## The Universe ‘May’ Have A Fractal Structure!

### August 24, 2011

In some ways it’s amazing… But in other ways, I wonder if I’m really surprised… ? I’ve been observing fractal patterns now for quite a few years in what many refer to as seemingly unrelated fields of occurrence i.e. hearing them in reverb simulations that I build within Max/MSP, OR while observing the patterns with which the Penicillium fungi grows on the bread that I want to avoid using in the toaster most mornings, through to markets and their ever shifting price-scapes… They’re everywhere. Yes… Everywhere.

They’ve even managed to naturally find their way into the experiential textures of my mind’s dynamic… Textures that the brain seems to weave together through strange attractor like eddies that occur between various nodes and hierarchical synaptic electrical discharges that fire so readily between various clusters within the brain’s overall structure… This in turn allows a type of consciousness to form i.e. myself, to perceive the material ‘aspects’ of the environment that I presently find myself in… ‘Aspects’ that are continually changing/moving/shifting. Most of these transformations are commonly seen as material changes i.e. day to night OR hot water turning into cold water… Changes that are forged from the same principles and materials i.e. atomic debris/fabric of the universe, that ‘I’ find myself a result of.

It’s amazing that ‘my’ five senses can somehow distinguish between these multifarious ‘aspects’ simply by observing the ever changing environmental interplay that unfolds in the world around me – and within me – allowing my body to cross-reference these abstractions (such as smell, sight, touch, taste, sound, etc…) into a functional braid of linear temporal registers that are plied together into a complex feedback loop of conscious awareness that correlates all of them into the fabric of experience. Through the natural evolution of this holographic image of universal dynamics – one that has been naturally selected for in most living organisms here on Earth in some manner or another – it’s pretty obvious that memetic evolution has given rise to – and certainly has benefited from – these unfolding fractal patterns of the mind, brain, body and environmental continuum… And, thus, so have I allowed myself – through much diligent study – to hang a myriad of meanings and socially accepted constructs onto the continuous flow of this biochemically experiential unfolding.

When I sit with this feeling, it seems very natural for everything to be just as it is… For us to be the way we are… Mortal, soft, delicate and changing… Prone to aging and death… Giving way to new progeny in an evolving loop of atomic re-awakening… And environmental readjustment/realignment… Suddenly it becomes okay to accept that one day I will die… And that my patterns of behavior will continue to ripple through the surrounding people I have met and the environment I once lived in, slowly being diluted, intermingling with other people’s activities, every evolving… Ever changing. Perhaps we don’t ever really die… ! Then I see that ‘I’ am not as free as many might imagine we are… Rather we are more willfully able to do whatever it is we choose to with the time we have here, acting within defined parameters of being… Operating to prolong our activities. I find acceptance in these limited modes… And I find true freedom in the limitless possibilities within my imagination. Just as chaos is limitless, and as the brain’s basis for functional ordering uses chaos to operate from… So I find myself not really being surprised that the universe ‘may’ have a fractal structure. When is see my lungs on a X-ray that had recently, there they are again… When I look at my arm closely and see the veins of blood flowing under my skin, fractal shapes come into focus… And I’m just amazed at the beauty of these patterns as they release their energetic uncoiling of potential energy into kinetic displays of wonder and marvel, spreading out over various timings into the delicately interconnected chaos of universal change.

So what I thought was originally surprise… Has in fact turned out to be more of a sense of discovery… A rediscovery of my connectedness… My roots… My interlinked existence to everything – absolutely everything – around me. In many ways it has been an important rediscovery for me because this feeling of interconnectedness seems to have been masked over, obscured from obvious sight, by the daily meanderings of advertising, fictional drivel (mainly in the form of film and pulp fiction), political discussion, religious debate, scientific enquiry and general distraction, all of which seem to come from the supposed “perks” of Western modern day living…

But, thankfully, while immersing myself in this tangled mess of experiential twine – mainly by reading many, many scientific journals/publications over the last fifteen or so years, ones that concern themselves with how universal structure and function came into being (whether on the astrological and/or microscopic levels OR within the dynamics of the mind, brain, body, environmental continuum) – I’ve been unwittingly reconnecting myself with this feeling of interdependence. While closely keeping my eye on how the present theories (yes, theories, in the plural, because there are many of them out there) are continually evolving and changing… I’ve been unintentionally observing another form of natural selection at work… Much like Darwin did. One that is occurring within our minds. And, on the whole, it’s doing exactly what any good evolving form/system does i.e. works through the plethora of memetic constructs that are being formulated from experience by scrubbing the obviously impractical and blatantly cumbersome theories, revealing only the ones that best fit the observations. Then, while subjecting these selected few to yet more stringent tests, each idea/theory is further developed… OR revealed to be a fraud. Eventually one idea/theory in particular is found… One that fits better than all the rest. One that can generate self-similar observed data by repeating the experiments over and over again. This idea/theory then becomes a sort of fact… One that can be expounded further into more developed and concise levels of understanding… Where each idea/theory can interconnect and interrelate to other seemingly unrelated areas of scientific inquiry. Time and again, further cross-referencing and testing ensues, scrutinizing each novel idea/theory/notion… If one doesn’t fit, it is then modified, tweaked, or reconfigured to work into the overall account produced thus far… OR EVEN, if an idea is so obvious, then the other areas might find themselves being revised. This continues ad-infinitum, moving even onwards into finer details… Heading towards the vanishing point of a complexity that knows no bounds… A sort tailor made fitting for a more concise scientific understanding that will never be found.

In fact… So to does the evolution of animal form work in much the same way… As Professor Armand Marie Leroi states, “Species give rise to other species, and as they do so, they change. The changes are minute and subtle, but given enough time, the results could be spectacular. And so they are!” So to do our mind streams change and evolve over time… Allowing us to see more clearly whatever it is we are looking at.

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What Darwin didn’t Know

Documentary which tells the story of evolution theory since Darwin postulated it in 1859 in ‘On the Origin of Species’.

The theory of evolution by natural selection is now scientific orthodoxy, but when it was unveiled it caused a storm of controversy, from fellow scientists as well as religious people. They criticised it for being short on evidence and long on assertion and Darwin, being the honest scientist that he was, agreed with them. He knew that his theory was riddled with ‘difficulties’, but he entrusted future generations to complete his work and prove the essential truth of his vision, which is what scientists have been doing for the past 150 years.

