An interesting study of polygonal motions (article by Ivars Peterson on Sciencenews, found via Radio):
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.
This picture is too funny:
(credit: Stan Wagon)
Some interesting points:
* A triangular wheel cannot roll smoothly on any surface
* Is there a road-worthy combination of wheel and surface in which the wheel and surface have the same shape ?