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: