Möbius strips and differential equations(hidden-phenomena.com) |
Möbius strips and differential equations(hidden-phenomena.com) |
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).
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?
Hence my statement about not being in a void.
>Is that point inside or outside?
Depends on which foot is inside and which is not. That's the beauty of reality. It's fully deterministic at the marco level.
Meaning, about an axis perpendicular to the strip (or at least it's like that at one point--see below) and running through the center of the "hole" in the middle?
You can do this, sure. But because of the shape of the strip, it won't be perpendicular to the rotation axis everywhere.
Try this experiment: make a Mobius strip and pick a small area of it that you will define to be in the plane of rotation. Draw a mark on each side of the strip pointing "outward", i.e., away from the rotation axis.
Now take a small arrow and start it on one side of the strip pointing in the same direction as the little mark. Move the arrow around the strip. You will find that it will reach a point on the other side of the strip, where the other mark you drew is--and when it's there, the small arrow will be pointing in the opposite directon to the other mark. In other words, the small arrow is now pointing at the "inside" edge of the strip, not the "outside". (Note that on an orientable surface, you would not be able to move continuously along the surface from one mark to the other at all without crossing an edge.)
In other words, while yes, there is a well-defined "direction of centrifugal force" arrow everywhere in the space the strip is embedded in, it cannot always point in the same direction relative to the strip. In some places it points "horizontally" relative to the strip, towards one edge (but it's impossible to continuously pick "the same edge" everywhere on the strip relative to this). In other places it points "vertically", out of the strip. So the "direction of centrifugal force" does not give you a well defined orientation everywhere relative to the strip. You can't use it to say, for example, "this is the outer edge" and have that work continuously everywhere.
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).
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.
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.
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.