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Local Topology

Somewhere between Butte and Bozeman I began to doubt that this local classification was going to yield a useful algorithm. It doesn’t even help much in the two-dimensional case (where each point has only two neighbors instead of four). A divide in the ant farm always lies at a peak, but the local properties of peaks will not tell you which of them is the divide. Finding the highest peak requires global information; you have to compare points throughout the entire array.

For the three-dimensional model, the situation is even stickier. Not only is local information unable to identify the divide, but in addition there is no simple global property that will settle the issue. You can’t just compare each point with all the other points, looking for a maximum of some kind. Instead, you need to consider multiple pathways through the array of points.

But if local configurations can’t solve the divide problem, maybe they can at least rule out lots of points that might otherwise be candidates for the divide. For example, it seems beyond question that a pit cannot form part of the divide, and so all pits can be crossed off the list. But as we rolled on beyond Bozeman toward Billings, I gradually realized that no other kinds of points could be excluded. Peaks and ridges and saddles are clearly allowed on the divide. It might seem that slopes and thalwegs would be ineligible, but this is not so. Think of a river delta, where streams diverge and bifurcate. If such a delta were to form at the outlet of a high alpine valley right on the continental divide, with channels flowing down either side, then the entire area of the upstream valley would have to be considered part of the divide, including the slopes and the thalweg.

Admittedly, a river delta in the headwaters is a pretty unlikely landform, but algorithms are supposed to cope with even the oddest cases. And natural landscapes do offer oddities. Some maps show the great divide itself dividing in Wyoming, where it envelopes a high basin. Even stranger are some unnatural landscapes. At the Big Thompson Project in Colorado a tunnel carries water across the great divide—or rather under it, or under the ridge where it ought to run—and thereby alters the topological genus of the earth’s surface.

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