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This video shows the visualization of the proof of inscribed rectangular problem. It elaborates the equivalence between a pair of points on a loop and a single point on the mobius strip. https://www.youtube.com/watch?v=AmgkSdhK4K8

My question, is that equivalence unique? I mean if we have another loop with different shape, will that still be equivalent to the same mobius strip or a different mobius strip? And if it is a different one, where will be the difference?

If, however, all loops can be represented on a single mobius strip, then there will be different assignment of any arbitrarily point on the strip to different pairs of points on different loops.

isaac
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1 Answers1

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I'm not sure I can answer all of your questions, but I can answer your questions about equivalence.

"Equivalence" in this context means homeomorphic. To be precise, there is an equivalence relation between topological spaces $X$ and $Y$, in which we say that $X$ is homeomorphic to $Y$ if there exists a function $f : X \to Y$ which is a homeomorphism (by definition, that means $f$ is continuous, and $f$ is a bijection, and the inverse function $f^{-1}$ is also continuous).

For example, any two Möbius strips are homeomorphic to each other. It follows (from the transitive law, which is part of the definition of an equivalence relation) that any topological space which is homeomorphic to one Möbius strip is also homeomorphic to every other Möbius strip.

In particular, since we know that the topological space $X$ consisting of pairs of points on given loop is homeomorphic to a Möbius band, it follows that $X$ is homeomorphic to every Möbius band.

Also, if $Y$ is the topological space consist of pairs of points on some other loop, then both $X$ and $Y$ are homeomorphic to a Möbius band, and so $X$ and $Y$ are homeomorphic to each other (although I'll say that it's probably easier to directly construct a homeomorphism between $X$ and $Y$, using the fact that any two loops are homeomorphic).

Lee Mosher
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