A special shape called a catenary curve, not a circle, is the curve that will give an absolutely level ride with square wheels. A road made with circles—as we’ve done in this Snack—is a reasonably close approximation, however, and is easier to build from commonly available materials.
To see a catenary curve, find a chain or heavy rope, hold one end in each hand, and let it hang upside down. Inverted, a catenary curve will also provide the greatest strength to an arch supporting only its own weight, such as the Gateway Arch in St. Louis, Missouri.
Calculating Wheel Size
In order to travel smoothly over the array of tubes, the sides of the square wheels must be 1.2 times the diameter of the tubes. The equations below explain how this relationship is derived; the diagram shows how the math applies to the square wheels and the “road.” Note that l is the side of the square, and d is the diameter of the circle (which represents the tube). The circumference of the tube equals 2πr. Click to enlarge the proof below.