If your bike’s like my bike it’s got circular wheels. You might just have one of them, but you’ve probably got two of them, but either way they are circular. However, you will probably be surprised to learn that it didn’t have to be like this. You can have a perfectly steady ride on a bike with square wheels, as long as you ride on the right type of surface.

The important feature of a rolling wheel for the cyclist is that you don’t keep jolting up and down as the bicycle moves forward. With circular wheels on a flat surface this is the case: the centre of the cyclist’s body moves forward in a straight line when the bicycle moves forward in a straight line without any slipping. Use square wheels on a flat surface and you are going to have a very uncomfortable up-and-down ride. But could there be a differently shaped road surface that results in a smooth ride when you are using square wheels ? All we need to check is whether there is a shape that leads to a straight-line motion of the cyclist when he has square wheels.

The answer is pretty surprising. The shape of the road surface that gives a steady ride on square wheels is created when you hang two ends of a chain from two points at the same height above ground. This is called the catenary. If we turn it upside down we obtain the shape that is used for many of the great arches of the world. But if you take a catenary and repeat if over and over again along a line you get a sequence of undulations of the same height.  This is the shape of the ground surface that gives us the smooth ride on square wheels. We just need the bottom corners of the square to fit into the sides of the successive “valleys” on the surface. The key feature of the catenary is that when one is placed next to another, the angle between the two sides of the adjacent arches as they come into the lowest point of each valley is a right angle, 90 degrees and that is the angle at each corner of the square. So the right angled wheel just keeps rolling along.

Tags: bike, square wheels