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Math Help - Euler charateristics of the Möbius strip

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    Euler charateristics of the Möbius strip

    Hello! I am trying to figure out why the Euler characteristic of the Möbius strip is 0
     \chi(Möbius)=1-1+0
    It indicates that the Möbius strip must have one vertix and one edge....but I dont see it....Can someone explain it to me?

    If we for example want to calculate the E.C for a loop. How can a regular loop have two edges? (inside and outside)???? How many vertices will it have then?

    Thank you
    Last edited by rayman; May 10th 2011 at 12:38 AM.
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    Quote Originally Posted by rayman View Post
    Hello! I am trying to figure out why the Euler characteristic of the Möbius strip is 0
     \chi_{(\text{M\"obius})}=1-1+0
    It indicates that the Möbius strip must have one vertix and one edge....but I dont see it....Can someone explain it to me?
    No, it indicates that the Möbius strip has one face and one edge (and no vertices).

    Quote Originally Posted by rayman View Post
    If we for example want to calculate the E.C for a loop. How can a regular loop have two edges? (inside and outside)???? How many vertices will it have then?
    If by a loop you mean a cylinder (without ends; in other words a surface like the Möbius strip but without the twist), then it has one face, two edges (namely the two sides of the strip) and again no vertices. So its Euler characteristic is –1.

    See the discussion here.
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    Quote Originally Posted by Opalg View Post
    No, it indicates that the Möbius strip has one face and one edge (and no vertices).


    If by a loop you mean a cylinder (without ends; in other words a surface like the Möbius strip but without the twist), then it has one face, two edges (namely the two sides of the strip) and again no vertices. So its Euler characteristic is –1.

    See the discussion here.

    Can you exaplain to me how it can have only one edge?
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    Quote Originally Posted by rayman View Post
    Can you explain to me how it can have only one edge?

    Trace round the edge of this Möbius strip, starting at the inside edge on the right of the picture, following the red line. By the time you have gone all the way round the red line, you are still tracing out the same edge but you have come round to the "other side" of the strip. If you keep going along that edge, you will go right round the strip once more and eventually get back to your starting point. You will have traced out the entire boundary of the Möbius strip, which consists of that single edge.
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    Thank you, this makes sense to me
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