This will be part one of three of a rather mathsy blog. We're going to talk about the Borsuk Ulam Theory and the Whitworth Three Plate Method because I believe that they're related to something I saw on YouTube. The claim is that if you've got a 4 legged table that wobbles then it can be corrected by simply twisting it. This is clearly wrong and I can prove it with a counter example: Image that the floor is a perfect plane and that the table has one short leg. It doesn't matter how you twist the table, it's still going to wobble. It'll stand on the two legs adjacent to the short one and wobble towards and away from the short leg.
I hate that there are (cheaply made AI) videos like this around, and I doubly hate that I had to prove or disprove their findings to my own satisfaction. However I think there is something in the claim that you stop a 4 legged table from wobbling by twisting it, and I'm exploring the limits of how far it's true.
Richard "Radial Lobing" B
A couple of hundred years ago Joseph Whitworth seemed to have worked out how to make accurately flat surfaces. There is great insight and also an oversight in this story. We had the technology to mark where one metal surface touched another (with blue dye) and the technology to remove tiny amounts of metal (by hand scraping). It was already commonplace to fit two metal parts together accurately using these techniques, but if you wanted to make something flat, you needed something even flatter to copy from. There's a bootstrapping problem with manufacturing the first flat reference surface.
Whitworth's insight goes like this. If Plate A fits against Plate B, then they match, but one might be concave while the other is convex. However if we introduce a third plate C, and C matches against both A and B then they can be neither concave or convex (he's right so far) and they must be flat (weirdly he's wrong here!).
The Whitworth 3 plate method was a huge leap forwards in manufacturing, and it was quickly corrected to deal with the oversight. The original problem is that there's a shape which is neither concave, nor convex, and which fits together with the other plates, but isn't flat. The problem was called "radial lobing". It's hard to visualise, and I sure as hell can't draw it, but I'll try to describe it. Think of an oblong plate which is mostly flat, but the top left corner is a little high, and the bottom right corner is a little low. Now imaging that every line you draw on the plate that goes through the middle is completely straight, not bent in the middle. You have radial lobing. All 3 plates like this can fit together perfectly, and they're not concave or convex, but they're also not flat.
I'm convinced that this is story relevant to my mathematical investigation into wobbly tables.
Richard "Moore Pattern" B
The next instalment in my investigation into wobbly tables is the Borsuk-Ulam Theorem.
The real formulation of the theorem is rather complicated, but the simple version is that if you squash the skin of a rubbery ball down onto a flat surface (without tearing it) then no matter how you stretch it, you always end up with two points on top of each other which were on opposite sides of the ball. If we say that barometric pressure and surface temperature on the earth vary continuously (no step changes from one point to the next), then right now there are two places on the surface of the earth which are exactly opposite each other and have the exact same pressure and temperature.
The real proof is rather hard to follow and involves tightening a symmetrical lasso around the origin of the Cartesian plane. However it's the first part of the easy proof that we need for wobbly tables. If we think about the barometric pressures around the equator of the earth, and pick two points that are opposite to each other, then either those points have the same pressure or they don't. If they don't have the same pressure then A has a higher pressure and B has a lower pressure. If we spin the points round the centre of the earth until they've swapped position, then by now B has a higher pressure and A has a lower pressure. Somewhere between where the points started and where they finished they must have gone through a place where they had equal pressure.
Richard "Are You Bored Yet?" B
My contention is that you can correct a wobbly 4 legged table by rotating it if two conditions are met: 1) The floor is continuous (no holes, no edges, no steps ). 2) The legs of the table are "flatter" than the floor.
I will first prove that this is true for a table where the ends of the legs are perfectly planar: If the table isn't wobbly, then we've already found an orientation where it doesn't wobble. If it does wobble, then there is one leg above the floor. If we were to rotate a square table by 90 degrees, then a different leg is above the floor. For an oblong table it would be 180 degrees but the same reasoning holds. In the same way that we prove the easy version of the Borsuk Ulam theorem, somewhere between the first position and the rotated position, which legs touch the floor must have changed. So somewhere along that rotation all the legs must have been touching the floor.
The same reasoning holds as long as the table legs are "flatter" than the floor, but "flatter" has a peculiar meaning. Any irregularity in the floor with 4-fold rotational symmetry would have to be discounted – the whole table would go up and down as it rotated, but it wouldn't change the gap under the short leg.
Richard "Q.E.D." B
