Is it possible to determine if you were on a Möbius strip?

Question

I understand that if you were to walk on the surface of a Möbius strip you would have the same perspective as if you walked on the outer surface of a cylinder. However, would it be possible for someone to determine whether they were on a Möbius strip or cylinder.

The possible answers depend heavily on the details of the question- are you actually a small three-dimensional person walking on a mobius strip, and if so, can you go "over the edge" or dig through the strip? Or are you a 2-dimensional being in a 2-dimensional mobius strip space? Are you allowed to leave things behind or otherwise "mark" where you've been? — MartianInvader, Sep 05 '17 at 22:59
To add to @MartianInvader's questions, is this strip embedded in $\mathbb{R}^3$, so that its curvature in the ambient space can be detected? Or is it a flat rectangle with edges identified without twisting and bending? — G Tony Jacobs, Sep 05 '17 at 23:23
@GTonyJacobs Note that a flat rectangle with edges oppositely identified can be embedded in $\Bbb R^3$ as well, it's just a bit more difficult. — Arthur, Sep 06 '17 at 07:50
@Arthur, can you sat more about that? I don't see how that's possible... — G Tony Jacobs, Sep 06 '17 at 15:11
@GTonyJacobs It's what happens when you take a strip of paper and perform the gluing assuming the paper doesn't stretch. There's a book about this called "The Mechanics of Ribbons and Mobius Bands" which has a proof that the Mobius strip is "developable." — Kyle Miller, Sep 06 '17 at 16:30
Is there a summary that can be provided, other than, "buy the book"? — G Tony Jacobs, Sep 06 '17 at 16:38
@GTonyJacobs https://arxiv.org/abs/1408.3034 This is in the book. If I understand it correctly, you can wrap a strip around three cylinders (like the recycling symbol) to get a developable Mobius band. — Kyle Miller, Sep 06 '17 at 16:55
@GTonyJacobs A cylinder is considered flat in the sense that a creature living inside the surface cannot do local measurements that tells it that its world isn't flat. In contrast, a creature living in a spherical surface can measure a circle's circumference and discover that it is not quite $2\pi r$. — Arthur, Sep 06 '17 at 18:09
Maybe a duplicate: https://math.stackexchange.com/questions/465594/can-a-2d-person-walking-on-a-m%c3%b6bius-strip-prove-that-its-on-a-m%c3%b6bius-strip — Niklas Rosencrantz, Sep 07 '17 at 13:31
I feel I must add that this question refers to a Möbius strip, but really is asking about a Möbius strip $\times I$. Which is an orientable 3-manifold with boundary, as opposed to the non-orientable 2-manifold. If you were on a Möbius strip and "made a mark," when you came around to the other side, you would be able to see the mark, as was mentioned somewhere. — N. Owad, Sep 12 '17 at 23:55

score 27 · Accepted Answer · answered Sep 05 '17 at 20:58

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If we imagine that we're walking on a broad walkway, and that we can't peek over the edge, either at the lateral surface or to the other side, then I don't think there's a way to tell. Suppose the walkway has a handrail, on "both" sides, and you start marking the handrail as you keep your right hand on it. After completing a full loop, you'll be underneath the point across the walkway from where you started. From there, you won't see any mark yet made, because you're underneath the path on which you started. If you keep going, you'll eventually, after another loop, return to the point where you started, and the only mark you see will be on your handrail, not on the handrail across the road to your left.

What has happened here is clearer if you consider what happens when we cut a mobius strip along its midline. Making that cut adds a second edge, producing a normal loop, one edge of which is the original edge of the mobius strip, and the other edge of which is produced by the cut. The walk described in the above paragraph is a walk along one edge of the resulting loop.