Evolutionary biologist Professor Armand Marie Leroi charts the scientific endeavour that brought about the triumphant renaissance of Darwin’s theory. He argues that, with the new science of evolutionary developmental biology (evo devo), it may be possible to take that theory to a new level – to do more than explain what has evolved in the past, and start to predict what might evolve in the future.

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As time has gone on, I’ve been fortunate enough to rediscover how similar basic patterns permeate almost every single aspect of our lives as human beings… This rediscovery – for me at least – occurred because I had the fortunate experience of studying many dynamical systems for musical analogy… That is, I studied them over and over again, looking at how to translate these natural never-ending patterns into sonic textures for art’s sake. When you see them, though, you begin to spot them everywhere you care to look. It’s almost like it’s so obvious that they’re there, just staring us in the face, that because of it, we just haven’t noticed them… They’ve always been there… In plain sight. So why would we notice them? In some ways it’s just like when the astronauts of Apollo 11 landed to the moon for the first time… When they got there, they couldn’t see any trace of the Earth around them anymore. Their home of a planet was now just a beautiful jewel hanging in the moon’s inky black sky, just out of their reach. Everything that they had taken for granted i.e. an abundance of air, all the trees, plants, life, all the oceans of water, our homes, the people they loved, movies, the abundance of food, animals, clouds, rain, wind, etc… They just weren’t there around them anymore… And it stood out like a soar thumb as to how fortunate they were to live on a planet that had all those things… Things that were so common on Earth. This voyage to the moon profoundly changed the way they i.e Neil Armstrong, Edwin “Buzz” Aldrin Jr and pilot Michael Collins saw the Earth afterwards. In fact it changed ever astronaut who ever went to the moon’s perspective… So that when they returned, they couldn’t help but wonder why people couldn’t see what they now could see so clearly i.e. how precious the Earth is and all the beings that live on it… How connected we all are to one another… To everything around us… How much we need our planet… And how futile all our wars and disagreements are in the greater scheme of everything.

Something similar is going on in science now… Over the last year or so I’ve been coming across many publications wherein scientists are seemingly wanting to let go of some of their earlier preconceptions about how the textbook ideals – one’s which their contemporaries wrote down with absolute certitude for their students to learn from – concerning universal flow and other areas of scientific interest, don’t really quite fit with what these students are actually observing in the “real world…” And along with how they are having to “pull-out-of-the-hat” seemingly bizarre concepts, such as dark matter, in order to balance their predecessors equations… Many are beginning to feel that it’s time to evolve again. Thus it can be noticed that many of the new generation of scientists are looking for novel ideas to re-evaluated what they have learned… And as the models get more and more complex, so to do we see that complexity needs to be better understood… Revealing many types of fractal structures and all sorts of non-linear dynamics residing within the natural flow of universal unfolding.

As I have mentioned before in several blogs contained in this website… Until fractal/chaotic dynamics are properly introduced and included into the equations of physicists, chemists, biologists, psychologists, etc… There will always be a thin vale of mist that detaches their efforts from discovering the true order of things. For, until this time, discrepancies and vague approximations on how universal flow actually functions will cloud the depth of understanding that lies waiting to be seen beneath this mist.

Saying that… There are those who are already daring to go beyond… As Francesco Sylos Labini clearly demonstrates with his intuitive proposition below… The universe may have a fractal structure…

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Largest Cosmic Structures ‘Too Big’ For Theories

Space is festooned with vast “hyperclusters” of galaxies, a new cosmic map suggests. It could mean that gravity or dark energy – or perhaps something completely unknown – is behaving very strangely indeed.

We know that the universe was smooth just after its birth. Measurements of the cosmic microwave background radiation (CMB), the light emitted 370,000 years after the big bang, reveal only very slight variations in density from place to place. Gravity then took hold and amplified these variations into today’s galaxies and galaxy clusters, which in turn are arranged into big strings and knots called superclusters, with relatively empty voids in between.

On even larger scales, though, cosmological models say that the expansion of the universe should trump the clumping effect of gravity. That means there should be very little structure on scales larger than a few hundred million light years across.

But the universe, it seems, did not get the memo. Shaun Thomas of University College London (UCL), and colleagues have found aggregations of galaxies stretching for more than 3 billion light years. The hyperclusters are not very sharply defined, with only a couple of per cent variation in density from place to place, but even that density contrast is twice what theory predicts.

“This is a challenging result for the standard cosmological models,” saysFrancesco Sylos Labini of the University of Rome, Italy, who was not involved in the work.

Colour guide

The clumpiness emerges from an enormous catalogue of galaxies called the Sloan Digital Sky Survey, compiled with a telescope at Apache Point, New Mexico. The survey plots the 2D positions of galaxies across a quarter of the sky. “Before this survey people were looking at smaller areas,” says Thomas. “As you look at more of the sky, you start to see larger structures.”

A 2D picture of the sky cannot reveal the true large-scale structure in the universe. To get the full picture, Thomas and his colleagues also used the colour of galaxies recorded in the survey.

More distant galaxies look redder than nearby ones because their light has been stretched to longer wavelengths while travelling through an expanding universe. By selecting a variety of bright, old elliptical galaxies whose natural colour is well known, the team calculated approximate distances to more than 700,000 objects. The upshot is a rough 3D map of one quadrant of the universe, showing the hazy outlines of some enormous structures.

Coagulating dark energy

The result hints at some profound new physical phenomenon, perhaps involving dark energy – the mysterious entity that is accelerating the expansion of space. Dark energy is usually assumed to be uniform across the cosmos. If instead it can pool in some areas, then its repulsive force could push away nearby matter, creating these giant patterns.

Alternatively, we may need to extend our understanding of gravity beyond Einstein’s general theory of relativity. “It could be that we need an even more general theory to explain how gravity works on very large scales,” says Thomas.

A more mundane answer might yet emerge. Using colour to find distance is very sensitive to observational error, says David Spergel of Princeton University. Dust and stars in our own galaxy could confuse the dataset, for example. Although the UCL team have run some checks for these sources of error, Thomas admits that the result might turn out to be the effect of foreground stars either masking or mimicking distant galaxies.

Fractal structure?

“It will be essential to confirm this with another technique,” says Spergel. The best solution would be to get detailed spectra of a large number of galaxies. Researchers would be able to work out their distances from Earth much more precisely, since they would know how much their light has been stretched, or red-shifted, by the expansion of space.

