River Math

*Edit: reading into this information, it appears the ratio is just a myth. Singh had cited a single researcher's paper in Nature magazine, and the results have not been replicated. It also only applied to a certain kind of low-land river. The actual ratio appears to be closer to 1.9- perhaps the value of pi/phi, which are two transcendental numbers*

    The average ratio of a river's length to its straight-line length (as the crow flies) is the approximate value of pi: 3.14 (Singh, 54-55). This is an example of a natural system's sinuosity being a transcendental ratio. The river is a perfect balancing act between the chaos of elevation that makes it flow outward, and the order of mass on the ground that redirects it inward. These two forces net to a graphical representation of the circle on a natural plane.
    We can see this by imagining the right triangle as a slice of our 3-dimensional Earth. Where the river starts, high on the y-axis, it will meander through the x and z coordinates. The forces at work in this viscosity scheme are gravity, represented by the y-axis, and mass, represented by the x and z axes. The net integral of the river's ratio will approach that of pi as it approaches the sea. But not always for steeper ones; the flatter the river is, the more this law applies.

    It's an interesting dynamic that plays out when the course of natural events portrays an idealistic geometry. In the natural world, there are no perfect shapes, but there are plenty of cases like this one that imply a perfect design.

Singh, Simon. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. 1998