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

[LightningInAJar]

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?

 

[Hornbein]

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?

It's a 2D object embedded in a 3D space.

 

[fresh_42]

Can anyone explain the meaning behind a mobius strip?

What do you mean by meaning? It is a standard example of vector bundles or manifolds. It exists.
My suggestion: make one and cut it along the long middle axis. Then explain the result mathematically!

What would be the meaning of Klein's bottle? It is an interesting object.

 

[robphy]

Possibly interesting:
https://math.stackexchange.com/questions/1942683/pacman-on-a-mobius-strip
http://www.thomasoandrews.com/klein/explanation.html

 

[LightningInAJar]

I will check out video. But isn't a "2D object" basically 3D because it is curved into 3D space? Does anything truly 2D even exist? Even a layer of graphene must be 3D.

 

[fresh_42]

I will check out video. But isn't a "2D object" basically 3D because it is curved into 3D space?

This is how we imagine it. But it can be defined without this embedding.

Does anything truly 2D even exist? Even a layer of graphene must be 3D.

Define existence.

 

[LightningInAJar]

This is how we imagine it. But it can be defined without this embedding.

Define existence.

Is there an object that isn't in any way 3D dimensional at least?

 

[fresh_42]

Is there an object that isn't in any way 3D dimensional at least?

No. Reality is four-dimensional, and in my opinion even discrete. But this is philosophy. As a Platonist, I consider everything existing what can be thought. Does a symphony exist in your understanding?

 

[pinball1970]

No. Reality is four-dimensional, and in my opinion even discrete. But this is philosophy. As a Platonist, I consider everything existing what can be thought. Does a symphony exist in your understanding?

Is this a co-ordinate thing, to define a point?

2D two numbers?
3D three numbers?

My intuition was 'volumes' rather than surfaces but co-ordinates seems better.

 

[fresh_42]

Is this a co-ordinate thing, to define a point?

2D two numbers?
3D three numbers?

My intuition was 'volumes' rather than surfaces but co-ordinates seems better.

You do the same as in real life: Cut out a square and glue it to a Möbius strip.

Now comes the difference: In real life, we cannot leave our three-dimensional space, and the paper remains three-dimensional. In mathematics, however, a square is $[0,1] \times [0,1]=[0,1]^2$ which is two-dimensional, and the gluing becomes an equivalence relation, i.e. identifying twisted opposite points.

Whether the abstract procedure qualifies for existence or not until we use paper is a matter of taste and in any case philosophy.

 

[Grelbr42]

The idea of a Mobius strip is a non-simply connected surface. That's a topology thing, meaning that the surface only has one side. You can continuously travel from any point on the strip and return to the point. You can demonstrate this making a paper one and drawing "the usual" path of going around the strip and coming back to the point directly through the paper from your start. If you were a 2-D object, you would be returning to your original location.

Another important aspect of a Mobius strip is that when you do this, you come back mirrored. You can see that with your paper model. Start with a symbol that you can see the difference when it is rotated. For example, the letter p. It comes back as a q.

The reason this is important is because it motivates notions of non-simply connected geometries of more than two dimensions. Things like 11-dimensions (as sometimes used in such things as string theory) are very difficult to diagram. It's often quite difficult to get started thinking about how such systems will look in three dimensions. So people will practice with Mobius strips and Klein bottles and such. This will help them develop the skill and analysis tools to deal with the higher dimensional things.

So you then get things related to Mobius strips when you start to consider higher dimensions being compacted down to 3 space and 1 time dimension. You wind up with various sub-dimensions being compacted into a variety of shapes.

Consider, for example, a regular old segment of a cylinder. Like a flattened hula-hoop. Then you give it a half twist and you have a Mobius strip. This twist cannot disappear continuously. In effect you have a topological charge number. A second twist and you have a related surface, now with two sides. This is potentially related to things like Fermions and Bosons. There are interesting effects when you combine two Mobius strips, or cut one strip down its middle.

Anyway, people get up to a lot of topology. They develop various topological charges and arrangements to produce various effects. It's an area of string theory and Kaluza-Klein type theories. https://en.wikipedia.org/wiki/Kaluza–Klein_theory

The general idea is, there are extra dimensions that are "curled up" into some sort of tiny manifold. The geometry, symmetry, and connectedness of these manifolds are hypothesized to produce various physical characteristics. It is a huge amount of fun playing with these things.

 

[LightningInAJar]

Any 3D object equivalent to mobius strip that allows traveler to return back to where they started along curved path and curved surface?

 

[fresh_42]

Any 3D object equivalent to mobius strip ...