Sylos Labini has made such a map using a subset of Sloan data. It reveals clumpiness on unexpectedly large scales – though not as vast as these. He believes that the universe may have a fractal structure, looking similar at all scales.

A comprehensive catalogue of spectra for Sloan galaxies is being assembled in a project called the Baryon Oscillation Spectroscopic Survey. Meanwhile, the Dark Energy Survey will use a telescope in Chile to measure the colours of even more galaxies than Sloan, beginning in October. Such maps might bring hyperclusters out of the haze – or consign them to the status of monstrous mirage.

by Stephen Battersby

Journal reference: Physical Review Letters, DOI: 10.1103/PhysRevLett.106.241301

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For some continued viewing on the subject, please watch the following BBC documentary entitled, “The Secret Life Of Chaos”.

The Secret Life Of Chaos

Chaos theory has a bad name, conjuring up images of unpredictable weather, economic crashes and science gone wrong. But there is a fascinating and hidden side to Chaos, one that scientists are only now beginning to understand.

It turns out that chaos theory answers a question that mankind has asked for millennia – how did we get here?

In this documentary, Professor Jim Al-Khalili sets out to uncover one of the great mysteries of science – how does a universe that starts off as dust end up with intelligent life? How does order emerge from disorder?

It’s a mindbending, counterintuitive and – for many people – a deeply troubling idea. But Professor Al-Khalili reveals the science behind much of beauty and structure in the natural world and discovers that far from it being magic or an act of God, it is in fact an intrinsic part of the laws of physics. Amazingly, it turns out that the mathematics of chaos can explain how and why the universe creates exquisite order and pattern.

And the best thing is that one doesn’t need to be a scientist to understand it. The natural world is full of awe-inspiring examples of the way nature transforms simplicity into complexity. From trees to clouds to humans – after watching this film you’ll never be able to look at the world in the same way again.

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## Koyaanisqatsi: Life out of Balance

### February 27, 2011

Life comes about through the unfolding of patternsFractal like patterns… Ones that repeat themselves over and over again… Subtly changing… Ever evolving… Into new rhythms and orchestrations of being…

For me, life seems to come about through the feelings that occur within these patterns of material structure… By using these structures – and the rhythms within these structures – to perceive with, more patterns from the outside world flood into our senses… And, somewhere down the line, our ‘minds’ couldn’t help but slowly evolve a schematic/memetic representation of the patterns that we perceive in the world/universe around us, which we now we learn off others through various channels of media funnelling… Which in turn allow the notion of our ‘self’ to build up a seemingly ‘real’ relationship between the outer and inner worlds. But regardless of what is ultimately real and what is only delusion/illusion, this process of perception that seems to make this world seem so real resides somewhere within a long, cascading chain of delicate and interrelated molecular mechanisms… Which we call our bodies… That were reliant on all the bodies before it, related to it, etc… in a long chain of cause and effect. With these bodies, we find our “selves” awash in an ocean of interrelated patternation… Of fabrication… Of perpetuation…

This fabric of perceived reality seems to be produced from within itself… As if each level were subtly stirring up the ones around it, so as to create in each other a unique, yet interrelated, turbulent, writhing flow… Each level emanates from the rest… Yet the rest emanate from it… Feeding back into and out of the others… Undulating across all levels of creation, perception of the universe seems to stem from a fractal-like feedback loop… From the smallest to the largest and all the way back again… Here, within this sea of chaos, arises our world… And what better way to become more familiar with it, than to watch it relayed back to us again… Through the lens of this magnificent film… One which stirred my imagination into a desire to understand my position within the natural order of things better… Clearing some of the slurry that had partially obscured my view during most of my adolescent years…

Here, the abstract patterns of creation that gave rise to the atomic matter that our bodies – and the material universe – seem to built from, now unfold in ever more complex ways… Ways that we seem to take for granted in an everyday, presumed, regular – almost clock work – reality…

In the following film, Godfrey captures a poetic sense of flow, viewing mankind’s life on Earth a sort of deluge of movement… It’s almost as if he’s giving us the chance to observe the daily patterns of our lives afresh, in a time-lapsed ‘sur-reality’… Providing us with an opportunity to re-evaluate them against the blank canvas of Earth’s barren and primordial landscapes. Perhaps one can clearly see that everyday life isn’t perhaps as obvious and as natural as it sometimes may seem… Especially to those of us locked in daily human importance… Through seeing this, I feel we are afforded a new insight into understanding how unbalanced our lives have become… How out of sync We – as human beings – are with the natural order of Heaven and Earth, and how oblivious we have become to nature’s delicate, chaotic dance. No doubt the contrast is obvious between those of us living in the ancient rain forests and those of us now residing within our concrete cities of self-obsession…

Besides this powerful warning, Godfrey never looses sight of the strange beauty – even synchronicity – of it all… And elegantly counters the weighty caution with a tapestry of visual delight that seems to beckon the viewer onwards, showing how evident even the idea self-similarity between two seemingly rather diverse concepts like a microchip and a city. Here we can see that the way we move even influences the way we think and work. Perhaps there is really no difference between the un-natural and natural… Save the ideas/memes/schemas that mankind uses to justify his beliefs with.

To find out more about these films, please read the official website’s press release by clicking here.

## The Secrets Of Intelligence Lie Within A Single Cell

### July 21, 2010

Having recently been to Dr Bruce Lipton‘s talk, entitled “The Biology Of Belief,” which was held in the Logan Hall of the Institute Of Education in London this last Saturday, the 17th of July 2010, I had reinforced the idea that we are nothing more than a bunch of atomic mechanisms, built from atomic polymers i.e. DNA, proteins, fatty acids, etc… all arranged into intricate cellular clusters, which – given the right circumstances – can function with amazingly natural flows of Being, demonstrating what we can only call, from a self referencing point of view, natural organic movements… And over the years we have – funnily enough – coined these flows to be “Life-Like.”

I really believe that when we begin to see Life in these terms i.e. that Life as we presently know it usually results from the complex interactions of the atomic machinery within an enclosed cellular body, which, if presented with more differentiated versions of itself, can build larger bodies from highly specialised cellular clusters… And then, once in place, out of all this unfolds a nonlinear biology/biochemistry of perceptive functions, all of which came about through the process of what we now know as ‘chaos’ – rather than the result of some divine intervention – and thus becomes nothing more than a complex, naturally occurring chaotic system that ‘intelligently’ reacts and responds, through effective behavioural patterns, to external environmental pressures and stimuli, precipitating survival habits that have been natural selected for… The behavioural patterns allow Life to survive in an ever changing environment, and the chaos inherent in our being affords us the ability to utilise the best survival traits that we can, one of which was the development of self-biased tendencies centred around a distinct notion of “self” and “body” that many of us seem to take for granted on a daily basis.

