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It turns out the "wobbly table theorem" is less of just a math curiosity than I thought.
The "wobbly table theorem" essentially says (ignoring https://haggainuchi.com/wobblytable.html ][caveats and different variants) that if you have a "table" with legs of equal length resting on a continuous surface with three legs touching the ground, you can find a rotation where the table rests on all four legs. I first encountered this years ago on Numberphile.
This investigation started with my frustration with the chair I am currently sitting on and me thinking that the wobbly table theorem couldn't apply to my use-case. Surely the chair was too poorly built, given how wobbly it was! I assumed the floor in the room I'm in to be relatively flat. But, when I tried turning the chair around, it worked!?
I thought once we have a table where one of the legs is shorter and the surface is flat, it's obvious that we cannot make the table stable. In my mind most wobbly chairs I encountered were broken like that (for example when one of their furniture glides gets lost).
After seeing this, I put the chair on another table (since I expected the table to be flatter) and it turns out the chair actually has much flatter legs than I had imagined.
So essentially my assumption that the chair was to blame was incorrect. In hindsight, I could have checked and realized this earlier.
Right before posting this, I realized that the chair was still to blame, to some degree. The chair I was sitting on was missing the black furniture glides at the back (those that you can see in the image above). So, at the back, the chair was essentially resting on multiple points out of pretty rigid metal. Switching to a chair with all furniture glides did in fact make the chair stable in any position I tried.
I had underestimated how much the problem had been motivated by actually touching the territory:
Many people eating lunch or drinking coffee on the terrace of the CERN cafeteria have had the following problem: the table is often not in a stable equilibrium position. It rests on three feet, and with very little energy, it can be made to wobble, spilling part of your coffee at best onto the saucer, or at worst onto the table. Why is this? Not because the table is poorly built, but because the ground is very irregular.
[...]
I carried out the experiment many times on the terrace, and even though the conditions of the theorem are not really satisfied - the feet are thick, the ground is sometimes discontinuous, but on the other hand, the legs of the tables have some elasticity - I have always succeeded in finding an equilibrium position.