This won't be a Möbius strip anymore, and so not an equivalent. They exist but are hard to imagine.
Do what I suggested: make a Möbius strip and cut it along the center into two parts.

... that allows traveler to return back to where they started along curved path and curved surface?

How about the Earth?

 

[Mark44]

But isn't a "2D object" basically 3D because it is curved into 3D space?

No. A "2D object" is an idealized mathematical object, just like points (0D), lines (1D), planes (2D) and so on.

 

[Hornbein]

Is there an object that isn't in any way 3D dimensional at least?

A shadow.

 

[Hornbein]

The idea of a Mobius strip is a non-simply connected surface.

You mean it's a simply connected surface yes?

 

[Hornbein]

I will check out video. But isn't a "2D object" basically 3D because it is curved into 3D space? Does anything truly 2D even exist? Even a layer of graphene must be 3D.

It's considered a 2D object because every point can be identified using two real numbers. In 3D all surfaces are 2D.

An Excel spreadsheet with ten columns can be considered ten dimensional.

 

[Tom.G]

Here is a way to answer the questions that you haven't even thought of yet.

https://www.google.com/search?&q=significance+of+möbius+strip

 

[Hornbein]

Here's a nice animation of a variant of the Mobius strip.



Author eklbr does very nice mathematical animations.

 

[LightningInAJar]

It's considered a 2D object because every point can be identified using two real numbers. In 3D all surfaces are 2D.

An Excel spreadsheet with ten columns can be considered ten dimensional.

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

 

[Hornbein]

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

Nope, only if you want the exact shape of the object. If you don't care about the exact shape and/or location you can identify every point with two numbers. So mathematicians say it is 2D. This is just a definition, a convention.

It remains 2D no matter how many dimensions you decide to embed it in.

And when mathematicians refer to a sphere, they usually mean the 2D surface. The interior is of less interest.

 

[LightningInAJar]

Nope, only if you want the exact shape of the object. If you don't care about the exact shape and/or location you can identify every point with two numbers. So mathematicians say it is 2D. This is just a definition, a convention.

It remains 2D no matter how many dimensions you decide to embed it in.

And when mathematicians refer to a sphere, they usually mean the 2D surface. The interior is of less interest.

What is the significance of using both sides of the strip to make it possible? Flatlanders don't have two sides. Even with 3D objects you don't really utilize the opposite side? I would think a real world object assumes barriers and the direction in which things flow in relation to the outside versus inside of the object. Otherwise isn't the inner surface basically virtual at best?

 

[Hornbein]

What is the significance of using both sides of the strip to make it possible? Flatlanders don't have two sides. Even with 3D objects you don't really utilize the opposite side? I would think a real world object assumes barriers and the direction in which things flow in relation to the outside versus inside of the object. Otherwise isn't the inner surface basically virtual at best?

You've hit the nail on the head. The Mobius strip does not really have two sides. An object outside of the strip might be considered to be on one side of the strip or the other. But a Flatlander that is inside the strip isn't aware of anything outside, so for it there are no sides at all to its environment.

Otherwise isn't the inner surface basically virtual at best?

Correct. 2D things are a completely imaginary concept. Those models they show you out of paper and stuff are only models.

 

[collinsmark]

What is the significance of using both sides of the strip to make it possible? Flatlanders don't have two sides. Even with 3D objects you don't really utilize the opposite side? I would think a real world object assumes barriers and the direction in which things flow in relation to the outside versus inside of the object. Otherwise isn't the inner surface basically virtual at best?

You're definitely on the right track and are asking the right questions. Here are a few things to consider:

Any Möbius strip that you can approximate in this real, physical world, is more akin to a strip with orientable double cover. From the start, you're considering two "sides" of a strip, not just a single side of a 2-dimensional surface as is normally done. But when discussing a Möbius strip in a strictly mathematical sense, it only has 1 side (unless otherwise specified). And it's not possible to tell which side of the strip that is, which makes it interesting; it's not orientable.

Let's take a step back. Maybe a few steps. Think back to the time in elementary school where you were first taught how to calculate area. Your teacher requests, "Calculate the area of 2 × 3 rectangle."

You'd get the wrong answer if you raised your hand and said, "twelve."

Your teacher would say, "No, the correct answer is six. Length times height. 2 × 3 = 6"

You could protest and say, "But look!" as you take out your scissors and cut yourself a 2 × 3 cm rectangle from a piece of paper. "There's six cm² on this side," and flipping the paper over, "and another six on this side. Six plus six is twelve."