If we can reach down and mark the lateral edge of our path, then there's a way to tell. We make regular marks (or a continuous mark) along the lateral edge to our right, and occasionally we check across the path on our left, to see if any marks are on the lateral edge of the path there. Halfway along the complete (2-loop) walk, you would notice marks on the left side, made from "underneath" the path. That would be evidence that you're on a mobius path.

answered Sep 05 '17 at 20:58

G Tony Jacobs

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How would you determine that someone hadn't made the check marks before you? – Griffin Sep 06 '17 at 14:28
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You use an ink composed of parts of your own blood, and do a DNA test. Then you can only be fooled by an identical twin, which is the usual risk level for mathematical hypotheticals. – G Tony Jacobs Sep 06 '17 at 14:30
This does not work if you cannot see underneath, as it should be the case – user Sep 06 '17 at 23:54
But can you see the outside edge? That's not underneath. – G Tony Jacobs Sep 07 '17 at 00:22
@DNA: you could also be fooled by someone having a sample of your blood - no need a twin. – Vincent Sep 07 '17 at 05:33
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@Vincent: Good point. Instead of a check mark, GTJ should write the ciphertext of the encrypted datetime with said blood, and place it in a lockbox which will open only when presented with a photon with the polarization of one he brings with him. – Gnubie Sep 07 '17 at 16:48
Is direction of polarization of light preserved on a circuit of nonorientable surface? I should probably have my twin wait at the start point while I take that walk. – G Tony Jacobs Sep 07 '17 at 16:52

score 12 · Answer 2 · answered Sep 05 '17 at 18:54

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The boundary of a cylinder is homeomorphic to two circles $S^1 \sqcup S^1$ and the boundary of a mobius strip is homeomorphic to one circle $S^1$, i.e. a mobius strip only has one edge.

answered Sep 05 '17 at 18:54

Kenny Lau

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True, but I hazard a guess that we need an algorithm for a person walking on a Möbius strip/cylinder to detect that the boundary is/is not connected. The OP didn't specify the rules of the game, but it often (?) is intended that the observer is allowed to make local observations only. Possibly leaving a trail of paint :-). I don't know for sure. We could wait for the OP to comment. – Jyrki Lahtonen Sep 05 '17 at 19:05
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For this specific example the observer can only move in 2d on the surface and therefore would not be able to determine that the two sides of the path they are traveling is actually a single edge. – Steven Wallace Sep 05 '17 at 19:11
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But I do think that this may be the best idea. We can traverse an edge, leaving a trail of paint. After completing a round trip, if we later stumble upon an unpainted edge we will know that it wasn't a Möbius strip. – Jyrki Lahtonen Sep 05 '17 at 19:31
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That would work assuming that you could see the edge itself, but for a person on the surface of the strip that can only interact with it in 2d then I am not sure if that would work. For example if you walked to the left until you get to the edge and then followed and painted it in red until it connects, you should then be able to walk right until you hit the edge again and it should be unpainted which should be true for a cylinder as well assuming you can't actually walk on the edge. – Steven Wallace Sep 05 '17 at 19:44
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@StevenWallace here's what is amazing about it: the right edge would actually be painted. – Kenny Lau Sep 05 '17 at 19:55
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I apologize, I don't think I made it clear enough in my explanation. The person walking this mobius strip would not be able to paint the edge itself but only the surface that touches the edge. If you could paint the edge itself this would work, however this is not valid for this situation. And to double check what I am trying to say, I have made a mobius strip and that does not work unless you can paint the edge, not the surface on the edge. – Steven Wallace Sep 05 '17 at 20:09
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If you can see the actual edge, that's one thing, but if all you can see is the surface adjacent to an edge, then @StevenWallace is correct. – G Tony Jacobs Sep 05 '17 at 21:01
What if it is an open cylinder or Mobius band with geometry so that the "edge" is infinitely far away? – Kyle Miller Sep 06 '17 at 06:29
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@KyleMiller if the observer is 'on' the strip and can't observe points on the 'other side' of the surface then they are actually on the (orientation) double cover of the surface, which for a Mobius band is the same as a cylinder. – ronno Sep 06 '17 at 07:15
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@StevenWallace What I was getting at in my suggestion that you don't have a positive ID of the type of surface after completing that paint job. But, you will have only painted a single connectivity component of the edge (or rather, paved a road hugging the edge). Then much later, after wandering randomly along the surface, if you somehow find an edge without that painted road, at that point you can deduce that it was not a Möbius strip (because the edge had more than one connectivity component). The problem is, of course, that no one has specified a strategy to take us to that "other edge". – Jyrki Lahtonen Sep 06 '17 at 14:46
@ronno That doesn't address what I was talking about: what if the space is a quotient of $[0,1]\times\mathbb{R}$? Then there are no boundaries, so this characterization by boundary components wouldn't work. – Kyle Miller Sep 06 '17 at 16:42
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This is really a question of whether you are walking on the surface of a thickened Möbius strip, which contains the oriented double cover--a cylinder, versus whether you are on the actual Möbius strip itself. If so, you can see the paint from "both" sides, so the distinction is not relevant. – Cheerful Parsnip Sep 07 '17 at 17:48

