Wobbly table

Why does a four-legged table rock on an uneven floor when a three-legged stool never does?

A three-legged stool has never wobbled in the history of stools. A four-legged table wobbles in half the cafes you sit in. The usual verdict is that the table is badly made, and the usual remedy is a folded beer mat.

Both are worth testing. A table whose legs are cut to identical length still rocks, on a floor that is merely gently uneven rather than damaged. And the mat is not the only remedy: there is a second one that costs nothing, and behind it sits a small theorem that is genuinely lovely.

Start with the stool, because the easy case tells you what the table is failing to do. Three points, provided they are not in a line, define exactly one plane. Give me any three feet and any surface at all, and there is a position where all three touch — not by luck, but because three is the number of points a plane is made of. A stool cannot rock. There is nothing to arrange.

Now the table. Its four foot-tips, if the legs are equal, also lie in a plane — a rigid one, fixed by the joinery. For all four to touch, the floor must supply four points, in that exact square arrangement, lying in one common plane. Ask yourself how likely that is. You are demanding one coincidence beyond what geometry hands you free, and a floor that is not exactly flat generally will not oblige.

So one foot hangs. The table rests on a tripod — three of its four legs — and when you press the hanging corner it tips onto the other tripod, the one built on the opposite diagonal. That is what a wobble is: not a vibration, but a switch between two tripods hinged along a shared diagonal. Which tells you why a table has exactly two resting positions, and what the beer mat does: it abolishes one of them.

Here is the part that surprises people. Leave the mat in your pocket and turn the table slowly on the spot. At each angle, imagine lowering it until three feet are down, and measure the gap under the fourth. Call the gap positive when that fourth foot floats in the air, and — this is the modelling step that a careful proof has to earn — negative when the foot would have to sink into the floor, which in the real world is the table rocking onto its other tripod instead.

Now turn a quarter of the way round. Each leg has moved to where its neighbour stood, the two diagonals have exchanged roles, and the foot that was floating is now the one that would have to sink. The gap has changed sign. But if the floor's height varies continuously — no steps, no missing tile — then the gap varied continuously as you turned. And a continuous quantity cannot pass from positive to negative without being zero somewhere in between.

At that angle, all four feet are down. A square table on a merely uneven floor can always be silenced by rotating it, and never by more than ninety degrees.

Four honest qualifications, because the result is easy to overstate.

It fixes the rock, not the level. At the angle you find, all four feet touch — and the tabletop may still be tilted, because a sloping floor slopes whatever you do. Your glass will sit still and your wine will not be horizontal. Those are two different complaints with two different fixes.

The clean argument assumes the feet sit at the corners of a square, which is what makes a quarter turn map the footprint onto itself. Rectangles are harder: the symmetry is gone and the known results need extra conditions on how steep the floor is allowed to be. My recollection of that literature is second-hand, so treat rotation on a rectangular table as an empirical trick rather than a guaranteed one.

It assumes the floor is continuous. A step, a lifted tile edge, or a gap between floorboards makes the gap function jump, and a jump can cross zero without ever equalling it.

And it assumes the legs are equal. If they are not, the four tips do not lie in a plane at all, the rock has nothing to do with the floor, and no amount of turning will help.

One further thing the picture hides: even when all four feet are down, the forces in them are not determined. Equilibrium gives you three equations — vertical force and two moments — against four unknown leg loads. What settles it is that neither the table nor the floor is truly rigid; both flex, and both creep. That is why a table that was quiet all morning starts rocking after someone leans on it.

Why put up with four legs at all, then? Because a stool's support triangle is small, and so is the load you can put on its edge before it tips. Lean on a table's corner and you want a foot near that corner. Four legs buy resistance to tipping at the price of never being sure all four are down.

• Why a four-legged chair on a hard floor is worse than a table, once a person's shifting weight is added.
• How statically indeterminate structures generally get their loads decided — by stiffness rather than by equilibrium.

Every term the collection defines is gathered in the glossary.