Mathematical table turning
This paper at the front for the Mathematics ArXiv describes a fascinating, unintuitive result: under certain (fairly lax) conditions, a four-legged rectangular table can always be rotated around its center so that all four legs are on the surface, that is, the table does not wobble. From the paper:
We prove that if the ground does not rise by more than arctan(1/sqrt(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.
The paper also discusses leveling tables in the real world. The result does not hold for tables with different leg lenths, and tables whose leg endpoints are not strictly rectangular. Still, it is a curious finding.
We prove that if the ground does not rise by more than arctan(1/sqrt(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.
The paper also discusses leveling tables in the real world. The result does not hold for tables with different leg lenths, and tables whose leg endpoints are not strictly rectangular. Still, it is a curious finding.
posted by db at 1:32 PM
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