The "positive" surface already contains all the necessary points. It's hard to prove that this surface on its own intersects with itself, but turning it into a Klein bottle makes the proof easy, since it's already known that the Klein bottle must intersect with itself when embedded in 3-D space.
It takes some rigor to ensure that mirroring the surface and turning it into a Klein bottle doesn't introduce a problem that would invalidate the proof, but the idea is this:
1) The surface exists only in the "positive" area above the x-y plane, and the mirror exists only in the "negative" area below the x-y plane.
2) The two surfaces only share the points on the original curve (on the x-y plane), and these points correspond only to the trivial cases where A=B. The surface and its mirror don't intersect anywhere else.
3) The resulting combined surface is a Klein bottle in 3-D space, which must intersect somewhere. Because of 2), that intersection must either be in the positive space or the negative space. Either way, that means there is an intersection in the original surface.
As briefly mentioned in the video, it's critical that the original constructed surface is only in the positive area, because otherwise when you mirror it and then turn it into a Klein bottle, the required intersection might just be the surface intersecting with the mirror, and not within the original surface itself.
> The "positive" surface already contains all the necessary points.
If this is the case then why are you allowed to duplicate the surface again on the "negative" plane? To me that gives the idea that you duplicate the whole original curve. He didn't motivate that step.
Or oh wait. I think I see it, a bit.
It's not so much duplication it's simply that by doing a negative visualization, you visualize it in a different way. But also in a way that relates to each other as the x,z plane are the same and the vertical (y) planes are inverted.
I guess one could say that the positive plane is for one midpoint and the negative plane for the other midpoint as the surface areas on those planes are a visualization of midpoints.
And then when you turn that into a klein bottle, you show how both midpoints relate.
If my understanding is correct enough, then I have to say, this is wild.
Thanks for explaining! This is really cool. Like I said, I'm not well-versed in math. To even have an understanding of following the main beats of it is really mindbending as most of it is new.
It takes some rigor to ensure that mirroring the surface and turning it into a Klein bottle doesn't introduce a problem that would invalidate the proof, but the idea is this:
1) The surface exists only in the "positive" area above the x-y plane, and the mirror exists only in the "negative" area below the x-y plane.
2) The two surfaces only share the points on the original curve (on the x-y plane), and these points correspond only to the trivial cases where A=B. The surface and its mirror don't intersect anywhere else.
3) The resulting combined surface is a Klein bottle in 3-D space, which must intersect somewhere. Because of 2), that intersection must either be in the positive space or the negative space. Either way, that means there is an intersection in the original surface.
As briefly mentioned in the video, it's critical that the original constructed surface is only in the positive area, because otherwise when you mirror it and then turn it into a Klein bottle, the required intersection might just be the surface intersecting with the mirror, and not within the original surface itself.