Robert Baillie writes: << On NPR's Saturday Weekend Edition, they did a story with "Math Guy" Keith Devlin. He described a recent (last year) proof that, if the slope of the ground is at most about 35 degrees (actually, ArcTan[1/Sqrt[2]]), then you can balance a wobbly table by rotating it. (This assumes a mathematically perfect table, with the ground being a continuous function of x and y).
A closely related old problem that seems to be a "folk theorem" (but *only* a folk theorem, apparently without a written proof), is this: Claim: Any simple closed curve in the plane contains the four corners of some square. ------------------------------------------------------------------------------------- I guess the Claim that Robert refers to can be stated rigorously as follows: Any continuous space curve of the form z = f(exp(i*theta)) (i.e., {cos(theta), sin(theta), z(theta), 0 <= theta <= 2pi}, where z has period theta) contains 4 points in space that form the corners of a square . . . as long as z = f(x+iy) is differentiable and satisfies (dz/dy)/(dz/dy) <= sqrt(1/2). -------------------------------------------------------------------------------------- A[n apparently as yet unproved] theorem that would imply both of these is this: (*) Claim: Any simple closed curve in R^3 contains the 4 corners of a (planar) square. --Dan P.S. I will say, smugly, that I believe I know how to prove (*).