I think you are saying something interesting. Could you elaborate a little though because I am not quite getting you yet hooked to understand what you are saying.
Spin gets you centripetal force. You know where inside and outside are globally because the forces will be opposite depending on which "side" you're standing, which directly contradicts their statement "..globally you cannot label one side as up and one side as down. If you try to label one side as up, and then you move continuously around the Möbius strip, you'll end up labelling that same side as down!"
> If you try to label one side as up, and then you move continuously around the Möbius strip, you'll end up labelling that same side as down!"
This statement is true (though it is using a different definition of "up" and "down" than you are), and your claim that spinning the strip will show a contradiction with it is false. See my post downthread in response to your "spin the strip as you would a wheel". I phrased it there in terms of which edge is the "outside" and which is the "inside" edge, but the same is true of "up" vs. "down" sides relative to the strip (note that I emphasized that phrase in my other post).
Note that the issue is not that spinning the strip does not define an "inside" (towards the axis) vs. "outside" (away from the axis) in the 3-space in which the strip is embedded; of course it does (and similarly for "up" and "down", if you want to use those terms instead, in the embedding 3-space). The issue is that there is no way to use this to construct a continuous orientation on the strip itself, the way you could on an orientable surface, such as an ordinary ring (a disk with a hole cut in the center).
I get what you're saying, but with the spin you need to introduce the notion of movement, mass and force. It adds complexity you might not be able to afford, especially since typically you want to investigate a whole bunch of shapes, not just one strip.
Also, since everything on your model is continuous, there must be a point on the strip where the centrifugal force is 0. Is that point inside or outside?
Ah. I think it's better we don't discuss this topic from a mathematical perspective and agree you can indeed do a lot of different things with a lot of different outcomes.
Comments
I think you are saying something interesting. Could you elaborate a little though because I am not quite getting you yet hooked to understand what you are saying.
Spin gets you centripetal force. You know where inside and outside are globally because the forces will be opposite depending on which "side" you're standing, which directly contradicts their statement "..globally you cannot label one side as up and one side as down. If you try to label one side as up, and then you move continuously around the Möbius strip, you'll end up labelling that same side as down!"
> If you try to label one side as up, and then you move continuously around the Möbius strip, you'll end up labelling that same side as down!"
This statement is true (though it is using a different definition of "up" and "down" than you are), and your claim that spinning the strip will show a contradiction with it is false. See my post downthread in response to your "spin the strip as you would a wheel". I phrased it there in terms of which edge is the "outside" and which is the "inside" edge, but the same is true of "up" vs. "down" sides relative to the strip (note that I emphasized that phrase in my other post).
Note that the issue is not that spinning the strip does not define an "inside" (towards the axis) vs. "outside" (away from the axis) in the 3-space in which the strip is embedded; of course it does (and similarly for "up" and "down", if you want to use those terms instead, in the embedding 3-space). The issue is that there is no way to use this to construct a continuous orientation on the strip itself, the way you could on an orientable surface, such as an ordinary ring (a disk with a hole cut in the center).
I get what you're saying, but with the spin you need to introduce the notion of movement, mass and force. It adds complexity you might not be able to afford, especially since typically you want to investigate a whole bunch of shapes, not just one strip.
Also, since everything on your model is continuous, there must be a point on the strip where the centrifugal force is 0. Is that point inside or outside?
Note that for a particle there is no discrepancy.
Hence my statement about not being in a void.
Depends on which foot is inside and which is not. That's the beauty of reality. It's fully deterministic at the marco level.
Ah. I think it's better we don't discuss this topic from a mathematical perspective and agree you can indeed do a lot of different things with a lot of different outcomes.
Got you.