(video no longer available)
Phillip Torrone rides the square-wheeled tricycle from the Math Midway, a traveling exhibition of mathematics. Figuring out what kind of catenary curves would be needed for differently shaped wheels is a branch of mathematics that I’m happy exists (as far as I know the problem dates back on some level to the 1960s, but for a good recent illustration of the math involved, check out this PDF from a St. Norbert College mathematical modeling class).
(Via Make)