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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.

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?

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:

1. 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.
2. 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.
3. They are astonishingly beautiful, especially using personal computer colour displays' ability to assign colours to selected points.

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.

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.

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

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.

I am busy compiling a list of real-world examples of fractals:

• Two or more colours of paint being mixed.
• High enlargement of the surface of a diamond.
• Waves on the sea and markings on the beach.
• Slices through cabbage.
• Wings of a dragonfly.
• Sunflowers.
• Mountains.
• Purkinje cells in the brain.
• Collagen at different enlargements.
• Diatoms
• Nautilus
• The Fibonacci sequence in pine, shells, flowers and phyllotaxy.
• Trees.
• Contourlines on a map.
• Drainage systems of rivers.
• Veins of a leaf.
• Bloodvesels.
• Amazon River.
• Lightning.
• Irish moss.
• Bundle of rope.
• Street map.

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