4.2.5 Applications Of Chaos Theory, Fractal Geometry And Self-Similarity

Critical Mass and in particular the butterfly effect interested me because it is connected with, and indeed arose from, studies that the meteorologist Edward Lorentz was doing on the weather, which is one thing in nature that I always feel is as unpredictable as human behaviour.

And it was while doing these studies that the rudiments of chaos theory were formulated, and connections were made by Lorentz with previous studies in physics in particular Heisenberg’s Uncertainty Principle.

It is also interesting to note, in considering systems theory – that is, how everything affects everything else – that the growth of interest in chaos (and indeed, an appreciation of chaos) was not confined to the world of mathematics and physics.

In music, chaos was evident in the growth in popularity of jazz.  It was around this time that it became acceptable to write poetry that didn’t rhyme, and modern art also had a far more chaotic aspect than the works of the old masters of the Renaissance, or even the impressionists.

Of relevance to this website, it was also around this time that Sigmund Freud [1] was formulating his groundbreaking theories on how unhappiness, discontentment and even mental illness in adults were linked to suffering as children – challenging many long-held beliefs about people being mad, or possessed by devils etc.

And perhaps Heisenberg and his colleagues’ unconscious minds were more open to uncertainty than previous generations of scientists.  Systems theory would suggest this. After all, uncertainty had been gathering some momentum in European society from the mid-19^th Century on. 

Looking back today on the late 19th or early 20th Century Western World, it is hard to imagine how different this must have been for people who always knew their place in society – and were virtually certain of it from birth to death.  It can’t be a coincidence that allowing every adult to vote (first all men, but some decades afterwards all men and women) that led to so much uncertainty paralleled the scientific, social and medical upheavals of that period.

(As an aside, it is worth noting that democracy, where the general public vote for candidates that put themselves forward to represent the ordinary citizen’s interest in a secret ballot – that we take for granted nowadays, is only about 140 years old – in the Western World anyway. But in what the Western World regards as primitive societies, the democratic selection of wise elders as leaders is, and always has been commonplace).

The Uncertainty Principle was initially concerned with the behaviour of sub-atomic particles but Lorentz applied the principle to the weather, and from it developed general theories on the incidence of chaos in nature.

Still further developments included applications in the general area of evolution of systems that can be deemed to be complex.  I define complex here to mean a system that has many variables, that is; things that change (vary) over time.

The uncertainty principle suggests that the evolution or the growth over time of a complex system cannot be accurately predicted.

When applied to humans, I suggest that the family is an example of a complex system, as it contains many elements that vary over time.

Uncertainty and chaos have relevance and/or implications in child protection within families affected by imprisonment – and their relevance and implications need to be taken seriously.

[1]. The father of modern psychotherapy, Sigmund Freud, who was a medical doctor, developed theories focused on the importance of healthy emotional development to ongoing mental health.  Much reading on Freud and his theories is available from a wide variety of sources.

Prior to exploring the implications of chaos in respect of human behaviour, and to assist in greater understanding, I will briefly introduce two subjects that are important in unpacking the world of chaos.

These are fractal geometry and its hallmark self-similarity.

(The Polish/French mathematician Benoit Mandelbrot introduced the notion of fractal geometry and a wide variety of literature is available on the rather complicated mathematical theories if you are interested).

For our purposes I will define a fractal as a pattern that repeats itself through time. Here is one explanation/demonstration of fractals on Youtube and there are many others if you want to search.

A simple example of the manifestation of both fractal geometry and self-similarity in nature is the growth of a tree.

We see a trunk, surrounded by sturdy branches, from which smaller branches emerge, with still smaller branches from those, and then twigs, smaller twigs, and then leaves, and within the leaves are capillaries from which smaller capillaries emerge.

The fractal part is the breaking up of the main trunk into smaller branches and still smaller and so on.  The self-similarity part is a (relatively recently worked out mathematically, but observed since the dawn of humanity) symmetry in nature where the overall pattern of the fractal is repeated at multiple scales.

That is, even a very small twig from which grows even smaller twigs will be similar to (but, and importantly, not exactly the same as) the trunk itself from which grows the branches [1].  The chaos part comes in when we consider that we cannot predict from looking at an acorn what exactly the oak tree will look like in a hundred years’ time.  But despite our not knowing what it will exactly look like, we know that it will follow a particular pattern as it grows – and not turn out like a sycamore tree, or a head of cabbage!

Like snowflakes, (which are actually more amazing as they are not growing – but that is another story) all trees are alike but all are different also.

Another example is blood flow in our body.  We observe the heart from which extends the aorta, the biggest and most important artery, surrounded by big arteries to the principal organs from which emanate smaller arteries and then even smaller ones again and then capillaries and then the tiniest of blood vessels close to the skin.

