-----Original Message----- From: math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission .com] On Behalf Of Daniel Asimov Sent: Monday, February 19, 2007 8:46 PM To: math-fun Subject: Re: [math-fun] You Can Balance a Wobbly Table by Rotating It

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).

...

...

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). -------------------------------------------------------------- ------------------------

I think this is a completely different conjecture than the "putting a table on the floor so it doesn't wobble" conjecture. In particular, most floors are surfaces, rather than curves, and most tables have a fixed distance between their legs. I think the "putting a table on the floor" theorem is a theorem of the form "Given a differentiable function of two variables, z = f(x,y), and a distance H, there exist four points in the plane, (x1,y1), (x2, y2), (x3, y3) (x4, y4) such that the four points in space, (x1, y1, f(x1, y1)), (x2, y2, f(x2, y2), (x3, y3, f(x3, y3)), (x4, y4, f(x4, y4)) form a square of side H.

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 (*).

I'd be interested to see a proof of this. But I don't see any connection between this theorem and the "placing a table on the floor so it doesn't wobble" theorem. Andy Latto andy.latto@pobox.com