Stan Wagon, a mathematician at Macalester College in St. Paul, Minn., had a bicycle with square wheels. It was a weird contraption, but he rode it perfectly smoothly. His secret was the shape of the road over which the wheels rolled.
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.
www.sciencenews.org/article/riding-square-wheels
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.
Maybe this is how they moved the huge stones way back in ancient times. Maybe the first invention wasn't the round wheel but the 'catenary trail' which the stones rode over.
