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A recent 3Blue1Brown video https://youtu.be/IQqtsm-bBRU?si=zYmk82ENp-5LLYEL [14:56] noted that a Möbius strip was homeomorphic to the space of two unordered points on a circle. The homeomorphism however was not explicit and not particularly intuitive.

Is there a clear and direct map from the space of two unordered points points on a circle to a Möbius strip in 3-space that elucidates what is going on? What does the Möbius strip having just one side have to do with the space of pairs of unordered points?

This issue was discussed in this stackexchange discussion: Various proofs: pairs of points in a circle = Möbius strip . However, that’s over my head mathematically. I was hoping for something more concrete.

RobPratt
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TLss
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  • Please explain in plain words in your question what an unordered pair of points on a circle is with the associated details of your question. Asking us to look at a video and a analyze it is not the appropriate way to ask a question. You have to make your homework first! – mathcounterexamples.net Jan 04 '25 at 20:44
  • I think a bit of pedantry is necessary here. You ask about a mapping to a Möbius strip in $3$-space. There are many awful subspaces of $3$-space that are homeomorphic to the Möbius strip, and mapping to them would not at all be elucidating. Can you specify a concrete embedding of a Möbius strip in $3$-space that would be elucidating to you? – Servaes Jan 04 '25 at 20:45
  • What have you tried? You would probably be better off thinking of the Möbius strip abstractly as an identification space (quotient space) obtained by identifying two edges of $[0, 1]\times(0, 1)$ rather than as some ad hoc subset of $\Bbb{R}^3$. Note that you need to exclude the boundary of the Möbius strip, because your space of unordered pairs is not compact. – Rob Arthan Jan 04 '25 at 21:01
  • @Servaes Well the embedding where say the midpoint of a small rod of length 1/100 is rotated around the unit circle in the xy plane. The rod is oriented parallel to a radius of the circle and its center traverses the circle at constant speed. The rod also spins at constant speed about its midpoint in such a way that it rotates 180° by the time its origin has rotated 360°. I think this should be a Möbius strip – TLss Jan 04 '25 at 21:41
  • @mathcounterexamples.net I added a link to another question that discusses this exact issue, but the answers there are not intuitive to me in the sense I had in mind – TLss Jan 04 '25 at 21:42
  • The paper by Tuffley cited in the MSE question you linked gives exactly what you looking for in Figure 1. – Rob Arthan Jan 04 '25 at 22:05
  • This question is similar to: Various proofs: pairs of points in a circle = Möbius strip. If you believe it’s different, please [edit] the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. – Rob Arthan Jan 04 '25 at 22:06
  • @RobArthan The paper by Christopher Tuffley, Finite subset spaces of $S^1$ in Alg. & Geom. Topology (2002) at https://arxiv.org/pdf/math/0209077 , is the same argument as the 3Blue1Brown video. I understand it theoretically but still can’t really visualize in a clear way how the pairs of points are transformed. I was hoping for something that makes the whole map seem more obvious somehow. – TLss Jan 05 '25 at 02:33
  • I think you should work hard on understanding the method of constructing surfaces from polygons by identifying edges. See https://en.wikipedia.org/wiki/Surface_(topology). Once you get the idea, it is a much simpler way of understanding topological structure than trying to find models in low-dimensional euclidean space. – Rob Arthan Jan 05 '25 at 21:19

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