| | Ivars Peterson's MathTrek |
July 13, 1998
However, a square wheel can roll smoothly, keeping the axle moving in a straight line and at a constant velocity, if it travels over a road with the right sort of profile. That corrugated surface must consist of evenly spaced bumps, each one the shape of an inverted catenary.
A catenary is the curve describing a rope or chain hanging loosely between two fixed points. At first glance, it looks like a parabola. In fact, it corresponds to the graph of a function called the hyperbolic cosine.
The Exploratorium in San Francisco exhibits a model of such a roadbed and a pair of square wheels joined by an axle to travel over it (http://www.exploratorium.edu/xref/exhibits/square_wheel.html). When Stan Wagon of Macalester College in St. Paul, Minn., saw that demonstration, he was intrigued. The exhibit inspired him to investigate the relationship between the shapes of wheels and the roads over which they roll smoothly.
Those studies also led Wagon to build a full-size bicycle with square wheels. "As soon as I learned it could be done, I had to do it," Wagon says. The bicycle is now at the Macalester science center, where it can be seen and ridden by the public.
A photograph of Wagon riding his square-wheeled bicycle can be found at http://stanwagon.com/. You can see an animated movie of a square rolling on a bed of inverted catenaries at http://home.att.net/~mathtrek/muse0299.htm.
It turns out that for every shape of wheel there's an appropriate road to produce a smooth ride, and vice versa.
Just as a square rides smoothly across a roadbed of linked inverted catenaries, other regular polygons, including equilateral triangles, pentagons, and hexagons, also ride smoothly over curves made up of appropriately selected pieces of inverted catenaries. Indeed, as the number of the polygon's sides increases, those caternary segments get shorter and flatter. Ultimately, for an infinite number of sides (in effect, a circle), the curve becomes a straight, horizontal line.
That's not all. One can find roads for wheels shaped like ellipses, cardioids, rosettes, and many other geometric forms. One can also start with a road profile and find the shape that rolls smoothly across it. A sawtooth road, for example, requires a wheel pasted together from pieces of an equiangular spiral.
Equiangular spiral on sawtooth road
Mathematics Magazine (Dec. 1992, p. 289, Fig. 8, left)
There's certainly more than one way to ride a bike!
Copyright 1998 by Ivars Peterson
References:
Gray, A. 1998. Modern Differential Geometry of Curves and Surfaces with Mathematica (2nd Ed.). Boca Raton, Fla.: CRC Press.
Hall, L., and S. Wagon. 1992. Roads and wheels. Mathematics Magazine 65(December):283.
Wagon, S. 1991. Mathematica in Action. New York: W.H. Freeman.
Comments are welcome. Please send messages to Ivars Peterson at ip@sciserv.org.
Ivars Peterson is the mathematics/computers writer and online editor at Science News (http://www.sciencenews.org/). He is the author of The Mathematical Tourist, Islands of Truth, Newton's Clock, Fatal Defect, and *The Jungles of Randomness. His current work in progress is Fragments of Infinity: A Kaleidoscope of Mathematics and Art (to be published in 1999 by Wiley).
NOW AVAILABLE The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics by Ivars Peterson. New York: W.H. Freeman, 1998. ISBN 0-7167-3250-5. $14.95 US (paper).