"You believe in a God who plays dice, and I in complete law and order
in a world which objectively exists, and which I,
in a wildly speculative way, am trying to capture.
I firmly believe, but hope that someone will discover a more realistic way,
or rather a more tangible basis than it has been my lot to do.
Even the great initial success of the quantum theory
does not make me believe in the fundamental dice game,
although I am well aware that your younger colleagues
interpret this as a consequence of senility."

- Albert Einstein, Letter to Max Born.

The emergent (mathematical) specifications of chaos have been recently recognised to be manifest in a great many interdisciplinary fields. Physics, biology, engineering, chemistry, physiology, astronomy and mathematics have all been effected by this "New Mathematics of Chaos".

The extracts below represent my notes made in the review of this work, and should entice the casual browser to search out the source material - to obtain and read the original book, which has established itself as one of the early reviews of the field of nonlinear dynamics, chaos and fractals.

Ian Stewart summarises the situation quite adequately when he observes:

"It is an entire new world,
a new kind of mathematics,
a fundamental breakthrough
in the understanding of irregularities in nature.
We are witnessing its birth.
Its future has yet to unfold.

In its unfolding - just as the unfolding of life from the fleeting wings of the present moment - it will represent a collaborative effort of research for the students of the field. The specifications of nature would thus now appear to be incomplete without this natural element of disorder or chaos, but just how this new vision is to be integrated into the future understanding of man - generic and individual - only time, research and engaged enthusiasm will discover the fuller journey of this far-flung path.

Until then, this resource is dedicated to the researchers and the students of whom each of us is part, and in particular to those who see themselves to be the students of life.

May your feet find their way to Journey's End,
And may the Age of Information dawn within.

All the best,

Pete Brown
Mountain Man Graphics,
Newport Beach, Australia
Southern Winter of 1997

E-Mail: prfbrown@magna.com.au

This dramatic discovery, whose implications have yet to make their full impact on our scientific thinking. The notions of prediction, or of a repeatable experiment, take on new aspects when seen through the eyes of chaos. What we thought was simple becomes complicated, and siturbing new questions are raised regarding measurement, predictability, and verification or falsification of theories.

In compensation, what was thought to be complicated may become simple. Phenomena that appear structureless and random may in fact be obeying simple laws. Deterministic chaos has its own laws, and inspires new experimental techniques. There is no shortage of irregularities in nature, and some of them may prove to be physical manifestations of the mathematics of chaos. Turbulent flow of fluids, reversals of the Earth's magnetic field, irregularities of the heartbeat, the convection patterns of liquid helium, the tumbling of celestial bodies, gaps in the asteroid belt, the growth of insect populations, the dripping of a tap, the progress of a chemical reaction, the metabolism of cells, chanages in the weather, the propagation of nerve impulses, oscillations of electronic circuits, the motion of a ship moored to a buoy, the bouncing of a billiard ball, the collisions of atoms in a gas, the underlying uncertainity of quantum mechanics - these are a few of the problems to which the mathematics of chaos has been applied.

It is an entire new world, a new kind of mathematics,
a fundamental breakthrough in the understanding of irregularities in nature.
We are witnessing its birth.

Its future has yet to unfold.

Pages 2 - 3

Some innate impulse makes humankind strive to understand the regularities in nature, to seek the laws behind the wayward complexities of the universe, to bring order out of chaos. Even the earliest civilizations have sophisticated calenders to predict the seasons, and astronomical rules to predict eclipses. They see figures in, and weave legends around, the stars in the sky. They invent pantheons of deities to explain the vagaries of an otherwise random and senseless world. Cycles, shapes, numbers. Mathematics.

Page 5

In the same way, mathematicians are beginning to view order and chaos as two distinct manifestations of an underlying determinism. And neither exists in isolation. The typical system can exist in a variety of states, some ordered, some chaotic. Instead of two opposed polarities, there is a continuous spectrum. As harmony and discord combine in musical beauty, so order and chaos combine in mathematical beauty.

