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.
Here, I have constructed a bicycle with square wheels. It's a weird contraption, but you can ride it perfectly smoothly. My secret is the shape of the road over which the wheels roll.
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.
Turn the curve upside down, and you get an inverted catenary--just like one of the bumps in my 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.
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.
In the end, I conclude with possible enhancements in the project that might take us to a whole new world.