ABSTRACT
The world around us contains elements of order and disorder. Crystals are regular, clouds are not. Sometimes regular behaviour becomes irregular: water pours steadily from a tap at low velocities, but if we turn it on full the steady flow becomes turbulent. Similarly, smoke from a cigarette first rises smoothly and then becomes turbulent. Physicists select for study just those phenomena that behave in a simple and regular way, such as the free fall of a particle or the motions of the planets, where any disturbing influences such as air resistance or the attractions of the other planets are either negligible or can be allowed for if high accuracy is needed. The astonishing success of these simple laws in accounting for many phenomena to great accuracy is deeply impressive, and inspired Laplace to declare that from a knowledge of the present a being with great mathematical ability could predict the future as well as the past. Nature seemed to be a rigidly determined system. It was then found that the time evolution of many systems of at least three
particles shows such a sensitive dependence on the initial conditions that prediction of future behaviour soon becomes impossible. If we add the belief associated with certain interpretations of quantum mechanics that the world is fundamentally fuzzy, then we conclude that the behaviour of the universe is in principle unpredictable. Even without considering quantum effects, the solar system apparently becomes chaotic on a timescale of a few million years. All this is a consequence of laws that prescribe exact behaviour. The world is thus ruled by deterministic chaos. Chaotic behaviour is found more frequently than the simple phenomena of
classical physics. Even in apparently simple systems we find chaotic behaviour that seems to be impossible to describe, and yet its average behaviour can often be described in a simple way. This applies also to the scale-invariant structures of Mandelbrot. Even in the midst of chaos, there is hidden order. The properties of the elementary particles show characteristic symmetries.
Sometimes a symmetry appears to have exceptions, the symmetry is broken, but is always found to be subsumed in a higher symmetry.
