

Pavlov established that the ability of a dog to resist heavy stress would fluctuate according to the state of its nervous system and its general health. But once protiective ‘transmarginal’ inhibition had been induced, some very strange changes in the functioning of the dog’s brain took place. And these changes could not only be measured with some precision by the amounts of saliva secreted in response to conditoned food stimuli, but were not liable, as when human beings have analogous experiences, to subjective distortions: there was no question, that is to say, of the dogs trying to explain away or rationalize their behaviour after having been subjected to these tests.

There distinct and progressive stages of ‘transmarginal’ inhibition were identified by Pavlov in the course of his experiments. The first he called the ‘equivalent’ phase of cortical brain activity. In this phase, all stimuli,of whatever strength, resulted only in the same amounts of saliva being produced. the observation is comparable to the frequent reports by normal people in periods of intense fatigue, that there is very little difference between their emotional reactions to important or trivial experiences. And though the feelings of a normal, healthy person will vary greatly, according to the strength of the stimuli experienced, nervously ill people often complain that they become unable to feel sorrow and joy as acutely as before. As the result of fatigue abd debilitation, in fact, a man may find to hs chagrin that the excitement at receiving a legacy of ten thousand pounds is no higher than if it were one of sixpence; his condition then probably approximates to the ‘equivalent’ phase of exhausted cortical activity identified by Pavlov in his dogs.

When even stronger stresses are applied to the brain, the ‘equivalent’ phase of transmarginal inhibition may be succeeded by a ‘paradoxical’ phase, in which weak stimuli produce livelier responses than stronger stimuli have done. The reason for this is not far to seek: the stronger stimuli are now only increasing protective inhibition; but the weaker ones still produce positive responses. Thus the dog refuses food accompanied by a strong stimulus, but accepts it if the stimulus is weak enough. This paradoxical phase can also occur in human behaviour where the emotional stress is heavy, as will be shown in a later chapter. On such occasions, the individual’s normal behaviour has been reversed to a degree that seems quite irrational not only to a detached observer, but to the patient himself – unless either of them happens to have studied Pavlov’s experiments on dogs.

In the third stage of ‘protective’ inhibition, which Pavlov called the ‘ultra-paradoxical,’ positive conditoned responses suddenly switch to negative one; and negative ones to positive. The dog may then, for instance, attach itself to a laboratory attendant whom it has previously disliked, and try to attack the master whom it has previosuly loved. Its behaviour, in fact, becomes exactly the opposed to all its previous conditioning.

The possible relevance of these experiments to sudden religious and political conversion should be obvious even to the most sceptical: Pavlov has shown by repeated and repeatable experiment just how a dog, like a man, can be condtioned to hate what it previously loved, and love what it previously hated. Similarly, one set of behaviour patterns in man can be temporarily replaced by another that altogether contradicts it; not by persuasive indoctrination alone, but also by imposing intolerable strains on a normally functioning brain.

Pavlov also showed that when transmarginal inhibition began to supervene in a dog, a state of brain activity similar to that seen in human hysteria might result. This can cause an abnormal suggestability to the influences of the environment. His case-histories frequently include reports on hypnodial or hypnotic states in dogs. Clinical reports on the behaviour of human beings under hypnosis, as well as in various conditions of hysteria, abound in description of abnormalities corresponding with those noted in Pavlov’s ‘equivalent,’ ‘paradoxical’ and ‘ultra-prardoxical’ phases of break down in dogs. In states of human fear and excitement the most wildly improbable suggestions can be accepted by apparently sensible people; as in August, 1914, a rumour that Russian soldiers were travelling through England ‘with snow still on their boots’ swept the country, and was so circumstantial that for a while it affected German stratagy; or as in the earlier stages of the Second World War, rumour continually reported the English renegade William Joyce (‘Lord Haw-Haw’) as having mentioned in a broadcast that the church clock of a particular village – the name of which always varied with the telling – was three minutes slow.

What I propose is that there are basins of attraction within the mind of us all. Let me draw on the analogy of the earlier idea of a point attractor, as I expressed within the definition of “attractor” at the beginning of this essay. For a moment, let’s think of the same pendulum. Now place three magnets as the vertices of an equilateral triangle with the pendulum at rest in the center. Each time you swing the pendulum it will come to rest either in the center i.e. the lowest point to the gravitational attraction, or will favor one of the three magnets. Each of these attraction points i.e. the three magnets OR the gravity, are point attractors. The image below shows the cumulative pattern of the pendulum within the attractor basins of this system. Each of the heart shaped regions in the center corresponds to the final state of the pendulum coming to rest favoring a particular magnet. The black areas correspond to the border between attractor basins…

Now let’s jump back to the human brain for a moment… I reckon similar patterns are hidden in the way the brain functions. Why? Well… We already know attraction basins exist within the mind, allowing the interplay of deeply instinctual drives, drives that have been naturally selected for over billions of years of evolution, to posit within behaviour, thus providing the organism with a better ability to survive, by using the best survival traits/techniques/actions available to them. Basically, these traits/techniques/actions, centered in the structure of the brain, ensure that the species carries on producing and surviving… So far, so good I hope. Current humans (and our distant relatives) survived natural selection for good reason… They were better adapted, both physically and mentally, to adapt and live/survive in their surrounding (and ever changing) environment. They developed habits and routines that allowed them to successfully (success = remain alive) with what ever came at them, whether is was a tiger, a disease or drought. This flexible mind, coupled with the constant, yet slow, mutation rate, allowed them to evolve and adjust the ever changing pressures of their surrounding environment. So the current hierarchy between the various controlling aspect of the body and mind, (a hierarchy that still governs us all deeply today) were all naturally selected for… And thus they are, by themselves, point attractors that exist around various structural nodes within the brain that provide particular modes of being to come into play and/or function for healthy, natural bodily activities and normal reflex human behaviour. An example of a point attractor within the brain is seen in the Right Temporal Parietal Junction (or RPTJ for short) that allows us to interpret what other people are thinking/gesturing/signalling/etc… Within these structures, certain varying patterns of neuronal firing occur as various inputs are received and interpreted… No doubt these are, as we have already seen, centered in a complex and chaotic dynamical feedback system of the whole body. After the system (the complete human body as a whole) interprets the stimuli, chemical cues manifest and cause further specific/relevant neuronal firing to posit relevant behavior for the organism… As this occurs, many other channels are triggered that will filter along into other areas of the brain, affecting other nodes… Producing subtle patterns that give rise to healthy OR (in the case of a malformed brain) unhealthy actions. This chemical and molecular interplay occurs much in the way that we’ve already seen in Bruce Lipton’s lecture entitles “The Biology Of Perception,” which can be viewed here.