Here are some pictures (nowadays called images for some reason) of self-similarity in nature.  The first is a fern leaf, the second portrays blood flow in our body, and the third is an elderberry twig. 

The self-similarity can be clearly seen in the pictures.

Once again let us look at our oak tree.  The tree, starting from an acorn, will get nourishment and sustenance from water and soil but once it breaks through the soil it will get energy, nourishment etc. from the sunlight and moisture from rain on its leaves.

When I consider fractals and self-similarity in nature, I always factor in growth!

As it grows bigger and bigger it is dependent on both to grow – if the ground dried up, or the soil turned to sand, or the sun stopped shining or the rain stopped falling, the tree would stop growing.

This implies that growth is dependent on the strength and solidity of the trunk, what I will call the centre of growth and the fragile, delicate and barely visible capillaries on the extremities of the leaves – what I will call the edge of growth.  I will link these terms to society and family in a following post.

[1]. The differences between the individual branches, twigs, leaves etc. can be described by a mathematical property called non-linearity.  If it was generated by a computer instead of growing in nature, the fractal could be exactly the same, as it could be given linear properties by the person who created the programme.  If you type computer generated fractals into a You-tube search you will find loads. They can have a hypnotic effect.

In the previous post I used an oak tree to describe self-similarity. I’d like to give a few more examples here, just so you can be more familiar with it.

As I live in a city where there is a big river (the Shannon) I often think of self-similarity as manifest in watercourses.

There are firstly, raindrops settling onto blades of grass which seep downwards into the topsoil and form tiny rivulets a few mm wide and deep, which then join together to form little gullies.  These eventually unite to form bigger flows which become small streams – then bigger streams – then smaller rivers, tributaries of larger rivers which then flow into big rivers which flow into a big estuary and then to the sea.  They are all unique and different but they are self-similar.

Waves are another example of self-similarity, whether they are in the ocean or cascading over a waterfall.  So are the flames in an open fire. Both form patterns that we seem to be drawn to and as such have an almost mesmerising effect on us.  How many of us have looked at waves or flames for a long time – transfixed by the constant motion and pattern of the ever-changing-but-still-staying-the-same movement.

Fractals also abound in the arts – probably more in songs than in any other art form. It’s the constant repetition of the melody in the different verses and, of course, the chorus that make songs attractive. Imagine a song with different meter and rhyme and/or melody for each verse – it would not be that attractive.

Getting away from nature and songs, a man-made example of the manifestation of fractal geometry would be a big highway or motorway from which main roads exit, then many secondary roads joining the main roads, still smaller roads extending from the secondary roads and still smaller lanes and boreens and then driveways to people’s houses and even paths to their doors.  Once again it is easy to see the self-similarity as the pattern of a small road with boreens extending from it is similar to but not the same as a big motorway with main roads exiting.

In the corporate/business world, it is easy to see self-similarity (though probably not as spectacularly as a motorway) in a big central headquarters with loads of staff, and smaller offices/premises with lesser numbers of staff with less responsibility, and then still smaller premises to deal with issues that arise at ground level. Most armies are organisationally even more self-similar than civilian businesses, with corps, divisions, brigades, battalions, companies, platoons, and sections.

In the next post I will discuss fractal geometry and self-similarity as they apply in the family – our main concern.

Upward and downward causation were described in the Chapter on Systems Theory – and in this post I will link both to self-similarity and fractals using the analogy of the tree to describe the phenomenon.

Consider, first, self-similarity and fractals in sport.  Let us take the example of soccer, which is a worldwide sport.  We could say that the main trunk is the World Cup.  The European Champions League and the Copa Libertadores are big thick branches and the premier leagues of the traditionally successful countries like England, France, Spain, Italy, Germany, Brazil and Argentina etc. are fairly thick branches too.

The lower divisions of leagues are smaller branches and local, small country leagues are the large twigs.  Street leagues, inter-firm and the like are the small twigs; and young lads and lassies kicking a football around a field are the leaves, nourishing the local leagues (arguably, the edges of growth in soccer) which in turn nurture the small country leagues, and so on up to the biggest leagues and international competitions. The praise and encouragement the children get from their Mams and Dads is like the sunlight and rain on the leaves of the tree.

Without the praise and encouragement there’d be no growth! But the growth is also dependent of the strength and solidity (and wealth) of the World Cup, (arguably the centre of growth) through which the game in promoted world-wide.

Looking at the worldwide popularity of soccer from the perspective of systems theory, there is downward causation from the great heroes playing in front of crowds of 80,000 shown on TV to tens of millions worldwide, and upward causation from the lads and lassies kicking a ball around the park getting praise and encouragement from Mammy and Daddy!

