Last year, Professor Steven Strogatz of Cornell University wrote a series of op-eds for the New York Times that discussed the presence of mathematics in unlikely places. I discussed one of these columns here.  Now, either those articles were well-received, or Professor Strogatz is well-connected, because this year he’s back in the Times with a much more ambitious series of articles.  This time around, Strogatz is attempting to “[write] about the elements of mathematics, from preschool to grad school, for anyone out there who’d like to have a second chance at the subject.”

Preschool to grad school is a significant amount of ground to cover, but thus far Strogatz has used his articles to assault this goal with gusto.  To date, he has tackled counting, patterns in addition, negative numbers, division, and basic high school algebra.  This doesn’t really do justice to his content, though.  Along the way he gives the reader some Sesame Street, and discusses a number of tangential topics, including the inability of Verizon employees to do math, the half-your-age-plus-seven rule, and pre-WWI European history.  The latter comes about in a discussion of that old adage which is familiar to anyone who saw the first Alien vs. Predator movie: the enemy of my enemy is my friend.

Predators must be awesome at math.

While some of Professor Strogatz’s explanations are a bit hand wavy (in particular, his explanation of why (-1) x (-1) = 1 is a lacking), on the whole they are quite good.  In particular, he offers a nice explanation of what it is for a mathematical argument to be “elegant.”  But even more impressive than his writing is its location – to have a discussion of mathematics with as wide an audience as the New York Times readership is commendable.  Even if people are not inspired to learn more mathematics after reading these pieces, hopefully they will have at least learned something.  As with exercise, a little mathematics is better than no mathematics at all.

Moreover, these articles highlight aspects of math not usually seen in popular discourse.  Much like Paul Lockhart’s A Mathematician’s Lament (which Strogatz references), these snack-size essays are focused on simple mathematical ideas, and the beautiful (and sometimes unexpected) results that follow.  Nowhere here does Professor Strogatz multiply two really big numbers together; in fact, he’s quite sympathetic to the fact that for many people, there is nothing more tedious than calculation.  By leading the conversation in this way, he’s hopefully able to give a taste of what makes math beautiful to an audience for whom such a statement might otherwise be labeled heresy.

I don’t know where this series of articles is headed, but I look forward to finding out, and hope you do to.  Professor Strogatz’s articles are grouped together here.

(Hat tip to dad for sending me a few of these articles.)

Let me highlight the first post, titled “Math and the City.” Professor Strogatz begins this article by describing Zipf’s law, an observation attributed to linguist George Zipf regarding the distribution of words in a language (for a linguistic motivation, you can check the Wikipedia article on Zipf’s law).

One of these things is not like the other.

In the context of cities, the law states the following: in a given country, if you rank the cities within that country by population, the largest city should be about twice as large as the second largest city, about three times as large as the third largest city, and so on. In other words, a city’s population is inversely proportional to its rank.

Using the power of the internet, it’s not too hard to find current population data to try and verify this observation. Here’s a table illustrating Zipf’s law for U.S. cities (I pulled the population data from here):

Professor Strogatz doesn’t provide heuristics for why Zipf’s law should be true, but he does observe that this phenomenon has been around for more than a century, and can be observed in countries throughout the world (with varying degrees of agreement). He then goes on to discuss more recent mathematical observations regarding urban development, including the observation that cities, as with many other things, enjoy economies of scale. For example:

If one city is 10 times as populous as another one, does it need 10 times as many gas stations? No. Bigger cities have more gas stations than smaller ones (of course), but not nearly in direct proportion to their size… the bigger a city is, the fewer gas stations it has per person…

The same pattern holds for other measures of infrastructure. Whether you measure miles of roadway or length of electrical cables, you find that all of these also decrease, per person, as city size increases.

In other words, the distribution of infrastructure is not quite the same as the distribution of the population – as population grows, so too does infrastructure, but it can grow more slowly. Further discussion can be found in the article.

Of course, there are many questions here. First of all, a little digging will show you that this trend is stronger in some countries rather than others. Why is this? Also, why must we break down our analysis by country to look at these trends? Why don’t we see this pattern emerge if we simply rank cities independent of country?

Moreover, this Zipfian trend depends of course on one’s definition of the word “city.” If one extends the notion to municipal areas, the trends become less clear. So, how should we define what it means to be a city?

As I learned from a recent article on The Daily Dish, Tim Gulden of George Mason University has recently tried to answer some of these questions. By using nighttime satellite data, Professor Gulden and his coauthors were able to more rigorously and consistently measure city sizes globally, and were able to use these measurements to compare rankings for population, economic activity, and patented innovations.

In their paper (the abstract of which can be found here), the authors give evidence supporting a Zipf-type distribution for all three statistics, and use this data to argue against the idea that globalization is making the world “flatter,” i.e. more equidistributed with regards to things like population or economic activity. Instead, they argue that the world of the future will feature global ranks that follow more of a Zipf distribution, and that one reason why this hasn’t happened already is because it currently can be difficult to migrate between the barriers of different countries.

For more math in the spotlight, I’d encourage you to read Dr. Strogatz’s other posts (here and here). Happy reading!