score 9 · Answer 3 · answered Sep 05 '17 at 19:18

Clearly the two surfaces are not homeomorphic, but from the perspective of the individual on the strip itself, I'm fairly certain it would be identical, though I'm not sure how to prove the general case. Both have 0 curvature so that's a no go, and no matter what type of markings you make, there is no way to differentiate having a single edge compared to two on a cylinder without allowing some type of puncture or extra dimension normal to the plane's surface. If you mark the ground using colors A and B at the edges, the spots where A overlaps B only happens on "opposite" sides, or where the surface is extended through the twist in connecting the sides, meaning you will have the perception of following two edges like on a cylinder.

Yeah. For a similar reason you cannot calculate the Euler characteristic of the surface by triangulating it. If you are on a Möbius strip, you end up triangulating the double cover. — Jyrki Lahtonen, Sep 05 '17 at 19:20

score 7 · Answer 4 · edited Sep 06 '17 at 11:19

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Leave a piece of bacon on the strip, and then walk around the strip, and then eat it. If your body processes it like normal, then you are on a cylinder. If your body treats like a reversed-chirality strip of bacon, you are on a mobius band.

edited Sep 06 '17 at 11:19

hyportnex

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answered Sep 06 '17 at 03:39

PyRulez

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This only applies if you were a 2d creature embedded in the 2d manifold or perhaps if you were a 3d creature embedded in a 3d mobius tube of some kind, not as a 3d creature walking on a 2d surface. – Shufflepants Sep 06 '17 at 15:45
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Is this the first use of bacon in an answer on Mathematics SE? If so, well done. – camden_kid Sep 06 '17 at 18:42
@camden_kid [Nope](https://math.stackexchange.com/search?q=bacon+is%3Aanswer) – PyRulez Sep 07 '17 at 18:48

score 7 · Answer 5 · answered Sep 06 '17 at 14:11

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If you drive long enough all of a sudden you notice that everybody is driving on the wrong side of the street!

answered Sep 06 '17 at 14:11

Mikhail Katz

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score 5 · Answer 6 · answered Sep 06 '17 at 06:26

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Let's talk about "flat" cylinders and Mobius strips, which means if you drew a triangle, the angles would add up to 180 degrees (versus curved space like a sphere or a Pringles(R) chip, where the sum of the angles are either more than or less than 180 degrees, respectively). Paper loops will give flat space, but it is worth thinking about it in a slightly different way:

A way to imagine this kind of space is like in the game Asteroids, where if you go off the screen you end up on the other side. A cylinder is a game of Asteroids where you can't go off the top or bottom sides, but if you go off the right side, you appear on the left side at the same height. A Mobius strip is the same, but when you go off the right side at height $y$, you appear on the left side at height $1-y$ (where the screen is $1$ unit high).

Light travels in straight lines. As a flatlander if you look in the $x$ direction, you would see the back of your head whether or not the space is a cylinder or a Mobius strip. However, the back of your head will be reversed if it is a Mobius strip.

