Why did the chicken cross the Möbius strip? ··gracefool |☺ 00:18, 12 Aug 2004 (UTC)

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Axeeeeeel. Nice work. One day I'll put some pictures with strip and some outstanding Maple code or something like that.FireJamXRasta 3 Wednesday [2002.02.27]

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Wouldn't it be cool to have a small java applet of the moebius strip in 3D?

Yes, (and Maple or Mathematica code too)

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I took out the R from the parametrization for three reasons:

• It was not explained.
• It looked as if it was a parameter, but it was in fact a fixed number representing the radius of the Möbius band.
• Not all values of R work (you can get self intersections if R is too small.

I explained the parametrization a bit better. AxelBoldt

Nice Axel. Yes R seems to be a constant and not a parameter. We should investigate for which R Moebius strip is really defined. I would like to say something more: I didn't mean that those presumptions about connection Universe<->Moebius strip come from SF - they come from science world (physics, cosmology) I guess. We should correct that fact somehow. Uh, Axel I don't want to be your student, ha, ha. I still owe to this page another picture of a strip... XJam [2002.03.25]] 1 Monday (0)

I haven't seen any serious cosmology suggesting a Moebius strip universe, but if you find anything, make sure to put it in the article. AxelBoldt

I assume by "generalized Moebius strip" something like a finite volume universe with orientation reversing paths is meant. My understanding of cosmology (very weak admittedly) is that such universes are possible. But for some reason most physicists suppose the universe to be orientable.--Chan-Ho Suh 10:09, Sep 7, 2004 (UTC)

I have checked 'very briefly' for R. As it seems strip degenerate near 0 and probably R must be positive or non-positive real number. Very intersting - how small should R be to get self intersections. We can also split R to R₁ and R₂. Does any self intersection appear if R₁ = - R₂. (I guess not - but how can you be shure?) Another output picture is coming...
XJam [2002.03.26]] 2 Tuesday (0)

The Klein bottle isn't a 3D analogue, since it's also a surface -- it's more an extension. -- Tarquin

The Klein bottle is actually two Moebius strips glued together along their edges. --Chan-Ho Suh 10:09, Sep 7, 2004 (UTC)

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I have a question regarding

Another equation for a Möbius strip is log(r)*sin(θ/2)=z*cos(θ/2).

I assume this is in cylindrical coordinates (r,θz)? This equation describes an unbounded figure though (you can enlarge r and z beyond all bounds), so I don't see how it can describe a Moebius strip. AxelBoldt, Sunday, June 2, 2002

It is in cylindrical coordinates. It describes an unbounded Moebius strip. If you want a bounded strip, you can take the part inside a torus, or restrict r and z. --phma

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moebius in fiction: there's also A Subway Named Moebius, AJ Deutsch.

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Am I the only person bothered by the phrase "Mobius strip is a topological object with only one surface"??? What does this mean??? I know what it is meant to mean, (i.e. that it's not orientable), and that this is the way to put it in as comfortable a language as possible, but the way it's phrased this way it seems too inaccurate (or meaningless) to be worth the benefit. Revolver 10 Nov 2003

I wouldn't say that's meaningless. It's implicit in that statement that the Moebius band under consideration is in 3-dimensional Euclidean space. There the band is one-sided. So I interpret "only one surface" to mean "one-sided". --Chan-Ho Suh 10:09, Sep 7, 2004 (UTC)

Harmless in the intro, I'd say. Anyone reading on is given a clearer idea. Generally speaking the first para of an article has some license to use looser language, and not to define everything with exactitude. There again, the surface link is probably unhelpful there.

Charles Matthews 18:00, 10 Nov 2003 (UTC)

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I removed this:

A family of 3D solids that closely relate to Möbius strips are the Sphericons. They are like a Möbius band, but without the hole in the middle. If you make a Möbius band out of a n-sided polygon sectioned strip, rotate it k amount and count the number of sides and edges created, more parallels can be found with the Sphericon.

I'm convinced at this point that the sphericon is that closely related; certainly not topologically.