While I will eventually get around to discussing the reality and validity of the “self” in a future blog (something that is taking me much longer than I had anticipated)… In this blog I’d like try to discuss why this idea of viewing ourselves as a machine is a lot more natural and effective a notion about our “selves” than any previous egocentric notion about what we really are i.e. we were created by one or several Gods, in their own images to be special, etc… Certainly Dr Bruce Lipton’s analogy about us being a group of living cells which function within the confines of this body as a “community” of beings, each performing their own specific roles within the body’s mechanism i.e. just as governments regulate countries and their home economies, while police men arrest criminals, so do certain parts of the central nervous system function as regulators of heart rhythm and bodily temperature, while white blood cells kill of infections from ‘maliciously behaved’ bacteria… This idea of self-similarity within the patterns of Life that we see unfolding here on Earth across all scales and modes of Being will provide us with a very deep and intuitive understanding about the subtle and – what we tend to call – divine aspects of our Being, as well as showing us all how we interconnect and relate to this universally unfolding discourse..

Bearing in mind this ‘rule’ of self-similarity that seems to present itself within and throughout the whole of this universal dynamic so pervasively… And by viewing Life as a type of mechanisation… I am curious as to where – or from which level of scale – the emotive force of Life actually originates from? Is it at the level of the body i.e. does it directly and uniquely come from the sum of all its parts, where each individual part would be able to do nothing whatsoever by itself? Or is this trait of the emotive Life force buried deep down with in the cellular – or even the atomic – matrix? Certainly when we try to address what this experience of Life actually is and how it comes about we can hopefully begin to see it does not only belong to the body as a whole unit, but also comes from the various levels of functionality within the body i.e. at the cellular and atomic levels.

Just as Jung is concerned as much with the individual within society, as the individual is him/her “self” the measure of society, so too we can apply this analogy to the cell and body. Without the individual, society cannot function, let alone exist… And without the cell, the body cannot function or even exist. Life and its dynamism directly stems from the units that comprise the whole. These units, just as much as the whole, are all subject to the same forces and methods of development as each other i.e. those of nonlinear evolution. This ‘Life,’ and its essence, relies upon the parameters of these nonlinear, fractal eddies with their dynamics. These cellular bodies that make up our own larger bodies are driven by and made from the same underlying principles of naturally occurring algorithmic phenomena… Even though at first glance it might not be obvious that they are… But they are. Thus, if these algorithmic patterns reside across all levels of scale, shape and form, why shouldn’t we expect similar ‘intelligences’ to reside across all scales of these naturally occurring systems, whether at the human body’s level or a cellular level? Ultimately it’s up to you what you believe… But to function better I personally would like to know a little bit more about the processes that give rise this “I”; the processes that drive all of Life here on Earth – and possibly beyond too – rather than giving into dogmatic nodes of parrot fashioned understanding.

As Jung once wrote in “The Undiscoverd Self“:

Human knowledge consists essentially in the constant adaptation of the primordial patterns of ideas that were given us a priori. These need certain modifications, because, in their original form, they are suited to an archaic mode of life but not to the demands of a specifically differentiated environment. If the flow of instinctive dynamism into our life is to be maintained, as is absolutely necessary for our existence, then it is imperative that we remould these archetypal forms into ideas which are adequate to the challenge of the present.

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Our denominational religions with their archaic rites and conceptions – justified enough in themselves – express a view of the world which caused no great difficulties in the Middle Ages but has become strange and unintelligible to the man of today. Despite this conflict with the modern scientific outlook, a deep instinct bids him hang on to ideas which, if taken literally, leave out of account all the mental developments of the last five hundred years. The obvious purpose of this is to prevent him from falling into the abyss of nihilistic despair. But even when, a rationalists, we feel impelled to criticise contemporary religion as literalistic, narrowminded and obsolescent, we should never forget that the creeds proclaim a doctrine whose symbols, although their interpretation may be disputed, nevertheless possess a life of their own on account of their archetypal character. Consequently, intellectual understanding is by no means indispensable in all cases, but is called for only when evaluation through feeling and intuition does not suffice, that is to say, with people for whom the intellect holds the prime power of conviction.

In order to emphasise this re-equation that we need i.e. to understand that we are part of a whole ecosystem of Earth, just as a cell is part of the body’s ecosystem, it is here that I’d like to present an article which I read not too long ago in the New Scientist magazine… One that tackles this issue of where emotive Life comes from. When we see that Life’s organic flow resides across all levels of being i.e. atomic, cellular, bodily, biospherically, or even within the planet and its solar system, we might begin to understand that some of our older religious notions of the divine state of existence that We – that is, all Life – experience no longer need to be fantasised over or marginalised in any inaccurate way whatsoever. Now, through the doors of science, we can directly see the mechanisms of Life at work, and thus ‘understand’ the essence behind their patterns and interdependent interactions, all through which we gain the essence of our Being. Natural ordering comes from the patterns of chance and chaos, which give rise to development and originality within all universal systems, whether biological or otherwise. These systems, if given favourable circumstances/environments in which to start, can then begin the arduous process of developing into complex systems of environmentally perceptive and adaptive systems. Human beings are even beginning to use these recursive patterns – which have been called the “Thumb Print Of God” – in their technological developments i.e. to develop semi intelligent robotic systems that can learn fast and develop effective solutions to presented problems in ways that surpass anything we’ve tried or known before.

Thus, with these many new observations, I believe it is time to re-write our archetypal programming. Just as when I first saw the Mandelbrot Set on a postcard from a friend while at school and immediately recognised its tortuous, writhing flow as something so familiar and deeply ingrained in my being… So too do all ‘Gods’ leave this same feeling of familiarity… Of spirituality… And of deep connection to the whole… Here lies an answer to a new understanding… That self-similarity resides within all units of the whole… If you find intelligence within the body… Then why not within cell too… Or even in the atom… After all, one essence is usually found within the other, and so permeates through the entire being. Certainly atoms are just as discerning as human beings are… We all choose what we will or won’t react/socialise/breed with. Does this intelligence then go deeper? Intelligence that can be found within the proton, neutron and/or electron… And, if so, then why not even in the quark… Or the God particle…. Etc, etc, etc… ?

