A Better Distorted View

By Ivars Peterson

A map can show much more than rivers, roads, and political boundaries. It can express an attitude. Saul Steinberg’s famous New Yorker cover illustration, “View from 9th Avenue,” shows a foreshortened map of New York City and its environs. Beyond the city’s avenues and the Hudson River, Steinberg’s map looks westward toward vaguely defined regions: New Jersey, the rest of the U.S., the Pacific Ocean, and a barely-visible Asia and Europe. The map neatly encapsulates a Manhattanite’s self-centered perspective on the world.

A map can also illuminate the way people live. It can incorporate census results, disease incidence, or the number of telephones in use. A simple color code, for example, can show where the incidence of a particular disease is high and where it’s low.

Such representations, however, can be misleading. Inevitably, cities would show a higher incidence than rural areas merely because the former have larger populations. Plotting per capita incidence takes care of that problem but discards information about where most of the cases occurred.

One solution is to take out variations in population density but still show how many cases occur in each region. This can be done on a distorted map in which the sizes of geographic regions appear in proportion to their populations, whether it’s people, goods, or other items. Such a map is known as a cartogram.

Constructing a usable cartogram is challenging. The computer methods currently available sometimes produce maps with overlapping regions or weird distortions. In some cases, the maps are so highly skewed that they’re difficult to read or interpret. Moreover, it can take hours of computer time to produce a single cartogram.

Now, two researchers have turned to the physics of diffusion to develop a new, speedy technique for generating cartograms by computer. “Our method is conceptually simple and produces useful, elegant, and easily readable maps,” say Mark Newman and Michael Gastner of the University of Michigan in Ann Arbor. They describe their procedure in the Proceedings of the National Academy of Sciences (1).

* * *

Newman and Gastner started with the observation that, on a population cartogram, the population is spread out evenly. The conversion of a geographic map into a cartogram reminded the physicists of diffusion-the process by which a gas spreads to fill available space until it has a uniform density throughout.

Applied to a map, there would be a flow from areas with high population densities to those with low densities. This flow would take any boundaries with it to create the new map.

Inspired by this analogy, Newman and Gastner turned to an equation that physicists use to describe diffusion. Starting with a mathematical description of the population density for a given map, they used a computer to solve the diffusion equation through a step-by-step process. The result: a map in which the population density is the same everywhere and which has no overlapping regions.

“The programming necessary to make this work is quite involved, so it took a while to produce the first maps,” Newman says. “Once we did, however, it rapidly became clear that this was a useful tool.”

It’s a little surprising that no one had thought to use this approach before. “Perhaps it’s reasonable to a physicist-diffusion is something physicists all study as undergraduates-but not to someone in cartography or geography, which is where most people working on this topic are coming from,” Newman says.

* * *

Newman and Gastner have tried out their diffusion method on population data from the 2000 U.S. census. To demonstrate the technique’s effectiveness and versatility, they created cartograms displaying the results of the 2000 U.S. presidential election, lung cancer rates among males in the state of New York, and the distribution of wire-service news stories by state.

In the close contest between Republican George Bush and Democrat Al Gore, Newman and Gastner’s cartogram based on population density reveals a fairly even split. In contrast, using a standard geographic map and simply coloring in each state according to which candidate had received more votes there produces a much larger area in the Republican color-even though Gore finished slightly ahead in the popular vote.

“Clearly, then, a simple map is a poor visual representation of the election results, in the sense that it is hard to tell which party got more votes by looking at the map,” Newman and Gastner say.

So far, Newman and Gastner have applied their technique only to standard examples. They plan to develop a software package that would permit geographers or cartographers to use the method on trickier, more interesting problems without having to write complex computer programs themselves.

Newman and Gastner are working to further increase the method’s speed. They are also considering the application of their tool on a global scale.

“We haven’t done anything larger than the United States, but that’s not because of scaling problems,” Newman says. “In fact, the U.S. maps were done on a 1,000-by-1,000 grid, so we already, in effect, had a million regions in those maps, and the calculations were still very quick.”

The challenge in creating maps of large portions of the world derives from Earth’s curvature. The method that Newman and Gastner used to flatten the area covered by the United States doesn’t work for the whole globe.

Going to larger areas “would involve rewriting our computer codes,” Newman says. Nonetheless, “in terms of running time, I see no reason why very large maps, including the entire world, should not be possible.”

“Creating [perfect] cartograms remains an algorithmic puzzle,” says geographer Daniel Dorling of the University of Sheffield in England. “I’ve been searching for 15 years to find a PhD student willing to take this problem on. We now have the computational power-we lack the brains. The solution has probably already been found in another area.”

“One of the repeated lessons of the field of complex systems in which I work is that there are an awful lot of good ideas out there, and many of them are well known, but few of them are well known in every field,” Newman says. “So, people can make significant progress by applying an idea commonplace in one area to a field in which it has less, or no, currency.”

1. Gastner, M.T. and M.E.J. Newman. 2004. Diffusion-based method for producing density-equalizing maps. Proceedings of the National Academy of Sciences USA 101(20):7499-7504.

About the Author

Ivars Peterson is a mathematics writer and online editor for Science News.

This article is adapted from Peterson, I. 2004. A better distorted view. Science News, Vol. 166, Aug. 28, 2004, p136-137 (www.sciencenews.org). Reproduced by permission of Science Service, Washington DC.

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