# Fractal geometry and chaos theory

# Fractal geometry and chaos theory:

## A short overview.

Discovery. Application. Impact. Discovery......

From Future Shock by Alvin Toffler (1970), p. 38.

Positive feedback leading to overstimulation.

The response: future shock.

Future shock is the distress, both physical and psychological, that arises from an overload of the human organism's physical adaptive systems and its decision making processes

From Future Shock by Alvin Toffler (1970), p. 297.

## Introduction.

Traditionally, scientists have looked for the simplest view of the world around us. Now, mathematics and computer power have produced a theory that helps researchers to understand the complexities of nature. The chaos theory.

If you watch from a bridge as a leaf floats down the stream, you may see it trapped by a small whirlpool, circulate a few times, and escape, only to be trapped again further down the stream. Trying to guess what will happen to a leaf as it comes into view from under a bridge is an idle pursuit in more senses than one: the tiniest shift in the leaf's position can completely change its future course.

Small changes lead to bigger changes later. This behaviour is the signature of chaos.

## What is chaos?

The term "chaos" is often used to describe the complicated behaviour of nonlinear systems. Chaos theory is an exciting, growing field which involves the study of a range of phenomena exhibiting sensitive dependence on initial conditions. Some natural systems, such as the weather, are so sensitive to even small local fluctuations, if you change a parameter ever so slightly, the results can be very different. This is called the butterfly-effect, meaning that a butterfly moving its wings on one continent can, theoretically, influence the weather on another.

Chaos and fractal geometry go hand in hand. Fractal geometry is a discipline named and popularized by mathematician Benoit Mandelbrot to describe a set of curves, many of which were rarely seen before the advent of computers. These sets have three important properties:

- They are generated by relatively simple calculations repeated over and over, feeding the results of each step back into the next, something computers can do very rapidly.
- They are, quite literally, infinitely complex: they reveal more and more detail without limit as you plot smaller and smaller areas. Furthermore, a magnified small section looks very similar to a large one over a wide range of scales. For example, consider the South African coast with its jagged irregularities. If you look at a map of a smaller area such as a bay, you find that the edge of the bay has the same kind of shapes and irregularities as the coastline. Move closer to examine a one foot section of the shore and you again find the same patterns. This repetition of shapes on smaller and smaller scales are called self-similarity.
- They are astonishingly beautiful, especially using personal computer colour displays' ability to assign colours to selected points.

It is the beautiful graphics associated with chaotic systems that have made the subject so appealing. This is a visual demonstration of complicated and beautiful structures which can arise in systems based on simple rules. And here we have an immediate link with nature, for trees and mountains are examples of fractals.

Some fractals exist only as shapes in abstract geometric space, but others can be used to model complex natural shapes and phenomena. This is a paradoxical combination of randomness and structure in systems of mathematical, physical, biological, electrical, chemical and artistic interest.

## The mathematics behind chaos.

Think of a number x, put it into a simple equation, and feed the equation to a computer. Put the answer back into the equation. Repeat the exercise and watch chaos evolve before your eyes.

The image is a set of mathematical points that have fractal dimensions. The richness of resultant forms contrast with the simplicity of the generating formula. Many fractals involve only the iteration of functions of complex numbers until some "bailout" value is exceeded, then colouring an associated pixel according to the number of iterations performed.

Although chaos seems totally "random", it often obeys strict mathematical rules derived from equations that can be formulated and studied.

The ideas of fractal geometry can be traced to the late nineteenth century, when mathematicians created shapes - sets of points - that seemed to have no counterpart in nature. Ironically, the "abstract" mathematics descended from that work, has now turned out to be more appropriate than any other for describing many natural shapes and processes. Experience shows that the mathematical beauty of today usually translates into the useful tool of tomorrow.

## The role of the computer.

Developments in the theory have issued from a coming together of abstract mathematics and one of the most important research tools today, the computer. The abstract mathematics contain a beauty and complexity which corresponds to behaviour no one could fully have appreciated or suspected before the age of the computer. As computers gained more graphic capabilities, the skills of the minds' eye were reinforced by visualization on display screens and plotters. Fractal models produced results - series of flood heights, or cotton prices - that experts said looked like the real thing.

## Add some chaos to your life and put your world in order.

At the moment, scientists cannot use the fundamental laws of nature to predict when drips will fall from a leaking tap, or what the weather will be like in two weeks' time. In fact, it is difficult to predict very far ahead the motion of any object that feels the effect of more than two forces, let alone complicated systems involving interactions between many objects. From chaotic toys with randomly blinking lights to wisps and eddies of cigarette smoke, chaotic behaviour is irregular and disorderly.

It is here that chaos theory steps in to shed some light on the way the everyday world works. It reveals how many systems that are constantly changing are extremely sensitive to their initial state - position, velocity, and so on. As the system evolves in time, minute changes amplify rapidly through feedback. This means that systems starting off with only slightly differing conditions, rapidly diverge in character at a later stage. Such behaviour imposes strict limitations on predicting a future state, since the prediction depends on how accurately you can measure the initial conditions. When modelling such a system on a computer, just rounding off the decimal points in a different way can radically change the future behaviour of the system.

Looking beyond their curious mathematical properties, chaos theory has an immense attraction because of the role it plays in understanding heart failure, meteorology, economics, population biology, neural networks, arrays of parallel processors, leukemia, brain rhythms and gold futures. Scientists now have an interpretive tool for describing many of the complexities of the world. Fractal geometry has been used to create images and models of many different areas. From three- dimensional landscapes in movies to accurate cross-sectional models of the heart, fractals are at the leading edge of the research in many fields. Most systems that confront the engineer are nonlinear dynamical systems. Even the flooding of the Nile is being investigated.

Many of the shapes mathematicians had discovered generations before are useful approximations of living tissue, clouds and galaxies. Visualization was extended to the physical world.

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Nature provides many examples of fractals, such as ferns, cauliflowers, broccoli or the course of a river, because each branch and twig is very like the whole. The rules governing growth ensure that small-scale features become translated into large-scale ones.

Chaos also seems to be responsible for maintaining order in the natural world. Feedback mechanisms not only introduce flexibility into living systems, sustaining delicate dynamical balances, but also promotes natures's propensity for self-organization. Even the beating heart relies on feedback for regularity.

Chaos theory has also been applied to a wide range of visual art. Random fractals became best known through the stream of forgeries of coastlines, mountains and clouds. Other examples are some of the scenes made for films such as Star Trek II. The complex graphics seem to teeter wondrously between order and randomness. Such colourful iterations have linked mathematics with art and nature in a stimulating way. Chaos has made mathematics come alive.

## In conclusion.

Chaos: the ability of simple models, without built in random features, to generate highly irregular behaviour. Chaos is exciting because it opens up the possibility of simplifying complicated phenomena. Chaos is worrying because it introduces new doubts about the traditional model- building procedures of science. Chaos is fascinating because of its interplay of mathematics, science and technology. But above all chaos is beautiful. This is no accident. It is visible evidence of the beauty of mathematics, a beauty normally confined within the inner eye of the mathematician but which here spills over into the everyday world of human senses.

circa 1996