answered Sep 06 '17 at 06:26

Kyle Miller

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This is only true of a person embedded in the 2d surface of the mobius strip, not of a 3d person walking on the surface. – Shufflepants Sep 06 '17 at 14:44
@Shufflepants I might have cheated by thickening the space so there was anything to draw in perspective (geometry of Cartesian product with real line), or I might have drawn it ambiguously and those are embedded stick figures. – Kyle Miller Sep 06 '17 at 15:29
I point it out because the question at hand is asking about an actual 3d person on the surface, NOT about 2d creatures embedded in the 2d manifold. I'm saying your answer doesn't apply to the question being asked. – Shufflepants Sep 06 '17 at 15:43
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@Shufflepants I don't think it is clear to take that interpretation of the question. If it were indeed about 3d people on the surface, then they could just look up and locate the twist (maybe? what does it even mean to be an actual 3d person on the surface of an abstract 2d space without specifying an embedding?). The asker mentions "have the same perspective as if you walked on the outer surface of a cylinder," and my interpretation is that they mean that an embedded creature would locally experience flat Euclidean space. – Kyle Miller Sep 06 '17 at 16:35
As I interpret this, the light from the back of your head would have to make a circuit of the entire strip (twice the length of the original construction in your "gluing patterns" diagram). At the point where right is left, up is also down and you're only halfway around the strip. By the time you come back to the original starting location right/left are restored, so the back of your head would appear not reversed but in the correct orientation. – Darren Ringer Sep 06 '17 at 17:05
I admit it's not super clear from the question directly, but it is from reading the OP's comments on some of the other answers. – Shufflepants Sep 06 '17 at 17:16
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@DarrenRinger If we must think about a 3d person, I was imagining that the gluing is done by a reflection through the XZ plane. If you do use the gluing you had in mind, imagine Mobius man is able to pass through the strip as if it were water, and if they had an eye on either side of the strip, one eye will see the back of their head reversed, and the other will see the back of their head with the usual orientation and twice as far away. (I believe http://geometrygames.org/CurvedSpaces/index.html have a mode illustrating what it would be like in such a 3d space but without the strip itself.) – Kyle Miller Sep 06 '17 at 17:24
@DarrenRinger Correction: with the double reflection gluing you were thinking of, that's just a 180 degree rotation, so you'll see the back of your head on both sides in the correct direction (neither reversed), but staggered as I said. – Kyle Miller Sep 06 '17 at 18:59

score 4 · Answer 7 · answered Sep 06 '17 at 01:40

The question is not well defined to me. What do you mean by determine, what is available?

You could mark your starting point, start walking, and after you have gone 360 degrees you either see your marking (cylinder) or you don't (Mobius strip). So, your question is equivalent to whether you can determine how many degrees you have rotated. If you have an external reference point, this is easy to do.

Another way, you could use a long band, white on a side, and white on the other side. You walk, leaving the band behind you, white face up. When you come back to the starting point, if it is still white, you are on a cylinder, otherwise, you are on a Mobius strip.

score 3 · Answer 8 · answered Sep 05 '17 at 22:11

If you can't see "both sides" of the surface simultaneously from any point on the strip then I question whether it really is topologically a Möbius strip.

When I make a "Möbius strip" from a strip of paper, for example, writing on one side does not show through. But the physical strip of paper that I held in my hands and whose ends I taped together has two faces separated by a non-zero distance; in reality, its topology is much more like a torus than like a Möbius strip.

If we add the constraint that we cannot travel or communicate in any way over the edge of the paper from one face to the other, each short segment of that edge effectively cuts apart the two parts of the surface of the torus near that segment. And if you actually cut the surface of the torus all the way along that edge, you end up with a strip with two twists. The boundary of that surface is two circles.

So no, with the constraints that you are not allowed to actually occupy or color the points of a true topological Möbius strip, then you cannot distinguish it from a cylinder using only local observations.

Duncan Ramage · Answer 9 · 2017-09-06T03:25:34.347

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Yes. When you return to the position of where you started, your right hand will be where your left hand was. Technically, this doesn't prove you're on a Mobius strip, only that you're on a non-orientable surface, but it's a step in the right direction.

edited Sep 06 '17 at 03:25

answered Sep 05 '17 at 19:00

Duncan Ramage

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You would be on the opposite side of the band – ignorance Sep 05 '17 at 21:56
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This was written assuming that you're a flatlander, and that "sides of band" mean nothing to you. – Duncan Ramage Sep 05 '17 at 23:48

score 2 · Answer 10 · answered Sep 07 '17 at 17:39

Assuming the person is held to the Mobius Strip or cylinder through a central gravitational force, yes.

