5 ms·
The Wobbly Table Theorem (2022)
- pontifier 2y agoThe biggest problem with this is that it doesn't work every time in real life. Sometimes you just have a table with one short leg.
- nighthawk454 2y agoIiuc I think it does work, but the only guarantee is the table will be stable. It very likely will not be _level_
- cperciva 2y agoCounterexample: Perfectly level floor, one table leg is 10 cm shorter than the rest. No way to make it stable.
- User23 2y agoThree legged table supremacy!
- cperciva 2y agoNote: 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.)
- ithkuil 2y ago"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.
- ncruces 2y agoOh but it can, as the sister comment alluded to.
- brianpan 2y agoNote: If your floor is wobbly- DROP, COVER, and HOLD ON.
- Willingham 2y agoInteresting choice of music for Harvard!
- alanbernstein 2y agoI 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.
- Cerium 2y agoA 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.
- matheist 2y agoThe linked proof doesn't work when the floor is too steep. I wrote about it here: https://haggainuchi.com/wobblytable.html https://haggainuchi.com/wobblytable.html 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.
- jansan 2y agoAlso 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.
- lloeki 2y agoIs 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.
- jansan 2y agoI think it is. The research paper is probably most useful when printed out, folded and placed under the short leg.
- contagiousflow 2y ago> criticisms of a pure math theorem being applied to physics Next you're going to tell me the Moving sofa problem isn't really about moving sofas!
- akubera 2y agoMathologer has a great video covering this: https://www.youtube.com/watch?v=aCj3qfQ68m0 https://www.youtube.com/watch?v=aCj3qfQ68m0 Visualizes a proof and talks about special cases and a little of the history.
- x3n0ph3n3 2y agoI'm always frustrated when I try to demonstrate this to someone, only to discover the table is missing a foot on a completely flat floor...
- gfan29234 2y ago[dead]
- kieckerjan 2y agoTangential, 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.
- lloeki 2y ago> 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.
- kieckerjan 2y agoFair enough. But I am talking about 50 years ago or so.
- lloeki 2y agoAh. 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.
- Y_Y 2y agoThe accompanying video is tremendous https://people.math.harvard.edu/~knill/teaching/math1a_2011/exhibits/wobblytable/table.m4v https://people.math.harvard.edu/~knill/teaching/math1a_2011/...