## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## March 20, 2007

### Square Wheel

Riding around on a flat tire is no fun. It feels really bumpy. But a square wheel may be the ultimate flat tire. There's no way it can roll over a flat, smooth road without jolting the rider again and again.

Stan Wagon, a mathematician at Macalester College in St. Paul, Minnesota, 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.

A square wheel can roll smoothly if it travels over evenly spaced bumps of just the right shape. That special shape is called an inverted catenary. A catenary is the curve formed by a chain or rope hanging loosely between two supports.

Turn the curve upside down, and you get an inverted catenary—just like one of the bumps in Wagon's road. Make the road out of a whole bunch of those bumps all in a row, and you can take your square-wheeled bike for a quick spin. Click on the image below to see the square wheel roll on its bumpy road.

It turns out that for every possible wheel shape, there's a road that produces a smooth ride. And for every road, there's an appropriate wheel.

Wagon has used mathematics and computers to investigate many wheel-and-road combinations without actually building them. Wheels shaped like pentagons (five sides), hexagons (six sides), and octagons (eight sides) also roll smoothly over bumps made up of pieces of inverted catenaries.

In fact, as the wheel you use gets more and more sides, the catenary segments required for the road get shorter and flatter. Ultimately, the wheel has so many sides that it looks like a circle and its road is practically flat. It rides just like a normal wheel.

That's not all. You can find roads for other wheel shapes, including an ellipse (which looks like a flattened circle), a cardioid (like a heart with a rounded tip), and a rosette (like a flower with four petals). You can also start with a road profile and find the wheel shape that rolls smoothly across it.

There's one little problem. A weird-wheeled bike has to travel in a straight line along its special road. You can't go left or right!

Muse, February 1999, p. 29.