3 ms·

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

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 https://arxiv.org/abs/math/0511490

jdddb 7y ago
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.
dzdt 7y ago
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.
michaelt 7y ago
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.
- mauvehaus 7y ago
  "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.

jdddb 7y ago

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.

dzdt 7y ago

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.

michaelt 7y ago

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.

mauvehaus 7y ago
"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.