Is a Mobius Strip Truly a 2D Object in a 3D Space?

[pbuk]

One might take the description in post #26 of the Möbius band as a square with two opposite edges identified with a reflection and ask how a flatlander living on it would discover that his world is non-orientable.

Flatlanders don't live on a 2D surface, they live in a 2D surface. A left-handed flatlander living in a cylinder will always be left-handed: if it were possible to be a left-handed flatlander living in a Möbius band then they could become right-handed by traversing the band (and hence there can be no such thing: at what point on the band would they switch hands?)

 

[lavinia]

if it were possible to be a left-handed flatlander living in a Möbius band then they could become right-handed by traversing the band (and hence there can be no such thing: at what point on the band would they switch hands?)

An interesting question.

One could imagine that the Möbius band is embedded in three dimensional space and that the reflection is actually the result of a 180 degree rotation. If the Möbius band has the standard shape of a strip of paper pasted at its ends with a half twist, the flatlander will rotate in three space as he moves along the equator of the Möbius band and become his mirror image at the point of return. He will not become his mirror image at any time before that.

He would not be aware of this rotation because he can only observe motion tangent to the Möbius band. When he notices that he is his own mirror image he is stunned since as far as he can tell, nothing has happened during his trip that would change his orientation.

 

[Jim Robison]

Can anyone explain the meaning behind a mobius strip? Basically just a means to travel on both sides of a flat surface? It's still a 3D object though since it uses 3D space for the twist to be possible?

I think it's a question of relativity. To people observing it from "outside" it is three-dimensional. But to a being trapped on it's surface, it would appear to be two-dimensional.

 

[lavinia]

So far the significance of the Mobius band has related to its topology, its non-orientability, its role in constructing other non-orientable surfaces, and as the simplest example of a non-trivial vector bundle. But it is also significant because it can be given a flat geometry. In this geometry, the world appears to a flatlander to be Euclidean in small regions. The sum of the angles of a triangle is 180 degrees and the Pythagorean theorem holds true. Until a Flatland Magellan sails around the world flatlanders would believe that their world is a flat plane.

Unlike on a curved manifold such as a sphere, on a flat manifold parallel translation of a tangent vector around a small closed curve (and in general any curve that can be continuously shrunk to a point) always returns the vector to itself. This is exactly what happens in the flat plane. For curves that cannot be shrunk to a point, such as the equator of the Möbius band, it is possible for a vector to return to a different vector. The flat Möbius band is the simplest example of a flat manifold where this happens. Therein lies its geometric significance.

Comments:

-Unlike on the equator of the Möbius band, parallel translation around the circles that are parallel to the equator does not return a vector to a different vector. The vector returns unchanged. If one excises the equator, then the resulting surface is still flat since removing the equator does not warp or stretch anything, but what is left over is no longer a Möbius band. Parallel translation around any closed curve always returns the vector to itself and the surface is now orientable.

-Interestingly, there are no flat closed surfaces that can be embedded in 3 space. It is easy to parameterize a flat torus in four dimensional space but not in three. I am not sure if the flat Klein bottle can even live in four dimensions. I suspect not. These are the only two closed flat surfaces.

-A physical approximation to the flat Möbius band is just the usual Möbius band made from a strip of paper; this because the strip is flat to start with and bending paper does not change angles or lengths. Any topological Mobius band made from a piece of paper by bending and twisting is geometrically flat. So any odd number of twists in the strip rather than just one is also a topological Möbius band with a flat geometry. Another nice example can be made from three strips of paper that are completely flat in the middle but wrap around three separate cylinders in order to turn and connect to each other. This one looks a lot like a triangle that has been widened into a strip. There is a picture of one in the technical article "The Dark Side of the Möbius band" which is online.

-The flat Möbius band also presents problems in the study of bending of non-stretchable materials. Bending imbues the material with potential energy called "bending energy". A difficult question is to find for a given rectangular piece of unstretchable material the Möbius band shape that it can be bent into whose bending energy is a minimum. The bending energy can be computed from the normal curvatures of the band and the variational problem is constrained to variations in which the nearby surfaces in the variation are flat Möbius bands.

 

[pinball1970]

Q: Why did the chicken cross the Möbius strip?
A: To get to the same side!

this is my first post at Physics Forums. I apologize to anyone who feels that it is inappropriate to post a joke.

There is joke section in "The lounge." This joke is relevant so I don't think mods will mind too much!

 

[Hornbein]

But with the twist it pushes into 3D space so requires the 3rd coordinate?

We say it is a 2D object embedded in a 3D space. Mathematicians seldom care about how the strip is embedded so they just use the two coordinates.