Square wheels can roll perfectly smoothly on roads made of precisely spaced catenary arches.

The catenary curve is the natural shape a chain or rope assumes when suspended between two fixed points. When inverted and arranged in sequence, these curves create a road surface that perfectly accommodates the geometry of a square wheel. As the square rotates, the distance from its center to the contact point varies continuously, but the catenary shape ensures the axle remains at a constant height, eliminating the vertical bouncing that would otherwise occur.The mathematics behind this phenomenon relies on the hyperbolic cosine function, which describes the catenary curve precisely. For a square wheel to roll smoothly, the length of one side must equal the arc length of one catenary hump. This principle extends beyond squares to any regular polygon—pentagons, hexagons, or higher—provided the road is designed with corresponding catenary curves.This concept was famously demonstrated at the Exploratorium in San Francisco, where a square-wheeled tricycle has been a popular interactive exhibit for decades. Mathematician Stan Wagon at Macalester College built one of the first successful square-wheeled bicycles in the 1990s to showcase these geometric principles in action. The square-wheel discovery illustrates how specific mathematical functions can solve engineering challenges that initially seem physically impossible, transforming abstract mathematics into tangible, working machines.