Tag Archives: math

On Puzzles

Puzzles are instructive, Mr. Gardner found, for they teach us to appreciate hidden structures of the world that are not owned by any particular discipline and are potentially useful to all. He saw the world as resembling not a magazine, where the subject of each section bears little relation to that of the next, but a well-written novel, where ideas introduced in one chapter are apt to reappear—transformed, modulated and extended—in others. He taught his readers to see the world in the same way, inculcating in them an openness and alertness to the often surprising possibilities of the world, and the desire to seek them out.

That’s from the WSJ. The story is about the reclusive mathematician who wrote Scientific American’s puzzle column between 1956 and 1981 and the cultish math-geek gatherings that now happen in his honor every two years. The entire story is worth reading and includes notes on cognition, neuroscience, and even references the indomitable Stephen Wolfram.

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Cities and the Efficiencies of Urbanity

The indomitable Paul Krugman has been discussing cities and geography lately in his NYT blog (a fitting subject for an economist who was awarded the Nobel for his work in economic geography, particularly what we call “New Trade Theory”; wiki here, Alex Tabarrok here). Some particularly interesting lines from different posts. First, the aesthetics of urban development in Hong Kong:

Hong Kong, with its incredible cluster of tall buildings stacked up the slope of a mountain, is the way the future was supposed to look. The future — the way I learned it from science-fiction movies — was supposed to be Manhattan squared: vertical, modernistic, art decoish.

What the future mainly ended up looking like instead was Atlanta — sprawl, sprawl, and even more sprawl, a landscape of boxy malls and McMansions. Bo-ring.

And some commentary on conservative attitudes about evironmentalism:

As I noted a while back, a lot of anti-environmentalism in America these days is about symbolism. And I think the same thing is true about pro-sprawl commentary. Consider the case of Portland, Oregon. Conservatives really, really hate on Portland; examples here and here. Aside from the tendency to engage in factual errors, the hate seems disproportionate to the cause. But it’s an aesthetic thing: conservatives seem deeply offended by anything that challenges the image of Americans as big men driving big cars.

My basic commentary is that I really really like cities. I like the idea of being able to walk most of the places I really need to go, I like the freedom from the responsibilities of suburbia and its trappings, I like the intellectual and cultural diversity that comes with putting lots of people close together. And from a sustainability standpoint, the basic premise of a city is that humans can gain much from exploiting the efficiencies of scale and synergies that close spatial organization allows. Specifically, I’d point you to this op-ed by mathematician Stephen Strogatz in the NYT; Strogatz is a very important mathematician for those following the fields of dynamical systems and non-linear analysis, among others. He wrote my introductory textbook on the subject, which is easily accessible for anyone with a working knowledge of differential theory. Some particularly good insights:

For instance, 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 number of gas stations grows only in proportion to the 0.77 power of population. The crucial thing is that 0.77 is less than 1. This implies that the bigger a city is, the fewer gas stations it has per person. Put simply, bigger cities enjoy economies of scale. In this sense, bigger is greener.

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. And all show an exponent between 0.7 and 0.9.

Now comes the spooky part. The same law is true for living things. That is, if you mentally replace cities by organisms and city size by body weight, the mathematical pattern remains the same.

For example, suppose you measure how many calories a mouse burns per day, compared to an elephant. Both are mammals, so at the cellular level you might expect they shouldn’t be too different. And indeed, when the cells of 10 different mammalian species were grown outside their host organisms, in a laboratory tissue culture, they all displayed the same metabolic rate. It was as if they didn’t know where they’d come from; they had no genetic memory of how big their donor was.

Freaky cool, innit? More later.

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