I was out of town (again) this past weekend, hence the posting shortage. Why in the world is it so much harder to find time to post during a nominal between-semester-break? I dunno, but it seems to be true. Free time doesn’t scale the way you’d like.

One of the fundamental skill sets a physicist or really just about any scientist needs is to understand how quantities change scale. This is especially true when things change scale at different rates. I first noticed this particular instance of scaling phenomena while sitting in traffic in the city of Houston, Texas whose map (via Google) is printed below:

Here at the same scale is the town where I spend most of my time: College Station, Texas.

Not counting the very dense central regions of Houston, it’s not to much of a stretch to say that a randomly selected patch of Houston’s sprawl will look a lot like the interior of College Station in terms of population density. In any case the density is usually of the same order of magnitude. But the developed area of College Station is tiny compared to the developed area of Houston. So as an approximation pretend that both are of equal and uniform population density, and we’ll try to use that fact to explain why traffic is so much worse driving through Houston than it is driving through College Station.

Such a simplification makes clear what the problem is. Pretend for simplicity that both Houston and College Station are circular. They aren’t, but this additional simplification is a very gentle one which will only cause an error which is O(1). Now let’s find a picture of a circle (this one from Wikipedia, with British spelling):

Now imagine that the circle encloses each city. The Houston circle will clearly be much, much larger. The roads entering the city must cross through the perimeter of the circle. Only so many roads per unit perimeter can fit, just as only so many people per unit area can fit.

But the perimeter is proportional to the radius, and the area is proportional to the radius *squared*. That is, for cities of larger and larger radius the number of people within the city increases much faster than the number of roads that can support them. Triple the radius of the city and you can triple the maximum number of roads entering. But the number of people in the city will have increased by a factor of 3*3=9, leaving you behind.

Of course there’s mitigating factors. College Station doesn’t have nearly the number of incoming roads that its perimeter can support, nor does it have any need for commuter lanes, toll roads, or mass transit. All those things can greatly increase the efficiency of the roads supporting a city. But in the final analysis, eventually you’ll have fought a losing battle and there will be some limit to the practical size of a city before traffic simply becomes unmanageable.

I imagine the traffic engineers reading this are unhappy with my very simplified model of the perimeter as the variable of interest, for obviously traffic and urban design are vastly more complicated. And that’s perfectly true. The mathematics is however entirely implacable and makes the accomplishments of traffic engineers that much more impressive.