Details

Given the road in the form , the wheel is given in polar coordinates by , where is the solution to the differential equation with . The details were first worked out by James Clerk Maxwell in 1849 [1], though his work seems to have been forgotten. A detailed modern investigation was carried out by G. B. Robison (1960) [2], who rediscovered the result found by Maxwell that a straight line rolls smoothly on a catenary. The important idea of truncating the catenary and line so as to get a rolling square is due to D. G. Wilson (1964) [3]. Further details and examples can be found in the paper by Hall and Wagon [4]. A photo of a working square-wheel tricycle can be found at the author's website [5].

References

[1] J. C. Maxwell, "On the Theory of Rolling Curves," Transactions of the Royal Society of Edinburgh, 16(5), 1849 pp. 519–540 (p. 537, Ex. 12). archive.org/details/transactionsofro16roy/page/518/mode/2up.

[2] G. B. Robison, "Rockers and Rollers," Mathematics Magazine, 33(3), 1960 pp. 139–144. (Result first in E1033 1952.) www.jstor.org/stable/3029034?origin=crossref.

[3] D. G. Wilson, Problem E1668, American Mathematical Monthly, 71(2), 1964 p. 205. Solution: 72(1), 1965 pp. 82–83.

[4] L. Hall and S. Wagon, "Roads and Wheels," Mathematics Magazine, 65(5), 1992 pp. 283–301. doi.org/10.1080/0025570X .1992.11996043.

[5] S. Wagon. stanwagon.com.