As simple as this argument may sound, however, proof was a long time coming. The first serious mathematical inroad against table wobbling seems to have occurred in the late s with Roger Fenn, a PhD student at the University of London.
One day Fenn and his graduate adviser ended up at a coffee shop faced with—you guessed it—an unsteady table. At his adviser's suggestion, Fenn wrote out a proof that for any smoothly curving floor that bulges upward like a hill, there is at least one way to position the table so that it is balanced and horizontal.
But he did not reveal how exactly to find that sweet spot, and he quickly tabled the subject. The season for proving the table turning hypothesis would not arrive for another 35 years. By then, the idea had become such a part of mathematical lore that two years ago mathematician Burkard Polster of Monash University in Australia included it in an article on neat math tricks for teachers.
He promptly received a letter pointing out that the idea would not work if a floor possessed sheer cliffs, such as between tiles. Polster rose to the challenge. So he and some of his colleagues ran through the appropriate calculus and satisfied themselves that if a floor has no spots that slope by more than They detail the proof in a paper accepted for publication by the Mathematical Intelligencer. Martin published a similar result within a few months of the Australians' version.
Polster's group even spells out a procedure for balancing the table [ see video above ]. First lift up the leg of the table diagonal from the wobbly leg. Make sure both legs are roughly equal distances off the ground and then begin rotating. When the stool has been turned through 90 degrees, leg A will be where B was originally, B will be where C was originally, etc.
So now, D has replaced A as the only leg that's not touching the floor. So, at some time during the rotation, leg A must have made contact with the floor and at a later time, leg D must have broken contact with it.
Why "later"? Because if D broke contact before A made contact, we would have a situation where only two of the four legs were touching the floor, contradicting the principle of three points always being in the same plane.
All that is needed is to find the moment during the rotation when A touches the floor, and the stool should be stable, with all four legs grounded. Then again, you could just use a folded-up beermat to wedge it. In fact, I have an ontological proof that such a beermat must exist - oh all right, I'll stop. The stool legs could be uneven in length so that they match the uneven floor. However, assuming the stool legs are even, the four legged stool will rock because one of the legs will not reach the floor.
The three legs that do reach the floor will hold the stool so the other leg cannot. If one shifts the stool to the formerly uplifted leg, the opposite leg will rise. For a three legged stool, the two other legs cannot prevent the third leg from touching the floor.
In other words, for any three points chosen on an uneven floor, there is an even floor with some slight tilt that passes through these points. A stool with legs touching the floor at these three points doesn''t know that it is not sitting on this even floor. But these three, these three legs here so we don't have this anymore.
These three points touching the floor. They would still determine a plane right now would be this one, and it would stabilize on that plane. See, on the chair. We didn't have that right? So if if we had this this sorry, this leg and this one perfectly in line with each other.
And we had these two, maybe one, a bit shorter and another bit taller than we would have one plane determined by these three and one plane being determined by these three. And they would be different planes, Right? So maybe if this one is the shorter one, then we have a plane that goes something like this, right? Because the chair wants to till this way. And if we have this one maybe being a bit shorter than we would have something.
A plane that wants to be like this because the chair wants to tilt to the right. So we have two planes here in this case, that kind of colliding with each other, right? The chair doesn't know where to go. So what kind of wobbles in between those two plans? Not for the stool. For stool, No matter what length you set the feet too. Are the legs too? You always have only one unique plane being determined on your floor.
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Luca A. Problem 47 Hard Difficulty Explain why a four-legged chair may rock from side to side even if the floor is level. Answer No; Three points define a plane, so the legs of the three-legged chair will always meet in the flat plane of the floor. View Answer.
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