Your teacher wouldn't buy it though. If you did that on a test your answer would be counted wrong.

That's because when we talk about a 2-dimensional surface, it's assumed that we're only talking about one side.

And the side we consider has a normal associated with it: a vector with the direction perpendicular to the surface, and only on that one side. If you know the normal vector of a small piece of a rectangle, you can figure out the normal of every other small piece that makes up that rectangle. You'll be able to "orient" that rectangle, knowing which side corresponds to the normal, and which "side" doesn't.

Now consider a mathematical Möbius strip (not one with double cover, but just a simple one). You can pick a small section of it and choose which "side" has the normal. But since there's no boundary, as you wrap around, that normal ends up looping back to the other side. But that doesn't make sense, because again, we're only considering one side. But it's not possible to tell where one side ends and the "other side" begins. The surface is not orientable.

That's the significance of a Möbius strip.

 

[snorkack]

This won't be a Möbius strip anymore, and so not an equivalent. They exist but are hard to imagine.
Do what I suggested: make a Möbius strip and cut it along the center into two parts.

Does Klein bottle count?

 

[fresh_42]

Does Klein bottle count?

Close, but not equivalent:

 

[lavinia]

One significant feature of the Möbius band is that all possible non-orientable closed surfaces can be constructed from it.

For example, pasting the boundary circles of two Möbius bands to each other makes a Klein bottle. Pasting the boundary circle of a disk to the boundary circle of a Möbius band makes the Projective plane.

Starting with any closed non-orientable surface, one can cut out a small disk from it and replace the disk with a Möbius band by pasting its boundary circle to the boundary circle of the hole made by removing the disk. This makes a new non-orientable closed surface. It turns out that starting with a Projective plane, any non-orientable closed surface can be constructed by a finite number of iterations of this process of replacing disks by Möbius bands.

Note: Non-orientable closed surfaces such as the Klein bottle cannot be embedded in three dimensional space. The pictures one sees are not embeddings because they have self intersections. If one keeps these self intersections in mind, it is not hard to see that these self intersecting surfaces have only one side.

Note: If in the diagram of the Klein bottle on the right in post #26, one draws two parallel line segments, one starting at the upper left corner and ending at the midpoint of the bottom edge, and the other starting at the midpoint of the upper edge and ending at the lower right corner, the Klein bottle becomes divided into two regions each of which is a Möbius band. The two lines form a circle that is the boundary of the two Möbius bands.

 

[lavinia]

Any Möbius strip that you can approximate in this real, physical world, is more akin to a strip with orientable double cover. From the start, you're considering two "sides" of a strip, not just a single side of a 2-dimensional surface as is normally done.

@collinsmark I love your observation (that is if I understand it correctly)that a physical Möbius strip is really a double cover. It was instructive to twist up a cylinder made of paper so that it looks like a double Möbius band.

If one imagines the strip of paper as being a continuous stack of infinitely thin rectangles, then smack dab in the middle, one of them gets pasted to itself and is a true Möbius band. So the physical Möbius band actually models a continuous stack of double covers over a true Möbius band sandwiched in the middle. Very cool.

Mathematically, the physical band is a shell surrounding the Möbius band that consists of all normal vectors of length less than some constant.

Now consider a mathematical Möbius strip (not one with double cover, but just a simple one). You can pick a small section of it and choose which "side" has the normal. But since there's no boundary, as you wrap around, that normal ends up looping back to the other side. But that doesn't make sense, because again, we're only considering one side. But it's not possible to tell where one side ends and the "other side" begins.

In the case of a closed surface embedded in 3-space, the notion of it having two sides can be taken to mean that it separates space into two disjoint regions, an exterior and an interior.

Equivalently a shell of normal vectors around the surface is split into two disjoint pieces when the surface is removed.

With the Möbius band the shell of normal vectors does not separate into two disjoint pieces when the Möbius band is removed. It remains in one piece. This can be taken to mean that the Möbius band has only one side.

 

[lavinia]

What do you mean by meaning? It is a standard example of vector bundles or manifolds.

The significance of the Möbius band might be in part that it is the simplest example of a non-trivial vector bundle

 

[lavinia]

The surface is not orientable.

That's the significance of a Möbius strip.

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.

There must be a way for the flatlander to figure this out because non-orientability is an intrinsic feature of the Möbius band and is independent of how it is realized whether in 3 space or in a Klein bottle or anywhere else.

The unit normal in 3 space is not available to the flatlander. Non-orientability must detected from within the Möbius band.

What even does this idea of intrinsic non-orientability mean?

 

[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.