You've redefined "the issue" but we agree here. The issue was there was no way for ants to know where they stand on a strip. Force will reveal where you stand. Topology makes it one (forever one) path (orientability). The ant will know where it stands, inner or outer path, based on how much grip it needs to cling to the surface. The outer edge is where the ant needs to start gripping the surface. The inner edge is where it does not. The twist is where it will need to apply grip force.
> The issue was there was no way for ants to know where they stand on a strip.
No, that's not the issue. The issue is that there is no possible continuous orientation on the strip. That's not a matter of "where they stand"; it does not preclude defining locations on the strip. Location is not the same thing as orientation.
> The outer edge is where the ant needs to start gripping the surface. The inner edge is where it does not.
But if the ant then starts walking along whatever edge it starts with, that edge will not stay "outer" or "inner". It will change as the ant walks. So it is impossible to label a particular edge "outer" or "inner" in a continuous way everywhere. Indeed, at some point while the ant is walking the edge, the force it needs to exert to "cling to the surface" will be zero (this must be the case by the mean value theorem because the direction of the force flips during the ant's walk).
No, I already understand what mistake you're making.
>1. Where there's surface paint, that is the inner surface.
2. Where there's no surface paint, well guess what, that's the outer surface.
Sure, go ahead, make yourself a Mobius strip and try to paint it this way, without stopping anywhere except an edge. There is no way to do it--you'll end up painting both "surfaces" that you see locally. The only way to not do that is to arbitrarily stop somewhere, not at an edge (and you'll have to choose an arbitrary stopping point this way in both directions from wherever you start painting). It will not work the way painting an ordinary two-sided surface would, where you can indeed paint the entirety of one side, not stopping anywhere that is not an edge, and never reach the other side.
You simply do not seem to understand the actual properties of a Mobius strip. Which is strange to me, since you can make one yourself and test whatever claims you want to make. I have one sitting on my desk.
> it is possible to label it the surface inner and outer
Not in a way that is continuous over the entire surface. You can do it locally on a small patch, but you can't extend any such local labeling over the entire surface.
Comments
You've redefined "the issue" but we agree here. The issue was there was no way for ants to know where they stand on a strip. Force will reveal where you stand. Topology makes it one (forever one) path (orientability). The ant will know where it stands, inner or outer path, based on how much grip it needs to cling to the surface. The outer edge is where the ant needs to start gripping the surface. The inner edge is where it does not. The twist is where it will need to apply grip force.
> The issue was there was no way for ants to know where they stand on a strip.
No, that's not the issue. The issue is that there is no possible continuous orientation on the strip. That's not a matter of "where they stand"; it does not preclude defining locations on the strip. Location is not the same thing as orientation.
> The outer edge is where the ant needs to start gripping the surface. The inner edge is where it does not.
But if the ant then starts walking along whatever edge it starts with, that edge will not stay "outer" or "inner". It will change as the ant walks. So it is impossible to label a particular edge "outer" or "inner" in a continuous way everywhere. Indeed, at some point while the ant is walking the edge, the force it needs to exert to "cling to the surface" will be zero (this must be the case by the mean value theorem because the direction of the force flips during the ant's walk).
Yes - it is possible to label it the surface inner and outer - see my previous comment about paint flying off the outer surface at high RPM.
Edit: I guess I have to make it extra simple.
1. Where there's surface paint, that is the effective inner surface.
2. Where there's no surface paint, well guess what, that's the outer surface.
3. There's no other, 3rd possibility. It exists or it doesn't. If you're the ant, you either feel a paint line on the surface or you don't.
> I guess I have to make it extra simple.
No, I already understand what mistake you're making.
>1. Where there's surface paint, that is the inner surface.
2. Where there's no surface paint, well guess what, that's the outer surface.
Sure, go ahead, make yourself a Mobius strip and try to paint it this way, without stopping anywhere except an edge. There is no way to do it--you'll end up painting both "surfaces" that you see locally. The only way to not do that is to arbitrarily stop somewhere, not at an edge (and you'll have to choose an arbitrary stopping point this way in both directions from wherever you start painting). It will not work the way painting an ordinary two-sided surface would, where you can indeed paint the entirety of one side, not stopping anywhere that is not an edge, and never reach the other side.
You simply do not seem to understand the actual properties of a Mobius strip. Which is strange to me, since you can make one yourself and test whatever claims you want to make. I have one sitting on my desk.
> it is possible to label it the surface inner and outer
Not in a way that is continuous over the entire surface. You can do it locally on a small patch, but you can't extend any such local labeling over the entire surface.