That person would begin to feel differences as they walked. Since the Mobius Strip has only one side, and the gravitational direction is towards the center of the strip, the person would eventually feel like they were walking on a slope. Eventually, they'd need to climb along the walls, then ceiling, and so on.

score 1 · Answer 11 · answered Sep 05 '17 at 19:05

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Get a can of yellow paint, punch a hole in the bottom, and start walking along the boundary. Assuming the traveler can measure angle, stop after 360 degrees. Did your yellow paint drips complete the circle? Or, do you see your paint trail across the road?

If you see it across the road, keep walking. After another 360 degrees, did your paint trail close the loop?

answered Sep 05 '17 at 19:05

Randall

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After 360°, you would not only be across the road, but also underneath it. Therefore, you wouldn't see your paint trail across the road. – G Tony Jacobs Sep 05 '17 at 20:21
I was assuming our band was clear :) – Randall Sep 05 '17 at 20:42

score 1 · Answer 12 · answered Sep 06 '17 at 12:16

For this purpose, I will assume that the 'strip' is seemingly straight walkway, with a barrier on either side, so you can't peer over the edge, and a fog or mist to prevent you seeing more than a couple of meters ahead or behind. 'Gravity' or and analogue always points towards the surface of the strip. (Alternatively, the structure is in space and you are attached to a metal surface via magnets)

Draw a line down the center of the 'strip' you are on, until you reach the start again. Then, drill a hole on the Right side of line through the entire depth of the strip, and follow the line until you find a hole.

If it is on the Right side of the line, you are on a cylinder. If it is on the Left then you are on a Mobius Strip.

The important aspect is that you require some form of indication that can be identified from both sides of the strip/surface

score 1 · Answer 13 · answered Sep 06 '17 at 12:51

To determine whether you are on a möbius strip you would need to change your position in a direction orthogonal to the shortest distance between the edges at your sides at the moment of measuring at your current location. If you are on a möbius strip, you will return to your original position after a traversal of 360 degrees times the number of turns in mobius where the number of turns is an odd number.

score 1 · Answer 14 · answered Sep 06 '17 at 19:40

Get surveying tools.

Mark your starting location.

Using them, work out your the relative position in 3-space as you move along the path.

If you first return to your starting location (or right next to it) without being able to see the mark, then next return to it and see the mark, you are on a mobius strip or similar non-orientable surface.

You can work out the exact geometry by extremely careful surveying the shape.

If you cannot get surveying tools, try finding a geometer.

score 1 · Answer 15 · answered Sep 06 '17 at 21:47

As a side note, there was an old Soviet SF novel about people living in a clearly artificial environment, where a social experiment is conducted. The place they live in is not necessarily a Möbius strip, but has some strange topological properties, such as there is a cliff on the one side and abyss on the other. People, who crash or are thrown down the abyss, "loop" and appear to have fallen from the cliff. There are some more, but that's a major spoiler.

And no, they did not notice.

Well, some did – of else we would not know it, as the story follows a protagonist, – but after an elaborate, arcane, and possibly out-of-that-world research which involved experiencing the non-mentioned further metric issues. The general populace never notices and did not bother.

score 1 · Answer 16 · answered Sep 07 '17 at 12:05

I don't think it's the same as walking on a cylinder. On the strip you should see at least 3 different stages: 1. walking as on the cylinder (i.e. not seeing too far away - low horizon). 2. turn to right 3. Go up (like being on the bottom of a hill). Repeat.

So it should be possible by measuring the vector of the forces if forces are available. In this case it's easy.

C Marius · Answer 17 · 2017-09-07T15:45:01.230

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I think that if there are two persons walking on a cylinder, one in front and one behind it at some distance, the person from back sees the person in front vertical, whereas on a Moebius strip, the person in back would see, at some point, the person in front ... off-vertical ?!

edited Sep 07 '17 at 15:45

answered Sep 07 '17 at 15:30

C Marius

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score 0 · Answer 18 · answered Sep 01 '19 at 17:16

I assume the Möbius band is a compact surface with boundary $(M, \partial M)$. The question is a local to global question. Triangulate the surface, record all the incidences and use this information to compute the relative homology $H_2(M, \partial M)$ with real coefficients. It should be zero telling you that you live on a non-orientable surface. By computing $H_1(M)$ you can then detect that this non-orientable surface is a Möbius band.

Is it possible to determine if you were on a Möbius strip?

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