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COMPUTING SCIENCE

Dividing the Continent

Brian Hayes

Dividing the Ant Farm

In a two-dimensional world, it’s easy to find a continental divide, if it exists. Think of an ant farm: a thin layer of soil sandwiched between two upright panes of glass. An ant walking along the surface of the soil from west to east will trace a one-dimensional profile, a graph of elevation as a function of longitude. If the profile has just one peak—that is, if the ant climbs steadily to some maximum elevation and thereafter descends continuously—then obviously that unique peak is the divide. ("It’s all downhill from here," the ant might well say.) If there are multiple peaks with valleys between them, the highest of the summits must be the divide.

Some pathological possibilities could spoil this easy analysis. The ant-farm profile could have several tallest peaks, all at exactly the same height, or a plateau might form a continuous line of highest points. In these cases there is no unique continental divide. But such landforms are unlikely. Ignoring them, the algorithm for finding the ant-farm divide is straightforward: Just look for the highest point.

Leaving behind the ant farm to consider a two-dimensional surface embedded in three-dimensional space, the divide problem gets more interesting. In particular, the find-the-maximum algorithm no longer works. Just try it for the case of North America! When you search out the highest point in the lower 48 states, you find yourself atop Mount Whitney, in California, elevation 4,418 meters. But Mount Whitney is nowhere near the continental divide, and all the water that falls on its flanks winds up in the Pacific, none in the Atlantic. (Indeed, much of it flows through the municipal water mains of Los Angeles.)

Figure 1. Global-warming algorithmClick to Enlarge Image

Thinking about this phenomenon on a larger scale raises doubts about the whole concept of a continental divide. Just as runoff can sneak around Mount Whitney, it can also find a path around the entire American Cordillera, which doesn’t really separate the Pacific from the Atlantic. After all, you can get from New York to San Francisco without climbing even the smallest hill: There is a sea-level route, around Cape Horn. From a topological point of view, a continental divide can exist only if a continent girdles the planet, so that the divide is a closed curve, with an inside and an outside.

Perhaps the best answer to this complaint is that the idea of a great divide belongs to the field of topography, not topology. Insisting on mathematical rigor is not necessarily helpful. In any case, we can rescue the concept of the divide, at the cost of making it somewhat artificial. The key step is to cut away a rectangular section of the earth’s crust, corresponding roughly to the lower 48 states, and put it in a high-walled glass box—a terrarium, not so different from the ant farm. Now a continental divide is either a closed curve that lies entirely inside the box, or else it is a continuous line whose endpoints are anchored to the glass walls. With this definition, the divide truly does divide the territory into separate regions.





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