Saturday, October 12, 2013

Let's do some exploring along The Catenary Trail

Stan Wagon, a mathematician at Macalester College in St. Paul, Minn., had a bicycle with square wheels. It was a weird contraption, but he rode it perfectly smoothly. His secret was the shape of the road over which the wheels rolled.

A square wheel can roll smoothly, keeping the axle moving in a straight line and at a constant velocity, if it travels over evenly spaced bumps of just the right shape. This special shape is called an inverted catenary.
A catenary is the curve describing a rope or chain hanging loosely between two supports. At first glance, it looks like a parabola. In fact, it corresponds to the graph of a function called the hyperbolic cosine. Turning the curve upside down gives you an inverted catenary—just like each bump of Wagon's road.


  1. Interesting.
    I have the urge to build a pyramid.

  2. I don;t know if you knew this or not -- the cartoon confuses the issue -- but it has been suggested that the large stones themselves were rolled on quarter-round pieces of wood (which have been found near some pyramids). I.e., the stones themselves are the wheels.