These regulatory areas of the brain provide critical responses that have been hardwired into the structure of our Central Nervous System (CNS) and allow us to perform the various functions that we perform daily in society. No doubt these functional groups of neurons in the brain, when triggered, will act in varying overriding strengths (as determined by just how critical the “input” signals are), making the organism depart from a normal stable stasis, and move into a different awareness/mode of being. It is my guess that these regulatory areas of the brain will interlink together via various channels, whether directly or indirectly, and will posit relevant interaction, much in the same way strange attractors operate i.e. they allow certain behavioural patterns to remain in a continual steady flux, but when triggered into chaotic oscillation, they allow us to glean advantages over old dogmatic methods of being, and thus provide us with new suitable solutions (OR actions) with which to advance with into these environmental changes OR “problematic” encounters. These attractors are at work in the minds of both healthy and “mentally unstable” individuals alike. The only difference being that the healthy mind has these attractors functioning within fairly stable and predictable parameters i.e. they respond to their evironment, their friends, family and relevant life experiences with healthy, “normal” and accepted behavioral responses/patterns…

Briefly then… As a graphical analogy to what we’ve been discussing so far… Let’s just take look at the chaos that can arise out of a nonlinear dynamical system… Here the chaos within the system is plotted on a Cathode Ray Oscilloscope machine to yield a Strange Attractor pattern within phase space.

So behavioural responses, which are results of basins of attraction centered around specific functional groups within the brain, are no doubt a result of two factors:

i) the structure of the brain itself within an individual – determined by nutrition provided during the formative years of the individual’s growth, and the stimulus provided during this growth, and type of stimulus i.e. positive or negative.

ii) the experiences an individual has throughout their formative years i.e. between birth and the ages of 15 to 18 – determined by the family’s religious disposition, social circles, education, etc…

These inputs of “nurture and nature” build the system. And thus, these factors directly have a role in determining how the organism will react in its later years to specific environmental stimuli i.e. whether it will allow an individual to function within the real world normally according to current social values, or whether they will be prone to negative and destructive behavior patterns. Either way, I think that there are literally patterns of mind that will be able to be observed; patterns that will determine whether they fall into Pavlov’s ‘strong excitatory,’ ‘lively,’ ‘calm imperturbable’ or the ‘weak-inhibitory’ character types. Because as the mind forms, the attractors at work will be molded and pulled into a finalized shape or rhythm that the individual will use, there-onwards in their life, to process all the input data for the rest of their lives.

No doubt if this is the case, then stable/healthy cycles between the “basins” of attraction (which should be able to be modeled, roughly, by mathematical simulations based on current observations) should provide a particular phase space pattern to demonstrate as much i.e. the nonlinear dynamical system has a steady and stable flow to it, similar to what is seen in diagram b) of the “Chaos and stability in a nonlinear dynamical system,” which was shown near the beginning of this essay. However, when undue stress is applied to the subject, and the ‘equivalent,’ ‘paradoxical’ or ‘ultra-paradoxical’ phases of transmarginal inhibition are approached, the steady dynamics of the “normal” human mind will perhaps begin to exhibit signs of instability, OR chaos. These states will no doubt start to manifest in behavioural anomalies i.e. non recognition of family members and/or friends, etc…, and might well provide plots in phase space like those seen in c), d) and maybe even e) of the “Chaos and stability in a nonlinear dynamical system” diagram. ?

Right… For now I would now like to return to Mac Cormac and Stamenov.

‘Mathematical chaos’ describes the properties of nonlinear dynamical systems, namely, algorithms that are complex and deterministic and yet unpredicatable. For some variables, these algorithms are stable with results clustered around point attractors. For other variables, the results are scattered or chaotic and unpredictable. For still other variables, the results unexpectedly cluster around strange attractors and may even oscillate between or amoung them. These strange attactors may have a dimension that is not an integer and hence are fractal.

Examples of mathematical chaos abound in the history of mathematics; nonlinear dynamical systems are not new. The relative newness of this field arises from the realization that these equations can be used to model physical phenomena allowing for the abrupt changes like the shift from laminar flow to turbulence in the flow of air over a wing. The ability of the modern digital computer to produce the results of hundreds of thousands of iterations of these nonlinear equations has also provided an impetus for an exciting and continuing investigation of chaos…

…Nonlinear dynamical systems offer the possibility of describing an interrelated network or neurons that move abruptly from chaos to stable patterns. Self-organized slight perturbations of initial values or the values of constants in the algorithms, force transitions from chaos to stable patterns and from stable patterns to chaos. These properties of nonlinear systems present an opportunity to give a mathematical description of neuronal processes. But which kind of nonlinear algorithm should we seek to model neuronal activation?

I think Markram and his team at L’Ecole Polytechnique Fédérale de Lausanne (EPFL) are well on the way to understanding which nonlinear algorithm is, OR set of nonlinear algorithms are, to be used for their computer model of the human brain.

Just to return one last time to what Mac Cormac and Stamenov have writen in their book “Fractals of Brain, Fractals of Mind: in Search of a Symmetry Bond”:

Fractals are related to mathematical chaos in an unusual manner. Mandelbrot claims that “every known ‘strange’ attractor is a fractal.” He expands on this claim by exploring the concept of “strange.”

“Whether or not all fractal attractors are strange is a matter of semantics. Increasing numbers of authors agree with me that for the most purposes an attractor is strange when it is a fractal. This is a healthy attitude, if strange is taken to be a synonym to ‘monstrous,’ ‘pathalogical,’ and other epithets once applied to individual fractals.” (Mandelbrot 1983:197)

Although the formal relationship between fractals and nonlinear dynamic systems remians unclear, some fractals seem to be a geometrical subset of nonlinear systems even though not all nonlinear systems are fractals. Amoung fractals, the property of resemblance would seem to be an extremely useful analogy in attempting to model the nonlinear activation of neurons. When fractals are scaled up or down with larger or smaller values, topological patterns recur unexpectedly over and over again. This poperty is called ‘resemblance.’ Since neuronal activations during various cognitive tasks involves at least some similarities in the areas of the brain activated, a property of resemblance in mathematical rational reconstrcutions would be desirable. Unfortunately, however, many fractals than can be used to model biological forms involve geometrical inversion of their forms (and are called self-inverse fractals) and do not possess the property of resemblance (self-similarity) (Mandelbrot 1983: 166-179). Even some of these fractals, howeverm are ‘nearly’ self-similar and perhaps it is those which we shall seek in our investigation of congintive neuroscience. Many random fractals possess the property of self-affinity; even though their patterns do not exactly resemble the each other in scaling, they are affine, meaning that they possess some similar properties but not exact resemblance. Richard Voss states: “This non-uniform scaling, where shapes are (statistically) invariant under transformations that scale different coordinates by different amounts, is know as self-affinity” (Voss 1988:44). This is the property necessary for a mathematical reconstruction of patterns of neuronal activation: statistical similarity with differences in detail at different scales.

A zoom into the Mandelbrot set...

Fractals also offer one advantage of representing temporal changes in geometrical forms for explaining cognitive neuronal processes. Neurons activate over time in various regions of the brain often physically separated. The dynamics of fractals allows for temporal and spatial changes. The Mandelbrot set, one of the most interesting and robust fractals, combines aspects of self-similarity with aspects of infinite change.

zn+1 = zn2 + c

where z is a variable and c complex number.

The Mandelbrot set can be displayed graphically by beginning z=0 and then iterating the algorithm for various values of c. If one fixes c and then varies z in the field of complex numbers, beautiful Julia sets are obtained (Douady 1986). While Julia sets are self-similar, the Mandelbrot set exhibits patterns that are affine. It is most likely that if neuronal processes can be represented by fractals, they will be rationally reconstrcuted by random fractals or fractals like the Mandelbrot set that are infinitely varied, complex and unpredictable, but still self-organized.

A Julia set plot showing julia sets for different values of c. The plot, when showing the distribution of c in the complex plane, resembles the Mandelbrot set.

The mathematics of chaos, especially fractals many of which are subsets of chaos, offer a ideal series of algorithms to rationally reconstruct the nonlinear self-organizing activation of neurons. In a search for the mathematical algorithms which reconstruct how billions of interconnected neurons are activated to account for both motor and cognitive actions, one must find representations that are nonlinear, that form recurring patterns, that are capable of self organization, and that move rapidly from order to disorder. Fractals fulfill all of these requirments. There are both deterministic and random fractals, with the latter and perhaps the former combining determinism and unpredictability. And all fractals posses a property either of exact resemblance or at least self similarity (affine) under scaling. But which fractals rationally reconstruct which neuronal processes? We must find an emprical method of discovering which nonlinear algorithms (fractals) will represent which neuronal processes.

And perhaps this is also something that Markram and his team will discover as they run and develop their software into an ever more similar/exacting model of the human mind… Perhaps they will literally see these almost kaleidoscopic patterns unfolding as sensory data are input into the complex structure of the central nervous system, interpreted and then output back into patterns that never repeat exactly the same way twice, always unfolding in new and original ways, giving the system freedom from stagnation… Making possible evolution.

A depiction of Lorenz's water wheel experiment.

Just a Lorenz discovered the chaotic patterns inherent in his equations by varying the input data to only a few decimal places i.e. ignoring several thousandths of a unit of accuracy, into his Royal McBee computer… Another system, that of a water wheel, show above demonstrates nonlinear dynamical fluctuations in its flow… At the top, water drips steadily into containers hanging on the wheel’s rim. Each container leaks steadily from a small hole in its base. If the input stream of water is slow, the top of containers never fill fast enough to overcome friction, but if the input stream is faster, the weight starts to turn the wheel. The rotation might become continuous. Or if the input stream is so fast that the heavy containers swing all the way around the bottom and start up the other side, the wheel might then slow, stop, and reverse its rotation, turning first one way and then the other.

As James Gleick mentions in his book entitled, “Chaos: The Amazing Science Of The Unpredictable”:

A physicist’s intuition about such a simple mechanical system – his pre-chaos iintuition – tells him that over the long term, if the stream of water is never varied, a steady state would evolve. Either the wheel would rotate steadily or it would oscillate steadily back and forth, turning first in one direction and then the other at constant intervals. Lorenz found otherwise.

Three equations, with three variables, completely described the motion of this system. Lorenz’s computer printed out the changing values of three variables: 0-10-0; 4-12-0; 9-20-0; 16-36-2; 30-66-7; 54-115-24; 93-192-74. The three numbers rose and then fell as imaginary time intervals ticked by, five times steps, a hundred time steps, a thousand.

To make a picture from the data, Lorenz used each set of three numbers as coordinates to specify the location of a point in three-dimensional space. Thus the sequence of numbers produced a sequence of points tracing a continuous path, a record of the system’s behaviour. Such a path might lead to one place and stop, meaning that the system had settled into a steady state, where the variables for speed and temperature were no longer changing. Or the path might form a loop, going around and around, meaning that the system had settled into a pattern of behaviour that would repeat itself periodically.

A Lorenz attractor 3D plot (a plot in phase space) demonstrating the way inwhich the data lines up. Here there are two attractor basins, demonstrated by the two points around which the circular patterns group.

