Chaos, Fractals and Self-Organization

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There has been something unconvincing about orderly shapes like triangles and squares: we find them in textbooks, in the way children in schools are made by teachers to assemble and disperse, but never in life itself. Life is actually wild, messy and labyrinthine, no matter what sort of  lens we view it through. The branches of a tree are bit like its roots – and also like the path of a river and the inside of our lungs. The question of form, of patterns of patterns, becomes a suddenly universal one, at once abstract and extremely realistic. You suspect there’s a revelation there, a momentous and fundamental epiphany, and yet there seems nothing to grasp. How does one get into all this? Do we all have to do PhDs in non-linear mathematics to know more about how leaves grow or how thoughts mutiny within us during meditation? Should we merely declare that nature works in mysterious waysand leave it at that? Meanwhile, Euclidean shapes continue to be taken for clarity and civilization, whether in Powerpoint, manicured gardens or in the normative ironing of clothes.

This is the sort of challenge that Dr. Arvind Kumar tries to help us with in a small book first published fifteen years ago. He introduces it as a ‘semi-popular book’; one feels that this is not to sit on the fence between the lay audience and the scientific, but because his own fascination for the subject has its source as much in his daily experiences of living as in  science. Kumar refers to chapatis and snowflakes as comfortably as to iterative formulae and thermodynamics. At the very beginning, he describes the book’s three dedicatees as “strange attractors of a chaotic existence” and that reveals , right-away, the charm of the subject for him: words like ‘chaos’ do mean something to us even before we roll up our sleeves for the details, but for a reader with serious exposure to contemporary science, these words, as are others like ‘string’, are charged with elusive secrets – indeed, there may be no experience that is not simultaneously scientific and subjective.

What perhaps gives the subject a special place among other equally fascinating concepts that scientists could write about for the lay reader are (1) that examples are all around us, if we care to pay attention and (2) high-powered computing has made possible projections that would have been left to guesswork only decades earlier, and in that sense it’s an exciting and revelatory time when we’re beginning to grasp what we might once have sensed intuitively at best. The radical ecologist or Gandhian is left with a troubling implication: that you might need to clear rainforests for an economy that allows for the technology that can help you appreciate how a rainforest emerges...

Chaos is introduced through the well-known example of the butterfly effect in  weather-prediction: a small change in initial conditions can, in a complex system, lead to greatly different outcomes – and of course initial conditions can never be known perfectly. We’ve seen how, in our own lives, that one move or moment of hesitance has changed everything,  eeded entire chapters in our stories; chaos, like the weather, is somewhat like that. Kumar helps us distinguish this meaning from that in common parlance: chaos describes not a random, free-for-all frenzy, but rather our inability to predict outcomes precisely despite knowing, more or less, how a system behaves – because even the very tiny things of the world matter.

At times, readers will be segregated into at least three categories: those who resign the moment they see an algebraic symbol, those who persist doggedly and even jot down the equations on  paper so as not to get derailed, and those for whom algebra is as much a language as is a simple text-sentence. This only reminds us that it’s difficult to be inclusive in such non-fiction. In this book, for instance, those who haven’t studied science may not understand how the swinging lamps that Galileo was watching have, after doses of bristly math embedded in English, “turned chaotic”, and may then approach with a renewed mistrust the more accessible points on chapati-making. Despite his enthusiasm, Kumar’s language periodically betrays his grounding in academic papers and textbooks. We must not expect the flair of a Bill Bryson – but it’s a respectable attempt nevertheless. Better illustrations would certainly have helped: barring the odd sketch, the book has the look and feel of an NCERT publication, and some of us have had enough of that sort of thing for a lifetime...

What Kumar does consistently is maintain the bridge between the old and the new: he does not dismiss linearity for non-linearity, or Newtonian physics for Quantum Physics, or Euclidean shapes for fractals. In that sense, he never asks for a leap of faith, but rather works through what we know. Even the progression from non-life to life is explained as within the same continuum, rather than as creation. But his respect for the classical seems limiting at times, in questions such as “can this paradigm...offer a scientific resolution of the enigma that is life?” that we must be careful with.

Another problem the professor hints at is the casual simplification of scientific methods in the minds of dilettanti, especially those looking for alternative views: that Newton was wrong, that Quantum Theory is finding out what Buddhists and Taoists already knew: that things have now been settled: holism is in, reductionism is out; faith is in, rationality, quantities and algebra are out.

Kumar questions this polarizing of the holism and reductionism, and this thrust is, if anything, understated in his book – it seems too important to be left to one chapter near the end. Opponents of reductionism, Kumar implies, create a straw man and then take it apart: reductionism is not merely ‘the whole as the sum of its parts’: what defines it, rather, is an assumption of upward causation. What he calls ‘sensible reductionism’ is, he says, not as different from holism as it is made out to be.

There are moments of great feeling in passages such as the tribute to the quest for the universal – this doesn’t, Kumar says, necessarily mean trying to control and predict everything – or the thrilled descriptions of Feigenbaum’s discovery of period doubling with a constant ratio of bifurcation-spacings and of how Prigogine’s opening up of Physics to Biology led to his work on dissipative structures.

The numerous examples of self-similarity he later provides, from earthquakes to the fluctuations of stock-prices, only heighten our interest: the idea of recursion begins to reveal the universe’s mind. There’s an uneasy awe that begins to flow beneath our reading, a sense of our lives dismantling, a confusion of things we’re taught to see as separate, as ‘it’ and ‘you’ and ‘me’. The bronchial network in our lungs has a fractal dimension of 3, while for granite and basalt it’s closer to 2.5; a line segment, if you split it perpetually, yields what is called ‘arithmetic dust’; and by now you begin to feel that everything is a metaphor for everything else, and to consider this is to feel that our histories, if you go back far enough, are also biology, chemistry, geology, all of that and...chaos

Kaushik stepped out of his management career to try and reclaim what it means to be alive. He has since been a wandering student with no affiliation or address, and many questions. You can write to him at