From: "G. A. Edgar" Subject: Re: Can a square wheel roll smoothly? Date: Thu, 22 Apr 1999 12:41:06 -0400 Newsgroups: sci.math In article , Tom Frenkel wrote: > To sci.math: > > Recently someone posed this problem to me, but we did not agree on the > best solution. > > If one has a square wheel, is there a shape that the ground could have, > that would allow the wheel to roll absolutely smoothly, i.e. with the > "axle" of the wheel not making any vertical movement? > > It's pretty obvious that the ground would have to have a shape something > like a succession of semicircles, but I don't know if this is the best > solution. Also (assuming a semicircle), what would be the ratio of the > side of the wheel to the diameter of the semicircle? And would the > semicircles be directly adjacent to each other, or would there be a gap > between them? It turns out the bumps should be catenaries, not semicircles. This can be solved using some differential equations. Stan Wagon (Macalester College) made a working square-wheel bicycle. There was a picture published in the newspaper last year that shows it actually working. Here it is... -- Gerald A. Edgar edgar@math.ohio-state.edu Department of Mathematics telephone: 614-292-0395 (Office) The Ohio State University 614-292-4975 (Math. Dept.) Columbus, OH 43210 614-292-1479 (Dept. Fax) ============================================================================== See also a "Math Horizons" article, and Ripley's "Believe It Or Not" ! -- djr ============================================================================== 94f:70002 70B10 Klein, Nelson H. Square wheel. (English. English summary) Amer. J. Phys. 61 (1993), no. 10, 893--896. Summary: "A solution to the paradoxical problem of a square wheel is presented. Using kinematics, it is shown that the correct roadbed for a square object rolling without slipping is a series of inverted catenaries. The dynamics of the square are revealed by the conservation of energy method. Remarkably, the square is shown to be capable of winning a downhill race against a sphere on a parallel inclined plane." _________________________________________________________________ 94e:26020 26A99 (00A69) Hall, Leon(1-MOR); Wagon, Stan(1-MACA) Roads and wheels. Math. Mag. 65 (1992), no. 5, 283--301. [To journal home page] The paper contains the following sections and subsections: Introduction, Building a wheel, Closed-form solutions, Polygonal wheels, Tilted roads, Cycloidal roads, A road that is its own wheel, Round wheels can roll on round roads, Off-centered elliptical wheels, Centered elliptical wheels, An elliptical road, Vertical scaling, Summary of some road-wheel relationships, Generating solutions numerically, Squaring the circle with Fourier series, Road approximations having closed wheels, A Mathematica wheel-building package. From the introduction: "San Francisco's Exploratorium contains an intriguing exhibit of a square wheel that rolls smoothly on a road made up of linked, inverted catenaries. 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 [Math. Mag. 33 (1960), no. 3, 139--144], 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. "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." The paper contains 21 figures. Reviewed by B. K. Lahiri © Copyright American Mathematical Society 2000