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 ?
if you think about the second one, the answer is yes. we use circular wheels on almost anything to get it rolling... cars, bicycles, you name it and the earth is in fact a sphere. a zorb is actually a better example because instead of a tire, which is really a cylinder on its side, a zorb is a sphere.
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