Now, to get down to more serious business – let us consider fractals and self-similarity in respect of violence and peace in the world, once again using the example of the tree.

Take, for example, the 20th Century.

In terms of violence, the tree trunk is, for example, World War One and World War Two.  The big branches might be thought to be the Spanish Civil War, Vietnam War, Korean War, Iran-Iraq, etc.

Then we have the various wars in the Middle East and lesser wars such as African wars of independence, lesser again are our own Troubles, the conflict in the Basque country in Spain, wars in Central America.  The smaller branches would be the so called drug wars in certain countries that are blighted by same, and the violence of criminal gangs. The twigs might be the nightly drink fuelled random violence on the streets of our cities and towns and the leaves could be the violence in our homes.

Like the sunlight gives energy to the leaves, and this energy passes through the twigs and small branches through to the trunk, the tension, stress, anxiety, emotional irresponsibility and violence in our homes, (mostly hidden from view), fosters the street violence through to the drugs wars and has upward causation to wars between countries.

Now, in respect of peace, and to counter all the above negativity, let me say that the love and compassion, empathy and gentleness in our homes and families (also, mostly hidden from view) has upward causation, i.e. nurtures friendship and camaraderie, cooperation and reciprocity in communities, and generosity, forgiveness etc. in society in general. 

This is the edge of growth of worldwide peace and harmony – which is why it is so important that the family is at the centre of endeavours to promote such desirable characteristics in humanity.

Curiously, there doesn’t seem to be a lot of in-our-face evidence of the equivalent thick branches and tree trunks of compassion or peace to mirror those of violence, e.g. the hugely destructive wars.

But digging a little deeper, away from the major news stories and history we learned in school, (which were all about wars – in my time in school anyway – maybe it is different now) got me thinking of what qualities of compassion in our Irish homes filtered upwards to communities and society in general, eventually leading to the Good Friday Agreement – which was a major peace treaty that brought 30 years of the Troubles to an end.

Another example of goodness in action was the generosity and goodwill that must have filtered upwards from ordinary people’s homes in the Soviet Union and the rest of Eastern Europe that first threw up a leader like Mikhail Gorbachev and then enabled him to dramatically change his own country and half a continent for the better. (Though sadly, in more modern times, that hasn’t been sustained).

And there are many others if we look hard enough. For example, Mahatma Gandhi in India in the 1930’s and 40’s, and Martin Luther King in USA in the 1960’s, where the compassion in ordinary people’s homes was a counterbalance to the opposite at volatile times in the histories of their countries.

And finally what about child protection?

Is there an element of downward causation in the way that the State, through its economic choices, neglects swathes of the population and the way that some parents, through their economic choices, neglect their children?

Just a thought!

Generally, thinking about upward causation of non-violence and harmony, my thoughts drifted to Martin Luther King’s inspiring statement about the subject; that those who love peace must learn to organise as effectively as those who love war.

And I totally agree with him, which is why I have a long Chapter on Organisational Matters later.

I am generally optimistic in my lookout but I acknowledge that this will be a slow process.  The peace movements of MLK’s time in Western democracies that ended the Vietnam War and promised much change in the world frittered away, swept away by the Pillars of the time.  They (whoever they were) were not going to allow upward causation from ordinary people to stop wars!

Through a mixture of prejudice, shallow thinking, apathy, selfishness, media control, promotion of fear (all well-organised by the way) and indeed violence itself they were far better able to organise themselves than the peace promoters.

Much the same happened with the Arab Spring some decades later.

So upward filtering of non-violence, respect and harmony whether it is initiated in a top-down way by leaders like Mikhail Gorbachev and Martin Luther King, or by the more ground-up nineteen sixties peace movements, is certainly work in progress.

Later on in the website I will propose that smaller sized self-similar units may be more effective in our efforts to up-scale a cultural movement than a large centrally led organisation.

In respect of the psyche, you might have come across the notion that progress in therapy is like peeling away the layers of an onion.  The tinier the area under investigation the more there is to be seen – in this way the exploration of the psyche can be compared to the discovery of subatomic particles in physics. 

There always seems to be one smaller (and having some significance) to the one that was last discovered.

To conclude, fractals are ever present in our world, in the microscopic, the macro and even in the cosmos.

And in respect of this website, I believe that supporting families in the Focus Group and protecting children growing up in them will bring only good to society at large, because, (as I explained above) there is significant and real upward causation of family characteristics like compassion, love, tolerance, forgiveness, hope and acceptance in extended family, community and thereby, gradually changing society at large for the better.

I am now going to apply principles of fractal geometry and self-similarity to the developing child.

To do this, I will get back to our humble tree, which is probably the example that will describe it best. In this, I am using symbolism to describe something that requires us to use our imagination a little!

