## Mandelbrot on Fractals

In his final TED talk, Benoit Mandelbrot spoke about roughness and fractals. The talk was appropriately titled Benoit Mandelbrot: Fractals and the Art of Roughness. In his talk, Mandelbrot explains how everything in nature is inherently rough (this is exemplified with close up images of things such as cauliflower). By looking at the distances between elements on an object or image, a roughness number can be given which can then be used to generate artificial landscapes and other images. This ability is being used in film making.

More generally, studying roughness allows patterns to be recognized in nature. These patterns can then be applied to studying other things in nature. For instance, fractals helped reconstruct the lung in a way that taught surgeons more about it. This in turn advanced their abilities in treating lung related diseases. Mandelbrot described it as, “A geometry of things which have no geometry.”

Interestingly, Mandelbrot actually began his career in financial mathematics. With the way the economy is today, I guess the stock market is a good place to start a career based on roughness.

Using this data, the team produced the equation $P(x)=Cx^{-\alpha}$, where $P$ is the probability, $x$ is the number killed, $C$ is a constant, and $\alpha$ is the slope of the line. The group theorized that this is a result of necessity when a group is fighting against a much stronger force. In order for their resistance to exist, it has to follow the discovered pattern.