Being able to see the Lorenz attractor with its 3D graphical nomenclature always helps, I've found...

Lorenz’s system did neither. Instead, the map displayed a kind of infinite complexity. It always stayed within certain bounds, never running of the page but never repeating itself, either. It traced a strange, distinctive shape, a kind of double spiral in three dimensions, like a butterfly with its two wings. The shape signaled pure disorder, since no point or pattern ever recurred. Yet… Within this disorder, a new order was discovered.

Here, it is my guess that Markram and his team at L’Ecole Polytechnique Fédérale de Lausanne (EPFL), as they begin running their computer simulation of the human brain, piecing it together bit by bit as they go along, will see strong evidence pointing towards the existence of “strange attractors” at work within the “neuronal forest.” Some of these “strange attractors” might be so complex in nature that they might even demand several dimensions of phase space to be plotted out. No doubt these attractors will begin to be linked to behavioural patterns i.e. sex drive, flight and fight responses, conginitive functions, etc… And eventually, as the simulation becomes more and more complete in structure, as well as tuned to function just like a real human brain does, the EPFL team will begin to see “strange attractors” within “strange attractors…”

Perhaps now might be a good time to have a look at what makes you, you…

The price we pay for being self-aware, is the understanding the ultimate demise of ourselves. As Spinoza once said, “A free man thinks of nothing less than of death, and his wisdom is not a meditation upon death but upon life…”

As you will have seen… When Professor du Sautoy goes out to the Wisconsin Psychiatric Institute, he observes that there is an important difference between the waking mind and mind “at rest/in sleep.” While awake, it seems that there is cross talk between various sides of the brain i.e. between the different centers of regulation within the neocortex. When du Sautoy has Transcranial Magnetic Stimulation (TMS) while awake (seen at about 42 mins into the documentary) he sees that the graphical recording of his own brain’s activity shows simulated areas of the brain as red. From this recording he can clearly see that when areas of his brain were stimulated by TMS, there were also other areas on the opposite side of the brain demonstrating stimulation… Almost as though there was some “secondary-stimulation” resulting from an interconnection between the different areas of his brain. However… When a subject was asleep, and the same proceedure was performed on their brain i.e. simulation of certain brain areas by TMS, the “secondary-stimulation” usually seen in subject who were awake, was NOT observed!!! Thus, as Professors Marcello Massimini and Marcus du Sautoy rightly observe, “consciousness is about the interconnectivity between the different elements of the brain.” It is my guess that various control centers of the brain cross-talk in chaotic ways; ways that are similar to “strange attractor” plots in phase space, producing chaotic patterns of mind… Perhaps if this “secondary-stimulation”, OR this cross-talk between various parts of the mind, was observed more closely, then nonlinear dynamics (OR chaos theory) would best describe the dynamical interactions between regulatory nodes of brain, which are posited as modes of mind, and behavioral modifications. The chaos inherent in our minds is the essence of what we perceive to be consciousness. But currently, I am awaiting the definitive proof of this conjecture from the world of science. Is there anyone reading this who can test my theory for me???

When, and even if, this is demonstrated… Perhaps these patterns will mimic ideas and shapes found in everyday life… ? Perhaps human consciouness will be able to be mapped in much the same way that a sea urchin’s shell can be described by a 4th degree polynomial graphical plot?

The beautifully patterned cases/skeletons of a sea urchin without their spines.

A fourth degree polynomial pattern trace. Sea resemblance? (Pun intended)

But this is something I will leave for time to reveal…

So… So far so good… I HOPE!!! But an important question that we’ve not yet tackled is… Why would the human body choose to use a central nervous system based on fractals??? Well… In many ways, the answer to that is pretty simple. Because fractals are everywhere! And what would be the most efficient system to recognize and sift through any self-similar data set? You guessed it… A fractal system.

If you don’t believe me that fractals exist everywhere, I’m now going to show you just the tip of the iceberg with a few photo collages that I’ve prepared.

The Mandelbrot Set

The (Mis)behaviour Of Markets” is the chronicle of Benoit Mandelbrot’s long lasting pursuit of understanding the financial markets. Instead of Brownian motion and Gaussian distribution, as the early 20th century French mathematician Louis Jean-Baptiste Alphonse Bachelier (March 11, 1870 – April 28, 1946) did (like flipping a coin, his idea was that the evolution of asset prices could be conceived of as purely random events, with a 50-50 chance of ticking up or ticking down as the time passes) Mandelbrot based his views on fractal geometry, a mathematical branch he himself originated.

Fractal comes from the Latin word “fractus” meaning broken. The idea is that a shape is broken down into smaller shapes, each echoing the large. Fractals can be found in many places in nature, like the British coast line, branches on a tree, or parts of a rock. The analogy to asset prices is the similarity between different frequencies of stock market data. We know that daily observations look very similar to lower frequency data, such as monthly observations. Hence the self similarity property of the parts to the whole.

Beniot Mandelbrot also wrote an article for Scientific American back in 1999, entitled “How Fractals Can Explain What’s Wrong with Wall Street,” which demonstrates how the geometry that describes the shape of coastlines and the patterns of galaxies also elucidates how stock prices soar and plummet.

Bifurcation diagram of a logistic map, displaying chaotic behaviour past a threshold.

Another fractal pattern occurs as dynamical system move towards chaotic behaviour. The Feigenbaum constant δ is a universal constant for functions approaching chaos via period doubling. It was discovered by Feigenbaum in 1975 (Feigenbaum 1979) while studying the fixed points of the iterated function f (x) = 1 – μ[x]r and characterizes the geometric approach of the bifurcation parameter to its limiting value as the parameter μ is increased for fixed x.

The plot above is made by iterating the equation f (x) = 1 – μ[x]r (where r = 2) several hundred times for a series of discrete but closely spaced values of μ, discarding the first hundred or so points before the iteration has settled down to its fixed points, and then plotting the points remaining.