If we consider, for example, influences on a child that is growing, we can analogise a main trunk as being a child’s own sense of self, the thick branches as being influences of parents and/or main carers, then siblings, then the smaller branches as being (depending on their proximity and/or influence) grandparents, school friends, teachers, cousins, then with less influence (twigs) are the community, town or city, the country, and so on.

Each element has an influence on the child but the more influential ones are the main trunk and thicker branches, that is, the interplay between himself and his immediate family.  In this, his edge of growth will be, in addition to his individual personality and character, the emotional experiences that impact on him every minute of every day in his home – from parents, siblings and extended family.

His centre of growth (nurturing his sense of self, identity, history, and feeling of whether or not firstly he is wanted, and secondly that the world is a safe place) will be the stability/security of good enough parental relationship(s), consistency in nurture, the secure base of his family, and the level of ease-of-being-in-the-world that there is in his family.

The self-similarity occurs in the relationships.

One way of identifying the differences between the edge and the centre might be to imagine the edges as constantly changing (all the little things that happen, ups and downs, incidents of hurt and joy etc. etc.) and the centre to be generally non-changing and steady (consistent familial relationships of parents/carers and extended family).

Similar dynamics will exist (and will have existed for a long time) within the systems that surround the child though they will not be exactly the same.  These dynamics provide a feedback loop that reinforces the behavioural patterns of the child over time. 

The child’s personality and/or identity, values etc. are all continually reinforced, and ultimately sedimented (that is, laid down like sediment in a slow-flowing river) by virtually endless cycles of re-entry (feedback) provided by influential others.

Indeed, the individual beauty of each child can be thought to result from the imperfections in his/her parents’ relationships, and also the imperfections in relationships of people of influence in his life as he grows and tries to make sense of the world.

Like the tree, a keen and experienced observer might make a good guess at how a child will develop if they have knowledge of extended family dynamics, general culture, school ethos etc. but the observer will not be able to predict exactly what will happen.

To get a handle on this it might be interesting to do a little exercise.

Try and think back on your childhood. Think of the impact of the general relationship of your parents or main carer(s) on you – and what you learned about life from their relationship. Much of this will be a felt sense, probably unspoken, but very powerful and influential nonetheless.

Now try and guess how your grandparents’ relationship (if you can remember them – or if they were known to you) had on your parents and then on you. (If you cannot remember any grandparents try and think of senior influential people in your parents’ or main carer(s)’ lives). Now try and identify patterns in the relationships (for example how one reacted if another was a certain way, if there was bad news, good news, stress etc.) that endured over the life course and influenced you. Also, the relationships with extended family and those with major influences like church, school etc.

Think of the learning from major relationship patterns as the centre of your growth.

Now try and remember some little incidents that happened – good and bad. Try and recall what emotional reaction your parent(s) or carer(s), and perhaps siblings had. Did they differ in different environments, i.e. if ye were on holidays, if ye were rushing out in the morning, if ye were relaxing in the evening, if there had been a recent bereavement, if ye heard good news, if it was a birthday or Christmas, if ye had visitors? Did they depend on what mood your parent/carer was in?

Consider the learning from these incidents to be the edge of your growth.

Fractals as they appear in nature imply the irregularity and change that accompanies evolution and growth.  (We came across something like this already when we discussed the robustness of self-organisation in a previous post).

This theory of the development of the personality of a child has resonances in the theories of social mirroring [1]. 

[1]. Briefly described, Social Mirror Theory (which dates from the late 19^th Century) proposes that what we display in public and what we experience in private are inseparably bound together.  We use our thoughts to draw out meaning in what we experience but the process of drawing out meaning is not complete until we have expressed it in a way that is understood by others.

The important thing to remember in all the above is that an awareness of fractal geometry and self-similarity can give us an understanding of how hidden, difficult to observe, (maybe even invisible) connections and associations create predictable patterns beneath the unpredictability that we come up against in our daily lives.

When considering the family whose existence we might describe as chaotic, chaos theory would propose that the minute-to-minute encounters, events, incidents, crises etc. can never be precisely anticipated.

Yet, beneath the surface of the ongoing chaos will lurk that difficult to observe, (or possibly invisible) order that determines the general, long term patterns of behaviour.

And, I’ll mention once again – because it is so important – the existence of the invisible order will also propose that it is not one big traumatic event, (e.g. one-off abusive incident, death of a parent etc.) that makes the difference, (important as a traumatic event will be) it is the innumerable and consistent feedback that takes place over many years.

The point for practitioners to note here is, of course, that the healing of trauma will not involve one big generous gesture, or one brilliant intervention, but consistent, less notable but more significant (in the long term) warm and loving experiences over a long time.

Just like what would be happening in a healthy extended family!