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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: 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
up vote 1 down vote accepted

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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One approach to visualizing a Möbius strip in Flatland, arguably more in the spirit of Cliff Stoll's glass Klein bottles, is to project an embedded Möbius strip from Spaceland:

Moebius strip in Flatland

The result is a "solid model" (to a Flatlander) rather than a "shell" (as a Klein bottle appears to us Spacelanders) because both the Möbius strip and Klein bottle are intrinsically two-dimensional.

A Flatlander might find the "shell" of a Möbius strip, i.e., the projection of the boundary (blue), easier to apprehend on its own. The point where the boundary crosses itself is analogous to the circle on a Spaceland model of a Klein bottle where the surface "passes through" itself. (As Neal noted shortly after the question was posted, the boundary curve is immersed in Flatland, just as a Klein bottle can be immersed in Spaceland.)

Mathematical expositors in Flatland would explain that, contrary to common sense and the incontrovertible evidence of one's eyes:

  • The Möbius strip doesn't actually collapse at one point, and the boundary doesn't pass through itself, but (in a way some Flatlanders would doubtless find mystical) passes "over" itself in the Third Dimension. (You just have to imagine "lifting one curve segment into the Third Dimension", and the crossing immediately vanishes.)

  • The boundary of a "real" Möbius strip in Spaceland encloses no "inside" or "outside"; points of the "solid" interior portion of the strip are accessible to Spacelanders who approach "from the Third Dimension".

  • The "real" Möbius strip has no "cavity" in the middle, i.e., does not separate Spaceland into two disjoint regions.

Ordinary Flatlanders would accept these "facts" with mild bemusement, gently rolling (or, rather, rotating) their eyes at those mathematicians: Any child can see the boundary of the Möbius strip crosses itself, and there's a cavity in the middle.

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