Simple systems can also produce chaos without relying on differential equations. An example is the logistic map, which is a difference equation (recurrence relation) that describes population growth over time. Another example is the Ricker Model of population dynamics.

Even these bifurcation diagrams exhibit fractal like properties, called Feigenbaum fractals.

A Bifurication Diagram is basically a probability map that accurately shows the path that a dynamical system can take at each cycle/oscillation at any one time.

This diagram shows "roughly" how the evolution of cyclical variations within a nondynamical system relate to the data shown in a bifurcation diagram...

These Bifurication Diagrams also demonstrate a surprising proportionate resemblance between the Mandelbrot Set.

The correspondence between the Mandelbrot Set and the logistic map.

Let’s move on to attractors, as these ideas are very pertinent to what we’ve been discussing above i.e. that brain function is, in its own way, based on fractal principles.

The Lorenz attractor, named after Edward N. Lorenz, is a fractal structure corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its lemniscate shape. The map shows how the state of a dynamical system (the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern.

An attractor is a set to which a dynamical system evolves after a long enough time. That is, points that get close enough to the attractor remain close even if slightly disturbed. Geometrically, an attractor can be a point, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory, as you may have already read in James Gleick’s book entitled “Chaos: Making A New Science.” The term “strange” was coined by David Ruelle and Floris Takens to describe the attractor that resulted from a series of bifurcations of a system describing fluid flow.

The human heart also exhibits chaotic patterns within its rhytmic cycles, again denoting a nonlinear dynamical system.

The human heart also has a chaotic pattern. The time between beats does not remain constant; it depends on how much activity a person is doing, among other things. Under certain conditions, the heartbeat can speed up. Under different conditions, the heart beats erratically. It might even be called a chaotic heartbeat. The analysis of a heartbeat can help medical researchers find ways to put an abnormal heartbeat back into a steady state, instead of uncontrolled chaos. For more information, please click here.

A pattern I noticed while calculating Mandelcube pictures... Notice the white lines? Don't they look veins!?

In studying fractal topography, I also taking a keen interest in generating images of the M set, and its related permutations. One of these is the Mandelcube. And just the other day, while generating the above image using FraxFlame, I noticed an uncanny similarity between vein patterns and the white fibral markings above…

Veins (from the Latin vena) are blood vessels that carry blood toward the heart.

Through out the body there are ever branching pipes, that get smaller and smaller, like trees trucks break into boughs, which break branches, which again break into twigs. This fractal physiology of the human body has been well documented in recent times. James B. Bassingthwaighte, Larry S. Liebovitch and Bruce J. West have done it great justice in their book entitled “Fractal physiology.” Even the structures in the heart are fractal…

Is there any wonder that these dynamical systems then keep arising through out the universe in many different ways, across many different scales???

Then there are general phenomenon that seem to be self-similar in an affine away… Spirals of weather systems roughly “match” those found existing within the cosmos on much larger scales. And it’s just mesmerizing to think that within each galaxy, there are near one thousand billions solar systems, most of which have planetary systems that will exude similar weather patterns to Earth’s own.

A satelite view of the Tibetan mountain range... Fractals shapes seen from high above.

Above, mountains “branch” over the surface of the Earth… Gravity pulling the carving forces for water, ice and rock down into basins (or oceans) of equilibrium. As these forces carve out the higher grounds, their random shapes elude to the divine geometry of Universal flow.

Even trees and plants use fractal structures. Notice the way the tree's branches flow... And how the leaves viens flow.

All plants arrange their structures in fractal ways… This maximises the surface area of their leaves exposed to sunlight over the day time.

Basic structures, when iterated over and over again, produce the complex patterns we see today... Think about the first single celled organisms, iterated into complex multicellular life...

Basic shapes and structures, when repoduced over and over again, form patterns out of the whole… This idea stretches from molecules in cells, cells in organisms… Even molecules in planets, planets in solar systems, all the way through to solar systems in galaxies.

Even the simple shape of a bacterium, when iterated over and over again, produces wondrous patterns of self-similarity.

Bearing this last idea about bacterial spread following fractal patterns of growth… Or should it rather be, producing fractal patterns in their growth… Is it any wonder that epidemiologists are finding it very convinient to understand the spread of disease in terms of fractal scaling laws?

4. Respiratory-circulatory interactions in health and disease by Steven M. Scharf, Michael R. Pinsky, Sheldon Magder

5. Fractals, graphics, and mathematics education by Michael Frame, Benoît B. Mandelbrot

Expanding this idea of spread and diffusion… In China, scientists have also noticed that the spread of fires is best described with the use of fractal geometry… This is handy as fire-fighters now understand that they should focus on the most recently ignited sites, as suggested by this new fractal mathematical model. One wall of flame may look like another to a fire-fighter on the ground, but the boundaries where a forest fire is growing fastest are more dangerous than the rest… So now better co-ordination for effective fire-fighting can be orchestrated between a ground and airborn team.

Even when one drops some ink onto blotting paper, there are fractals unfolding before your very eyes.

The fine filaments within the structure of paper provide fine holes woth which ink is absorbed by “capillary action.”

And even as we zoom in, the pattern remains self-similar...

These branches and patterns also occur throughout other systems. A common one being ground water draining through top soil after is rains.

L. Sander & T. Witten made this CGI of a Diffusion Limited Aggregation (DLA) system in 1981... Another fractal structure commonly found in the natural (and man made) world.

Diffusion-limited aggregation (DLA) is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by Witten and Sander in 1981 [1], is applicable to aggregation in any system where diffusion is the primary means of transport in the system. DLA can be observed in many systems such as electrodeposition, Hele-Shaw flow, mineral deposits, and dielectric breakdown.

Diffusion Limited Aggregation of iron flowers that formed naturally in bedrock...

A DLA cluster grown from a copper sulphate solution in an electrodeposition cell.

DLA can even be seen in the structures of moss and lichens.

A picture of moss growing on a stone.

A picture taken from a Mandelbrot zoom that I created in FraxFlame. Notice similarities between the moss covered stone?

