So something just arrived in the mail. I'm very excited about it. It's this huge humongous thing here some of you may be familiar with it - It's a Klein bottle. It's a humongous Klein bottle cost me a couple of thousand dollars Definitely worth it as you can see I'm pretty nuts about these things I've got like a pretty huge collection of Klein bottles I'll just put a couple here on the table. You know as many as could fit in this frame! Basically here, and so I want to talk a little bit about Klein bottles. Why I'm nuts about them? but also We want to do something. You know do something related to what we've been doing before so I've put Rubik's cubes in bottles and today the final thing I'm going to do is I'm going to put some Rubik's cubes in Klein bottles. It's going to be something all right, so I think before we start talking about these these various bottles, I better give you an explanation of what it's all about mathematically. Let me tell you a little bit about what makes these Klein bottle interesting, Mathematicians are really nuts about them, so what we do to build one, We just take a sheet of paper well not paper, some flexible material and do this, okay? So we'll bring the red edges together make them into a tube And then we connect up the yellow circles in a special way, okay? If you just bring them together like this we get a like a donut shape But we can do this in a second way and that gives the Klein bottle. Let's have a look, so we extend out like this and then extend and together comes the Klein bottle okay? Alright, so that's one way of building it the other way actually focuses on the yellow edges first Okay, so let's just go straight to that So what I'll do here is I'll take a rectangle that's a bit thinner to start with okay, and then the way the yellow edges are brought together is Well as there's just two different ways and so you could just do like this and you get a nice ring Or you could give it a twist and bring it together like this And that's of course something that pretty much everybody who watches this I guess knows This is a Mobius strip all right, and what's special about a Mobius strip? I mean you start out with two sides, and when you connect like this When you connect like this, white gets connected to white and yellow is connected to yellow and you get two sides again But when you kind of switch over like that you actually connect the two sides So you connect the white side to the yellow side the yellow side to the white side and actually now what you get here is a one-sided surface okay, so if you take two different points like one point here and Like on the opposite side you can actually travel from here to there by walking along the surface, okay? So just walking along the surface So that's that's something very special a one-sided surface one sided surface Okay, now the Klein bottle is actually just a mobius strip, extended in such a way that eventually the the two edges come together and merge together Okay, let's just have a look at how that works so first we connect this up into a Mobius strip in a special way So what we have to do is We have to connect The yellow point to the green point and the green point to the yellow point, and we do this in a special way here So with the twist is kind of hidden a little bit okay, so goes like that just extend it out Bring it around and now you can see if I just extend this out those two points are going to get Merged those two points going to get merge and you get a Mobius strip okay, and now let's just go into our real starting position Which is basically this so there's your Mobius strip, okay? And now we kind of extend things out, So we extend things out we extend further we extend further we extend further and if you just extend a little bit further It's all going to come together and you're going to get your Klein bottle What's nice about the Klein bottle is that doesn't have any edges whatsoever? It's like a sphere basically It's like a sphere doesn't have any edges It's a closed surface as opposed to one with an edge the mobius strip. So that makes it somewhat special, okay? Okay, now Klein bottles they actually don't live in our world. They really live in four dimensions, and they're really I mean these sorts of guys they really feel uncomfortable here They're really kind of crammed into three dimensions, and you can see it here kind of they're kind of clash into it themselves They don't want to do this, they want to be like like donut shapes. Donut shapes, they don't clash into themselves They want to be like that and in four dimensions. They can be like this. I just want to show you How you actually get the fourth dimension in here? So if you've got a figure eight living in a plane, all right? Now there's got a clash here If you want to get rid of this clash what you do is you kind of just take a little bit of this This figure eight here and kind of raise It slowly up into the third dimension and bring it back in like we've kind of built a bridge and all of a sudden the whole thing doesn't clash into itself you can do the same [sort] of Thing here with the Klein bottle so what you do is at this stage here You kind of push things off slowly into the fourth dimension And you kind of just go across like in the fourth dimension kind of building a bridge in the fourth dimension and then well