## 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.

## Math and War

In the TED Talk Sean Gourley on the Mathematics of War, using mathematics to track and interpret war is discussed.

Sean Gourley, a physicist from New Zealand, began his project by assembling a team of scientists, economists, and mathematicians. They then used various media sources to obtain information on the war in Iraq, and then used a computer to filter all of it and pull out the bits in which they were interested. Using this data, the distribution of attack sizes in Iraq was produced and graphed. The vertical axis was frequency of attacks, and the horizontal axis was number of deaths. For instance, the ordered pair (47,1) would mean there were 47 attacks with 1 casualty.

They then did the same technique for other wars, and surprisingly, the same distribution emerged. Expanding their study further and further, every war produced a similar distribution. Furthermore, each war had a slope that was within .75 of the mean (which was -2.5).

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.

Gourley concludes that we may be able to use this model to interpret the progress of a war, and in theory try to push it in the right direction, whatever that may be.

## Wolfram Alpha

In Stephen Wolfram’s talk Computing a Theory of Everything, he describes an incredible project that he has been working on called Wolfram Alpha. The talk begins with Wolfram illustrating how relatively simple programs can produce infinitely complex results. This inspired him to make all knowledge computational, and led to the creation of Wolfram Alpha.

Wolfram Alpha is a remarkable website that allows its users to type in any question, and generates a seemingly well-informed answer. It can even generate code based on the queries of its users. What I found to be most amazing about Wolfram Alpha is its capability to be used for simulation, experimentation, and discovery. Wolfram explains, “We can use the computational universe to get mass customized creativity. I’m hoping we can, for example, use that even to get Wolfram Alpha to routinely do invention and discovery on the fly, and to find all sorts of wonderful stuff that no engineer and no process of incremental evolution would ever come up with.”