I learned something cool from the book I’m readingright now.

I did not know this, but apparently there is a reasonably consistent ratio between the length of rivers from source to mouth, and their direct length as the crow flies.

In the diagram below, the green line shows the course of a river from its source to its mouth.
The black line shows the distance between source and mouth as the crow flies.

The ratio between the green and black lines varies slightly from river to river, but remains reasonably consistent. In fact the average value is approximately 3.14, which is close to the value of π.

Einstein was the first to suggest that rivers have a tendency towards an ever more loopy path because the slightest curve will lead to faster currents on the outer side, which in turn will 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, 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, creating an ox-bow lake. The balance between these two opposing factors leads to an average ratio of π between the actual length and direct distance between source and mouth.

Of course this doesn’t apply for all rivers, but the ratio of π is commonly found in rivers under ideal conditions (gentle slope on an homogeneously erodible substrate) such as those found in Brazil or Siberia.

Isn’t that cool and awesome?