from our perspective it just seems to vanish here right From the perspective of somebody who lives in here the curves just seems to vanish and then to reappear, but it's still there It's still there in three-dimensional space same sort of thing is possible here for a Klein bottle So here it just kind of starts vanishing But it's actually still there in the fourth dimension goes across and you don't get this running into itself So this is really a four dimensional creature, and it's a one sided surface And there's a couple of other things that are really amazing about it. So one of them Which not just mathematicians find amazing, but just about anybody finds amazing is that they provide You know a hint that the universe we live in might actually have a very strange property and I just want to show you this By looking at a mobius strip real quick. So here's a really really big mobius strip okay, and Let's say there is a creature living in here So there's like our two-dimensional man lives in here, and that mobius strip that it doesn't have any third dimension it's just that it's there. But it doesn't have any third dimension. So what we can do is this man Here is we can let him travel around the Mobius strip, okay? so let's just let him travel around the Mobius strip and now he kind of just turns on the spot and Now something amazing has happened but just walking around a mobius strip this man has actually turned into his mirror image, so Something similar is theoretically possible in our universe that we kind of go off for a trip through the universe We come back and we come back mirror reversed It could theoretically Happen, and it could happen in a kind of a three-dimensional counterpart of of the Klein bottle so that's what makes Klein bottles interesting For new Mathematicians and also you know I hope Everybody watches here. It gives you basically a space a Flat analog of of the three-dimensional universe we live in That has this special property that if you kind of just walk around in the right way You kind of turn into your mirror image, and you know you won't feel it You won't feel it that you know you'll feel perfectly fine all the way throughout the trip, but you know I come back it's strange So let's get serious about putting some Rubik's Cube in Klein bottles and talking just in general a little bit about about my collection here, so What have I got here, I've got Different sizes, so this is kind of just straightforward shape basically four different sizes here Well really five if I've got all of these really small ones here, so there's another one So five different sizes you can use them for in earrings or pendants or whatever now. They're all made from glass by real glass blowers Pretty much all of these come from a place in California I'm going to put a link somewhere here So they're all coming from California. I've got one here this one came from the UK so, They actually attached something here, so they can use the Klein bottle as a Drinking vessel and then I've got something very strange here three Klein bottles kind of merge together like this very strange Then I've got a Klein bottle opener so you can actually open beer bottles like this Which is actually 3D printed. And it's in metal and there's another 3D printed version just from plastic, and this is something that I use on a daily basis So this is my Klein bottle mug So if you actually have a really really close look This is actually a real mug, and you can see that water's kind of sitting in here in a strange way And that's basically you know the talos space. Get some water in every once in a while, okay now how do we get a Rubik's Cube in a Klein bottle Okay, so we want to get maybe this one here into this Klein bottle I mean from the movie you maybe remember well this thing actually only has one side, right? So it actually doesn't have an inside and an outside Mathematically speaking, so pretty much you could just say well, It but nobody will be impressed also. It's not really true. It's not really true because while we're talking about a Klein bottle kind of squeezed into our 3D space It would be true if we had what kind of you know doing this video in four dimensions Then I mean there would really be no point to do anything with this thing You know just be inside whichever way you look at it, right? But because it's kind of squeezed into three dimensions. You can actually see well. There is really an inside here somehow, right? There's really an inside so how do we get it into this space inside it looks hard and actually it would be completely impossible Were it not for a strange flaw Okay, let's let's be kind and call it feature of all these glass Klein bottles, okay? So when you look at them And when you remember so there's basically a tube going through to the inside here And then there's a surface kind of cutting across here, so I mean here we've got this kind of circular intersection so a circular intersection and really What should be there it should two surfaces in two intersecting right, so this surface And that surface here there should be intersecting and actually that hole should be closed But it isn't you can actually get in through the tunnel here and move things to the inside It's a floor. I guess. It's just very hard for for glassblower to get any sheet of of glass in here So that's actually the