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.

Tom Lehrer’s New Math

Most of you have probably heard the song New Math by singer-songwriter-mathematician Tom Lehrer. The song explains how to do “new math”, and uses subtraction in base-10 as its first example. This is how I learned to do subtraction in elementary school. Lehrer then moves on to a more abstracted way to do subtraction: base-8. Before singing the calculation, he humorously comments, “Base eight is just like base ten really – if you’re missing two fingers.”

This portion of his song reminds me of Abstract Algebra because it explains how to do a fairly simple calculation ($342-173$) with an unfamiliar binary operation. To understand this section, I had to read the lyrics and do the calculation myself. That is when I understood the irony of this song. The calculation described is incredibly simple, but the explanation itself is difficult to follow, making the techniques of “new math” hard to understand. Below is a the video and a copy of my work ($LaTeX$ was not cooperating).

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.

Deceptive Mathematics (Statistics)

In the video, Peter Donnelly Shows How Stats Fool Juries, Peter Donnelly delivers a talk on how statistics can be deceptive. After a few jokes about the social awkwardness of statisticians, Donnelly moves on to an example. He describes a scenario where you flip a fair coin until the patter HTH emerges (H being heads and T being tails). You then flip the coin again until the pattern HTT emerges. Then he asks the audience what they believe to be true:

a) The average number of flips for HTH to emerge is less than the average number of flips for HTT to emerge.

b) The average number of flips is equal for both.

c) The opposite of a) is true.

Most people in the audience answered b). The correct answer, however, is a). This is because, as Donnelly explains, the pattern HTH can repeat itself in five flips (HTHTH), where HTT cannot (HTTHT).

This is the first example he gives of how topics in statistics can often times be deceptive. He then moves on to a more relevant example (one covered in MATH 341) of the probability of having a disease given a positive test result for a test with 99% accuracy. He illustrates that while a positive test result may make it seem that there is a 99% probability that you have the disease, the true probability depends on how many people have been tested, as well as the actual probability of having the disease. If a million people are tested and there is a .01% probability of having the disease, then there will be a much larger number of false positives (9999) than people who actually have the disease (100). Furthermore, the number of the number of people tested who have the disease, only 99% of them will have a positive test result. This makes the probability that one actually has the disease given a positive test result considerably small (less than 1%).

This example relates to how statistics can be used to deceive a jury. As a matter of fact, in the wrong hands, statistics can be incredibly dangerous. This was illustrated by a true story about a pediatrician who testified against a woman accused of killing two of her babies. The pediatrician mistakenly claimed that the chances of having two infants die from Sudden Infant Death Syndrome (SIDS) is 1 in 73,000,000. One of the many mistakes made by the pediatrician was that the probability of having two children die from SIDS is independent. The woman was convicted, and was not released until her second appeal.

See the video of Peter Donnelly’s talk below:

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.”