Different Roads for Different Wheels
It turns out that wheels in the shape of other regular polygons, such as pentagons and hexagons, also ride smoothly over bumps made up of pieces of inverted catenaries.
The number of sides on the polygon affects the road's shape: as you get more and more sides, the catenary segments required for the road get shorter and flatter. Ultimately, the wheel has so many sides that it looks like a circle and its road is practically flat.
Triangular wheels don't work, though. As a triangle rolls over one catenary, it ends up bumping into the next catenary.
A triangular wheel doesn't work because its points get stuck in the roadbed.
Mathematicians have found roads for other wheel shapes, including an ellipse (which looks like a flattened circle), a cardiod (like a heart with a rounded tip), a four-lobed rosette (like a flower with four petals), and a teardrop.
Roadways for a four-lobed rosette (above) and a teardrop shape (below).
You can even start with a road profile and find the wheel shape that runs smoothly across it. A sawtooth road, for example, requires a wheel pasted together from pieces of an equiangular spiral (sort of a cross between a flower and a star shape).
So far, no one has ever found a road-wheel combination in which the road has the same shape as the wheel. That's an intriguing challenge for mathematicians.
Various combinations of wheel shape and road profile would give you a smooth ride.
NEXT: Rolling with Reuleaux
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