Bike with Square Wheels

Figure 1

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).

Presentation Suggestions:
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. <http://www.math.hmc.edu/funfacts>.

References: L. Hall and S. Wagon, "Roads and Wheels", Math. Magazine 65(1992), 283-301. S. Wagon, Mathematica in Action. Keywords:    bicycle Subjects:    geometry, calculus, analysis, other Level:    Easy Fun Fact suggested by:   Stan Wagon Suggestions? Use this form. 4.21 current rating Click to rate this Fun Fact...     *   Awesome! I totally dig it!     *   Fun enough to tell a friend!     *   Mildly interesting     *   Not really noteworthy and see the most popular Facts! Get the Math Fun Facts iPhone App!

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