case for all of these all of these Handmade glass bottles and strangely enough it's also the case for this plastic one. I mean for this plastic one There's actually no excuse for not putting anything But I guess whoever made this he just looked at the glass Klein bottle copied it and didn't realize that He was actually making a mistake there now the person who made this one here very famous 3d printing artist, Bathsheba Grossman She got it right, so if you actually look in the hole here So there's the surface cutting across so among all those things that I have here. It's really this is really the only one That's you know perfect right in that sense, but I mean we're not really worried about this We're actually pretty happy about this because at least there's a way in you know, so otherwise yeah, there's no hope Okay, so how do we get something in? Well was this large one it's actually pretty easy, so I got to just kind of take it like this and I get a small cube, maybe my favorite cube the 1x1x1, and I just kind of pop it in alright, and it's in Nobody will be impressed, well I think you'd still be impressed, by me holding this thing So anyway, so you can get a cube in like this, but of course you want to want a real 3x3x3 okay, well that's that there's still a really really tricky way of doing it because I've got this really really small 3x3x3 So that's actually you know that's actually the smallest mass-produced 3x3x3 cube. That's just as big as One of the little cubes that make up an ordinary one and you can really you know twist the sides of this thing and can you scramble it up And you can solve it and do all sorts of things. It's called a Nano Cube So this one let's put it in probably a world first There, a 3x3x3 in a Klein bottle just like that Okay, but again I mean you've never seen this before and I'm sure it's going to be very memorable you'll never forget this Image of me throwing this thing in but still you know it's too easy. I mean, I could have done it I mean this this one here You only really impressed because I got this thing in that doesn't really fit through the hole alright So we have to somehow get this a cube like this into one of those Klein bottles Now I'm not going to use this one here because I mean if you've actually watched the video where I put this one in you know It's dangerous, sometimes I destroy these things in the process right, so I'm not going to use the several thousand dollar Sample here of which only like two exists in the whole wide world So I'm going to use one of the other ones right, so I've got quite a few of the other ones and so What I thought I'd try is, I'm going to try and put Something into this one here Okay, so this one here and well I mean this this 3x3x3, definitely doesn't fit in all right. So can't can't fit that one inside and even larger one would be this 2x2x2 right so that that would I think just just just fit inside and Well, I guess we could do another world first since I'm here we could put a Klein bottle inside a Klein bottle since we've got It here. We might as well do it I've got this little Klein bottle here Okay, so we're going to put a Klein bottle inside a Klein bottle, there it is, never seen before, world-first All right, okay. Let's leave this in there. I think I might just leave this in there permanently and Now I'm going to go away and then And then I'm going to see what I can get this thing inside, okay? alright. I'm back. I'm wearing my Klein bottle hat Which you can get also for the place where we get these glass Klein bottles from so it's a very neat hat So if you go inside, you pull You can actually see the two kind of sliding inside so very nice anyway, so I succeeded as you can see The Rubik's Cube is in alright. The Rubik's Cube is in the Klein bottle and let me just show you a little bit, so Here it is right. I just kind of move it around a little bit here Wow, okay, anyway That's as far as we'll go. It's actually pretty tight fit. This way, so you know great stuff?
If the universe were shaped like this, sure. We have no current reason to believe that it is though.
That guy really likes his bottles
how did he get the cube in there?
Wouldn't the 2D figure remain unchanged if he actually walked it through the entire Mobius strip? He only goes through half of it and then brings it back to the beginning.
As he said, this only holds in four dimensions. The bottles he has are 3 dimensional models, the necks physically intersect the bodies. The neck of a real Klein bottle would go from the outside to the inside without actually touching the bottle. If you take a Mobius strip and flatten it into 2 dimensions, all its fun properties disappear and it just becomes a ring. A 2 dimensional being moving in the surface would just go through the fold and come to the original position unchanged. Similarly, blowing a giant Klein bottle model and walking into it, at the part where you'd "flip" through 4 dimensions you'd just walk through the hole.
It's easier to think about with Flatland. A 3-D creature could turn a 2-D creature left-handed by using the 3rd dimension to flip them. The analogy holds if there's a 4th dimension in which we could be flipped.
I personally think this is why the time travellers in "Primer" couldn't write properly. I imagine it would be horribly confusing to suddenly become your mirror image.