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The mathematician Benoit Mandelbrot coined the phrase fractal and brought together the ideas of fractal geometry. Hank Morgan / Getty Images
The mathematician Benoit Mandelbrot coined the phrase fractal and brought together the ideas of fractal geometry. Hank Morgan / Getty Images

Forecasting weather or financial markets all begins with a cauliflower

Scientists now believe that the geometrical entities known as fractals may hold the answers to a variety of problems. That's why they are studying one vegetable in particular.

Scientists now believe that the geometrical entities known as fractals may hold the answers to a variety of problems, such as more accurately forecasting the weather, or the world financial markets. That's why they are studying one vegetable in particular.

Within academia, theoretical physicists have a reputation for being clever but cock-sure. They certainly talk a good game. They'd have us believe they're on a quest to find the Theory of Everything - which sounds impressive until one learns that such a theory embraces just the basic forces and particles in our universe.

That's still ambitious, of course, but it sure won't give us the answers to Everything - such as where best to invest our money, or what the weather will be like, or why cauliflowers are the shape they are. Yet while the heirs of Einstein pursue their esoteric quarry, a rather more modest community of scientists has been quietly honing a theory with an astonishing variety of applications, ranging from investing to weather forecasting, and - it now seems - understanding the shape of cauliflowers.

At the heart of these advances is one of the simplest yet most powerful mathematical inventions of recent times, geometrical entities known as fractals.

The name might be unfamiliar, but we've all seen examples of fractals. An example is the jagged coastline of a country. Looked at on a map of the world, the coastline has a few major features that only become clearer on a national-scale map. Examined on a smaller-scale local map, yet more details appear, with an actual visit to that area of coast revealing even more. This is the essence of a fractal shape: no matter how much you magnify it, yet more detail appears.

Mathematically, many fractals also have a property called self-similarity, meaning the "jaggedness" revealed at deeper scales always has the same general appearance.

In the real world, this property always breaks down at some point: after all, there is a limit to how much one can keep magnifying things before their fundamental constituents reveal themselves.

But that doesn't detract from the value of fractals in allowing us to capture precisely that otherwise vague concept of jaggedness.

And there is no lack of demand for that in the real world. Take a look at the ebb and flow of, say, the currency exchange markets, or the price of commodities. They zig and zag in ways that seem to defy description. Yet to some fractal researchers - including the late Benoit Mandelbrot, the French-American mathematician who coined the term (from the Latin for "fractured") - it's possible to extract valuable insights from the fractal nature of those charts.

For example, using fractals to capture the volatility of the markets reveals the degree to which they breach long-cherished assumptions about the behaviour of markets.

Put simply, markets are often assumed to be "efficient", in the sense that, say, a company share price reflects all the publicly available knowledge about that company.

Any variation is thus assumed to be the result of just random jitters. That, in turn, means the volatility will follow the well-known "bell curve", which allows so-called "rocket scientists" in financial institutions to calculate the risk of huge swings and take suitable countermeasures.

But the very fact that the zigs and zags follow a fractal law shows this to be a very dangerous assumption. Unlike the bell curve, which has a well-defined spread of values, fractals have no such property - making a mockery of attempts to put numbers to the risk of getting huge changes.

The ability of fractals to capture the fine detail of market movements is now being put to use by more enlightened quantitative analysts (the proper name for rocket scientists).

It's hard to tell whether they're succeeding or not: for obvious reasons, they're likely to keep the results under wraps, either way. But we should hope it's working, as the recent financial crisis shows the alternative doesn't.

Another burgeoning application for the "fractal theory of everything" is in weather forecasting. The laws governing the weather are notoriously complex, which is why providing precise forecasts is so challenging.

Some meteorologists think fractals can be used to cut through this complexity. At McGill University in Montreal, Prof Shaun Lovejoy has put together an international team to explore the idea that the weather is really just a vast collection of cascade-like processes, each feeding the next layer down.

In a series of papers now appearing in major research journals, Prof Lovejoy and his colleagues have built a compelling case for this "multi-fractal" view of the weather.

Evidence from satellite and ground data sources shows the existence of fractal-like systems affecting the weather on scales ranging from 10 kilometres to 10,000km or more.

These might serve as simple substitutes for the appallingly complex mathematical models currently used by meteorologists to predict the weather - and they might even prove more accurate.

Now comes news that the power of fractals can even cast new light on that great scientific mystery: why do cauliflowers look like they do?

These humble vegetables are clearly fractal: the closer you examine their bobbly surface, the more of the same kind of bobbles you see.

And as the authors of research just published in the New Journal of Physics point out, such shapes appear elsewhere in nature - such as roiling clouds and combustion fronts inside engines.

What the team led by Dr Mario Castro of the Universidad Pontificia Comillas, Madrid, wanted to know is why; what leads to these disparate phenomena showing fractal characteristics? Using a combination of theory and experiment, the team thinks it has identified the origin of these fractal structures.

As with the bell curve, random chance plays a role, but competition for space and accumulation through growth are also crucial.

When plugged into their mathematical model, Dr Castro and his colleagues were able to replicate the shapes of different cauliflower-like phenomena on a huge range of scales, from the molecular to the, well, culinary.

This quiet revolution in science might seem less glamorous than the quest for the Theory of Everything. Certainly some theorists working on the forces behind the cosmic Big Bang would dismiss it all as trivia.

Yet for anyone wanting to know how our world works here and now, fractals are proving to be anything but.

Robert Matthews is visiting reader in science at Aston University, Birmingham, England.

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