Even lightening displays a type of DLA fractal structure…

But enough of this demonstration. Find your own fractals! And remember… They might not be as obvious as these visual displays, as we saw with the functioning of Strange Attractors within the human mind.

Now I will have my say…

Just as all life originated from the single celled organisms called stramatolites… Single cells which eventually came together to join and work as one multicellular organism… So, in these early multicellular sheets of algae, each cell initially performing exactly the same tasks as one another… Thin enough so that nurtients and sunlight might be readily available and waste might dufuse easily away into the environment. Similar natural laws that caused the atomic star dust of accretion to assemble into molecular balance in order to meet energetic needs/requirments, providing a beautifully elegant solution via atomic sharing and symmetry, eventually began to work on these very molecules themselves. Basins of chaotic attraction, bestowed upon these molecular communities by their own structural and physical necessitarianism, started to arrange themselves into masses. Characteristics of their energetic dispositions dictated how each molecules unique phsical and chemical properties would impinge upon its own destiny of intermolecular interaction. Some, such as the fatty acids, formed vesicles… Others formed protein-base pair coupled reactions that cycled in these vesicles like Beluzov-Zhabotinsky marvels.

Simple Belousov-Zhabotinsky swirls in naturally occuring reaction. Waves of instability propaagating outwards...

After much time, chance and probablity bequeathed stability within these basins of attraction. And so these Life forms developed a stable reproducability that allowed them to become successful enough to multiply over the surface of this novel and wondrous “petri dish”, suspended in the inky back void of time and space around its central star.

However, as chance is a fickle thing, even crashing in on stability’s own new founded molecular party… Somewhere within these environmentally isolated vesicles (introduced by disfusion, turbulent, or concentration inadequacies) errors were caused by the chaotic instabilities, and new functions were either formed… Or, much more commonly, old functions were terminated. Either way, these “stromatolites” were at times given the opportunity to distiguish themselves from the rest and separate, even mutate, the original plan… So that subtle differences occured within their single celled bodies. Successful ones even developed advantages over the others and began racing ahead with a new found impetus for spreading and consuming.

After many millions of years of steady, vibrant gowth, chemical by products of life i.e. oxygen, built up in the environment… And as all good chemists and physicists now, as partial pressures of gases increase, so do the ability to which they dissolve into solution. So, no doubt, oxygen began to build up in the primordial soups of life… And as we all know so well from experience, as ‘polutants’ build up, so they have an effect on the environment and the creatures living within that environment. As this eponymous oxidating agent ever increased in volume, the pressure it began to place on evolution’s cause began to impact selection. Where as mutations that bestowed abilities to handle the “king of all oxidating elements” in earlier organisms might have gone un-noticed, due to anaerobic evironmental pressures; they were now able to capitalize on the build up of “waste” oxygen from earlier generations. Specializing in “O two” consumption, these organisms were given new chances to interact around new basins of attraction. This idea of evolving didn’t stop there… Chaos wanted to span the eons majestically with new iterative designs. As Life moulded the environment, so the environment moulded Life.

Eventually the limitations of the single celled organisms came about, as sheets of algal cells lnked sides and became the first “crude” multicellular organisms… And as with each new generation of modifications, the odds of mutation stacked higher and higher. And so cells began to clump together, forming algae like sheets that became larger nd larger. Sooner or later, as these clumps of cells began to mutate/differentiate in their own ways within their multi-cellular bodies, they might have discovered symbiosis… Aliances between these differentiations arose, where by one cell’s waste, became another cell’s food. Now these chaotic attractors developed new internalized basins to modulate their own survival/needs with. Eventually radical differences came into being, whereby certain cells could begin to specialize in particular functions for the Whole… Functions that others could not i.e. eyes with which to see, kidneys with which to purify the body with, etc… This ever changing flow eventually gave rise to complex organisms like ourselves, humankind.

Bearing in mind mankind followed similar social ideals to these of the cells journey i.e. was born into a family, would leave the family and then come together again in a new family… We can see that these patterns are repeating. Living organisms are generally following an “affine” fractal pattern. The idea of the whole i.e. survival, is the key… And the patterns used to create survival are essentially the same as they always were, across many different scales… And across time.

Man, as they spread… Originally started out as family groups of hunter gatherers, which eventually grouped together in social groups, aiding in grooming, mating and providing protective measures from predators… Even further down the line these ideals developed into villages, towns, cities, and countries… And even, as we are seeing today, Global Communities. So… Isn’t it becoming obvious? We’re doing the exact same things that all life on Earth has always done, since the day it got its very first, and very sucessful, foot hold here… We’re coming together, and operating in new ways that only large groups of organisms can do… We’re evolving new patterns of thought, like little bodily cells of the Universe, perceiving the majestic wonder in which we’ve ‘suddenly,’ and rather randomly, appeared in. What a lottery ticket We (the molecules on Earth) have been given… That chance to experience Life!

This idea of self-similarity, which we use every day of our waking, sleeping, eating, talking, partying, thinking, studying, dancing lives, is showing us something that we’ve been trying to express through Religions, mystical dances, etc… for EONS! And now we’re beginning to see that nature is showing this obvious pattern of being… Via memes, via scientific research, via mathematical, chaotic and fractal research even… And most deffinitely via artistic expression… We’re copies of copies, no longer originating, but simply evolving ideas and techniques, using the colors of past happenings, with the shading of present experiences, to paint a brighter and more deeply textured future with…

And we should heed the leasons that this ancient process is showing us… That the Butterfly Effect, where small seemingly insignificant gestures can generate instabilities in current flows, which are then amplified further up the line through a chain of turbulent features, so that they become unpredictable eddies of wondrous new possibilities… We should understand the sensitivity of our “petri dish” life here on planet Earth, and know that the way we are consuming and heating out homes, and running our machines with fuels, eating foods that we no longer grow ourselves, are all having effects on the surrounding environment… The period of stable weather systems that we have enjoyed for so long, which we have taken for granted even i.e. summer to mean warm sunny days and winter to mean snowy cold landscapes of white wonder (if you live in the Uk, that is), ar phasing away from our presumptions… Change is certain. Stability never endures for long. Along with the consequences of pollution throughout the food chain, over watering, deforestation, etc… this will all feedback into the chaotic system of Life on Earth. And we should be aware of this! For this is part of the Tao and the way it flows…

To ignore this Wisdom is to ignore the essence of Life itself!!!

