| > 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. |