Similar question here, but it doesn't quite address what I'm hoping to have illuminated.
I've recently started watching a series of lectures on topology. The material I've gone through thus far is mostly presented geometrically. Frankly, I'm not sure that's the technically correct term, so to clarify, I mean "not very rigorous" and "lots of drawings are used". Therefore, I'd like any answers or interpretations to be presented in a similar light, in - dare I say - layman's terms.
In this video (between 0:34 and 11:50), prof. Tokieda outlines the process of
- considering a sphere
- drilling a hole into it by cutting a disk from its boundary
- filling in that void with a Möbius strip and connecting it to the rest of the sphere
which is permitted since the boundary of a disk and the boundary of a Möbius strip are homeomorphic (to circles, as I understand it), thus putting a "lid" on the sphere, and calling it the (real) projective plane, $\mathbb RP^2$. This manifold is then referred to by a symbol that strongly resembles the Death Star.
In the question linked above, one of the comments directed me to this interesting animation that seems to project a point onto $\mathbb RP^2$, and tracks its position along $\mathbb RP^2$, but the surface itself is split into a disk that is homeomorphic to the drilled sphere and the Möbius lid.
What I would like to do is somehow be able to visualize this completed "Death Star" in the same way I can visualize the Klein bottle as a self-intersecting surface in 3 dimensions, like in this image. Is this possible?
Inspired by the first 3 or 4 videos in the playlist, I embarked on a brief investigation into making this "Death Star" out of paper. Here's what I attempted, step-by-step:
- apply a twist to strip 1 and tape its ends together; leave strip 2 alone
- starting at an arbitrary point along strip 1, begin lining up one of the ends of strip 2 with the boundary of strip 1
- after a few moments of this, the boundary of strip 1 exceeds the length of the boundary of strip 2, so I "deform" strip 2 by cutting another strip of paper with the same dimensions, then taping it to the end of strip 2 and effectively doubling its length
- continue until the free end of strip 2 can be taped to the end taped to strip 1 at the start
If this process is unclear, I'll be happy to upload a sequence of pictures I took at various points showing its progress.
At the end of this admittedly frustrating process, I end up with strip 3, what appears to just be a wider (by a factor of $3$) Möbius strip than strip 1 was. If it were any wider, I'm certain the paper would tear immediately, which makes me think completing the process is impossible or requires a material much stronger than cheap notebook paper. (Unfortunately, I'm all out of infinitely pliable, self-permeable rubber.)
Is there a way to visualize this complete surface?