Last month, the world lost one of the most brilliant mathematical minds of the past few decades: Benoit Mandelbrot. This mathematician was born November 20, 1924, in Poland, grew up in France, and spent most of his adult life working in the United States. Mandelbrot made contributions to numerous fields, including economics, information theory, and fluid dynamics, but he is perhaps best known for his work in theoretical mathematics, specifically, fractal geometry. A fractal is a shape that possesses the property of self-similarity——zooming in reveals that the border forms patterns similar to the whole, making the shape never-endingly complex. Computer programs can be used to generate fractals that have color-coded areas, giving rise to beautiful, trippy images.
However, a fractal is more than just a pretty picture. The equations that give rise to these fractals can actually be used to describe various phenomena in nature.
In fact, Mandelbrot was first inspired to explore the concept of fractals when pondering how best to measure the coast of Britain. This seemingly easy endeavor is actually quite difficult. The more precise the measurements are, the longer the coast becomes. Mandelbrot realized that these complex borders can be described according to simple formulas. For instance, the mathematician’s eponymous fractal, the Mandelbrot set, is defined by a very simple equation: z → z^2 + c. But zooming into the border of the fractal reveals consistent, infinite complexity.
Fractal geometry can be applied to a variety of fields. Fractals have been used for image analysis software, musical compositions, image compression, and even computer games. They can also be used in more scientific fields, such as seismology or medicine. Perhaps most importantly, fractals have shown that seemingly random and confusing things can actually be described and predicted very precisely with elegant formulas. Thanks to Mandelbrot, it is now understood that all sorts of phenomena, from occurrences in nature to the stock market, are full of intricate patterns that reveal a hidden order and beauty in the world.
Rest in peace, Benoit Mandelbrot. You will be missed.
Watch this TED talk by Mandelbrot himself a few months ago:
Fractal wrongness
