If Flatlanders walk around the Mobius string and we put a new one behind them every few steps, at which point they will be "reversing"?

Question

Let's say that we put a Flatlander on the Mobius strip and make them move forward. Every X feet they travel, we put another one X feet behind them so that they can see each other. Then, after both of them move forward another X feet, we add another one and repeat the whole process.

Now, let’s keep doing it until the first Flatlander almost goes through the whole Mobius string, then we put the last one just in front of them.

Will the first Flatlander be mirrored already from the perspective of the last one (I suppose so) or will they have to go another few feet before they suddenly appear to be "magically" mirrored?

If they're already mirrored, then what about the one behind them? And then about Flatlander before that one, and then the previous one?

---

In other words, will there be a point, where one Flatlander will suddenly be seen as their mirror image by their neighbour? If not, will the change be continuous as they keep walking?

Assuming continuous change, does it mean that, if looking at any of their neighbour, every single Flatlander should see them as being "distorted" in some way, depending on how far away they are? Like squeezed horizontally or something like that? (i.e. can we assume that path before them appear to not rotate, thus they are being squeezed in just one dimension?)

Answer 1

Let's imagine a slightly different experiment which illustrates the same issue you are trying to highlight.

Alice and Bob are standing next to each other on a Möbius strip, facing each other. Bob walks backwards, tracing his way around the Möbius strip. Alice maintains eye contact as Bob backs away; we assume that Alice can actually see all the way around the Möbius strip, due to light itself bending along the strip.

When Bob has completed a full journey around the Möbius strip, will he appear the same as he started, or reversed? One explanation is that he must appear reversed, since he made a full journey around the strip. On the other hand, Alice was watching Bob throughout his whole journey, and he looked the same throughout the entire journey. It seems as if he discontinuously switched orientations just before returning, which is clearly impossible.

The paradox is resolved by realizing there are really two Bob's being discussed. When Alice and Bob are next to each other, and Alice looks far away, she will see an image of herself and Bob standing next to each other at a distance equal to the circumference of the strip. After making his journey, the real Bob, standing next to Alice, will be reversed, but the image of Bob in the distance will appear normal.

To make this even weirder, consider Alice's perspective when Bob is half way around the strip. If Alice looks forward, Bob will appear normal, but if Alice turns around, she will see Bob at the same distance, but appearing reversed! This just goes to show that there is no notion of orientation on a Möbius strip; you could look at the same clock from two directions, and it will appear to rotate clockwise in one, and anti-clockwise in the other.

Answer 2

Let circumference length be L. If I understood right, equivalently let the second flatlander be stationary, only the first one moves in relative motion.

When there is no twist, first flatlander will be mirrored by the second after traversing length L.

Walking on a twisted Moebius Band the mirroring takes place after first flatlander traverses a length 2L.

Answer 3

The last one will see the first one as mirrored, and will also see the visible ones behind the first one (the first several) as mirrored, to the limit of their vision. At the same time, the first one will see the last one mirrored (as well as the second-to-last one, etc.).

You can do this experiment yourself! with a Möbius band and some items that you can distinguish left from right on—maybe bits of tape with one edge marked with a sharpie.