Morphing a "physical' Möbius Strip (one with non-zero thickness) into a Toruspic.twitter.com/2SzJzUNqbN
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At first i thought they were topologically equivalent.. but now i'm not so sure
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Here’s a decent way to see that they’re homeomorphic: the Möbius strip can be viewed as a line bundle over a circle. However, giving the Möbius strip a finite “thickness” is equivalent to compactifying the fiber (the line bit) into a circle...
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The resulting shape is a fiber bundle whose base and fiber are both circles. There are two options up to homeomorphism: the torus or the Klein bottle. Since the physical Möbius strip can be embedded in 3 dimensions, we know it must be the torus.
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No they aren't, a torus is orientable and a Möbius strip isn't ;-) https://en.m.wikipedia.org/wiki/Orientability …
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maybe it's a twisted torus, and the space by itself might be topologically equivalent to a torus, but if you put a flat metric and then construct such a torus by glueing it together w/ a twist, you might get sonething weird going on as you parallel transport around it. not sure
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That just demonstrates that the torus and the “physical Möbius strip” aren’t diffeomorphic. They’re definitely homeomorphic, however.
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Sooo, it depends on definition/limits. It is safer to say they are not, but, your caveat ("physical model" of Mobius strip) makes the question meaningful, and I would argue it is still more NO than YES.
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From a completely lay point of view, a non twisted strip can be turned into a torus too, no?
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Here's another thought. When we look at tori w/ complex structure (C/some lattice), isn't the torus you get from cutting a torus and giving it a 180° twist before glueing together, a torus w nontrivial complex structure bc the two circles are now at an angle?
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