Whenever I look at a tree or a coastline, I think of Benoit Mandelbrot. Mandelbrot’s work poked a hole in the universe of mathematics through which everyone could see the beauty in it and experience the profound connection between nature and mathematics. Although the foundations of Mandelbrot and his famous Mandelbrot set are beyond the reach of many laypeople, they apply to seemingly simple problems like measuring the length of a coastline and maybe even understanding the movement of the stock market. When graphed and colored these Mandelbrot sets are also visually beautiful.
The fact that a function using complex numbers can generate something so applicable to nature is a wonderful mystery. For the record, complex numbers are a combination of a real or “normal” numbers and a so-called imaginary number which is a multiple of the square root of -1. Using a simple function, the Mandelbrot set is constructed in the complex number plane which can be thought of as the space in between the purely real and purely imaginary number systems. These sets generate points that, when plotted, show repeating patterns that have a kind of “infinite zoom,” – if you zoom in you see more of the same pattern, and the same happens if you zoom out. Thanks to widespread availability of computers, most anyone has been able to play with fractals and marvel at their complex simplicity and appreciate the power of the underlying mathematics.
Mandelbrot was unusual in the math world for being expert at both pure and applied mathematics. Most are good at one but not the other. This led him to apply his theories directly to problems like “How long is the coast of Britain?” which he published in 1967. The understanding and use of what came to be known as fractals expanded over the next ten years, and once computers made it easy to generate visual representations of them they started showing up everywhere.
Mandelbrot’s work is also directly related to Chaos theory, which has helped us understand that seemingly random patterns are not so random after all, but often have repeating patterns and some reliable properties buried in the noise and roughness of what we see.
Mandelbot even got around to applying his theories to financial markets in The Misbehavior of Markets: A Fractal View of Financial Turbulence, which was published in 2004. The book makes a compelling case for looking at markets and price movements very differently and underscores the need for more thoughtful risk management and investment strategies. Like many, we focus on fundamental valuations which, over the long term, are predictable; but other markets, like commodities, currencies, and debt have been known to “not trade on fundamentals” for a long time. Now that assets are showing much higher correlations in the market, I’d argue that nobody can afford to ignore these ideas and the action of less fundamentally-driven markets because they can have deep and lasting effects on company equity valuations.
Mandelbrot was a deep and lateral thinker which is what brought him the success and notoriety he achieved. There’s a lesson there for the more mediocre thinkers out there, which is to expand across boundaries to apply knowledge and ideas from one area to another. The fertile ground for breakthroughs often lies in between the plowed fields.
In a world that has millions of protesters against having their retirement age extended by a couple of years (effectively defining a working lifetime of 41-42 years), Mandelbrot had a 60-year career and, at 75 years old, accepted a full professorship at Yale. He was 85 when he passed away on October 14th.
- Benoit Mandelbrot, RIP (boingboing.net)
- The Beautiful Fractals of the Late Benoit Mandelbrot [Video] (gawker.com)
- Celebrating Benoit Mandelbrot, the Man Who Made Math Beautiful (popsci.com)