I enjoyed reading this. I have been making my way through Needham's Visual Complex Analysis (https://global.oup.com/academic/product/visual-complex-analy...) book which takes a similar visual-focused view of teaching all of complex analysis. Chapter 2 in particular covers multivalue functions and branch points to make a similar point about how some paths will yield different values at same points.
Yes im reading it now. I find that the total lack of formalism makes things difficult from time to time just like his insistence on using basic geometry when there are easier ways. That being said, its an excellent book.
Here's the reason you need enginners in the universe. Yes, the strip has non-orientability. Howwever, just introduce spin about the center, as you would a wheel, and you get all the same benefits of orientation as long as you're not in a completely empty void in space. Upside sees constantly changing, inside sees the other side of the strip.
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.
Spin the strip as you would a wheel. The forces tell you where you're standing on it; Just measure them: If you painted a line down the center, the paint would fly-out from half of the surface (at high RPM) before it could fully adhere to the surface. The no-orientation topology only makes sense if you're concerned with topology, the universe has other ways of showing you real orientation.
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.
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).
> 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.
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.
