Ride a bike with round wheels and it rolls smoothly on
a flat road (i.e., the axle remains level).
Can you get a bike with square wheels to
roll smoothly on a road of some other shape?
Surprisingly, yes. A road made up of
inverted catenaries will do the trick!
A catenary is a portion of a cosh curve.
Figure 1 shows how such a bike would roll
(thanks to Stan Wagon for providing this photo of
Wayne Roberts on one at Macalester College).
You can demonstrate this to your class using
Mathematica code available in the reference.
The Math Behind the Fact:
You may wonder:
what about bikes with other shapes for wheels?
What road shape would allow the bike to roll smoothly?
The reference by Wagon and Hall discusses the
fundamental differential equation that solves this problem.
How to Cite this Page:
Su, Francis E., et al. "Bike with Square Wheels."
Math Fun Facts.