Thanks to all of you who steered me towards Gardner's article "Wheels" (reprinted as the first chapter of "Wheels, Life and Other Mathematical Amusements"). It's rare for Gardner to get anything wrong, but he may have done that in one or two places in this piece; what do you all think? Speciifcally: (1) Gardner writes (in the second to last paragraph of the article proper, on page 8): "There are varied and perplexing problems that involve non-circular "wheels." For example, suppose a square wheel rolls without slipping on a track that is a series of equal arcs, convex sides up. What kind of curve must each arc be to prevent the center of the wheel from moving up and down? (In other words, the wheel's center must travel a straight horizontal path.) The curve is a familiar one and, amazingly, the same curve applies to similar tracks for wheels that are regular polygons with any number of sides." But my understanding is that when the polygon is an equilateral triangle, the inverted catenary curve doesn't work: the corner of the two triangle tries to cut into the catenary arc that it's purportedly moved beyond. (2) Gardner writes (in the Answers section): "If a wheel is an irregular convex polygon, the track must have arcs that are differently shaped catenaries, one for each side of the wheel." But will that work for EVERY convex polygon? I'm guessing "no", if only because of triangles and things with similar sharp corners. But my intuition tells me that for a generic convex polygon, the answer is "no" (unless you discontinuously change what you mean by the "center" of the wheel as you roll it). I looked at "Rockers and Rollers" ( http://steiner.math.nthu.edu.tw/disk5/js/cardioid/6.pdf), but it seems to me that at best, Robison proves that the only curve that might work is one made of catenaries; he does not explore conditions under which this necessary condition is also sufficient. Jim Propp