Eh, it’s charming but honestly, for someone who doesn’t already know the maths it’s still just edutainment, as it leaps from trivial to incomprehensible in the blink of an eye.
I don't think he's trying to make people understand the proof, rather to show them that topology really has an application for problems that aren't themselves topological in nature, and it is comprehensible enough for that purpose.
Well that's the "edu" part of edutainment. Sometimes you've gotta rewind or pause and think about what's being said to make sense of it. I do understand that sometimes videos go way too fast and leave tons of stuff out and that's very frustrating but 3b1b is a pillar of the community for very careful and complete descriptions of things. But then also the "tainment" part would signal that there's no need to watch if you're not interested.
But all this could be my bias of having some math background, though never having studied topology or even analysis from anything like a class or textbook. Felt like the video was aimed directly at people like me
I disagree, I'm not well-versed in math but I felt I could follow most of it.
What I don't get though is the jump from the mobius strip to the klein bottle.
He just goes and does it and duplicates the surface to reflect it to the original one. I do understand to some extent that once you have to assume the klein bottle is the shape you're looking for that because it's self intersecting, it must mean that you have 2 different points on that same surface and therefore 2 lines of equal length with the same midpoint.
The point of the jump is that if you want to track an extra coordinate and visualize it with the restrictions he mentions, then the klein bottle is the correct topology (the correct "visualization").
1. The positive surface is for tracking one midpoint for coordinates A and B
2. The negative surface is for tracking another midpoint for coordinates C and D
Together it's a klein bottle. Klein bottle's always intersect, so therefore there's always an intersection of the two midpoints, which is why there's a set of points A, B, C and D such that line segments A and B are equally long as C and D going through the same midpoint.
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.
The surface with the interior of the loop added forms something called a projective plane. A Klein bottle is just two projective planes glued together. Neither can be embedded in R^3 without intersections.
In many ways, I agree. I have an engineer's understanding of math for my discipline but topology is most definitely not one of them. Through his graphics, I could most follow the gist of what he was attempting to get across but when it was over, I honestly had to ask my self, what did I just watch. Perhaps watching it again, really concentrating on it, and trying to understand, might help, but, in reality, it is so far out of my interest zone, I'll never do it.
