Square Wheels, Part 1

Bicycle Riding on Square Wheels

Before starting our long working week, let's relax with this story of a bicycle with square wheels. No, it's not a joke. And it even rides smoothly. But there is a trick: the road must have a specific shape. The Math Trek section of Science News Online tells us more about this strange bicycle -- actually a tricycle with front wheel and two back wheels.

Stan Wagon, a mathematician at Macalester College in St. Paul, Minn., has a bicycle with square wheels. It's a weird contraption, but he can ride it perfectly smoothly. His secret is the shape of the road over which the wheels roll.

Here is Stan Wagon riding his tricycle (Credit: Stan Wagon).

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 -- just like each bump of Wagon's road.

In fact, the idea is not new, and Wagon picked it after seeing an exhibit about square wheels at the Exploratorium in San Francisco. But Wagon went further by exploring the relationship between all kinds of wheels and road shapes.

Just as a square rides smoothly across a roadbed of linked inverted catenaries, other regular polygons, including pentagons and hexagons, also ride smoothly over curves made up of appropriately selected pieces of inverted catenaries. As the number of a polygon's sides increases, these catenary segments get shorter and flatter. Ultimately, for an infinite number of sides (in effect, a circle), the curve becomes a straight, horizontal line.

Here is the conclusion of the article.

So far, no one has found a road-and wheel combination in which the road has the same shape as the wheel. That's an intriguing challenge for mathematicians.

So why don't you try to solve this math puzzle?

Source: Ivars Peterson, Science News Online, Week of April 3, 2004

technorati tags: science, mathemathics, weird stuff