Stan Wagon, a mathematician at Macalester College, St. Paul, Minn., designed and tested a tricycle with square wheels. However, you won't see him riding it on sidewalks because he can only pedal it on surfaces with evenly spaced bumps having just the right shape.
Mathematically speaking, these special shapes are a series of inverted catenaries. Catenaries are curves formed by a sagging cord strung between two points, resembling a hyperbolic cosine. Turning the curve upside down produces an inverted catenary, just like each bump in Wagon's road.
Riding surfaces for bikes with other wheel shapes must be profiled accordingly, following the respective shape's inverted catenary. As the number of wheel sides increases, catenary segments get shorter and flatter. For an infinite number of sides — a circle — the curve becomes a straight, horizontal line.
