When a man dies and his obituaries mention the discovery of geometric shapes with a Hausdorff dimension that exceeds its topological dimension, you might think he’s just another hardcore mathematician.
But when a man dies and his life’s work encompasses cauliflower, the stock market, and paisley shirts, you know there’s something special.
That man was Benoit Mandelbrot, a French-American mathematician who died late last week at the age of 85. He discovered (or at the least defined) fractals. Though a wildly complex subject, a fractal is effectively a shape that can be split into small pieces, each of which is (to a greater or lesser degree) similar in shape to the whole thing. One statistician gave the example of a romanesco cauliflower, with each piece being cauliflower-shaped.
Fractals appear in many areas of nature, most notably in snowflakes, but also in some coastlines. Other natural examples include the way rivers break down into tributaries, which matches the way lightning bolts fork. In effect, many seemingly random natural shapes in fact follow a mathematical pattern.
Mandelbrot’s work largely involved working on ways to model fractals in mathematics, which in turn made it possible to generate fractals. This led to the Mandelbrot set which, if you are a mathematician is (cue Wikipedia) “the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomialzn+1 = zn2 + c remains bounded” and if you are a non-mathematician is a set of numbers that can be plotted into a pretty pattern which, when zoomed in looks like the image above.
Most importantly, of course, fractals can be used to generate patterns in video games: none more explicitly than in the 1984 game Rescue on Fractalus! where players had to navigate a space fighter around an ever-varying mountain range.
There’s even an argument that fractals appear in human behavior, most notably in stock market charts. Mandelbrot argued that although it was not possible to use fractals to predict when market prices would rapidly change direction, it was predictable that in the big picture they would — and that it in turn it was foolish to make investment decisions on the assumption that markets would always behave in a stable manner. He likened traditional stock market strategy to sailing in seas where the weather was calm 95% of the time and thus deciding it not worthwhile preparing for a typhoon.