How does one measure the volume of a Mobius Strip mathematically?

[Tomoyo]

I got into a really heated argument with my friends at school about this.

Here're the important events:

1) In math class, we had a substitute teacher, so my friend decided to cut a mobius strip in half and see what happened (the books never tell you!).
2) We then cut the cut mobius strip again and are awed.
3) We go show some of our other friends at lunch and we start talking about mobius strips.

We spent a long time arguing whether a mobius strip was three dimensional (well, it's 3D because it has been twisted, right?) and then we used the argument that "if it's 3D then you can measure its volume".

Of course, we could just dump it in water. But my friend insisted that we find a mathematical way to do this. I think the formula is [(area of the strip)/2]*Height, but my friend said that there was no height since both sides were the same side.

Assuming that the strip has been a Mobius strip for all eternity, how does one mathematicall calculate the volume of it using mathematical methods? I also proposed flattening the strip and measuring the triangles, accounting for overlaps, but the response was "that wouldn't make it a Mobius strip".

So I'm confused...

 

[madviking]

wait, are you trying to find the volume of the stuff inside? or the volume of the paper?
OR you were you trying to find the area?

OH! i think i know. you chop once and then its just a strip of then find the volume. i think. don't forget to get the tape.

 

[Tomoyo]

The volume of the paper...

 

[madviking]

i added an edit

 

[North King]

Measure how long it was from a line drawn on it going around the mobius strip to that same line. Divide by 2, because you've gone over "both sides" of the one sided paper. Multiply by paper width, multiply by paper height.

 

[newfangle]

A mobius band is a two-dimensional object. You may only measure its SA.

What happens when you cut it in half anyways? You get two more of them, don't you?

 

[Tomoyo]

Measure how long it was from a line drawn on it going around the mobius strip to that same line. Divide by 2, because you've gone over "both sides" of the one sided paper. Multiply by paper width, multiply by paper height.

Yeah, that's what I originally thought, and it still seems correct. The only potential problem that was pointed out was that there is no height - how can there be height when there is no inside and no outside?

EDIT:

@newfangle: No, you don't get two mobius strips. You get one double-twisted connected band.

 

[Babbler]

I talked to my Dad; he is a math professior. A Mobius strip is 2-D; it has no volume. It has a surface area, but it is hard to calculate.

Edit: Too slow!

 

[Bluemofia]

Yeah, they are 2-D. I read in a book. They have 2 dimensions, but it involves moving into the 3rd dimension to make it possible to do. It itself doesn't have 3 dimensions. A Klein (sp?) Bottle, is a 3D version of the Mobius Strip. It has 3 dimensions, but it is only creatable if it moves into the 4th dimension.

 

[North King]

Yeah, that's what I originally thought, and it still seems correct. The only potential problem that was pointed out was that there is no height - how can there be height when there is no inside and no outside?

EDIT:

@newfangle: No, you don't get two mobius strips. You get one double-twisted connected band.

There is a height--you just take the distance between the side and itself.

 

[newfangle]

There is no normal way to take the surface area of a Mobius band since it is not an oriented surface.

 

[North King]

There is no normal way to take the surface area of a Mobius band since it is not an oriented surface.

Correct me if I'm wrong: assuming the distance between the two "edges" are constant, couldn't you multiply that by half the distance around the strip to get SA?

 

[newfangle]

No because then you have to ask yourself whether you're measuring the top or the bottom of the thing.

 

[Bluemofia]

Well, to measure it, just measure the length of the thing, then tape it togeather.

 

[thetrooper]

So what's the conclusion?

The mathematical object you call Mobius strip/band have no volume only surface area.

 

[IglooDame]

If you insist on treating it as a 3D object I'd suggest finding out the thickness and multiplying that by the width (assuming it is constant throughout) and multiply that by half the length of a line drawn all the way through it (i.e. "through both sides").

I think it would be easier to cut this particular Gordian Knot, though.

 

[Timko]

Surely its very easy to measure the surface area. Ok, so it only has one surface, but then so does a sphere.

 

[Mise]

If it can be defined as a function, can't you just integrate it? It might not yield analytical solutions, but you could do it on a computer.

 

[Stegyre]

I talked to my Dad; he is a math professor. A Mobius strip is 2-D; it has no volume. It has a surface area, but it is hard to calculate.

Are we talking about a theoretical Mobius strip or a physical object?

Of course, we could just dump it in water.

I’ll wager that if we take a plastic strip 48”x1”x0.25” and twist it into a Mobius strip, it will displace 12 cubic inches of water (if I’ve done my math right).
This would be the same, mathematically, as taking the (width x length x height)/2. Argue all you wish about the theoretical difficulty in taking these measurements (“but the edges I am measuring the width between are really the same edge”), practically, we have no difficulty doing so.

This seems like one of those cases, like Xeno's paradox, where "theory" creates a problem that does not exist in reality.

 

[pboily]

Surely its very easy to measure the surface area. Ok, so it only has one surface, but then so does a sphere.

It is not like a sphere as it cannot be oriented. (see Newfangle's post)