Perfect squares do not exist in nature. They are a math construct.
Under ideal conditions (uniform gentle slope on a homogeneously erodible substrate), the sinuosity of a meandering river approaches π. The sinuosity is the ratio between the actual length and the straight-line distance from source to mouth. Faster currents along the outside edges of a river’s bends cause more erosion than along the inside edges, thus pushing the bends even farther out, and increasing the overall loopiness of the river. However, that loopiness eventually causes the river to double back on itself in places and “short-circuit”, creating an ox-bow lake in the process. 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.
As far as our limited knowledge is concerned, Pi is infinite and unpredictable. Math helps us build our world. But the world which built us, cannot be measured by us.
