Pi and the meandering paths of rivers

Einstein, fluid dynamics, and the beautiful mathematics of nature.

Pi and the Meandering Paths of Rivers

“Rivers run through our civilisations like strings through beads,” Olivia Laing wrote in her stunning meditation on life, loss, and the meaning of rivers“There is a mystery about rivers that draws us to them, for they rise from hidden places and travel by routes that are not always tomorrow where they might be today.” But this chaotic, civilization-strewing mystery may be underpinned by one of the most elemental mathematical truths of nature.

In her splendid ode to pi, the Nobel-winning Polish poet Wisława Szymborska extolled it as “the admirable number… nudging, always nudging a sluggish eternity to continue.” On the second day of spring in 1996, the Cambridge University earth scientist Hans-Henrik Stølum published a paper announcing his astonishing finding that pi is also nudging, always nudging, the bendy paths of the world’s rivers to continue their seemingly chaotic meanderings — in a mathematically predictable pattern. His simulation, using empirical data and fluid dynamics modeling, found that the oscillating paths of rivers — their sinuosity, calculated by dividing the river’s actual meandering length by the length of the direct line drawn from source to sea — average 3.14.

Although crowdsourced data failed to replicate Stølum’s average, the crowdsourced sample was too small to prove or disprove a theoretical model addressing the total average of all the world’s rivers. What is more interesting than the finding itself is the very notion that pi — or any number — can undergird seemingly chaotic patterns in nature. Simon Singh takes up this question in his endlessly fascinating book Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem (public library), which also gave us Pythagoras on the purpose of life and the meaning of wisdom.

Citing Stølum’s finding, Singh writes:

The number π was originally derived from the geometry of circles, and yet it reappears over and over again in a variety of scientific circumstances. In the case of the river ratio, the appearance of π is the result of a battle between order and chaos. Einstein was the first to suggest that rivers have a tendency toward an ever more loopy path because the slightest curve will lead to faster currents on the outer side, which will in turn result in more erosion and a sharper bend. The sharper the bend, the faster the currents on the outer edge, the more the erosion, the more the river will twist, and so on. However, there is a natural process that will curtail the chaos: increasing loopiness will result in rivers doubling back on themselves and effectively short-circuiting. The river will become straighter and the loop will be left to one side, forming an oxbow lake. The balance between these two opposing factors leads to an average ratio of π between the actual length and the direct distance between source and mouth.

Complement with the fluid dynamics of Van Gogh’s “Starry Night” and mathematician Lillian Lieber on what Euclid can teach us about social justice, then revisit the story of how a Hungarian teenager revolutionized mathematics and equipped Einstein with the building blocks of relativity, without which he would not have been able to make his prediction about the curvature of rivers.

Central photo: The Tetons and the Snake River by Ansel Adams, from Ansel Adams in the National Parks.