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There are 3D representations of Klein bottles that give people in our 3D universe a pretty good idea of how one is constructed:

enter image description here

We can sort of see how this thing needs to be 'twisted' in the fourth dimension.

But how do Flatlanders create 2D representations of Möbius strips? The only thing I can imagine them representing a Möbius strip as is a line segment.

Or can they somehow represent that it's twisted in the third dimension?

And if not, why not? That is, why is there not an analogue in Flatland of our representations of Klein bottles in the 3D world?

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I'd imagine that a Flatlander's representation of a Mobius strip looks like any 2D drawing of a Mobius strip: mathworld.wolfram.com/images/eps-gif/MobiusStrip_1000.gif It's a surface that has to cross over itself - it can be immersed in the plane, but not embedded, much as a Klein bottle can be immersed in 3-space but not embedded. –  Neal Nov 30 '12 at 16:34
But we can only see that by looking down at it from above. A Flatlander can't do that. –  Paul Reiners Nov 30 '12 at 16:43
Wouldn't that be a 4-lander's reaction to our Klein bottles? –  Neal Nov 30 '12 at 17:01
A Flatlander trying to represent a Möbius strip is dealing with a codimension zero object, while a Spacelander trying to represent a Klein bottle is dealing with a codimension 1 object. It'd be more like us trying to visualize $\Bbb{R}P^3$ or something (which we usually do by identification, as Peter Smith suggests). –  Micah Nov 30 '12 at 17:16

2 Answers 2

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You're asking how a 2-D flatlander would represent a Möbius strip. The right analogy is not how we 3-D beings represent Klein bottles, but how we represent various compact 3-D spaces.

A useful method for representing these spaces is to start with some compact region, like a rectangle in 2-D or a cube in 3-D, and to glue some of the boundaries together. A Möbius strip is what you get when you glue one pair of opposite sides of a rectangle together with opposite orientations, as in the first picture below. The second picture represents a cube, with one pair of faces glued together with opposite orientations.

Mobius Strip $\phantom{spacefiller...}$ Glued Cube

The words I used to describe these spaces, without any pictures, are perfectly good representations of the spaces, both for us and for flatlanders -- I could give a similar description in words (or equations) of a 4-D space defined by gluing specific boundaries of a hypercube together. I could tell you if it's orientable, what kind of loops exist in that that space, and many other things; but I have no way of drawing the space in all its 4-D glory.

For our 3-D space constructed by gluing opposite edges of a cube together, we have the picture above (which is a 2-D image of a transparent cube), and we could also take a block of wood and paint arrows on opposite faces to indicate the gluing. We would have to rotate the cube, or walk around it, to see the gluing pattern. This is how a flatlander would deal with our rectangle representation of the Möbius strip. A flatlander could also draw a 1-D image of the rectangle, by making the sides transparent. It wouldn't look like much to us, but it would make perfect sense to the flatlander.

Finally, a flatlander could paint a picture of what it would be like to live in a Möbius strip. Below is a picture of what it would look like to live in our 3-D space formed by gluing opposite sides of a cube:

enter image description here

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Well I create a 2D representation of a Möbius strip by taking a rectangle, and identifying opposite sides with the opposite orientations. I guess a Flatlander does just the same ....

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