As you may have noticed above… We must understand that our minds are sensitive hosts for memetic parasites… Hosts that are based on nonlinear dynamical systems… Systems which can be manipulated and reprogrammed quite a lot easier than any of us would care to imagine. The illusion of “Self,” something that I have blogged about earlier in “Another Take On Reality – Meme, Myself and I“, is an illusion of defining ourselves in a consumerist world. Remember, the term ‘illusion’ is not derogatory. It simply means illusion.

We should be aware of this, and take heed of the divine in ourselves… For the magic of Life itself, is locked within the free flow of chaos. Once we see this, we will begin to ‘See’ deeper into being than we ever have before… And come of this will a new revolution of thought and considerate being… A world where we can affect eachother in new positive ways of being, while maintain a balance with one another, AND the rest of life here on Earth.

No doubt this tipping of the scales was necessary to arrive at these new wondrous discoveries and Knowings. But the burden of mankind on Gaia has not gone un-noticed. We will feel Gaia begin to give way, as our needs crush the delicate balance that has arrisen over the epochs. For now… Chaos is slowly buckling with instability, poised and ready to “reiterate” a new change over the new “butterflies” of mankind. This is, without a doubt, the calm before the storm.

One question that still bothers me deeply is… Why is so hard for mankind to know this truth??? Have these “consumer chains” bound us to their availabilty like Crack or Methamphetamine does to their “addicted” users???

Postscript

I would like to re-iterate to the reader my own intentions by writing this blog… It is not my aim to disclose a “hidden meaning” to life’s eternal flow. Neither is it to procure new scientific or religious standings. Nor is it my aim to put into disrepute current world views OR Religious ideals. Rather it is to ‘suggest,’ using analogies recently disclosed through science’s accolades (most of which have so far been reviewed within these Blogs) new modes of possible understanding about ‘what’ We, as human beings, are and ‘why’ We came about in this Garden Of Eden that orbits around a bright star of light… One that, like all the others before it, forges all the known matter into what Life has become today. It is an intricate process, with seemingly unrelated parts not fitting in at time… However, all of it, no matter how bizarre or strange, plays a part in the whole, and becomes applified through Chaos’ own design.

The sages of old spoke about this wisdom… And some of us today can see how that wisdom is now based around scientific truths. It’s a puzzle, and a marvelous one at that… One that we should all cherish dearly… And, if given half the chance, it is something we should become ultimately aware of.

But I cannot tell people what to do. It is for you to decide what you do with your lives here on Earth… And it is your choice to decide what to believe. Besides… This could just be nothing more than an egocentric bias that I have for a particular meme that is lodged in my head… A bias that came to mind many moons ago… And just the other day it was once again presented to me while reading an excert about the late Dr Albert Hoffmann, the father of LSD, where he is reportedly quoted as saying:

“I am a figment of my own imagination…

I am a part in this universe…

Ergo…

I am the universe experiencing itself…

I am the universe questioning itself.”

Taken from the New York Times.

EVERY STRUCTURE, EVERY PATTERN FOR MODULATING YOUR MIND – AND EVEN YOUR HEART – BOTH IN STRUCTURE AND IN RHYTHM, AS WELL AS THE TREES, RIVERS AND MOUNTAINS AROUND YOU, KNOW IT!!! WE ARE ALL THE SAME… USING STRUCTURES OF SELF SIMILARITY TO SIMPLY BE!

When you See and Feel this for yourself… You will begin to Know truth.

The ‘Idea’ Of Infinity…

March 17, 2009

“If any philosopher had been asked for a definition of infinity, he might have produced some unintelligible rigmarole, but he would certainly not have been able to give a definition that had any meaning at all.” Bertrand Russell

In this brief essay on the ‘infinite,’ I do not want to ramble on about uncertainties or truths. My aim here is not to lecture. Rather it is to encourage… So instead of joining the dots together in an obvious proclamation of basis, I am happy to quote certain others’ works that have more pertinently and eloquently touched aspects of the ‘infinite’ over the years, with a hope that the reader’s mind will naturally settle on the splendor lying behind the complex and distracting facades of catechism.

1. William Blake

William Blake (28 November 1757 – 12 August 1827) was an English poet, painter, and printmaker. Largely unrecognized during his lifetime, Blake is now considered a seminal figure in the history of both poetry and the visual arts of the Romantic Age. His prophetic poetry has been said to form “what is in proportion to its merits the least read body of poetry in the English language”. His visual artistry has led one modern critic to proclaim him “far and away the greatest artist Britain has ever produced”. Although he only once journeyed farther than a day’s walk outside London during his lifetime, he produced a diverse and symbolically rich corpus, which embraced ‘imagination’ as “the body of God”, or “Human existence itself”.

William Blake in an 1807 portrait by Thomas Phillips.

In one of his most insightful poems “The Auguries of Innocence”, he states:

“To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.”

For the full poem, please visit: http://www.artofeurope.com/blake/bla3.htm

What could such majestic tapestry mean? Well… Perhaps to the bovine logician, or the unenlightened literary reader, this idea might fall short of the lofty missive prescribed by Blake’s godly eye. But thankfully, with today’s scientific awareness of all things great and small, this notion may be better ‘understood’ with only a lax dedication towards ‘knowing’ the world around oneself better.

2. Niels Fabian Helge von Koch

Niels Fabian Helge von Koch (January 25, 1870 – March 11, 1924) was a Swedish mathematician who gave his name to the famous fractal known as the Koch snowflake, one of the earliest fractal curves to be described.