The Secrets Of Intelligence Lie Within A Single Cell

Late at night on a sultry evening, I watch intently as the predator senses its prey, gathers itself, and strikes. It could be a polecat, or even a mantis – but in fact it’s a microbe. The microscopic world of the single, living cell mirrors our own in so many ways: cells are essentially autonomous, sentient and ingenious. In the lives of single cells we can perceive the roots of our own intelligence.

Molecular biology and genetics have driven the biosciences, but have not given us the miraculous new insights we were led to expect. From professional biologists to schoolchildren, people are concentrating on the minutiae of what goes on in the deepest recesses of the cell. For me, however, this misses out on life in the round: it is only when we look at the living cell as a whole organism that wonderful realities emerge that will alter our perception not only of how single cells enact their intricate lives but what we humans truly are.

The problem is that whole-cell biology is not popular. Microscopy is hell-bent on increased resolution and ever higher magnification, as though we could learn more about animal behaviour by putting a bacon sandwich under lenses of increasing power. We know much about what goes on within parts of a cell, but so much less about how whole cells conduct their lives.

Currently, cell biology deals largely with the components within cells, and systems biology with how the components interact. There is nothing to counterbalance this reductionism with a focus on how whole cells behave. Molecular biology and genetics are the wrong sciences to tackle the task.

Let’s take a look at some of the evidence for ingenuity and intelligence in cells that is missing from the curriculum. Take the red algae Rhodophyta, in which many species carry out remarkable repairs to damaged cells. Cut a filament of Antithamnion cells so the cell is cut across and the cytoplasm escapes into the surrounding aquatic medium. All that remains are two fragments of empty, disrupted cell wall lying adjacent to, but separate from, each other. Within 24 hours, however, the adjacent cells have made good the damage, the empty cell space has been restored to full activity, and the cell walls meticulously realigned and seamlessly repaired.

The only place where this can happen is in the lab. In nature, the broken ends of the severed cell would nearly always end up remote from each other, so selection in favour of an automatic repair mechanism through Darwinian evolution would be impossible. Yet something amazing is happening here: because the damage to the Antithamnion filament is unforeseeable, the organism faces a situation for which it has not been able to adapt, and is therefore unable to call upon inbuilt responses. It has to use some sort of problem-solving ingenuity instead.

We regard amoebas as simple and crude. Yet many types of amoeba construct glassy shells by picking up sand grains from the mud in which they live. The typical Difflugia shell, for example, is shaped like a vase, and has a remarkable symmetry.

Compare this with the better known behaviour of a caddis fly larva. This maggot hunts around the bottom of the pond for suitable scraps of detritus with which to construct a home. Waterlogged wood is cemented together with pondweed until the larva has formed a protective covering for its nakedness. You might think this comparable to the home built by the testate amoeba, yet the amoeba lacks the jaws, eyes, muscles, limbs, cement glands and brain the caddis fly larva relies on for its skills. We just don’t know how this single-celled organism builds its shell, and molecular biology can never tell us why. While the home of the caddis fly larva is crude and roughly assembled, that of the testate amoeba is meticulously crafted – and it’s all made by a single cell.

The products of the caddis fly larva and the amoeba, and the powers of red algae, are about more than ingenuity: they pose important questions about cell intelligence. After all, whole living cells are primarily autonomous, and carry out their daily tasks with little external mediation. They are not subservient nanobots, they create and regulate activity, respond to current conditions and, crucially, take decisions to deal with unforeseen difficulties.

“Whole living cells are not subservient nanobots, they respond and take decisions”

Just how far this conceptual revolution about cells could take us becomes clearer with more complex animals, such as humans. Here, conventional wisdom is that everything is ultimately controlled by the brain. But cells in the liver, for example, reproduce at just the right rate to replace cells lost through attrition; follicular cells create new hair; bone marrow cells produce new circulating blood cells at a rate of millions per minute. And so on and on. In fact, around 90 per cent of this kind of cell activity is invisible to the brain, and the cells are indifferent to its actions. The brain is an irrelevance to most somatic cells.

So where does that leave the neuron, the most highly evolved cell we know? It ought to be in an interesting and privileged place. After all, neurons are so specialised that they have virtually abandoned division and reproduction. Yet we model this cell as little more than an organic transistor, an on/off switch. But if a red alga can “work out” how to solve problems, or an amoeba construct a stone home with all the “ingenuity” of a master builder, how can the human neuron be so lowly?

Unravelling brain structure and function has come to mean understanding the interrelationship between neurons, rather than understanding the neurons themselves. My hunch is that the brain’s power will turn out to derive from data processing within the neuron rather than activity between neurons. And networks of neurons enhance the effect of those neurons “thinking” between themselves. I think the neuron’s action potentials are rather like a language neurons use to transmit processed data from one to the next.

Back in 2004, we set out to record these potentials, from neurons cultured in the lab. They emit electrical signals of around 40 hertz, which sound like a buzzing, irritating noise played back as audio files. I used some specialist software to distinguish the signal within the noise – and to produce sound from within each peak that is closer to the frequency of a human voice and therefore more revealing to the ear.

Listening to the results reprocessed at around 300 Hz, the audio files have the hypnotic quality of sea birds calling. There is a sense that each spike is modulated subtly within itself, and it sounds as if there are discrete signals in which one neuron in some sense “addresses” another. Could we be eavesdropping on the language of the brain?

For me, the brain is not a supercomputer in which the neurons are transistors; rather it is as if each individual neuron is itself a computer, and the brain a vast community of microscopic computers. But even this model is probably too simplistic since the neuron processes data flexibly and on disparate levels, and is therefore far superior to any digital system. If I am right, the human brain may be a trillion times more capable than we imagine, and “artificial intelligence” a grandiose misnomer.

