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LightningInAJar
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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.LightningInAJar said: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?
What do you mean by meaning? It is a standard example of vector bundles or manifolds. It exists.LightningInAJar said:Can anyone explain the meaning behind a mobius strip?
This is how we imagine it. But it can be defined without this embedding.LightningInAJar said:I will check out video. But isn't a "2D object" basically 3D because it is curved into 3D space?
Define existence.LightningInAJar said:Does anything truly 2D even exist? Even a layer of graphene must be 3D.
Is there an object that isn't in any way 3D dimensional at least?fresh_42 said:This is how we imagine it. But it can be defined without this embedding.
Define existence.
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?LightningInAJar said:Is there an object that isn't in any way 3D dimensional at least?
Is this a co-ordinate thing, to define a point?fresh_42 said: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?
You do the same as in real life: Cut out a square and glue it to a Möbius strip.pinball1970 said: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.
This won't be a Möbius strip anymore, and so not an equivalent. They exist but are hard to imagine.LightningInAJar said:Any 3D object equivalent to mobius strip ...
How about the Earth?LightningInAJar said:... that allows traveler to return back to where they started along curved path and curved surface?
No. A "2D object" is an idealized mathematical object, just like points (0D), lines (1D), planes (2D) and so on.LightningInAJar said:But isn't a "2D object" basically 3D because it is curved into 3D space?
A shadow.LightningInAJar said:Is there an object that isn't in any way 3D dimensional at least?
You mean it's a simply connected surface yes?Grelbr42 said:The idea of a Mobius strip is a non-simply connected surface.
It's considered a 2D object because every point can be identified using two real numbers. In 3D all surfaces are 2D.LightningInAJar said: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.
But with the twist it pushes into 3D space so requires the 3rd coordinate?Hornbein said: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.
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.LightningInAJar said:But with the twist it pushes into 3D space so requires the 3rd coordinate?
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 said: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.
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.LightningInAJar said: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?
Otherwise isn't the inner surface basically virtual at best?
LightningInAJar said: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?
Does Klein bottle count?fresh_42 said: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.
Close, but not equivalent:snorkack said:Does Klein bottle count?
@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.collinsmark said: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.
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.collinsmark said: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 significance of the Möbius band might be in part that it is the simplest example of a non-trivial vector bundlefresh_42 said:What do you mean by meaning? It is a standard example of vector bundles or manifolds.
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.collinsmark said:The surface is not orientable.
That's the significance of a Möbius strip.
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 said: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.
An interesting question.pbuk said: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?)
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.LightningInAJar said: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?
There is joke section in "The lounge." This joke is relevant so I don't think mods will mind too much!Bombu said: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.