Cut Klein bottles and topological music


♪ ♪ As you can see, we’re back in Klein bottle land. Well, after the recent rush of klein bottle videos I think everybody knows what a Klein bottle is, right? And also pretty much everybody knows that if you cut a Klein bottle along its symmetry plane, then you get two Möbius strips. These guys. What it also means, of course, is that if you take two Möbius strips, and stitched them together along their boundary, you get a Klein bottle. So that’s one way of making up a Klein bottle. Umm, okay. Now It’s one thing to know that you can cut a Klein bottle like this in half; it’s a totally other one to actually do it and see one of those things, so I actually got a model for you here. This here is a Klein bottle — glass Klein bottle — cut in half. And here are the two different parts. And, if you really have a close look, you can see the Möbius strips. Okay? Okay, well it’s one thing to have a cut Klein bottle like this, it’s a totally different one to actually find a practical application for this, and I think we were actually the first ones to do this. And we incorporated that practical application in our first two Klein bottle videos, so if you haven’t seen them yet, check them out there. Watch the first one see whether you can see the uh cut Klein bottle. It’s a bit of a puzzle. That’s what the whole thing is really about. If you can’t, there’s a second video that explains everything. Alright. Now this was one way of cutting a Klein bottle, and one way of making up a Klein bottle. There’s another way of cutting the Klein bottle which I guarantee nobody here has seen, nobody in the whole wide world has seen. Because. My eleven-year-old invented it. [Laughter] So I was actually working on, uh, slides for the first Klein bottle video. And I was describing, how you usually make up a Klein bottle. So what you do is, and this is just a reminder, is you take a sheet of flexible material — rectangular — and you bring the opposite sides together, like that, and you get a cylinder. And then you have two circular holes, you just bring them together like this to get a torus. To get a Klein bottle you kind of go like that. Okay? So let’s just look at this again. So here we’ve got the first stage of this bringing around, the second stage, and then the third stage is the complete Klein bottle, okay? Now my eleven-year-old was sitting next to me, and he said: “stop. stop.” Before I actually went to the third stage, “stop. That looks like a trumpet to me.” You know? So we’ve got the mouth piece there, you can see the mouth piece. And, you can see the funnel. Right? So you can actually kind of imagine you can make a trumpet like this if you make it cut like that around the Klein bottle. And so, since he’s actually playing the trumpet I thought “well, I’ll make him a Klein bottle like this. A Klein bottle trumpet like this. And surprise him for his birthday.” And I actually did that. So what I did is I got two of those here off eBay Okay? And I cut one up, and recombined it into this thing here. Uh, and, so you can actually see, you know, there’s the flared bit, and it comes around, and it pierces through itself, and then the mouthpiece comes out like that, and then, you know, the third stage would just bring it — be bring it together. Okay, and now, let’s see Karl in action playing this thing, alright? Here you go. Here is the trumpet, (Klein bottle trumpet) [Laughter] and here is the trumpet master. [Laughter] Karl! Karl: so… Mathologer: so… actually did we forget something? Karl: oh, yeah. Mathologer: oh yeah. [Laughter] Wait. You really need these. Because otherwise your ears are going to fall off. Okay? Karl: yeah. Mathologer: okay. Ready to go? Karl: yeah. ♪ ♪ Karl: yeah! Mathologer: there we go! [Laughter] So maybe just to finish off this first part of the video, a little magic trick. Not bad, huh? [Laughter] Okay, but now since I’m talking about mathematical, musical instruments, I might as well show you something else. So this one I’ve actually — I think invented it — you know, a couple of years ago. Maybe 15 years ago? It’s something to do with me coming to Australia. So when I first came to Australia I was kind of hunting around for a nice, new, mind and motor skill to master, and I found a didgeridoo: an amazing Australian instrument. And, well, have a look at this. This is my knotted didgeridoo, and this is my other knotted didgeridoo. And I actually got some old footage that I want to show you of me actually performing on this guy here. This is a real didgeridoo. I’m going to show you what it sounds like. ♪ ♪ It’s really long, really heavy, and really expensive. Now let me show you the opposite. This is the knotted didgeridoo. You can play exactly the same thing: ♪ ♪ This is a lot of fun. Give it a go. So maybe just one homework assignment for you is: what sort of knot is this, and what sort of knot is this? I’m sure you’ve all watched the Numberphile videos on knots, so you should know. These are special knots. And just in case you’re wondering where I got my cut Klein bottle from, well I got it off Cliff Stoll a couple of years ago, and, well, he just describes it [how he does it] in the most recent Numberphile video. ♪ ♪