Roads and wheels

San Francisco's Exploratorium contains an intriguing exhibit of a square wheel that rolls smoothly on a road made up of linked, inverted catenaries (see FIGURE 4). That exhibit inspired us to generate a computer animation of a rolling square and further explore the relationship between the shapes of wheels and roads on which they roll. In a sense, we are bringing up to date the paper by G. Robison [4], showing how much more can be done, both numerically and graphically, with modern computer hardware and software. The problem of the square wheel has been rediscovered and solved several times; see [5, 7]. All the diagrams and animations were prepared in Mathematica. Our package that generated the diagrams and the associated animations (see Section 5) can be obtained by sending a Macintosh disk to one of the authors. It is noteworthy that some of the results of this paper, in particular the discovery of a cycloidal locus generated by a noncircular wheel, were discovered only after viewing certain graphics. Mathematica was also used to do all the symbolic integrations that occur. For further applications of symbolic and graphic computation to wheel/road problems, in particular, a complete discussion of the cycloid, see [6, Chapter 2]. The paper is organized as follows. Section 1 discusses the theory and the fundamental differential equation. Section 2 contains many closed-form examples. Section 3 shows how numerically approximating the solution to the differential equation is an excellent approach to diverse examples, even those solvable in closed form. Section 4 squares the circle by considering Fourier approximations to the catenary. And Section 5 discusses the Mathematica package that we built.