1 9 N ov 2 00 5 Mathematical Table Turning Revisited

We investigate under which conditions a rectangular table can be placed with all four feet on a ground described by a function R 2 → R. We start by considering highly idealized tables that are just simple rectangles. We prove that given any rectangle, any continuous ground and any point on the ground, the rectangle can be positioned such that all its vertices are on the ground and its center is on the vertical through the distinguished point. This is a mathematical existence result and does not provide a practical way of actually finding a balancing position. An old, simple, beautiful, intuitive and applicable, but not very well known argument guarantees that a square table can be balanced on any ground that is not " too wild " , by turning it on the spot. In the main part of this paper we turn this intuitive argument into a mathematical theorem. More precisely, our theorem deals with rectangular tables each consisting of a solid rectangle as top and four line segments of equal length as legs. We prove that if the ground does not rise by more than arctan 1 √ 2 ≈ 35.26 • between any two of its points, and if the legs of the table are at least half as long as its diagonals, then the table can be balanced anywhere on the ground, without any part of it digging into the ground, by turning the table on the spot. This significantly improves on related results recently reported on in [16] and [21] by also dealing with tables that are not square, optimizing the allowable " wobblyness " of the ground, giving minimal leg lengths that ensure that the table won't run into the ground, and providing (hopefully) a more accessible proof. Finally, we give a summary of related earlier results, prove a number of related results for tables of shapes other than rectangles, and give some advice on using our results in real life.