If we take a 2D projection of Möbius strip's edge, and then create an object by rotating it around the indicated axis in 3D for 180 degrees until it connects with itself, do we get a type of Klein bottle?
I apologise for the asymmetrical image. :)
2026-03-25 16:03:53.1774454633
Turning 2D projection of edge of Möbius strip into Klein bottle
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Nope!
First off, when you create the planar diagram, you are forgetting that you started with a nonorientable surface (with boundary), the Mobius band. You could have just started out by asking what happens when we rotate that given planar diagram without any reference to where it came from. This way we are not in any way prejudiced by our knowledge of the Mobius band.
Secondly, when you rotate the diagram, that point of intersection will become a problematic point on the resulting figure. The resulting figure will not technically be a surface, because that figure, locally around that point, does not look like a disk. (A surface is a $2$-manifold, which means that every nonboundary point on it has a neighborhood around it one can draw which is homeomorphic to a disk.) The resulting figure however will be the image of a continuous map taking a certain surface into three-dimensional space. What is this surface?
Think about the outer curve rotating to form an outer sphere, and the inner curve rotating to form an inner sphere. One can go between the outer and the inner sphere by traveling along the curve itself, or any of its rotations (which are cross-sections of the resulting figure). One can think of it as a sphere with an ingrown sphere, or something like that.
Make a cut around the top of the inner sphere that goes counterclockwise (from our outside, top-down-looking perspective). Then you can take the inner sphere totally outside of the situation (think of it as a ghost that can pass through things). The inner sphere will actually be a sphere with a hole cut counterclockwise at the top. Then the remains of the outer sphere one can stretch out the top until you get a normal-looking sphere also with a hole cut at the top, but running in a clockwise direction now. Take the remains of the inner sphere, turn it upside down so that its hole cut runs clockwise too, and you can connect the two components back together again.
You get a plain old sphere!
This "cutting and pasting" process for surfaces is perfectly valid, and as long as you retain information on how the cuts were made (i.e. which direction), you will ultimately be preserving the surface. Whenever you make a cut, you can think of the two resulting edges as orange and blue line segments or circles with arrows. Why orange and blue? Well, have you ever played portal? It's the same principle (except with beings living on a two-dimensional surface, the portals themselves are one-dimensional line segments and circles).
Indeed, the classification of (compact) surfaces relies on this procedure. One can triangulate any compact surface (a highly nontrivial fact), cut the triangles out, then put them back together again to form a rectangle with "portal" edges, and then perform an algorithm involving cuts, pastes and simplifications in order to achieve a "normal form" fundamental polygon that is of a certain type prescribed by the classification.
Alternatively, Conway's "zipper" proof (as I understand it) basically says if we take the surfaces with boundary prescribed by the classification and draw oriented colored line segments on the boundary circles (two pairs of each), then this class of "surfaces with zippers" is closed under the operation of zipping pairs of same-colored edges up as their orientations dictate. Since every surface is equivalent to a polygon with zippers, this means every surface is covered by this classification.