Charles Matthews 13:48, 7 Apr 2004 (UTC)

I apologise, I think I was wrong. I was regarding a mobius band as not just a 2D rectangle cuved round into a circle and twisted, but also as 3D prisms curved round and twisted. These result in shapes with one side and one edge just as a mobius band does, and I believed they all fell under the same name. At simplest you can have a triangular prism, but a prism with many sides approaches a torus. Similarily, the simplest sphericon is based on a sixty degree apexed cone split and twisted, and a more complex sphericon approaches a sphere. What page would you reference it to?

You could add it to List of polygons, polyhedra and polytopes, perhaps as a 'see also'; also to the list of geometry topics under the 3D shapes.

Charles Matthews 20:01, 9 Apr 2004 (UTC)

Thanks, it's done.

Proberts2003 20:50, 9 Apr 2004 (UTC)

An error

Don't know how to edit this -- the second to last paragraph

A cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. The term 'cross-cap', however, often implies that the surface has been deformed so that its boundary is an ordinary circle. This cannot be done in three dimensions without the surface intersecting itself.

is wrong. This error is repeated in the cross-cap entry as well (word for word!). It is possible to embed a Mobius Band in <math>\mathbf{R}^3<math> with boundary a perfect circle. Here is the idea: let C be the unit circle in the xy plane in <math>\mathbf{R}^3<math>. Now connect opposite points on C, i.e., points at angles <math>\theta<math> and <math>\theta + \pi<math>, by an interval which is an arc of a circle. For <math>\theta<math> between <math>0<math> and <math>\pi/2<math> the arc lies above the xy plane, and for other <math>\theta<math> the arc lies below (there are two places where the arc lies in the xy plane).

I don't know what this embedding of the Mobius strip is called. sam Mon Aug 16 17:49:53 CDT 2004

Thanks for pointing this out. I have adapted what you wrote and added it to the page. Charles Matthews 10:20, 7 Sep 2004 (UTC)

Unfortunately, I can't make sense of the given description. I think an improved version is needed. The way I personally visualized this is hard to verbalize, but I found some nice descriptions in George Francis' A Topological Picturebook. They rely extensively on some key pictures however. Because of copyright I can't just scan them in, but maybe I could draw my own versions. At least, for now, we can give Francis' book as a reference (for a doubting Thomas).--Chan-Ho Suh 11:45, Sep 7, 2004 (UTC)

I'm having some problems with this description as well. It is a little ambigouous. A picture would be worth a thousand words here. If someone can give me a more concrete description, I'd like to plot this in Mathematica. -- Fropuff 17:10, 2004 Oct 20 (UTC)

Perhaps these photos help. I made an actual physical model, with a wire coat-hanger and a tee-shirt. In the first picture, quadrants 1, 3, and 4 are just a single layer. Quadrant 2 inside the circle has three layers; the middle layer connects to the boundary circle, while the top and bottom layers make the corner-shaped pouch in the upper left. The rope demonstrates that there's really a hole. Travelling from the end that goes off the edge (towards the yellow taped end), the rope passes up through the circle twice: it wraps around the circle inside the pouch.

The second picture is just the first one flipped over.

Fropuff, does that help at all? I've been fortunate enough not to have to learn Mathematica, so I can't make a plot of it. dbenbenn | talk 11:49, 29 Jan 2005 (UTC)

Thanks dbenbenn, I appreciate the effort. Though I must admit that I've stared at this picture for quite awhile and I'm afraid I still can't picture it. How can the rope pass twice through the circle? Where exactly is the circle in this picture? -- Fropuff 07:16, 2005 Feb 1 (UTC)

Tonight, I'll cut a "window" in the model and upload a new photograph. Perhaps that'll help. I'll also try highlighting the boundary with the Gimp. dbenbenn | talk 23:43, 3 Feb 2005 (UTC)

Okay, I procrastinated ridiculously, but it's here now. Does that help? dbenbenn | talk 07:24, 19 Feb 2005 (UTC)

Handedness of Möbius strip

Is the photo at the top of Möbius strip a right or left handed strip? I can't tell from reading the definition in the article. Perhaps someone who knows the convention could improve the description. dbenbenn | talk 23:42, 14 Mar 2005 (UTC)