How can they exist?

http://www.cut-the-knot.org/do_you_know/moebius.shtml

Thanks, but that was useless. I know how to make one, I want to understand how it works.

Magic.

[quote author=Hitmen link=board=2;threadid=3202;start=0#msg25122 date=1066870375]
Magic.
[/quote]
You misspelled "Math".

In an effort to make this post not as useless I will add a link explaining some of this math. Check out the gears animated gif!

I already saw that, it didn't explain much. I'm looking for an explanation of how a three-dimensional one-sided object can exist.

The math explains it, fool. If you're looking for a visual, then follow it, it's all one side.

No, fool, the math doesn't explain how a one-sided object can exist in three dimensions. I'm looking for a practical explanation.

Why don't you have a problem with a sphere, or a donut?

Because a sphere has length, width, and depth... As does a donut. Mmm... Donut. A mobius strip doesn't.

A mobius strip doesn't necessarily exist; it should exist, by theory. The three-dimensional plane is often denoted by a coordinate plane of an intersecting x, y, and z-axis, with the z being the "through-dimension" axis. Now, let's say that when you plot the mobius strip, it's coords (in (x,y,z)) would include points such as (0,0,8), (0,0,4), (0,0,-2), and (0,0,-8). This way, the mobius shape itself can actually be flat in every dimension it exists in, yet still span through multiple dimensions. This is all theoretical, ofcourse. Obviously I may be wrong, but I think that's the way it works. Hopefully that helped atleast a bit. :)

[quote author=CupHead link=board=2;threadid=3202;start=0#msg25162 date=1066882965]
No, fool, the math doesn't explain how a one-sided object can exist in three dimensions. I'm looking for a practical explanation.
[/quote]

Isn't it obvious how it exists after you make yourself one?

And yes, a mobius strip has length and width and depth depending on what coordinate system you pick.

[quote author=CupHead link=board=2;threadid=3202;start=0#msg25169 date=1066884365]
Because a sphere has length, width, and depth... As does a donut. Mmm... Donut. A mobius strip doesn't.
[/quote]

A donut is no different from a mobius strip in dimensions, other than their thicknesses. They both share the same general shape in one axis, and differing thicknesses along the angular path (circumference).

Unless you're talking about a non-real mobius strip with a 0-thickness? If so, that's little different from a 2-dimensional plane that you twist a little bit to make it appear 3-dimensional. But a mobius strip just gets twisted pi radians.

How can an object with only one side have any thickness though?!

There's your problem, you're the one defining it to have only one side and no thickness. So I ask you again, are you talking about a physical model of a mobius strip, or the theoretical one?

The physical model will exist in 3 dimensions and have a positive thickness.
The theoretical one will exist in three dimensions and have zero thickness.

The physical model will be characteristically the same as a donut.

I am talking about the physical model, how can it have a thickness if it only has one side?

[quote author=CupHead link=board=2;threadid=3202;start=15#msg25222 date=1066926616]
I am talking about the physical model, how can it have a thickness if it only has one side?
[/quote]

The physical model doesn't have one side. Ergo,
[quote]The physical model will exist in 3 dimensions and have a positive thickness.[/quote]

Um... Explain how it has two or more?

say you had a moebius strip constructed out of a tube. originally how many sides did the tube have ? i don't see how it should be different once you construct a moebius strip. cuphead you said you don't understand how there can be one side on a 3-dimensional object, and tubes are 3 dimensional, with 1 surface to begin with, so that category shouldn't apply. if it was originally created by a tube, logic should tell you it'd result in one side. if it has no 'thickness' then it wouldn't be 3 dimensional, so that's excluded too. if you started with a rectangle, it wouldn't be a 1 sided shape, it'd have 2 sides.

Since no one can seem to comprehend this:

[img]http://cuphead.valhallalegends.com/images/illus.gif[/img]

First of all, we can plainly see that the tube has all three dimensions, not just one surface. I have no idea where you got that from. Next to it is a piece of tape, or paper, from which to make a Moebius strip. Clearly, it has a front and a back, no? However, when fashioned into a Moebius strip, one of those sides disappears, thus losing it a dimension. Explain that.

cup, we know there are 3 dimensions. when you make the tube into a torus it has more than 1 surface ?? i'm revising

And simultaneously making no sense.

i'm telling you you wouldn't use a tube or a 1 dimensional object with no 'thickness' to make a moebius strip in the first place, so forget them.

