"Zipf's Law" is an intriguing empirical regularity that appears in a number of physical and social situations. Zipf's Law applies in situations where the frequency of the occurence of an element is inversely proportional to its rank in a frequency table of all elements in the set.
I've seen two interesting posts recently on Zipf's Law and its application to urban populations, one by Scott Sumner and one by Ed Glaeser. They remark on dependence of our calculation of Zipf's Law on where we set the physical boundaries of an urban center. New research shows that when you don't pay attention to politically determined boundaries, and instead compute Zipf's Law on a geographic grid, the law doesn't hold nearly as well. Both writers go into more detail on why.
I have a soft spot in my heart for Zipf's Law and these sorts of questions because of a relatively random engagement that I had with it as an undergraduate. Somehow - I don't recall exactly - I became fascinated with the "spatial economics" literature, including Zipf's Law. I explored this literature more deeply in a paper on economic geography in a History of Economic Thought class that I took. I traced this relatively obscure field from Johann Heinrich von Thunen in the early 19th century, through Zipf's Law and the Hotelling model (applied spatially), and ultimately to Paul Krugman's New Trade Theory and his work on economic geography (recently recognized with a well-deserved Nobel Prize). It also first got me hooked on complex adaptive systems and emergent behavior. At the time, this was all just really fascinating stuff for me. My roommate that year was in the class with me, and he wrote on business cycle theories. I thought that was a very dull topic to chose, but of course now the business cycle holds considerable interest for me!
This initial flirtation with spatial economics took me down two important roads that have shaped how I think today. First, my reading on the Hotelling model and New Trade Theory took me down the "imperfect competition" road in the traditional Industrial Organization sense. That reading later introduced me to a range of people like Joe Stiglitz, Edward Chamberlain, and Joan Robinson - Keynesians who put a lot of emphasis on market imperfections. The other road it took me down, of course, was Krugman's Keynesianism. Ever since I first read some of Krugman's work on economic geography, I've been captivated by him. I don't always agree with him, but I've always been impressed. I never paid that much attention to Keynesianism as an undergraduate, but keeping up with Krugman always kept it in the back of my mind. So after I graduated, when I had more time in the evenings for free reading, I decided I ought to read The General Theory cover to cover because after all - the economist I most respected thought fairly highly of it. Even after reading it, I still didn't grasp everything there was to grasp about Keynes - but I was hooked. Whereas before I had an ambiguous identity as an economist, after reading The General Theory I knew I was a Keynesian of at least some stripe. Since then, keeping up with Stiglitz, Mankiw, DeLong, and Krugman, reading a couple more pieces by Keynes, and some additional books by Milton Friedman and Lawrence Klein has really helped me to triangulate what kind of Keynesian I am. But it all started with a fascination with Zipf's Law and economic geography!