How Mandelbrot's fractals changed the world
In 1975, a new word came into use, when a maverick mathematician made an important discovery. So what are fractals? And why are they important?
During the 1980s, people became familiar with fractals through those weird, colourful patterns made by computers.
But few realise how the idea of fractals has revolutionised our understanding of the world, and how many fractal-based systems we depend upon.
End Quote Benoit Mandelbrot
Clouds are not spheres, mountains are not cones, coastlines are not circles”
On 14 October 2010, the genius who coined the word - Polish-born mathematician Benoit Mandelbrot - died, aged 85, from cancer.
Unfortunately, there is no definition of fractals that is both simple and accurate. Like so many things in modern science and mathematics, discussions of "fractal geometry" can quickly go over the heads of the non-mathematically-minded. This is a real shame, because there is profound beauty and power in the idea of fractals.
The best way to get a feeling for what fractals are is to consider some examples. Clouds, mountains, coastlines, cauliflowers and ferns are all natural fractals. These shapes have something in common - something intuitive, accessible and aesthetic.
They are all complicated and irregular: the sort of shape that mathematicians used to shy away from in favour of regular ones, like spheres, which they could tame with equations.
Mandelbrot famously wrote: "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."
What are fractals?
- Geometrical objects that are self-similar when the distance at which they are viewed is changed
- Concept is helpful in allowing order to be perceived in apparent disorder
- Eg, in the case of a river and its tributaries, every tributary has its own tributaries…
- ...so that it has the same structure organisation as the entire river except that it covers a smaller area
Source: McGraw-Hill Concise Encyclopedia of Science and Technology
The chaos and irregularity of the world - Mandelbrot referred to it as "roughness" - is something to be celebrated. It would be a shame if clouds really were spheres, and mountains cones.
Look closely at a fractal, and you will find that the complexity is still present at a smaller scale. A small cloud is strikingly similar to the whole thing. A pine tree is composed of branches that are composed of branches - which in turn are composed of branches.
A tiny sand dune or a puddle in a mountain track have the same shapes as a huge sand dune and a lake in a mountain gully. This "self-similarity" at different scales is a defining characteristic of fractals.
The fractal mathematics Mandelbrot pioneered, together with the related field of chaos theory, lifts the veil on the hidden beauty of the world. It inspired scientists in many disciplines - including cosmology, medicine, engineering and genetics - and artists and musicians, too.
The whole universe is fractal, and so there is something joyfully quintessential about Mandelbrot's insights.
Fractal mathematics has many practical uses, too - for example, in producing stunning and realistic computer graphics, in computer file compression systems, in the architecture of the networks that make up the internet and even in diagnosing some diseases.
Fractal geometry can also provide a way to understand complexity in "systems" as well as just in shapes. The timing and sizes of earthquakes and the variation in a person's heartbeat and the prevalence of diseases are just three cases in which fractal geometry can describe the unpredictable.
Another is in the financial markets, where Mandelbrot first gained insight into the mathematics of complexity while working as a researcher for IBM during the 1960s.
What is a Mandelbrot set?
The most famous computer-generated fractal is called the Mandelbrot set - a swirling, feathery, seemingly organic landscape that is reminiscent of the natural world, but is nonetheless completely virtual. It is infinitely complex, but it is built from an extremely simple equation repeated endlessly. In the same way, natural fractal forms really are built up by simple rules - ultimately, the interactions between atoms.
Mandelbrot tried using fractal mathematics to describe the market - in terms of profits and losses traders made over time, and found it worked well.
In 2005, Mandelbrot turned again to the mathematics of the financial market, warning in his book The (Mis)Behaviour of Markets against the huge risks being taken by traders - who, he claimed, tend to act as if the market is inherently predictable, and immune to large swings.
Fractal mathematics cannot be used to predict the big events in chaotic systems - but it can tell us that such events will happen.
As such, it reminds us that the world is complex - and delightfully unpredictable.