Fractal Power

Share prices, the shape of the coastline, the strength of metals and the latest technique for compressing computer images have one thing in common – the maths of fractals

On Monday 19 October 1987, the world’s stock markets crashed overnight. Shares lost a quarter of their value; many people went bankrupt. The day has entered the history books as Black Monday.

Yet why did it happen? There were no unusual market conditions the previous week; and no political calamities. Many analysts now believe that Black Monday was the result of Chaos Theory.

According to Chaos Theory, a tiny effect can produce an effect out of all proportion. Make a pile of sand by dropping grains one at a time on the ground. The sand builds up into a shapely cone; but at some point it will become unstable and collapse. You can’t say in advance just which grain will cause the heap to fall apart – nor exactly how the collapse will happen. That final grain has caused an effect out of all proportion. Similarly, an unnoticed change in trading conditions in the middle of October 1987 set in train a series of share dealings that fell through the floor on 19 October.

Chaos theory has meant a whole new analysis of economics. If we look into the performance of the Dow-Jones or the FTSE-100, we might expect to find a reason for all their ups and downs. Sometimes this is the case: when a war is looming, or oil prices rise. But most of the fluctuations happen for no ostensible reason. This is the hallmark of Chaos Theory.

But that jagged line which shoots up and down showing the course of the market was also responsible for the discovery of another feature of chaos: the fractal.

Mandelbrot sets the agenda

In France in the 1960s, mathematician Benoit Mandelbrot was looking at the behaviour of share prices. He found that the stock market fluctuated in much the same way over months as it did over hours: in other words, however detailed a view you take, the fluctuations always look similar. Mandelbrot went on to discover that this invariance of scale occurred all over the natural world.

It’s true of the coastline of an island like Great Britain: you have headlands and estuaries when you look from the air; if you look at a few centimetres of the shore under magnification, it looks much the same.

The growth of plants also shows an invariance of scale. Keep cutting off pieces from a cauliflower and the smaller and smaller pieces look just like shrinking cauliflowers. The branches on a fern or a snowflake look very similar as one increases the magnification. The structure of the lung also demonstrates this characteristic invariance.

Mandelbrot invented a new term – a fractal - to describe this shape for something that looks the same, no matter how you magnify it.

How can all continents have an infinite coastline - apart from Africa? Read on to find out more....