Here's what a moebius strip, constucted of a material with no width, would look like:

He seems to have a built-in assumption that something with 1 face cannot exist in 3 dimensions, which is why I brought up a sphere, or much closer to his mobius--a donut. Both only have one face and exist in 3 dimensions.

Grok: Assume they're hollow. They now have two surfaces. This is an object without any sides, just one continuous surface.

Aha, you just proved my point.

Why can't the mobius strip be hollow as well? The physical mobius strip has a thickness, and thus can be hollow. This is why I'm equating it to a donut.

Grok, it can't be hollow because there aren't two sides between which it can be hollow. It only has one side.

[quote author=CupHead link=board=2;threadid=3202;start=15#msg25258 date=1066944156]
Grok, it can't be hollow because there aren't two sides between which it can be hollow. It only has one side.[/quote]I'm pretty sure that anything with a positive thickness could theoretically be hollow. :)

If it doesn't have two sides, how does the thickness exist?

[quote author=CupHead link=board=2;threadid=3202;start=15#msg25265 date=1066947668]
If it doesn't have two sides, how does the thickness exist?
[/quote]
I think you could safely call the edge of the piece of paper (or whatever thin material) that you're making the strip out of a side, because however small, it's thickness is >0.

I don't get it either... a sphere has an outside wall and an inner wall, that's two sides. How can you possibly have a sphere that has no inner wall?

[quote author=Soul Taker link=board=2;threadid=3202;start=30#msg25317 date=1066974717]
I don't get it either... a sphere has an outside wall and an inner wall, that's two sides. How can you possibly have a sphere that has no inner wall?
[/quote]

By not making it hollow?

How can it be a sphere, then, because it would have to have no thickness for there to be no chance of it being hollow.

You guys seem to be making this way more difficult than it is...

[quote author=CupHead link=board=2;threadid=3202;start=15#msg25258 date=1066944156]
Grok, it can't be hollow because there aren't two sides between which it can be hollow. It only has one side.
[/quote]

A moebius strip has two sides at every point. It just doesn't have two sides total. You can cut off a piece anywhere you like and you'll find two sides, so you can treat it as having two sides for the purpose of thickness etc.

There's a difference though. Once you cut it, it's not a Moebius strip anymore. :P

Your definition of 'side' is flawed.

[quote author=Soul Taker link=board=2;threadid=3202;start=30#msg25317 date=1066974717]
I don't get it either... a sphere has an outside wall and an inner wall, that's two sides. How can you possibly have a sphere that has no inner wall?
[/quote]Cull its backfaces, then itt'l have no inner wall.

[quote author=CupHead link=board=2;threadid=3202;start=30#msg25352 date=1067001047]
There's a difference though. Once you cut it, it's not a Moebius strip anymore. :P
[/quote]

Of course not, but you're measuring its thickness at some point, and then you can cut off a piece containing that point and you'll realize that you have no problem!

But then it's not a Moebius strip anymore, so it is a problem.

Why would that be a problem? Do you think the thickness of part X changes because you cut it off?

Also, a theoretical moebius strip has no thickness while a "real world" moebius strip has two sides, at least.

[quote author=Adron link=board=2;threadid=3202;start=30#msg25368 date=1067011260]
Why would that be a problem? Do you think the thickness of part X changes because you cut it off?

Also, a theoretical moebius strip has no thickness while a "real world" moebius strip has two sides, at least.
[/quote]

That's why I asked him to confirm he was discussing a physical model, not a theoretical one. The theoretical one has no thickness in one dimension.

[quote author=CupHead link=board=2;threadid=3202;start=30#msg25366 date=1067010936]
But then it's not a Moebius strip anymore, so it is a problem.
[/quote]

That's like saying if we take out a person's heart from his body, it's no longer a part of a person; therefore, if we study a heart that's been removed from a body, it's no longer relevant to its workings inside the body, which clearly isn't completely true.

Some fun to do with a mobius strip
http://www.exo.net/~pauld/activities/mobius/mobius.html