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Usually, the pop explanations of the properties of the Möbius strip say that if you take a pencil and start drawing a line in the middle of the strip, you'll have to make two complete loops to return to the original position. But if you look at the parametrization of the strip, after one loop you are again at the same position, albeit with the opposite chirality: it's like to say that the ink passes through the paper and colors both "faces" of the strip.

Which is the correct way to look at the strip?

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    $\begingroup$ The first sentence is "partially" true because we're thinking about the "two" sides of the surface, that is, the fact that the Möbius strip is not orientable. If you consider points with no orientation, then of course you'll return to the same spot. $\endgroup$ – Pedro Tamaroff May 20 '16 at 19:39
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    $\begingroup$ When the mark bleeds through, then you only have to go around once to complete the loop. but the pop description is referring to pencil marks that do not bleed through, but stay on one side only. So after one loop, you are back to the same point, but on the opposite side of the strip. You have to go around again to close the loop. The pencil mark is not just position, but also local orientation (chirality if you prefer), because it exists on only one side. $\endgroup$ – Paul Sinclair May 20 '16 at 22:25

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