Built on Facts

Roads and Radii

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.


  1. #1 Hawke
    January 5, 2009

    I’ve never thought about it that way…

    I first encountered the same idea in microbiology with the explanation of why cells can only grow so big before they start becoming too inefficient.

  2. #2 Rhett
    January 5, 2009

    Did I miss the part about the mirrors? Anyway, my favorite part was “here is a circle from wikipedia.” Classic (because I have done that as well).


  3. #3 Matt Springer
    January 5, 2009

    Good catch. I was going to work in an optics comparison but decided it was best left for another day. Looks like I forgot to change the post title, which I have now fixed!

  4. #4 Judy
    January 5, 2009

    …Yes, but I think it’s scarier to drive in College Station than Houston, especially when all the students get back to town! :)

  5. #5 Jolly Bloger
    January 5, 2009

    Very interesting interpretation. I hadn’t thought about it that way before either. However, your model assumes any given person inside each circle wants to leave the circle with equal probability. As the population of a city grows, the possible destinations for residents grows approximately proportionally, reducing the desire to leave the circle.

    If you assume that a city hits a maximum density, then begins spreading outwards, then traffic congestion within the city stays constant, and traffic into/out of the city is less than proportional to the square of the radius. Whether or not it ever becomes less than proportional to the radius itself is unknown.

    Of course, cities don’t hit a maximum density, they start growing upwards as well, which makes things all the more complicated!

  6. #6 Jolly Bloger
    January 5, 2009

    Oops, I guess I forgot that as ‘destinations’ in a city increase, traffic going in will increase, and of course everyone who goes in must come out and vice versa… This may render my entire comment moot.

  7. #7 Bob Sykes
    January 6, 2009

    The whole business of perimeter-limited growth needs some rethinking. A few years ago, biologists discovered Epulopiscium fischelsoni, a fish parasite. This eubacterium is about 0.5 mm in diameter, about 500 times the size of a typical eubacterium. Other giant bacteria have also been discovered. Then there are cities. There are several Third World cities with populations in excess of 10 to 20 million. So perimeter limitation doesn’t seem to kick in as soon as we believe.

  8. #8 Eric Lund
    January 6, 2009

    The perimeter per se cannot be the limiting factor, because real cities (and cells) can form shapes other than circles/spheres, for which the ratio of perimeter to radius is much larger than for a circle. In the extreme case, a fractal perimeter has infinite length but still encloses a finite area. Radius may still enter in, but as Bob correctly points out there are many cities in the world with much larger populations than Houston–and not all of them are in the third world (examples like Tokyo, London, and New York come to mind).

    I suspect the issue is one of infrastructure design. As cities grow, they tend to expand, and preferentially increase density, along their primary transportation corridors. Whether those corridors are rivers, rail/subway lines, or highways makes little difference in the argument. The point is that the corridors have a finite capacity (implicit in the case of rivers, explicit in the case of highways), and once a corridor is saturated it is difficult to add capacity to it since any additional land required for it is precisely the land most likely to be occupied with dense development. Houston has grown significantly over the past 60 years, but many of its roads were designed for the city as it was expected to be years (or more likely decades) ago. So you sit in traffic in Houston (or other city) because roads intended for the Houston of circa 1980 (in some cases more recent, but in other cases it may be more like 1970) have to handle the traffic of today’s Houston, for which they are entirely inadequate.

    Here’s another observation about city traffic, which may or may not be related. Often the circumferential highways will be among the city’s most congested. I don’t know if that specifically applies to Houston, but I know it to be true of Boston, Washington, Los Angeles, Seattle, Atlanta, London, and Beijing. (San Francisco is a counterexample, but mainly because there many trips require crossing the bay, which can only be done in a handful of places.)

  9. #9 Uncle Al
    January 6, 2009

    How does blood perfusion of tissue limit the size of a mammal, say mice to elephants to blue whales?

    Horrible traffic is politically desirable. Increase expenditures for fuel and other consummables (then more taxes collected). Belch pollution (then more regulation levied). Cry out for more construction (then more corruption enjoined). Exacerbate road rage (then more traffic citations collected). The worst possible circumstance is a polity with good roads and synchronized traffic signals.

    The only power any government has is the power to crack down on criminals. Well, when there aren’t enough criminals, one makes them. One declares so many things to be a crime that it becomes impossible for men to live without breaking laws. Ayn Rand.

  10. #10 Kaleberg
    January 11, 2009

    There is a political impact as well. Roads are government spending. The ratio of road to population, that is, government spending to population, is highest in rural areas which is why rural areas are more likely to be anti-government.