I think it is time to acknowledge fully that living cells make us what we are, and to abandon reductionist thinking in favour of the study of whole cells. Reductionism has us peering ever closer at the fibres in the paper of a musical score, and analysing the printer’s ink. I want us to experience the symphony.

by Brian J. Ford

Despite the authors final sentiments, I still feel that this reductionism does provide us with certain, otherwise unobtainable, clarities for understanding the similarities between the processes within and without… After all, one needs to know how to make paper and ink, and understand something about the musical scoring technique before they can write a symphony down for the future enjoyment of others…

## 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.

## What Was The Beginning Of It All?

### November 26, 2009

While I was watching a film the other day, I happened to come across a quote at the end among the credits… And having read it, I found that it touched a deep resonant chord within me about the nature of our reality and Being… And from this “chord” posited a beautifully simple, yet wondrously elegant, understanding about why We came into being…

For… When given the chance to do something… Surely it is in our nature to do, rather than not? Probability is everything… And as chance sometimes provides that possibility, so atoms take a chance on complexity, coming together in a gradually evolving divine countenance of planetary habituation… A habituation that many of us still see as “every-day” Life.

Perhaps as and when we choose to depart from this mundane acceptance of existence, and clearly stride towards a better realization of self, we might begin to See and understand what it is that Sri Aurobindo talks of here…

Sri Aurobindo (Aurobindo Ghosh) (August 15, 1872 December 5, 1950) was an Indian nationalist and freedom fighter, poet, philosopher, and yogi. He joined the movement for India’s freedom from British rule and for a duration (1905 10), became one of its most important leaders, before turning to developing his own vision and philosophy of human progress and spiritual evolution.

The central theme of Sri Aurobindo’s vision is the evolution of life into a “life divine”. In his own words: “Man is a transitional being. He is not final. The step from man to superman is the next approaching achievement in the earth’s evolution. It is inevitable because it is at once the intention of the inner spirit and the logic of Nature’s process”.

The principal writings of Sri Aurobindo include, in prose, The Life Divine, The Synthesis of Yoga, Secrets of the Vedas, Essays on the Gita, The Human Cycle, The Ideal of Human Unity, Renaissance in India and other essays, Supramental Manifestation upon Earth, The Future Poetry, Thoughts and Aphorisms and several volumes of letters. In poetry, his principal work is “Savitri – a Legend and a Symbol” in blank verse.

## 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 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.

## Self-Similarity ~ An Idea For “Knowing” and “Understanding”

### September 9, 2009

In mathematics, a self-similar object is exactly or approximately similar to a part of itself i.e. the whole has the same shape as one or more of the parts. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals.

Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually iterated and magnified infinitely without changing shape.

However, the Mandelbrot set is a symmetrical but scale-varient. As one zooms into it’s rough surface it changes shape in beautifully diverse ways and yet still has “memories” of the whole locked deep within itself. The M-Set is a common example of a type of affine fractal i.e. the images of the whole locked within are slightly altered via ‘scaling,’ ‘rotational’ or ‘shearing’ types of transformation.

On that note, I would like to present to you, the reader, an idea that is based on scientific observation:

Now… Keeping in mind what we have just seen… AND remembering that the above image is derived from only two fractal-like structures found in nature i.e. it could hardly be used as absolute conclusive evidence… I would like to pose a question to you, the reader. Are we using self-similar structures and processes of the “immense whole” to ponder and perceive the universe around us with? After all, the brain itself is a self referential system, that uses previous memories of experience to understand itself and the surrounding environment with…

“The question of the self has intrigued philosophers and psychologists for a long time. More recently, distinct concepts of self have also been suggested in neuroscience. However, the exact relationship between these concepts and neural processing across different brain regions remains unclear. This article reviews neuroimaging studies comparing neural correlates during processing of stimuli related to the self with those of non-self-referential stimuli. All studies revealed activation in the medial regions of our brains’ cortex during self-related stimuli. The activation in these so-called cortical midline structures (CMS) occurred across all functional domains (e.g., verbal, spatial, emotional, and facial). Cluster and factor analyses indicate functional specialization into ventral, dorsal, and posterior CMS remaining independent of domains. Taken together, our results suggest that self-referential processing is mediated by cortical midline structures. Since the CMS are densely and reciprocally connected to subcortical midline regions, we advocate an integrated cortical-subcortical midline system underlying human self. We conclude that self-referential processing in CMS constitutes the core of our self and is critical for elaborating experiential feelings of self, uniting several distinct concepts evident in current neuroscience.”

Northoff Georg; Heinzel Alexander; de Greck Moritz; Bermpohl Felix; Dobrowolny Henrik; Panksepp Jaak. “Self-referential processing in our brain–a meta-analysis of imaging studies on the self.”
NeuroImage 2006;31(1):440-57.

So… Perhaps the idea of Grand Universal Structures being used to perceive Themselves with, is not as “outlandish” an idea as one might at first think… When one looks around in nature, we continually find self similar patterns/structures:

Ferns are in fractal shape.

A Romanesco Cauliflower exuding self-similar structures.

It has even been postulated that “many real world network systems obey a power-law scaling, just as if they were fractal shapes.”

Drainage patterns on the Earth's surface, taken from over 100 miles up.

Have you ever wonder why careful geologists always include a scale or reference when taking a picture of geologic interest? The reason is that, if they didn’t, the actual size or scale of the object pictured could not be determined. This is because most geoforms are self-similar, i.e. a fold 1 cm long looks quite the same as if it is 10 meters or 10 kilometers long. The same with most fault systems, layering, foliations, coastlines, topographic features, drainage patterns (like the one above), etc…

Self-similarity is a clue to the origin of many physical systems whose internal dynamics may be complex, in the sense that the system is at a critical state between chaos and order; a condition that has been called a self-organized critical state. A very readable account of Self-organized criticality is in the recent book by Per Bak, entitled “How Nature Works.”

For many years geologists have complained that classical math and physics are too simplistic in their representation of nature to be useful to the geologist. Now there are no more excuses for that kind of drivel. Fractal geometry, chaos theory and the science of complex systems do accurately represent many geosystems. Learning about fractals and chaos theory will considerably extend your understanding of geosystems and the workings of nature.

Even evolution is involved… Steve Jay Gould has this to say about fractals and evolution: “Finally, this pattern of long stasis, with change concentrated in rapid episodes that establish new equilibria, may be quite general at several scales of time and magnitude, forming a kind of fractal pattern in self-similarity” (Sc. American, Oct 1994).

And, as if to end this meditation on a high note… I would like to bring to the attention of the reader the following video, compiled by the late Sir Arthur C. Clarke, entitled “The Colors Of Infinity” which discusses the ideas of fractals in a novel light.

Postscript

Again… I would like to bring to the reader’s attention 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, most of which have so far been reviewed within this Blog, new modes of possible understanding about ‘what’ We are and ‘why’ We came about in this Garden Of Eden that orbits around a bright furnace, one that, like all the others before it, forges all the known matter into what we are today.

