Chaos theory tamed

Clockwork universe

The book was published at a time now known as the Age of Reason, when philosophy and science were developing very rapidly in Europe. Through the work of Newton and others there arose the idea of the 'clockwork universe'. The proper name for this idea is determinism. Determinists believe that by measuring things as they are now, we can accurately predict the future using the laws of science.

This works well for predicting the positions of the planets over the next hundred years or the timing of high and the low tides for the next ten years. After Newton's theory, many scientists became convinced that they knew everything there was to know about the universe:

"We are probably nearing the limit of all we can know about astronomy," wrote American astronomer Simon Newcom in 1888. 'The more important fundamental laws and facts of physical science have all been discovered… Our future discoveries must be looked for in the sixth place of decimals," opined Prof Albert A Michelson at the University of Chicago in 1894. And Lord Kelvin, in 1900, concluded that, "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement".

The butterfly effect

Even as these speeches were being made they were proven wrong by the discovery of X-rays (1895), radioactivity (1896), the electron (1897) and the beginning of quantum physics (1900). Einstein's theories of Special Relativity (1905) and General Relativity (1915) presented even more challenges for great minds. When the predictions of Einstein's theory were confirmed by a solar eclipse, in 1919, he became world famous and the idea of a clockwork universe was dead.

In the 1960s, a scientist named Edward Lorenz was using computers to build a mathematical model of the way air moves within the Earth's atmosphere. In order to predict the weather, data about the atmosphere was entered before starting the computer program. He thought that if data was entered more accurately and precisely, then weather predictions would become more accurate as well. He was surprised to find something completely different! Small differences in the data led to divergent patterns in weather, so predictability was difficult. For example, when air pressure was entered as 1.222223 instead of 1.22222 a completely different prediction was given.

Lorenz wrote about this in 1963, explaining how a butterfly flapping its wings in Beijing could affect the weather thousands of kilometers away a few days later. This became known as the Butterfly Effect.

Stated scientifically, we say that deterministic systems (like the weather) which are sensitive to tiny changes in initial conditions (like butterfly wings flapping) can behave unpredictably. Tiny inaccuracies in our recording of today's weather make predicting more than a few days ahead impossible. Although we may be accurate to six decimal places in our measurements, over time the unknown seventh decimal place can have a large and unpredictable effect.

The chaos theory is born

The word chaos is sometimes defined as a state of complete confusion and disorder. The name "chaos theory" comes from the fact that the systems that the theory describes are apparently disordered, but chaos theory itself is really about finding the underlying order in apparently random data.

Systems such as the one described above are called chaotic and show that simple laws can lead to seemingly random and unpredictable behavior. It also shows that situations that seem random and unpredictable can be caused by certain events, and they can be understood through the use of simple laws. This is what the chaos theory tries to do.

It turns out that the orbits of planets are in fact chaotic because they never quite follow the same path. Over long periods of time tiny changes in the orbits can lead to unpredictable behavior. Tiny variations in the orbits of asteroids around the Sun can happen when they pass near to Jupiter. From time to time, this can lead to asteroids hurtling into the Sun.

If a meteor or asteroid hits the Earth it could cause the end of life on the planet. This is unlikely to happen any time soon, but it is not impossible. It could very well be the reason behind the extinction of dinosaurs millions of years ago.

Our heartbeat occurs as a result of the contraction of many individual muscle fibers in the heart. Small changes to the heart can, over a period of time, lead the fibers to contract in a random and chaotic way. This is called fibrillation and if it is not controlled it can make the heart stop beating (cardiac arrest). However, the good news is that modern doctors can revive the heart once again by using a controlled electric shock delivered by a machine called a defibrillator.

Our brain waves - the pattern of electrical impulses in our brain - also show signs of being chaotic. Many people think this could be the origin of consciousness, free will, imagination and creativity. After all, if we really lived in a clockwork universe how could these phenomena be explained?

Try visualizing an egg on the tip of a pyramid. When you let go of the egg it could roll down any of the four sides. Why do these variations take place when it is the same egg and the same pyramid? The reason is that there are minute differences that change the outcome each time - a sudden breeze, the placement of weight, slight alterations in an angle, a heartbeat, etc.

Fractals: Nature's patterns

There is a problem in the chaos theory known as the coastline problem. It goes like this: How long is the coastline of the island of Great Britain? The correct answer is it depends how you measure it.

If you measure it on a map, you get one answer. If you walk around the island and measure it in meters, you get a much larger result. This is because you measure all the little details that weren't shown on the map. If you measure it in feet, you get an even bigger calculation. If you continue to measure the coastline using smaller and smaller increments, the result gets bigger and bigger.

In other words, the coastline gets longer the more precisely you measure it. However, the area of the island doesn't change much, however you measure it. This doesn't make sense. An island with a longer coastline must surely have a bigger area.

A whole new type of geometry had to be invented to deal with this paradox. The coastline is known as 'fractal'. Fractal comes from the word fractional. Fractal has come to mean any image that displays the attribute of self-similarity. The more we zoom in on it, the longer it gets, as more intricate and fine details are revealed.

There are many fractals in nature. Clouds are fractal, and so is the way tree branches and blood vessels split up, getting smaller and smaller. When we zoom in we see the same kinds of patterns being repeated over and over. This is called self-similarity, and it is a property of fractals.

Computers have been used to create fractal images, some of which look just like structures seen in nature. A simple calculation is carried out by the computer, and a dot is added to the image if the answer is above or below a certain value. The answer is then used to start the next calculation. This is repeated millions of times. By starting with a slightly different number, very different images can be obtained. Two of the most well known examples of fractal images are known as the Julia set and the Mandelbrot set.

The chaos theory is now being applied to problems as diverse as traffic flow, stock market fluctuations, disease control and climate change. It is also thought that the information encoded in our DNA may be fractal, since it would not be large enough otherwise to contain all the detail needed. Being fractal, it would consist only of simple rules and the starting numbers, which require much less storage space than a detailed plan.

Next time we'll examine the connections between Archimedes, the Titanic and how one should stay afloat.

Corrin Funnell is a laser physicist with a specialty in laser spectroscopy. He has taught in the UK, Egypt, at Thailand's own Harrow International School, where he became head of the physics department. Currently, he is head of Physics at Island School, Hong Kong.