Points on opposite sides a paper Mobius strip.

[Spinnor]

Say I pierce a paper Mobius strip with a pin and call the point on the side the pin entered the paper the point A and call the point where the pin comes through the paper the point B. In an idealized Mobius strip are these points different? Can they be the same?

I would like a closed surface where I can travel once around with a surface normal, come back to where I started (but on the "other" side), but have the surface normal pointing in the opposite direction, this happens with a Mobius strip?

If I were to make a Mobius strip out of a thick strip of paper points on opposite sides of the paper are not near each other, do we still have a Mobius strip? It seems points on opposite sides of a Mobius strip ( I know there is only one side for a Mobius strip) can be very near each other with thin paper or farther away with thick paper or be the same point where the thickness of the paper goes to zero? I want points on opposite sides of the Mobius strip to be the same point, can they be?

Sorry if I'm not making myself clear.

Thanks for any help!

 

[quasar987]

When you make a Mobius strip out of paper, you are not making a Mobius strip in the mathematical sense because a Mobius strip has no thickness and your paper has some thickness.

In a Mobius strip in the mathematical sense, when you go around the strip and come back "where you started (but on the other side)" as you say, you actually come back to the same point.

Also, I don't know if this level of preciseness matters to you, but the Mobius strip is not a closed surface (i.e. it is not connected, compact and without boundaries). For a closed surface with similar properties as the mobius strip, check out the Klein bottle.