Bends in the river

Oxbow lakes speak of the sinuositythe 'bendiness' if you likeof rivers. The narrower a river, the more bendy it is

A pleasant thought I have held on to for many years, without thinking much about it, got torn to shreds recently. Let me explain.

Several years ago, while flying somewhere, I looked out of the window and spied some curious-looking bodies of water on either side of a river. Long and thin, each was shaped like the letter “C". A mystery indeed: why these similar-shaped lakes in close proximity?

Got home and did some research, to learn that these are “oxbow" lakes. They owe their existence to the river that runs nearby. Here’s how: especially in the plains, rivers never run in straight lines. (Pretty much nothing in nature forms straight lines, in any case). They form curve after curve, and the flow of water itself makes those curves get more curvy. Imagine a section of the river that’s shaped like a shallow “S". Over time, the “S" gets deeper; the two tongues of land that it marks out get more pinched at their waists; eventually the river breaks through at the waist and takes the shorter, straighter path. If you picture this in your mind, you will see that what’s left after the breakthrough are these “C"-shaped bits of water: oxbow lakes.

The science of how these bends form and deepen is intriguing by itself. But for me, oxbow lakes speak of the sinuosity—the “bendiness" if you like—of rivers. The narrower a river, the more bendy it is. The more bendy it is, the more likely it is to form oxbow lakes. Now if you divide the actual curved length of a river by how long it would be if it flowed in a straight line from source to mouth, you get a reasonable measure of such bendiness. It’s called the river’s sinuosity.

Think about it: the higher this ratio, the more the river tends to bend; the lower it is, the straighter the river is; at 1, the river is a perfectly straight line. (Not that you’re going to find a river like that). Also, the higher the sinuosity, the more likely oxbows will form. And as soon as one forms, that part of the river straightens, lowering the sinuosity.

For many years now, guys like me who like fiddling with numbers have known something remarkable about this particular measure. In a 1996 paper, Hans-Henrik Stølum of the University of Cambridge described how he used both data and some sophisticated simulations of “freely meandering rivers" to consider how rivers, well, meander. He found that sinuosity varied between a low of 2.7 and a high of 3.5. Perhaps those numbers already suggest something to you? Stølum spelled it out in his paper: on average, the sinuosity of a river is pi.

Yes, pi.

This left guys like me astonished. Here we are, wandering peacefully among the abstract patterns and contours of geography, and suddenly one of the fundamental numbers in mathematics leaps out at us. Yes, at one level it is astonishing. At another level, perhaps it tells a story of exactly how fundamental that number really is.

Still, why the connection to the bendiness of rivers? In his research, Stølum simulated a river’s meanderings with fractal geometry. He characterizes this geometry with this delectable line: it is “idealized in the form of a perfectly symmetrical hierarchy of bends": large bends, beyond which are smaller bends, and so on. Photographs of rivers from space actually suggest such a hierarchy—though, naturally, more chaotic than symmetrical.

The bends can be thought of as semicircles or other arcs of circles. And where you have circles, you have pi. Shorn of Stølum’s mathematics, which is a little beyond the scope of this essay, that’s why sinuosity tends towards pi.

Or so Stølum suggested.

But since he published his paper, some people have tried to match this theory against actual data, now more easily available via such tools as Google Maps. The website PiMeARiver.com asked people to submit data about rivers around the world, calculating sinuosity for each one. It now lists nearly 300 rivers, including India’s Krishna, Yamuna, Narmada and more. And what it suggests comes as a small knock for pi-obsessed people like me.

The Krishna’s sinuosity? 1.73. The Rhine? 1.82. The Volga? 2.25. The Godavari? 1.57. The Yellow River? 2.62. Perhaps those numbers already suggest something to you? For the nearly 300 rivers on PiMeARiver, the sinuosity works out, on average, to 1.91.

Pretty far from pi.

The thing is, on the face of it, there is not a lot to fault in Stølum’s methodology and reasoning: the circle segments, the hierarchy of bends. But somehow the data now doesn’t seem to support his conclusion.

But that’s ok, because it will just spur mathematicians to do what they love doing: search for an explanation for an intriguing observation. Did they just happen to stumble on 300 less-bendy rivers? Is there a significance to 1.91 that they haven’t figured out yet, perhaps even a connection to oxbow lakes?

Or will pi manage to astonish us again, by popping up somewhere else in all this?

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers explores the joy of mathematics, with occasional forays into other sciences.

Comments are welcome at dilip@livemint.com. To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza

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