## Romanesque Networks

### March 30, 2009

I read this article back in 2005… And as I feel it brings credence to some of the ideas I have discussed thus far, I am now posting it here for all to see:

## Self Similarity, Diversity, Divergence and Evolution

### March 30, 2009

Having introduced the idea of infinity, I would like to now introduce and develop four other important ideas. Namely ‘Self Similarity’, ‘Diversity’, ‘Divergence’ and ‘Evolution’. These ideas will become the back bone to most of the future writings found on this blog.

1. Self Similarity

In mathematics, a self-similar object is one that exactly or approximately looks the same on any scale i.e. the whole has the same shape as one or more of the parts. Many objects in the real world, such as coastlines and fluctuations in stock market prices, are statistically self-similar: parts of them show the same statistical properties at many scales.

It is also known that self-similarity is a typical property of fractals. A fractal is generally “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole, as stated in Mandelbrot’s The Fractal Geometry of Nature (1982 W. H. Freeman and Company). The term “fractal” was coined by Benoît B. Mandelbrot in 1975 and was derived from the Latin word fractus meaning “broken” or “fractured.” A mathematical fractal is based on an equation (in the case of the Mandelbrot set, the complex quadratic polynomial zn+1zn2c ) that undergoes iteration, a form of feedback based on “recursion” i.e. an expression, such as a polynomial, each term of which is determined by application of a formula to preceding terms.

When a fractal equation is iterated ad infinitum OR infinitely, they are then considered to be infinitely complex. Through out this infinitely complex structure self similar patterns abound through out the whole on all scales.

Perhaps now would be a good time to develop the idea of fractals further. Below I have provided a link to an hour documentary narrated by the late Sir Arthur C. Clarke:

Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, lichen and snow flakes. Some man made systems are fractal too i.e. stock market prices have been shown to be fractal in nature.

i. Coastlines:

http://en.wikipedia.org/wiki/How_Long_Is_the_Coast_of_Britain%3F_Statistical_Self-Similarity_and_Fractional_Dimension

ii. Stock Market Prices:

http://www.sciam.com/article.cfm?id=multifractals-explain-wall-street

iii. Lichen

Here is a picture of a lichen pattern:

Below is a picture that I generated using Corel Draw’s ‘Fract Flame’, where I’ve used a similar color scheme to that found on the lichen above:

Notice anything similar between the the two of them?

iv. Bacterial Growth

Professor Herbert Levine discovers bacterial fractal patterns!

While the above eddies and flows of growth might not be quite as stable a near perfect mathematical models (possibly due to imperfections in the surface structure of the agar), there is none the less a similarity shared with the Mandelbrot set.

A zoom into the M set.

v. Obviously if one was to zoom into one of these real life objects, such as a snow flake for example, one would eventually loose sight of the self similar cascade of iced tips, as molecules of atoms, then atoms themselves, then electrons, protons and neutrons, and eventually quarks became visible. It’s almost as if there are sudden disparities between two structural scales of observation; a boundary where self similarity flips from one larger field of view into another smaller field.

However, even then, self similar structures that are seemingly unrelated to one another can jump out at an observer from the most unlikely of places:

Taken from the New York Times.

vi. Cosmological Phenomenon

The Cat’s Eye Nebula (NGC 6543) is a planetary nebula in the constellation of Draco. Structurally, it is one of the most complex nebulae known, with high-resolution Hubble Space Telescope observations revealing remarkable structures such as knots, jets and sinewy arc-like features.

Above is an outline of the Julia set, another very complex mathematical structure exuding infinite complexity, which structurally resembles the Cat’s Eye nebula.

vii. Generating Fractals

If you would like to view a fractal on your home computer, you can download a freeware program called GNU Xaos (which has both a PC and Mac version available) from the following link:

http://wmi.math.u-szeged.hu/xaos/doku.php

Note: obviously the above pictures are only meant as suggestions to coax one into keeping an eye out for new ideas and patterns observed in nature’s mesmerizing flow.

2. Diversity

In the English dictionary, the word ‘diversity’ makes a point about difference. It is defined as:

a. the state or fact of being diverse; different; unlikeness

b. variety; multiformity

c. a point of difference

It denotes separateness, division, discrepancy, fluctuation, heterogeneity, incongruity,  inconsistency, even mishmash. For example, when we talk about biodiversity, we often look as the variation of life forms within a given ecosystem, biome, or for the entire Earth, at given time. The biodiversity found on Earth today consists of many millions of distinct biological species, which is the product of around and about 3.5 billion years of evolution.

But where did all these distinct biological species come from? Were they always here? Was difference always present? Or did they evolve, as Darwin’s theory of evolution states, from one ‘point’ i.e. one common ancestor? Did they transmutate over ages, slowly being reinvented as iterated nuances of the original form, redefining their habits into separate niches that gave them greater domain to sustain their lively needs and modes of being, modified ad infinitum into a distillation of harmonious countenance with the surrounding and ever changing environment, so as to separate inordinately competitive struggles into a slightly more relaxed interplay?

3. Divergence

In “The Vestiges Of The Natural History Of Creation”, written by the Scottish journalist Robert Chambers, though anonymously published in October of 1844 for fear of ridicule, the idea that natural phenomena arise and evolve through natural laws of development. It also “boldly” postulated that there could therefore be some sort of rational explanation as to how everything in the universe came into being… “The whole train of animated beings, from the simplest and oldest, up to the highest and most recent, are then to be regarded as a series of the principles of development. It has pleased providence to arrange that one species should give birth to another, until the second highest gave birth to man.” However, due to the orthodox views of the time, it suffered greatly at the hands of many critics.

Darwin, having been discouraged by the cruel obloquy that the “The Vestiges Of The Natural History Of Creation” had received, decided to postpone publishing his ideas on evolution until he had bolstered the theory with near irrefutable evidence. One thing in particular troubled him about his concept… And in hind sight he wrote, “At that time I overlooked one problem of great importance. The problem is the tendency in organic beings descended from the same stock to diverge in character as they become modified.” Darwin further noted in his quest for refinement of the basis of evolution, the principle for divergence, which turned out to be the missing piece of his great puzzle. In November 1854 he wrote, “And I can remember the very spot in the road whilst in my carriage, when to my joy, the solution occurred to me. The solution, as I do believe, is that the modified offspring of all dominant and increasing forms tend to become adapted to the many and highly diversified places in the economy of nature.”

