ebw said:
When's the last time you tried embedding a Möbius strip in two-dimensional (Euclidean) space? True Möbius strips can only exist in 3D, or higher. It is a terrible, terrible question relying on popular misconceptions, but it's not terrible for the reason you claim. The Klein bottle has the same dimension as the Möbius strip: they're both two-dimensional surfaces, so the analogy is very poor. One of them has a boundary (an exposed edge), while the other is closed. One can be realized in R^3 while the other can only be immersed in R^3 and embedded in R^4.
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I didn't even think about Möbius strips and what dimension they could be embedded in, but now that you mention it, yeah I see why it's pretty obvious that you need R^3 for it to exist. Considering that, the question makes even less sense. But my point still stands, a trur Klein bottle can only be embedded in R^4.
Anywho, I'm not gonna go any further here, I'm hardly an expert lol. While I love pure math, I don't want to become a mathematician.