Page 20

Page 188

Scales of Measurement

[ ... ]

... The mechanism of phase transitions, where a mass of billions of atoms suddenly changes its gross physical characteristics, tends to spread itself across a rather large range of scales, mixing up the microscopic and the macroscopic. This is one reason why the mathematics of phase transitions has proved very difficult.

One of the newer techniques for dealing with this kind of problem has just made an entrance: renormalisation. As we've seen, this is a method for finding the limiting infinitisimal structure of a self-similar object or process, by repeatedly magnifying smaller and smaller parts of the whole. Self-similar objects, by definition, dont have a characteristic length scales: they look much the same on many different scales of measurement.

The orthodox shapes of geometry - triangles, circles, spheres, cylinders - lose their structure when magnified. We've seen how a circle becomes a featureless straight line when viewed on a large scale. People who think the Earth is flat do so because that's the way it looks to a tiny human being. Mandelbrot invented the term fractal to describe a very different type of geometric object: one that continues to exhibit detailed structure over a large range of scales. Indeed, an ideal mathemtical fractal has structure on an infinite range of scales.

Page 216

With hindsight, you can often see things that weren't anything like as clear at the time. The trick is not so much to know something, but to know you know it. That is, to appreciate that it's important, and to have a context in which to put it.

Earlier ages saw parts of this picture - but never put them together. They didn't have the motivation to ask the right questions, the techniques to find the answers. They saw isolated details, but never the Big Picture.

But it's clear that Poincaré, in particular, saw more than his contemporaries appreciated. To establish this, I'm going to give a rather long quotation from one of Pointcare's essasys. You'll find much of the above discussion within it, even though it's almost a century old. Its title:

Chance.

"A very slight cause, which escapes us, determines a considerable effect which we cannot help seeing, and then we say this effect is due to chance. If we could know exactly the laws of nature and the situation of the universe at the initial instant, we should be able to predict exactly the situation of this same universe at a subsequent instant. But even when the natural laws should have no further secret for us, we could know the initial situation only approximately. If that permits us to foresee the subsequent situation with the same degree of approximation, this is all we require, we say that the phenomenon has been predicted, that it is ruled by laws. But this is not always the case; it may happen that slight differences in the initial conditions produce very great differences in the final phenomena; a slight error in the former would make an enormous error in the latter. Prediction becomes impossible and we have the phenomenon of chance.

Why have the meterologists such difficulty in predicting the weather? Why do the rains, the storms themselves seem to us to come by chance, so that many persons find it quite natural to pray for rain or shine, when they would think it ridiculous to pray for an eclipse? We see that great perturbations generally happen in regions where the atmosphere is in unstable equilibrium. The meteorologists are aware that this equilibrium is unstable, that a cyclone is arising somewhere; but where they cannot tell; one-tenth of a degree more or less at any poinr, and the cyclone bursts here and not there, and spreads its ravages over countries which it would have spared. This we could have foreseen if we had known that tenth of a degree, but the observations were neither sufficiently close nor sufficiently precise, and for this reason all seems due to the agency of chance.

The game of roulette does not take us far as might seem from the preceding example. Assume a needle to be turned on a pivot over a dial divided into a hundred sectors alternately red and black. If if stops on a red sector, I win; if not, I lose.

The needle will make, suppose, ten or twenty turns, but it will stop sooner or not so soon, according as I shall have pushed it more or less strongly. It suffices that the impulse vary only by a thousandth or a two thousandth to make the needle stop over a black sector or the following red one. These are differences the muscular sense cannot distinguish and which elude even the most delicate instruments. So it is impossible for me to foresee what the needle I have started will do, and this is why my heart throbs and I hope everything from luck."

"When we wish to check a hypothesis, what should we do? We cannot verify all its consequences, since they would be infinite in number; we content ourselves with verifying certain ones and if we succeed we declare the hypothesis confirmed."