Note: This is really the wobbly floor theorem. It applies to any continuous floor but only to a subset of tables. (Even "square tables which are perfectly level" isn't sufficient unless you allow the tabletop to intersect the floor.)
"wobbly" means that it moves unsteadily so it cannot be an attribute of a floor. It's the table the one that wobbles if the tips of the legs are coplanar and the floor is not a plane.
Your point of the extent of deformation the floor can have for the theorem to hold is valid though.
I always had trouble envisioning this. I think part of the reason is the "angle x", which, according to the explanation and video on this page, is vector-valued, not scalar. IVT applies to scalar functions, is there an equivalent for a function with a vector domain?
The table does not rotate around its own axis, but rather it rotates in "such a way that three legs stay on the surface", i.e. moves around in 3d with a surface-contact constraint, which seems like a motion with more than one degree of freedom to me. Is such a rotation always possible? Is the motion somehow effectively 1D?
These questions don't seem to have "obvious" answers to me, and they're only addressed as "assumptions" on this page.
A three legged table will always sit stably on an uneven surface. Adding a forth table introduces wobble. To remove that wobble, we need to find a location where the fourth leg is exactly the distance between the table top and the ground under the table top.
As we rotate the table we know that in some locations a particular leg will be too short (wobble when we press in that location) or become too tall (source of wobble for another leg).
As long as the ground is continuous and does not have any cracks to introduce discontinuity, we know that there must be a location that the leg is the exact length. By the intermediate value theorem the length cannot go from too short to too long without a solution in between.
I first ran across the Wobbly Table when my wife was studying the Borsuk-Ulam theorem. It is fun since you can effectively solve wobbly tables on many patios.
In terms of the math: the table legs are assumed to be equal length, and the wobble is caused by variations of the surface. Specifically the feet of the table are in the same plane. So you could rotate your mathematical table until all feet are secure on the plane, then cut the legs to make the top flat again (legs will not be same length, but top and bottom remain planes).
As for @Cerium's real-life usage, you have possibility of uneven legs and uneven floor (and discontinuities, like a raised floorboard) so it's obviously not guaranteed, but if the floor is warped and smooth enough, you can try.
Yes, the math assumes the table is perfect and the floor uneven. In my experience this is frequently true with outdoor patio furniture - think decent glass table on a concrete patio.
even then the table will stit stably on the slope. The tabletop will not be level with regards to the horizon/gravity but that is not what the theorem requires. It only requires stability (i.e. no "wobbling")
Yes this might not be entirely practical if you try it on one of the famous sloped roads in San Francisco but for most other parts of the world (restaurants, cobblestone sidewalks, etc.) it will be good enough from a practical standpoint :)
Tangential, but I was once told a story that seems fitting here. It was told to me by a mechanical engineer who was educated at Eindhoven Technical University in the Netherlands.
He claimed that in the early days there was a lecturer or professor there that, at least in Eindhoven, was very important to his field of expertise. If I understood him correctly, this prof's ideas about engineering mechanical systems revolved around restricting the degrees of freedom as much as possible. A three legged table cannot wobble, but a four legged table can and usually does because it is overdetermined. In mechanical systems (for instance sensitive optical mechanics) reducing "wobble" is key. And the best way to reduce wobble is to make sure it cannot occur.
Here it gets interesting. My source claimed that this professor had laid down his ideas in a standard work in Dutch, which was never translated in another language, restricting its influence to Dutch mechanical engineers. He also claimed it is not a coincidence that Philips and later ASML took an early lead in designing optical systems.
Not sure if it is true, but an interesting story nonetheless.
This sounds like the principle of "designing for exact constraint". You learn about under-constrained and over-constrained typically in a strength and materials course in ME undergrad. An simple case of exactly constrained is a 2D beam with a pin/hinge at one end, and a roller support at the other. The pin removes the 2 translational DOF's and the roller removes the third rotational DOF. A beam with a pin/hinge at both ends is over-constrained and is "statically indeterminate" (or over-constrained), which requires you to use other techniques to find the stresses in that beam, such as principle of "virtual work", or some others I've forgotten. For more complex structures, there is a formula which has as its inputs the number of members, number of joints, type of joint, etc, and will give an integer output which will tell you over, exact, or under constrained. Although a 3 legged table will never wobble, they will more easily tip, and the surface of 4 legged table can be considered somewhat flexible, and provide an additional DOF that keeps the legs in good contact with an irregular floor.
> this prof's ideas about engineering mechanical systems revolved around restricting the degrees of freedom as much as possible. A three legged table cannot wobble, but a four legged table can and usually does because it is overdetermined. In mechanical systems (for instance sensitive optical mechanics) reducing "wobble" is key. And the best way to reduce wobble is to make sure it cannot occur.
This is how I was taught mechanical engineering (France, 2000-2005) and not a Dutch in sight.
Ah. I was looking for a time reference but the only one I could find was that Philips thing, which I failed to relate to a specific time period through a cursory search.
My preferred proof goes through a theorem of Dyson that any continuous real valued function on the sphere attains equal values on some square on a great circle.
Also it does not work if the floor is even and one leg of the table is shorter than the others, which I suspect is the reason for the "wobble" in many cases.
IMO this proof is a good example of science taking itself much too important.
Is it? My experience is that most of the time in practice uneven legs of four legged objects are within a small enough tolerance margin that they can be considered mostly equal, but floors are surprisingly less even than we initially suspect, their vertical variability being perceptually dwarfed by their overall surface.
