# Order in Chaos

## Prof. Dr. Heinz-Otto Peitgen

HENN Akademie, November 26, 2009

Order and chaos are two sides of the same coin. They form the two sides of a revolutionary mathematical way of describing nature, which was developed during the second half of the 20^{th} century. The very fact that nature can be described mathematically was the basic idea, which marked the start of modern sciences in the 17^{th} century. Galileo Galilei was a prominent supporter of this idea and, for example, asked the question what the unalterable relationship is between the frequency of a pendulum and the varying length of the pendulum cord. Formulating such “laws of nature” with the help of mathematics followed the ideal of a radical, which means: a description which starts at the “root” of the phenomena and which turned against a purely phenomenological approach, which stops at the phenomena.

If today in research we talk of chaos, we do not mean a muddle which would come about against a background of order. The theory of chaos is at the same time a theory of order – it investigates cracks in the scientific world view, which likes to see either order or chaos. This not simply nullifies the validity of natural laws but puts them into perspective.

With regard to the theory of chaos, the question of how well we can predict events with the help of natural laws is of great importance. It is revealed that chaos is the loss of practical predictability in the presence of a strictly valid law – but this loss is not absolute. Chaos does not spread without discrimination. In the way it spreads chaos reveals a very definite order. An experiment in 1991 illustrates this. To do this, one blue, one red and one yellow magnet are attached in the shape of an equilateral triangle on a plate and a pendulum consisting of a steel ball and thread is suspended over its centre. An initial experiment shows that in spite of the fact that the pendulum is deflected equally and in spite of the same release position, the ball is “captured” in a sequence that appears to be random, sometimes by one and sometimes by another magnet.

With the help of the computer, we can simulate this process across the whole area, precisely establish the release positions and draw the pattern of these positions in an ever tighter mesh. If we give each release position the colour of the magnet over which the ball finally comes to a halt, an astonishing picture is revealed, which no longer appears to be random: the colours are arranged as in the case of marbled paper; monochrome islands of colour are surrounded by constrictions in which the colours seem to run into one another. For the pendulum system described, the picture produced illustrates how chaos and order belong together: it shows “islands of order in a sea of chaos”. What is important is that the pendulum law in this case, which Galilei was already on the track of, is nowhere invalidated but rather the predictability of events is variously marked and distributed with the help of this law. Events can be clearly predicted within the “islands of order” with the help of the law. However, where the colours run into one another, the system behaves practically randomly. In these areas, the slightest differences – such as a vibration when the ball is released – can produce a great effect and lead to a completely different result.

Today the belief is widespread that with the help of mathematical descriptions and vast computing capacities, the range could be even further extended, in which reliable forecasts can be made concerning events in the world. Research into chaos explains with mathematical precision, why this is not correct and why there are fundamental limits to forecasting events – and where they are to be found. In times, where hunger appears to be growing ad infinitum, according to forecasts, this should give us food for thought.

In 1992, Prof. Dr. Heinz-Otto Peitgen set up the Centre for Complex Systems and Visualisation, CeVis, of which he is director, at the University of Bremen. In 1995 he founded the interdisciplinary Center for Medical Image Computing, MeVis Research GmbH, in Bremen, which became an institute of the Fraunhofer Society at the beginning of 2009. In 1992 Prof. Dr. Heinz-Otto Peitgen was appointed to the European Academy of Arts and Sciences and in 2008 to the Academy of Sciences in Göttingen.