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

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.


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.


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.

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General Topology

Mobius Band

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