Mandelbrot’s fractals are not only gorgeous – they taught mathematicians how to model the real world

At the beginning of my third year at university studying mathematics, I spotted an announcement. A visiting professor from Canada would be giving a mini-course of ten lectures on a subject called complex dynamics.

It happened to be a difficult time for me. On paper, I was a very good student with an average of over 90%, but in reality I was feeling very uncertain. It was time for us to choose a branch of mathematics in which to specialise, but I hadn’t connected to any of the subjects so far; they all felt too technical and dry.

So I decided to take a chance on the mini-course. As soon as it started, I was captured by the startling beauty of the patterns that emerged from the mathematics. These were a relatively recent discovery, we learned; nothing like them had existed before the 1980s.

They were thanks to the maverick French-American mathematician Benoit Mandelbrot, who came up with them in an attempt to visualise this field – with help from some powerful computers at the IBM TJ Watson Research Center in upstate New York.

A fractal – the term he derived from the Latin word fractus, meaning “broken” or “fragmented” – is a geometric shape that can be divided into smaller parts which are each a scaled copy of the whole. They are a visual representation of the fact that even a process with the simplest mathematical model can demonstrate complex and intricate behaviour at all scales.

How the fractals are created

The system used by Mandelbrot was as follows: you choose a number (z), square it and then add another number (c). Then repeat over and over using the sum total from the previous calculation as z each time.

Starting, for example, with z=0 and c=1, the first calculation would be 0² + 1 = 1. By making z=1 for the next calculation, it’s 1² + 1 = 2, and so on.

To get a sense of what comes next, you can plot the value of c on a line and colour code it depending on how many iterations in the series it takes for the sum total to exceed 4 (the reason it’s 4 is because anything larger will quickly grow towards an infinitely large number in subsequent iterations). For example, you might use blue if the series never exceeds 4, red if it gets there after 1-5 iterations, black if it takes 6-9 iterations, and so on.

The Mandelbrot set is actually more complicated because you don’t plot c on a line but on a plane with x and y axes. This involves introducing several more mathematical concepts where c is a complex number and the y axis refers to imaginary values. If you want more on these, watch the video below. By plotting lots of different values of c on the plane, you derive the fractals.

This idea of visualisation from Mandelbrot, who would have turned 100 this month, led mathematicians to accept the role of pictures in experimental mathematics. It has also led to a huge amount of research. On five out of eight occasions since 1994, the Fields medal – among the highest accolades in mathematics – has been awarded for work related to his conjectures.

Mandelbrot in the real world

For centuries, mathematicians had to live with the uncomfortable thought that their existing tools – known as Euclidean geometry – were not really suitable for modelling and understanding the real world. They all produced smooth curves, but nature is not like that.

For example, one can sketch the shape of the British coastline with a few continuous strokes. But once you zoom in, you can see lots of small irregularities that were previously invisible. The same holds true for the beds of the rivers, mountains and the branches of trees, among many others.

When mathematicians tried to model the surface of anything, these small imperfections were always in the way. To make their work fit reality, they had to introduce additional elements which superimposed “noise” on top. But these were ugly and absurd, compensating for their inadequacies by creating an illusion.

Mandelbrot’s revolutionary philosophy, presented in his 1982 manifesto, The Fractal Geometry of Nature, argued that scientific methods could be adapted to study vast classes of irregular phenomena like these. He was the first to realise that, scattered around the research literature, often in obscure sources, were the germs of a coherent framework that would allow mathematical models to go beyond the comfort of Euclidean geometry, and tackle the irregularities without relying on a superimposed mechanism.

This made his theory applicable to a wide range of improbably diverse fields. For example, it is used to model cloud formation in meteorology, and price fluctuations in the stock market. Other fields in which it has application include statistical physics, cosmology, geophysics, computer graphics and physiology.

Mandelbrot’s life story was just as jagged as his discovery. He was born to a Jewish-Lithuanian family in Warsaw in 1924. Sensing the approaching trouble, the family first moved to Paris in 1936, then to a small town in the south of France.

In 1945 he was admitted to the most prestigious university in France, the École Normale Supérieure in Paris, but stayed only for a day. He dropped out to move to the less prestigious École Polytechnique, which suited him better.

Following an MSc in aerodynamics at California Institute of Technology and a PhD in mathematics at the University of Paris, Mandelbrot spent most of his active scientific life in an IBM industrial laboratory. Only in 1987 was he appointed Abraham Robinson Adjunct Professor of Mathematical Sciences at Yale, where he stayed until his death in 2010.

It is no exaggeration to say that Mandelbrot is one of the greatest masterminds of our era. Thanks to his work, visual images of fractals have become symbolic for mathematical research as a whole. The community recognised his contribution by naming one of the most famous fractals the Mandelbrot set.

In the epilogue of a 1995 documentary about his discovery, The Colours of Infinity, we see Benoit addressing the camera:

I’ve spent most of my life unpacking the ideas that became fractal geometry. This has been exciting and enjoyable, most times. But it also has been lonely. For years few shared my views. Yet the ghost of the idea of fractals continued to beguile me, so I kept looking through the long, dry years.

So find the thing you love. It doesn’t so much matter what it is. Find the thing you love and throw yourself into it. I found a new geometry; you’ll find something else. Whatever you find will be yours.

Polina Vytnova has been previously supported by EPSRC.