# Euler charateristics of the Möbius strip

• May 9th 2011, 11:53 PM
rayman
Euler charateristics of the Möbius strip
Hello! I am trying to figure out why the Euler characteristic of the Möbius strip is 0
$\displaystyle \chi(Möbius)=1-1+0$
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
• May 10th 2011, 01:33 AM
Opalg
Quote:

Originally Posted by rayman
Hello! I am trying to figure out why the Euler characteristic of the Möbius strip is 0
$\displaystyle \chi_{(\text{M\"obius})}=1-1+0$
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
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.
• May 10th 2011, 01:39 AM
rayman
Quote:

Originally Posted by Opalg
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?
• May 10th 2011, 02:46 AM
Opalg
Quote:

Originally Posted by rayman
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.
• May 10th 2011, 07:51 AM
rayman
Thank you, this makes sense to me;)