# 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. ♪ ♪

I want to make a knotted didgeridoo now.

"if you really have a look you can see they're both mobius strips"

blocks the closeupcan you please tell me what the speed of smell is??

The knotted didgeridoos look like an Interment that my Fave band the Blue Man Group would use in there stage performances.

Awesome video! Were you inspired by numberfile?

Good stuff. I was amused by the handing of ear protection to your son before he played the "Klein Bugle".

Did you get the challenge i gave you?

Your son has picked an awesome instrument! I'm a little biased since I played it, but still – super cool. 🙂 Also – I may need to look up some videos on how the didgeridoo actually makes the noise it does. Such a unique sound. 🙂 And Klein bottles give me headaches – so do mobius strips. 😛 Awesome video!

Very interesting video! Side note: what is the name of the song at the end of your previous video?

Are those technically knots? Do the mouthpiece and bell not count as loose ends?

How much was that big Klein bottle?

Right on sync with the latest numberphile video!

That trumpet is absolutely amazing!

You might be interested in this: https://www.passdiy.com/pdf/KleinHorn.pdf

what if you create an instrument that is just like a trumpet or any wind intruments but you do not use botton or something but a rubik's cube that if turned change the sound

This video was rushed. The sound is terrible. Do you really have to follow numberphile's footsteps?

seems like you haven't quite mastered circular breathing 😉

they aren't really knots since the ends don't connect! 😉

if they did though, the white one would be a figure 8 knot and the gray one a trefoil knot

Just ordered 3 other klinebottles

I now have 6 🙂

But a half klinebottletrompet is still missing in my Collection

Oh and this arent knots becsuse they have lose ends

Mathematical knots cant have lose ends

I think Vi Hart would enjoy this. Have you been in touch with her? I would love to see what would come from a collaboration between the two of you.

I have nothing to say except I love your videos! Would love to see some more on topology!

Ring ! Ring ! Ring ! Hello there – Your 11 year old is H-I-R-E-D ! 👍

your channel ist very sympatic

I do enjoy the great fun that is the single sided thing I can hold between opposing fingers.

It's cool how you take a Numberphile video idea and shed light on a different aspect of it. I always like it when you do that. 😄

Are you and Numberphile a team/cooperative? Or is each an inspiration for the other?

This link is 15 pages of Klein bottle diagrams including some references.

http://www.cs.berkeley.edu/~sequin/PAPERS/2013_JMA_Klein-bottles.pdf

For some it may give some insight into the properties of the bottles.

I love the Klein bottle.

I like the raw PVC look of your knotted didgeridoo, but this Instructable gives another fun option for finishing it: http://www.instructables.com/id/Make-PVC-Look-Like-Wood/

So, how would you get 3.4 gallons of water in the die hard challenge?

Your son is a talented little guy, intuitive with shapes and music. Thumbs Up.

You can also make a didgeridoo the traditional way – just bury a stick in a termite mound.

Aboriginal technology is pretty easy to replicate with no skills or experience.

Father of the Year!

In principle it should be possible to determine the sound of a "perfect" (mathematical) trumpet simply via harmonics, right?

Two questions about that:

– How close would the mathematical model be to the real thing? – like, is the sound match actually decently similar or are there too many extra factors in the real world such that the sound becomes incomparable?

– Given that the sounds are comparable: How would a "perfect" mathematical

KleinTrumpet sound like? – That is, one that actually resides in 4 dimensions somehow. This should still be simulatable, I think…OH and Bonus Question: At least up to this point in these videos you always went with the more well-known Klein-Bottle which is actually bottle-shaped. What about the other one? The figure-eight Klein-bottle? Would it sound different from the one shown in this video? – That's both the "real-life" analogue and the actual mathematical one. Could the figure-eight one even be made into an actually playable trumpet?

The gray one looks like an overhand knot, and the white is a figure-eight knot

I think it's almost time for you to rename your channel, "Fun with Klein Bottles: My Favorite Thing in the World"

both of them are no knots cause knots only exist if it is A loop and not possible to bring into A circle shape

probably would need to use ear protection while playing it

edit: oh wait they do, just got to it.

Good job kiddo

I don't see any knots here… ends are not connected…

But those a not knots. By definition a knot is connected into a loop. These are topologically untyable.

The white one seems to be a figure-eight. If it is a figure-eight then I believe it would be a quatrefoil. That being said its got loose ends so technically no knot.

the expensive digeridoo sounds beautiful

To answer your question, they are not knots. Lol. They are open ended, so topologically, they aren't knots.

A trumpet that can play music directly into your own face. Truly a revolutionary development in mathematics.

Good call on the headphones. I was cringing from the moment the kline-trumpet came on screen until the headphones came out.

that's a horn mouth piece!:-)

Here's a connection between maths and music which I thought was quite interesting:

https://www.youtube.com/watch?v=xUHQ2ybTejU

We need a mobius piano

i was thinking about möbius bands and endet up making one out of paper . after that i took another stripe of paper and i craftet the band out of it glueing the both ends together without twisting . so i had a normal- and a möbius band in my hands and i thaught to myself … how many edges would the object have that would be created if i bound the edges together and how would such a thing look like. i cant think of the object that would be created by combineing those two bands . but i still want to have a answet to my questions .

As soon as I saw that design, I was like, "Earplugs? Please?" So happy you gave your kid those.

And then you brought out the didgeridoo. MATH. MY GOD, MATH.

what a cool kid, you must have a beautiful wife. ok, ok, its offtopic, sorry

Why did the chicken cross the Mobius strip? To get to the same side.

V.good trumpet playing!!! Like the concept of how it can be developed into a Klein bottle.

This is knot a didgeridoo, got it.

I'm a fairly new viewer, and I'm not even sure if you're going to see this comment, but I just wanted to thank you for being an awesome teacher and the world's coolest dad. Keep up the good work.

Interesting, the knotted didgeridoo! It produced far better overtones than the original.

How

knotto like your videos?do you have a video on topological musical analysis?