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Above: Picture from Sandpile Cellular Automat by Paul Gough
Left: Mountains by Anish Kapoor (1995)

Until quite recently, sciences like physics emphasised the deduction and confirmation of the laws and regularities of the world.  The teaching of science was built around simple, soluble, problems that could be dealt with using pencil and paper.  During the last decade, there has been a change. The advent of small, inexpensive, powerful computers with good interactive graphics has enabled large, complex, and disordered situations to be studied observationally - by looking at a computer monitor.

Experimental mathematics has been invented. A computer can be programmed to simulate the evolution of complicated systems, and their long-term behaviour observed, studied, modified, and replayed.  One can even construct virtual realities obeying laws of Nature that are not our own, and simply explore the consequences.  By these means, the study of chaos and complexity has become a subculture within science.  The study of the simple, exactly soluble, problems of science has been augmented by a growing appreciation of the vast complexity expected in situations where many competing influences are at work.   Prime candidates are supplied by systems that evolve in their environment by natural selection, and, in so doing, modify those environments in complicated ways.

As our appreciation for the nuances of chaotic behaviour has matured by exposure to natural examples, novelties have emerged. Chaos and order have been found to coexist in a curious symbiosis.  Imagine a very large egg-timer in which sand is falling, grain by grain, to create a growing sand pile.  The pile evolves in an erratic manner.  Sandfalls of all sizes occur, and their effect is to maintain the overall gradient of the sand pile in equilibrium, just on the verge of collapse.  This self-sustaining process has been dubbed 'self-organising criticality', by its discoverer, the Danish physicist, Per Bak.

At a microscopic level, the fall of sand or rice is chaotic.  If there is nothing peculiar about the sand, which renders avalanches of one size more or less probable than others, then the frequency with which avalanches occur is proportional to some mathematical power of their size (the avalanches are said to be 'scale-free' processes).   There are many natural systems - like earthquakes - and man-made ones - like stock market crashes  - where a concatenation of local processes that appear to combine to maintain a semblance of equilibrium in this way.  Order develops on a large scale through the combination of many chaotic small-scale events that hover on the brink of instability.  The sand pile is always critically poised, and the next avalanche could be of any size; but the effect of the avalanches is to maintain a well-defined overall slope of sand.  The course of life on planet Earth might even turn out to be described by such a picture.  The chain of living creatures maintains an overall balance despite the constant impact of extinctions, changes of habitat, disease and disaster, that conspire to create local 'avalanches'. Occasional extinctions open up new niches, and allow diversity to flourish anew, until equilibrium is temporarily re-established.  A picture of the living world poised in critical state, in which local chaos sustains global stability is Nature's subtlest compromise.  Complex adaptive systems thrive in the hinterland between the inflexibilities of determinism and the vagaries of chaos.  There, they get the best of both worlds: out of chaos springs a wealth of alternatives for natural selection of sift; while the rudder of determinism sets a clear average course towards islands of stability.

The sand pile's behaviour is characteristic of a situation where there is a steady build up (here, of sand) followed by a sudden jump (an avalanche) to another nearby equilibrium. The meandering of rivers is another such system where increased flow makes the river bends get sharper unti lthey are cut off by the sudden formation of an oxbow lake, leaving the river a little straighter than before. In this way the meandering river organises itself.

Text from ‘Sand piles and Self-organising criticality’ by John Barrow