Mobius Strip? What's all the fuss about?

[Swapnil]

I don't really get what makes mobius strip so special? Yeah, sure you can get from one from of the strip to another without touching its boundary but so what?

BTW, I am not saying the mobius strip is useless. I just want to know how it helps you get a deeper understanding of other dimentions.

 

[radou]

The Moebius strip is so special because it is a popular example of a nonorientable surface. You may want to do some google-ing or mathworld-ing for further information.

 

[James R]

The mobius strip is a one-sided surface, even though it initially looks like it has two sides.

 

[Dcase]

Even more interesting are Mobius transformations.
http://en.wikipedia.org/wiki/Möbius_transformation

 

[George Jones]

I don't really get what makes mobius strip so special? Yeah, sure you can get from one from of the strip to another without touching its boundary but so what?

Suppose that the strip is a closed 2-dimensional universe for 2-dimensional beings. 2-dimensional Bob and Betsy, both with their hearts on the left, stand side-by-side. Bob goes around the strip once while Betsy stays put.
After this, what is the relationship between their hearts?

 

[Swapnil]

Suppose that the strip is a closed 2-dimensional universe for 2-dimensional beings. 2-dimensional Bob and Betsy, both with their hearts on the left, stand side-by-side. Bob goes around the strip once while Betsy stays put.
After this, what is the relationship between their hearts?

Well, the same, ofcourse. Unless, you are talking about Bob being on the "backside" of Betsy. Then Bob's heart would be upside down and to the right side. Eitherway, they wouldn't really meet each other since they are on the reverse side.

Anyways, how is this significant?

 

[robphy]

This may be of interest
http://arxiv.org/pdf/gr-qc/0202031
http://www.iop.org/EJ/abstract/0264-9381/19/17/308
"The orientability of spacetime" by Mark J Hadley

I haven't read it myself.

 

[cesiumfrog]

Well, the same, ofcourse. Unless, you are talking about Bob being on the "backside" of Betsy. Then Bob's heart would be upside down and to the right side. Eitherway, they wouldn't really meet each other since they are on the reverse side.

Anyways, how is this significant?

Imagine you'd drawn Bob and Betsy on with the kind of pen that soaks through the entire thickness of the paper.

It's significant because, if you leave your left shoe at home and take a journey across this physical universe, it is conceivable (according to GR at least) that it will fit your right foot when you return. (And if you weren't already dyslexic..)

 

[arivero]

Going to more practical thinks, we want to be able to proof Gauss law and Stokes theorem. The proof relies on orientability.

Indeed it is not a big fuss because locally you still have orientability. The bussiness becomes more complicated if you want to proof an assertion for an integration about the whole surface, and this proof depends on dividing the integral in two halfs and relying on Stokes theorem.

 

[arivero]

Or supposse you want to claim that whenever a bidimensional surface is limited with only one closed line, you can deform it into a circle. Moebius strip should be a counterexample because its border is a single closed line too. To a topologist, the real important point is not that it has a single side (there is not such thing as sides for topologists) but that it has a single border. A usual strip has two borders.

Then here comes the surgery classification of two-dimensional surfaces: as the border of the Moebious strip can be deformed to be a circunference, I can paste (sew across the border) a circle to a Moebius strip to build a closed figure which is different of the one I get by pasting the borders of two circles (there I get the surface of an sphere, as usual).

Now this is really mind-blowing, leave Bob and Betsy ****ing in the grass and put yourself about cutting a circle out of the closed surface you got before, and staple there another moebius strips. Is it equa to the sphere? Is it a new, different surface?

Ah yeah, you can not do it in 3 dimensions. But do not worry because Nash (do you remember the film? Paranoid guy about the martians, the russians and a small girls always following him?) got to proof that you can always do it in 5 dimensions, and even in 4 with a little effort.

 

[raybuzz]

Hi all,
I was reading One two three to infinity book, In pg62, fig 23 the author describes a donkey going around a mobius strip in a 2 dim world. He says that when it comes back to the original position the donkey gets inverted (i.e. heads down, legs up). However when I tried it practically it didnt work,
Suggestions please,

Vinay

 

[yenchin]

Hi all,
I was reading One two three to infinity book, In pg62, fig 23 the author describes a donkey going around a mobius strip in a 2 dim world. He says that when it comes back to the original position the donkey gets inverted (i.e. heads down, legs up). However when I tried it practically it didnt work,
Suggestions please,

Vinay

A mathematical surface does not have a front and a back. When you do this in practice of course it doesn't work as the donkey, when it comes back to the position, is on the back side of the paper. That's what cesiumfrog meant by "Imagine you'd drawn Bob and Betsy on with the kind of pen that soaks through the entire thickness of the paper" in the previous post.

 

[raybuzz]

Thank you yenchin, I now understood it.

Vinay

 

[mathwonk]

try imagining two mobius strips glued together along their identical borders. what surface do you get?

 

[Bacle]

Hi:
Just some followup questions:

1)What is the relation between Moebius transformations and the Moebius
strip, if any.?. Maybe the same guy worked on both.?. I know the
Moebius Maps are the automorphisms of the Riemann Sphere,aka,
S² (as the 1-pt compactification of the complexes).
But I don't see a relation.

2)What fails if we do integrate along the Moebius strip.?. I mean, how do
Stokes' theorem and Gauss' Law go wrong , when integrating.?. I believe
orientation allows us to determine the sign of the value of the integral.
Is there something else.?