A square wheel with its axle at the center will roll smoothly (that is, the axle will stay level) on a road made up of pieces of an inverted catenary. A catenary is the curve formed by a hanging chain, and a famous structure based on an inverted catenary is the Gateway Arch in St. Louis. The catenaries forming the road for a rolling square meet at right angles. These corners can be smoothed out if the corners of the square are appropriately rounded off at the same time. The image above shows such approximations to the square and the road. The approximations were obtained using Fourier series. Details can be found in:
Leon Hall and Stan Wagon, Roads and Wheels, Mathematics Magazine 65 (1992), 283-301
This paper also discusses how roads for other odd-shaped wheels can be found. Elliptical wheels are interesting. If the axle is at a focus, the road can be described in terms of the cosine function, and if the axle is at the center, the road involves the Jacobian elliptic functions. Here are some more pictures of roads and wheels.
The rolling square and its catenary road.
Wheels can be any shape and the axle can be anywhere.
The road for this ellipse, with axle at the focus, is y = cos x - sqrt(2).
Leon Hall / lmhall@umr.edu / January 30, 1997