On what kind of surface would a square wheel function the same as a round wheel?
(In reply to
How's this for a solution? by levik)
Consider a surface with a cross-section of 180 degree arcs (convex up) with the same radius, touching. One side of the square wheel will roll along one arc until its corner is wedged between two of these arcs. When I say wedged I mean suspended without touching except at two points, one on each semicircle. Then it will continue to roll along the next arc with its next side until the next corner is wedged between the next two arcs, etc. A circle will do the same thing. In order to roll rather than slip, the length of the side of the square should be equal to the part that rolls (which will be 1/4 of the circle or π/2 * R) plus two times the part that gets wedged (which will be R * √2 - R). I would call this a solution except that I can't explain without a picture.
When I say a circle will do the same thing, I mean it will roll until it is touching two semicircles, so part will roll and part will just be wedged between semicircles momentarily. If it is to roll along the same part of the path as the square wheel it should have radius equal to R * √2 - R.
I think this is the answer.
