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X and the City: Modeling Aspects of Urban Life

Princeton University Press
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John Adam is a professor of mathematics at Old Dominion University in Norfolk, Virginia, and he likes to walk. While walking, he, like many of us, thinks about mathematics. But he doesn’t simply think of adding numbers or counting steps. Rather, he sees the world and asks himself how parts of the world can be modeled. In his A Mathematical Nature Walk, for example, Adam walks through the outdoors and asks questions such as: Why is the sky blue? (An obvious question and it is not, as some have told me, because sunlight reflects off the blue ocean!) He questions why leaves are arranged as they are on a plant, or how can the shape of an egg be modeled geometrically. Those questions, and that book, dealt with nature. Now Adam takes a walk in the city. And what does he see there?

He sees planes that take people to the city and asks: what is the flight path of a plane landing and how can I represent it as a mathematical curve? Or. how many taxi rides would it take to circle the Earth? These are questions to get us started and wondering about a city. If you read just the beginning chapters, I think you would be disappointed; the mathematics is rather pedestrian. Read on, because Adam is just getting started.

Further in the book we find ourselves in Central Park wondering how many squirrels live in that famous part of the world. This is a question I’ve never considered but Adam does; with some reasonable approximations his estimate is 10,000. To get a flavor for the book, let’s follow his work.

Central park has an area of approximately 2 square kilometers. That’s a large area so Adam looks at something smaller, say, a football field. There are about 200 football fields in a square kilometer. So now we are breaking the problem into a more reasonable estimate. But how many squirrels are in a football field? Here, we take a guess and consider that there is at least one, but no more than a 1000. So, for such a large disparity, we use the geometric mean of 30; Adam explains that for a large range of data values, the geometric mean is best suited as an estimate.

We can compute the number of squirrels as:

Number of squirrels = 2-square kilometers × 200-football fields per square kilometer × 30 squirrels per football field = 12,000 squirrels. Or, with a little rounding, about 10,000 squirrels in Central Park.

The number is not exact, of course, but it gives you a good idea. You might want to ask yourself how many bags of peanuts you should bring to feed the little guys.

Adam addresses an old problem as well. If it’s raining outside, how fast should you walk to stay as dry as possible? The answer requires various approximations and a reasonable model for the person, but in the end, Adam recommends that when the rain is coming from behind you, walk as fast as the rain is falling along the horizontal direction. That will keep you “dry” in the back without sweeping rain across you to your front. If, however, the rain is falling in front of you, “walk as fast as you safely can” so you minimize your time to get soaked.

Naturally, there’s a chapter titled “Sex and the City” where the book discusses population growth. Here, we find a little history and, I think, Adam has some fun at the expense of earlier prognosticators. He quotes from the November 4, 1960, journal Science, an article titled “Doomsday: Friday 13 November, AD 2026.” And here’s the prediction:

At this date human population will approach infinity if it grows as it has grown in the last two millennia.

It’s now 2013, so that’s only 13 years away! Adam shows that the author of the article used a model for population growth that has a singularity at that date. When we reach the singularity, we’re doomed. Adam derives the model for us and shows us why such a catastrophic prediction could be made. Of course, that’s just a model and no one expects such dire consequences 13 years from now. But it does show the limits of a model and just how careful we have to be in extending results beyond the data. Interpolation is one thing, extrapolation quite another.

In the end, you’ll find this book quite extensive in how many different areas you can apply mathematics in the city and just how revealing even a simple model can be. The mathematics is easy to follow and, while Adam does not include every (or even most) of the steps, you can fill them in if you like. What is more, many of the models lend themselves to computer implementations so you can play with them at your leisure.

A Mathematical Nature Walk opened my eyes to nature and now Adam has done the same for cities.

David welcomes your feedback and can be contacted at mazeld at gmail dot com.

Date Received: 
Wednesday, April 4, 2012
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John A. Adam
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David S. Mazel
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Tuesday, May 22, 2012