COMPUTING SCIENCE

# Up a Lazy River

Meandering through a classic theory of why rivers meander

The Best of All Possible Meanders

Perhaps the strongest rationale in support of Leopold's theory of
meanders is simply that meanders look more like sine-generated
curves than like other common objects from the mathematical
cupboard. But why should we expect meanders to have *any*
simple mathematical form?

The explanations based on bending stress and directional variance rest on principles of global optimization. The favored path is one that optimizes some property measured over the entire course of the river. By choosing the path with the smallest total squared curvature, for example, the river minimizes the energy it invests in turning through sharp bends.

The physical sciences are full of such optimization laws. Optics,
for example, has the principle of least time, which explains the
geometry of refraction by saying that light always follows the path
that can be traversed fastest. This manner of reasoning has proved
very successful, and yet it can be tricky to apply. *Why*
does light take the path of shortest travel time? And how does a
photon know what angle of refraction will get it through a
window-pane most quickly?

In the case of the meandering river, it's not obvious which variables ought to be optimized. Minimizing energy cost seems plausible enough, but what about directional variance? Leopold himself points out that it might make more sense to minimize the variance in curvature, so that the work of turning the river would be spread out as uniformly as possible. But that choice would favor the circle over the sine-generated curve.

It's also hard to know where to *stop* optimizing. The curves
under discussion here are the best possible curves only if one
accepts a number of constraints or assumptions, some of which seem
rather arbitrary. For example, as the experiment with a steel spring
reveals, bending stress can be further reduced by converting a
series of little meanders into a single big one. Thus if minimal
bending stress were the only criterion governing the river's plan
form, all meanders would be as large as possible—but they
aren't. Most meanders have a characteristic scale, proportional to
the width of the river. An even more critical assumption is the
fixed length *L*. We could make the meander problem go away
altogether just by shortening the river.

Finally, to have much explanatory power, a global optimization principle needs to be linked to some local mechanism that puts it into effect. We may well calculate that a certain shape of bend minimizes energy loss, but what are the forces at each point along the river channel that create and maintain that shape? The river can't think globally; it can only act locally.

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**Of Possible Interest**

**Letters to the Editors**: Nautilus Biology

**Technologue**: The Quest for Randomness

**Feature Article**: Twisted Math and Beautiful Geometry