So, in short, Darwin realized that the more individual species differed from each other, the better able they would be to take advantage of the particular environment in which they all shared. Just as importantly, species would adapt even more as they adjusted to each other. And it is this interdependence, this balancing out of supply and demand, that has a parallel in what would eventually become the Victorian factory system. As can be seen from Adam Smith’s ideas about industry, a wonderful resemblance between the notion that one can produce more wealth if one has people who are specialists i.e. instead of everyone being farmers, if some people became tailors, while others become leather workers, you can produce more wealth of better quality, than if everyone was to do everything themselves i.e. be a jack of all trades and a master of none.

After all, using a notion of Self Similarity, a branch of a tree comes from the main tree trunk, which stemmed up from seed and nut… And no two branches ever precisely overlap. For what would be the efficiency of leaf upon leaf upon leaf, stacked one a top another with only the upper most being exposed to the Sun’s light?

So perhaps the ‘point’ of life’s diversity (the main trunk) could have ‘seeded’ from this simple idea of non living matter transmuting into a living form:

http://blog.wired.com/wiredscience/2008/09/biologists-on-t.html

But perhaps I should let a “grand” documentary describe how basic tenets of these ideas gave rise to the intermingling of Self Similarity and Diversity. After all, man’s folly for almost time immemorial was to separate himself from nature’s base of animal like bewitchery. How could ‘He’, descendant of Adam, be linked to the surrounding unenlightened, earthy, distinctly corporeal and depraved natural world? Darwin, having been slammed by his colleges from Cambridge for his heretical ideas about man’s ‘obvious’ links to the natural world, found new angles of commonality within the real world which provided a solid and sure footing for his theory of evolution to remain in the fore front of scientific review, regardless of theistic doctrine…

Darwin’s Struggle – The Origin Of Species

Diversity, seen in pigeons… Natural selection, seen in corporate pressures of the industrial world… What more does one need to see that Self Similarity flows unifyingly through the Diversity of life and all of creation, man-made or natural? A Diversity that originated through Self Similar patterns branching out and away from the ‘seed’ of life’s origin, chaoticly evolving through time’s eternal languid flow, using fixed laws of physics, chemistry and biology, iterating subtilties into ever more refined complexities of balanced spread, till entwined ecosystems abound throughout the world!

But what perhaps brings me closest to Darwin was his compassionate and humanistic direction through the comparison of his children to the orangutan babies he had seen in London. We know from his diaries that he loved his children with all his heart, and so this comparison is not a debasement of their being. Rather he is ‘painfully’ aware of the similarities between both human and orangutan, as well as mankind’s own denial of the obvious truth.

Clearly it can be seen just by zooming into the Mandelbrot set that Diversity and Self Similarity abound in beatific balance. And, as M. C. Escher duly noted in the endeavors of his life’s work, these patterns also abound in the natural world, and flow from one another’s essence…

M. C. Escher's complete Metamorphose print.

Darwin noticed that even the hexagons created by honey bees in their hives emanated from instinct rather than divine providence. So what he realized was that, while divine origin gave credence to the existence of these sturdy and structured forms which Euclid had disclosed and discussed in his Elements, it was innate in their being to do so. Much in the same way, as the previous photographs of a brain cell and the universal arrangement of galaxies demonstrates, the idea of the whole is used to know itself.

This notion that we are part of universe knowing itself is perhaps not such a divine myth as one might initially presume. Rather when empirical knowledge is in place, it becomes evident. After all, we are all comprised of atoms, which nearly everything in the universe is made from. So would we surely not use the same forces and methods that the universe uses to simply be?

4. Evolution

But perhaps it is What Darwin Didn’t Know that really solidifies the ideas of Diversity and Self Similarity…

What Darwin Didn’t Know:

As one can probably see, the exquisite diversity found here on Earth stems back from time immemorial. Having been a molecular geneticist myself, I too have seen good cause for the ‘evidence’ that DNA has illuminated. Namely that the beautiful unfolding of species throughout the eons of time slowly distill back into one basic precursor of life, that central ‘point’ if you will, where we all came from.

Obviously not every branch will survive all the way through to the present day. Resources are limited and space is finite here on our spherical world. The “Terrible Lizards” were the dominant vertebrate animals of terrestrial ecosystems for over 160 million years, from the late Triassic period (about 230 million years ago) until the end of the Cretaceous period (65 million years ago). However, Dinosaurs eventually became extinct. What happened to them, no one exactly knows. But the fossilized remains of their bones are shinning evidence that these creatures once had the upper hand, as humans now do.

When I zoom into the Mandelbrot set, it’s almost as if similar patterns to what we vaguely know about our past, can be found writhing and rippling through it’s sublime topography. Subtle changes that allow greater efficiency and diversity, so that life settles into a new nest of chaotic equilibrium. As the first law of thermodynamics mandates the conservation of energy… So how better would life conserve energy, other than to diversify and specialize, rather than engage in the draining aspects of unnecessary competition?

But what astounds me most is that this rather obvious (but much over looked) beauty of fractals also gives credence to the idea that self similarity has been repeating throughout the ever evolving structures of organic life here on earth… Over the course of billions of years, basic structures have been reused again, and again… Eyes, teeth, brains, stomachs, bones, limbs, etc… All these anatomical textures of interwoven molecular weaves have ‘worked’ for life in some way or another, and over the years have simply been reworked/refined into better ideals, so that better ‘results’ occur. Some of these organs occur in very successful combinations, while others in fleetingly strange and unviable anomalies that only serve to demonstrate nature’s ‘bugged out’ chaotic approach to this complex chemical reaction called life.

Obviously it can be seen that universal modes of being have simply been reused, recombined and refined over and over again to give rise to what we now know and see around us today. I would hazard a guess that the eyes of Dinosaurs and man do not differ that much from one another. And I’d bet that they too would have had very similar internal organs to our own. And yet the outer bodies only vaguely (if at all) resemble one another’s. Just in this way, as I delve into the M set, I see islands of similarity in an eternal sea of change. Could there be some credence to my mode of thinking?

That I will leave for you to decide…

Postscript

Again, I would like to bring to the reader’s attention 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 that appeal deeply through intuition to my reasoning, new modes of possible understanding about ‘what’ We are and ‘why’ We came about in this Garden Of Eden.

## 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