Points on opposite sides a paper Mobius strip.

In summary, a Mobius strip is a surface with only one side and one edge, and it has no thickness in the mathematical sense. Points on opposite sides of the strip can be the same point, but when using materials like paper that have thickness, these points will not be near each other. The Mobius strip is also not a closed surface, but a similar surface with these properties is the Klein bottle.
  • #1
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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!
 
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  • #2
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.
 

1. What is a Mobius strip?

A Mobius strip is a one-sided surface with only one boundary. It is created by taking a strip of paper, giving it a half-twist, and then joining the ends together.

2. How is a Mobius strip different from a regular strip of paper?

A regular strip of paper has two distinct sides, while a Mobius strip has only one side. This means that if you were to draw a line down the middle of a Mobius strip, it would end up on both sides of the strip.

3. Can a Mobius strip be made with any type of paper?

Yes, a Mobius strip can be made with any type of paper as long as it is flexible enough to be twisted and joined without breaking.

4. How many points are on opposite sides of a Mobius strip?

There are no points on opposite sides of a Mobius strip because it only has one side. Any point on the strip can be considered to be on the opposite side of itself.

5. What is the significance of points on opposite sides of a Mobius strip?

The concept of points on opposite sides of a Mobius strip is often used in mathematics and topology to demonstrate the properties of one-sided surfaces. It can also be used to explain the concept of infinity as there is no clear distinction between the inside and outside of the strip.

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