This is a bit creepy, maybe creepier than the UFO in a tunnel.  


   "π also appears as the average ratio of the actual length and the direct distance between source and mouth in a meandering river. Hans-Henrik Stolum, a geologist at Cambridge University, calculated the ratio between the total length of a river to the direct distance between its source and end point.^[6] He found the average ratio to be a bit larger than 3. It is actually around 3.14, which we recognize as an approximation of π. Coincidence?
Rivers have a tendency toward a loopy path. A slight bend will lead to faster currents on the outside shores, and the river will begin to erode and create a curved path. The sharper the bend, the more strongly the water flows to the outside, and in the consequence the erosion is in turn the faster.
The meanders get increasingly more circular, and the river turns around and around in semi-circles. It then runs straight ahead again, and the meander becomes a bleak branch. Between the two reverse effects a balance adapts. The process is demonstrated in Figure 4-3.
We have created a model where a fictitious river is superimposed by semicircles to represent the eventual curve of the river flow. Now by labeling the source point of the river A, end point B, the midpoints of every semicircle M_i (i=1,2,3,4), and the radius of each semicircle r_i, we can check if the ratio of the length of the path of the river (C) to that of line segment AB (l) is actually π."