Is a Möbius band homeomorphic to a torus?
It seems like it would be, but if someone tracks the points of either surface, it doesn't seem to share homeomorphism; however genus-wise, they appear homeomorphic. Which one is it?
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Catwheezle
It depends on your definition of "band". If you mean "a flat thing with zero thickness", then no, because zero thickness (a surface with two sides) is an attribute in itself. But you can use that band to slice the torus into two interlocking rings, which would be useful in parties.
If by "strip" you mean "a flattish thing with nonzero thickness", then yes, they're the same thing: blow air into it, inflate it fat enough, and they're visually identical.
To a 3D rendering program though, it would be as if someone had sliced the donut and rotated one side of the slice 180 degrees, then re-glued it.
But since the location of the "vertices" in that 3D program is arbitrary, just as the "tracked points" are, you can find points on that inflated strip that will have a 1:1 mapping between the torus and the flat strip when you deflate it.
If by "strip" you mean "a flattish thing with nonzero thickness", then yes, they're the same thing: blow air into it, inflate it fat enough, and they're visually identical.
To a 3D rendering program though, it would be as if someone had sliced the donut and rotated one side of the slice 180 degrees, then re-glued it.
But since the location of the "vertices" in that 3D program is arbitrary, just as the "tracked points" are, you can find points on that inflated strip that will have a 1:1 mapping between the torus and the flat strip when you deflate it.
TetrisGuy · 26-30, M
See that's what I was wondering. I'm not sure if the homeomorphism requires a perfect transformation of the points. It still seems... off to me, though. I'm talking about a physical Mobius band (in which case, I'd have brought forth rings, instead of toruses.
Catwheezle
But that would be a different question, with a different answer.
A flat ring and a mobius strip can't be homeomorphic, because they have edges, and you have to take your points relative to those.
A torus (whether inflated, flattened so it looks a bit like a ring, flattened so it looks like a mobius strip, or dimpled and stretched into a coffee cup) doesn't have those edges, so you can place your points arbitrarily. It's homeomorphic with any volumetric solid that has a single hole through it.
A flat ring and a mobius strip can't be homeomorphic, because they have edges, and you have to take your points relative to those.
A torus (whether inflated, flattened so it looks a bit like a ring, flattened so it looks like a mobius strip, or dimpled and stretched into a coffee cup) doesn't have those edges, so you can place your points arbitrarily. It's homeomorphic with any volumetric solid that has a single hole through it.
BluCrystal
Girl, you give me headaches thinking about the stuff you do. Is there anything else in your life that catches your interest?
TetrisGuy · 26-30, M
Fûck no it doesn't five me headaches. It turns me on if anything. I like a lot of things, but math is my favorite thing... EVER.
fredcs
I think of it more as a 2 dimensional object in 3 dimensional space
TetrisGuy · 26-30, M
Well yeah, but that doesn't say anything about whether or not it's homeomorphic to a torus.