He was born into a family of Swedish nobility. His grandfather, Nils Samuel von Koch (1801–1881), was the Attorney-General of Sweden. His father, Richert Vogt von Koch (1838–1913) was a Lietenant-Colonel in the Royal Horse Guards of Sweden.

Von Koch wrote several papers on number theory . One of his results was a 1901 theorem proving that the Riemann hypothesis is equivalent to a strengthened form of the prime number theorem.

He described the Koch curve in a 1904 paper entitled “On a continuous curve without tangents constructible from elementary geometry” (original French title: “Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire”).

The Koch snowflake (or Koch star) is a mathematical curve and one of the earliest fractal curves to have been described. (Actually Koch described what is now known as the Koch curve, which is the same as the now popular snowflake, except it starts with a line segment instead of an equilateral triangle. Three Koch curves form the snowflake.)

The Koch curve is a special case of the Cesaro curve where:

$a=\frac{1}{2}+\frac{i}{\sqrt{12}}$,

which is in turn a special case of the de Rham curve.

One can imagine that it was created by starting with a line segment, then recursively altering each line segment as follows:

1. divide the line segment into three segments of equal length.
2. draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.
3. remove the line segment that is the base of the triangle from step 2.

The Koch snowflake/star is generated using the same recursive process but starting with an equilateral triangle rather than a line segment. After doing this once for the Koch snowflake, the result is a shape similar to the Star of David.

The Koch curve is the limit approached as the above steps are followed over and over again.

The Koch curve has an infinite length because each time the steps above are performed on each line segment of the figure there are four times as many line segments, the length of each being one-third the length of the segments in the previous stage. Hence the total length increases by one third and thus the length at step n will be (4/3)n: the fractal dimension is log 4/log 3 ≈ 1.26, greater than the dimension of a line (dimension 1) but less than Peano’s space-filling curve.

Iterations of the von Koch curve...

Ever smaller and smaller… As one zooms into the Koch curve, steady self-similarity is exuded infinitly:

3. Karl Menger

In mathematics, the Menger sponge is a fractal curve. It is the universal curve, in that it has topological dimension one, and any other curve (more precisely: any compact metric space of topological dimension 1) is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge. It is a three-dimensional extension of the Cantor Set and Sierpinski Carpet. It was first described by Austrian mathematician Karl Menger in 1926 while exploring the concept of topological dimension.

A Menger sponge, iterated four times...

Each face of the Menger sponge is a Sierpinski cerpet; furthermore, any intersection of the Menger sponge with a diagonal or medium of the initial cube M0 is a Cantor set.

The Menger sponge is a closed set i.e. it contains its own boundary (unlike the Mandelbrot set); since it is also bounded, the Heine-Borel theorem implies that it is compact. Furthermore, the Menger sponge is uncountable and has Lebesque measure 0.

The topological dimension of the Menger sponge is one, the same as any curve. Menger showed, in the 1926 construction, that the sponge is a universal curve, in that any possible one-dimensional curve is homeomorphic to a subset of the Menger sponge, where here a curve means any compact metric shape of Lebesgue covering dimension one; this includes trees and graphs with an arbitrary countable number of edges, vertices and closed loops, connected in arbitrary ways.

In a similar way, the Sierpinski cerpet is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not planar, and might be embedded in any number of dimensions. Thus any geometry of quantum loop gravity can be embedded in a Menger sponge.

Interestingly, the Menger sponge simultaneously exhibits an infinite surface area and encloses zero volume. This idea of the infinite held within the finite is perhaps not such a revelation as it might initially seem…

A ‘simpler’ more visual way to understand the complexity of Menger’s idea can be seen in the follow animation:

http://www.pure-mirage.com/html/MillersMengerSpongeFastPlay.htm

4. Benoît B. Mandelbrot

Benoît B. Mandelbrot (born 20 November 1924) is a French mathematician, best known as the father of fractal geometry. He is Sterling Professor of Mathematical Sciences, Emeritus at Yale University; IBM Fellow Emeritus at the Thomas J. Watson Research Cente; and Battelle Fellow at the Pacific Northwest National Laboratory. He was born in Poland. His family moved to France when he was a child, and he was educated in France.

In mathematics, the Mandelbrot set, named after Mandelbrot himself, is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1zn2c remains bounded. That is, a complex number, c, is in the Mandelbrot set if, when starting with z0=0 and applying the iteration repeatedly, the absolute value of zn never exceeds a certain number (that number depends on c) however large n gets.

In other words… Part of the charm of the set is that it springs from such a simple equation: z2 + c. The terms z and c are complex numbers, which consist of an imaginary number (a multiple of the square root of –1) combined with a real number. One begins by assigning a fixed value to c, letting z = 0 and calculating the output. One then repeatedly recalculates, or iterates, the equation, substituting each new output for z. Some values of c, when plugged into this iterative function, produce outputs that swiftly soar toward infinity. Other values of c produce outputs that eternally skitter about within a certain boundary. This latter group of c‘s, or complex numbers, constitutes the Mandelbrot set.

When plotted on a graph consisting of all complex numbers, the members of the set cluster into a distinctive shape. From afar, it is not much to look at: it has been likened to a tumor-ridden heart, a beetle, a badly burned chicken and a warty figure eight on its side.

A closer look reveals that the borders of the set do not form crisp lines but seem to shimmer like flames. Repeated magnification of the borders plunges one into a bottomless phantasmagoria of baroque imagery. Some forms, such as the basic heartlike shape, keep recurring but always with subtle differences.

The Mandelbrot set shows more intricate detail the closer one looks or magnifies the image, usually called “zooming in”. The following example of an image sequence zooming to a selected c value gives an impression of the infinite richness of different geometrical structures, and explains some of their typical rules.

The magnification of the last image relative to the first one is about 10,000,000,000 to 1. Relating to an ordinary computer monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometres. Its border would show an inconceivable number of different fractal structures…

And here I will leave you with a quotation…

“Pure mathematics is, in its way, the poetry of logical ideas.”  Albert Einstein