Numberphile made a video on this: https://m.youtube.com/watch?v=OuF-WB7mD6k

However, I tried doing this many times, but it was hardly ever successful.

most likely due to the caveats I assume:

"Two caveats. This only works for a table with equal legs, where the wobble is caused by an uneven floor. If the table has uneven legs, you probably need the folded napkin. Also, the floor can be bumpy but has to be free of steps"

Spherical cows rearing their ugly heads once again

How does that work? I guess the sphere just expands?

Old physics joke about using approximations to model a farming problem... "Start with a spherical cow in vacuum, ..."

thfuran's comment extends the joke by inquiring how a cow might raise its head while being a perfect sphere.

Gotcha - 'rearing their ugly heads'... I should slow down and think the thread through every now and then lol

It also requires a table with multiple legs, instead of a single pillar table (common outdoors). If the top->pillar->base connection chain has a weak point that's wobbly no amount of rotation or napkins will fix it.

> This only works for a table with equal legs

Which is bullshit because tables in a public space are abused.

I think the proof forgets that legs A and C are locked in a struggle over being on the ground. There is no guarantee that C is still on the ground when A touches.

I've had a great deal of luck getting the table to wobble less by turning it, but have also met with infrequent success. And square tables are more constrained to orientation.

For decades, I have been well known among my colleagues, friends and family for rotating tables any time there’s even the slightest wobble, not giving up until the deed is done, and nearly always doing so successfully unless a leg is clearly missing its foot. Most of them tolerate it.

This was my thought too. The other legs don’t just stay the same as you turn it. What if one of them is now off the ground or too far into the ground?

Isn't this a topology problem?

On an uneven surface, are there always 4 points that define a square of sides length x that are always [in the same plane]? That has to be no, right?

Best solution: replace it with a table with 3 points of contact with the ground. Three points define a plane; it's guaranteed not to wobble.

But then it's easier to topple when the load it is bearing isn't equally distributed

Not if your table is a triangle

And yet triangular tables are a fairly poor shape for restaurants etc.

Clearly, graceless barbarians and children cannot be afforded the luxury of a sturdy table, then.

Better solution: create a startup to use computer vision to scan the ground and find the optimal rotation for your table. Raise millions in investor capital. Exit before they figure out your company can't make a profit.

Ultimate solution: use a tatami mat.

You'll need to incorporate AI somehow to boost media interest, investment and moxie.

Me: Alexa, unwobble my table Alexa: "Scanning ground..."

> Three points define a plane; it's guaranteed not to wobble.

This gets repeated a lot. I think if you stop to consider the common case of non-planar ground, you will see that this mathematical factoid isn't actually super-relevant.

(I will concede that it's not trivial to add a fourth leg and make it co-planar to the other three. But this often isn't the cause of wobble.)

It's more relevant that three legs have three degrees of freedom (ignoring rotating or translating the table), and thus can each sit at an independently chosen height.

(It is conceptually satisfying that once you throw rotation in, the added degree of freedom permits four legs to meet the ground!)

Do you carry a spare table when you go to restaurants?

I deal with analogous situations, on a larger scale, all the time. The solution most designers choose is to make the legs adjustable (like via screws) so you get the strong broad support of multiple legs while gaining the ability to deal with imperfections.

Only if they are non-collinear

Hence tri-pods

Possibly dumb question: Will this work for tables where there is no point of rotation that the equal-length legs are equidistant from? I.e. there's no point of rotational symmetry as far as the legs are concerned.

It seems that the argument is predicated on the idea that after rotating the table 90 degrees, there's a leg in the same location on the ground as there was before rotating the table 90 degrees.

The linked paper [0] goes so far as to cover the case of rectangular tables, but it's quite common to have trapezoidal arrangements of legs, on e.g. a demilune table. Chairs also often have their 4 legs arranged in a trapezoid.

If the linked paper addresses this, it's in mathematical language beyond my reading comprehension :-/

[0] https://arxiv.org/abs/math/0511490

For sure it couldn't work with a diamond shaped table. A diamond table would have two pairs of legs at the same radii on opposite sides of the center. Imagine the floor is bowl shaped radially symmetrically around the center of the table. The pair of legs on a longer radius always stand on a higher part of the floor as the table spins. If the table is tipped so the top is level, both of the shorter legs are above a lower part of the floor. Tipping one short leg down to the floor tips the other one up: it is wobbly. Since everything is radially symmetric spinning the table makes no difference.

There'll be a point of rotation the legs are equidistant from for any _symmetrical_ trapezoidal arrangement of legs - it just might not be under the table. After all, a demilune table is a circular segment!

Of course, as most demilune tables are put against walls, rotating them into the wall is probably a better idea in theory than in practice.

"Of course, as most demilune tables are put against walls, rotating them into the wall is probably a better idea in theory than in practice."

Depending on how the wall and floor are framed, there might not even be a well-defined "floor" if you try. Or the floor may not have a smooth curve.

Also: is there a word or phrase for the situation of getting 2 informative answers to a question and being even more confused than when you asked? Because I'm there thanks to you and dzdt.

There is a trivial failure case if the legs are not on a circle around the rotation axis: One leg is closer to the center (rotation axis) of the table than the other three. The floor is flat except for a ring, centered on the rotation axis, under the odd leg. In this case all 4 legs will never be on the floor at the same time.

By far the most interesting part to me was this aside:

>> But getting it to work proved much harder than some other equally cute, real-world applications of the IVT [Intermediate Value Theorem], such as the fact that at any moment in time, there is always at least one location on the earth's surface where the temperature is exactly the same as at the location diametrically opposite on the other side of the globe.

How to find one (two): The temperature at the spot (B) on the other side of the world from you (A) is either the same (Finished) or different. If different, head towards B, (any direction is fine!) and at some point before you get there and the A vs B temperatures are swapped, the two will be the same. Since the direction you go doesn't matter, evidently there's a path "around the world" where this true, not just a point! (Ok, just a point if they're initially the same.)

I suspect this is a local thing these folks experienced. I've never eaten at a place with a floor so uneven it makes tables wobble, but I've been to places where the endcap for the legs had fallen off pretty often.

> a place with a floor so uneven it makes tables wobble

Many outdoor paved patios have exactly this property, especially as the slabs settle over time

Slabs?

Paving slabs - UK term for the individual 'stones', typically large squares or rectangles.