Water Cataracts and the “Shortest-Time Descent” Curve (brachistochrone) As One Natural Phenomenon

This article unveils the connection between design in nature and a classic mathematics problem from 1696 to 1697: the brachistochrone. Some flow designs seem to act as obstacles to flow (cataracts, waterfalls, and roll waves), in contradiction with the natural tendency to facilitate flow (round ducts, bifurcated channels, animal speeds, and frequencies). The brachistochrone problem is to determine the curve of shortest-time descent without friction. The connection communicated in this article is that cataracts, roll waves, and the curve of shortest-time descent are about one natural phenomenon, which is predictable. This is demonstrated by comparing two ways for water to flow downhill: (i) on a stepped path (free fall over a dam, followed by accumulation in a large and nearly stagnant pool) and (ii) on a straight incline. We find that (i) is faster than (ii). In conclusion, brachistochrone-like paths are naturally occurring flow configurations, in accord with common observations of natural flow configurations that facilitate access.