They’ve taken their model past the chalkboard, out into the field and, more recently, across the tumultuous, noisy totality of Earth’s river basins. Now they’re looking even farther afield to Mars, and perhaps Saturn’s moon Titan, each of which hosts its own mysterious branching channels. Their basic math doesn’t work all the time, but it does work widely. And where it doesn’t work, the team believes, that breakdown provides its own hint to underlying environmental conditions.

Beyond all that, their recipe for river networks also offers a certain aesthetic quality. “The mathematics is beautiful,” said Christopher Paola, a geologist at the University of Minnesota, who was not part of the research. “It’s just gorgeous.”

From Bump to Branch

If Rothman is right, his team’s work would add river networks, or at least some of them, to a class of eerily similar branching patterns found throughout nature. These systems all follow what mathematicians call Laplacian growth, named for the 18th-century French mathematician Pierre-Simon Laplace. Snowflakes, analyzed up close, seem to sprout their symmetric-looking crystalline structures through Laplacian growth. The process also predicts the branching pattern that electric current takes when it leaps across a gap, how bacterial colonies spread in petri dishes, and how minerals grow into veiny, dendritic patterns that look like fossils in rocks around the world.

In each, patterns grow when a bump develops from an imperfection on an otherwise smooth boundary. Consider the surface of a newborn snowflake, a frozen edge creeping out into unfrozen ambient water. Invariably, what starts out as a smooth edge will have some little bump on it — even just a few out-of-place atoms. That bump will jut a little bit into the liquid. Out there, the bump loses heat to the surrounding water a little faster. It cools, and a bit more water freezes on top of it. In time the bump grows, forming a bigger bump. The process continues, and soon enough the atomic imperfection extends into a crystalline branch.

The details vary in different Laplacian systems, but the rule is the same: Growth begets growth. Bumps make branches. Branches keep growing at their tips. Eventually, the branches may spawn their own bumps through the same process. That can make new branches that copy the same shapes as their parent branches, only at smaller scales.

Rothman’s team has long argued that certain river networks — the granddaddy of all obvious natural branching patterns — belong to this illustrious group. But the rub, for pattern hunters, is to show that simple rules really do carry over into messy reality.

A River Grows

Rothman’s group found their proof of concept near the town of Bristol in the Florida Panhandle. There, a vast network of channels feeds water into the Apalachicola River.

The network itself, ending in dendritic channel tips, is slowly extending away from the river. As the channel tips grow, they cut into 2-million-year-old sand. At each growing tip, groundwater burbles to the surface. Just like the cold water around a growing snowflake, it’s the kind of environment that lends itself to Laplacian growth.

Building on work on groundwater-driven erosion by Thomas Dunne, a geomorphologist at the University of California, Santa Barbara, the Rothman team set out to test whether simple math could describe this situation. They flew to Florida and sloshed through these streams, measuring the rate at which water flows through individual channels. Then they used ground-penetrating radar to check the height of the water table below.