You're watching a Mathologer video and that that probably means you're eating Klein bottles and Mobius strips for breakfast and you know that these tasty mathematical surfaces have just one side. Except, and only real mathematical connoisseur seem to know this, they are Klein bottles and Mobius strips that have two sides. Let me explain. Quick revision: this strip of paper has two edges and two sides. To make it into a Mobius strip what I have to do is to bring the ends together such at the edges combine into one long edge. Every Mobius strip has just one edge but as you can see something else happens here. As I bring the ends together also the two sides combine into just one side, so this is a Mobius strip that has one side. Now there are actually a couple of different ways to bring the ends together by twisting them to make this strip into a Mobius strip. You can just do one twist and glue, that gives you a mobius strip, three twists or five twists or any odd number of clockwise or counterclockwise twist that will yield a Mobius strip. Now all these Mobius strips have just one side. If you do an even number of twists and then glue you get one of these surfaces. They all have two edges and two sides. Now these are not Mobius strips these are called topological cylinders or just cylinders. Now I claim there are ways to bring together the ends into Möbius strips that are two-sided. Hard to imagine how is that miracle possible? Well, it turns out that the number of sides of a Mobius strip or actually of any 2d surface depends on which 3d space which 3d universe it is contained in and how exactly it is contained in this 3d universe. Now most people think that 3d just means xyz space what you deal with in school or at university. But there are actually infinitely many 3d universes, mathematical 3d universes and we actually don't know which one of these mathematical universs describes the universe that we live in. Now quite a few of you will actually have heard that our universe may be a pacman universe which means that there may be a direction, special direction. If i head off in this direction and just keep going straight I'll get back to where I started from. So let's just assume we are inside a mathematical universe that has this property and let me introduce you to my math cat maskot the QED cat. Well, actually, there's a bit of dispute at home whether it is a cat or chihuahua but no matter let's just launch it in this special direction on its space surfboard and see what happens. So as the cat travels along the surf board actually generates a strip. Ok now keep going, keep going. Eventually it gets back to where it started from, there. And now it wants to turn the strip into Mobius strip and you can see that to create this one long edge what we have to do is have to kind of flip upside down and keep moving forward. Ok, so we're creating a twist like this and you can see the Mobius strip that we've actually created here is just one of those one-sided Mobius strips, so nothing new here. Now, in a more fancy universe something else can happen, so let's just do this again. So QED cat heads off again, we leave one of those ghost images behind. It gets back to its starting position but now something else has happened, it's actually turned into its mirror image. That's unusual and now you see to create this one long edge, to create the Mobius strip the cat has to just keep on going, it doesn't have have to do any of those upside-down acrobatics. So just like this and we've created a Mobius strip and obviously that Mobius strip is two-sided. So if you put a anti-cat on the other side and have QED and the anti-cat run around on those two sides they'll never meet. Once you've got one of those strange mirror reversing trips all sorts of other nice things start happening so for example the cat gets back to the starting position, it's mirror reversed and it wants to eat some of its cat food but actually that's no longer possible because the mirror reversing happens at a molecular level and so the food and whatever processes the food in the stomach won't match anymore, won't happen. So to unscramble itself the cat actually has to either backtrack or just do a second round and then it can eat. Also once you have a two-sided Mobius strip like this you can extend it into something solid and this solid corridor is actually a real 3d counterpart of a 2d Mobius strip. A lot of you may have wondered whether something like this exists, well there you go. Finally what I want to show you is how you can use a mirror reversing path like this to create a one-sided cylinder. Now, usually, cylinders are two-sided, right? So now let's just look at this situation again, we can now turn this strip here into a cylinder by just doing a twist, a twist like this creates two edges, we're dealing with a cylinder but as you can see this is a one-sided surface. So at this point what you really want me to show you is one of those mirror versing path in our real universe or at least in a mathematical universe that i can hold in front of you. But that's actually very hard to do because no matter what you doing in xyz-space round-trip wise you'll never mirror reverse yourself. That also means that we cannot have a copy of one of those reversing universes inside xyz- space, makes it hard to describe. But what I can do is I can show you the analog of one of those one-sided cylinders, a 2d analog and for that I need a flat cat. Now what's a counterpart of a cylinder in a 2d world, it's just a circle. So what I want to show you is one-sided circle. Now, usually, circles are two-sided right. Two-sided circle with respect to this 2d world that the cat is living in. Now instead of off using this off-the-shelf 2d universe I use a Mobius strip universe, that's a 2d universe. The cat's living inside it, the circle is part of this universe and I'm going to chase to cat around it but what's really important here is actually too emphasize that a real mathematical surface has zero thickness just like the xy-plane inside xyz-space has zero thickness. So that Mobius strip has zero thickness, the cat sliding around in it has zero thickness. Let's just see what happens when it runs around the circle. Ok so it's completed its roundtrip and as you can see with respect to this 2d universe its living in it's actually mirror-reversed itself and it seems to be locally on the other side of the circle but, as you can see, when we do a second trip around it actually gets back to the beginning and what this means is that this circle here has just one side. On the other hand, if I take away the Mobius strip and surround this circle by this ring here, then the circle is actually a two-sided circle. So what that also tells you is that without the 2d context it actually doesn't make any sense to ask how many sides this circle has. Just kind of floating with in 3d space it doesn't make any sense to ask how many sizes this thing has and similarly if you've got a surface you need a 3d context to be able to ask and to answer how many sides one of these surfaces has. Otherwise it just doesn't work. For example, we could put something like this in four-dimensional space and just have it floating there, it doesn't make any sense to ask how many sides one of those things has. Now I also promised you some 2-sided Klein bottles. How do we get those? Have a look at this. So QED this flat so it can't really see a Mobius strip but it wants to play with it anyway, so it can do this a la pac-man. It's not ideal but it's good enough to visualize what's going on. Ok, so what you do is you just kind of draw a flat rectangle and QED can run around in there and then you just indicate how the ends are going to be glued together with arrows like this so. The arrows here basically tell you that these two points get glued together and these two points get glued together and so on and now a Klein bottle is actually just a Mobius strip whose edge has been route to itself in a certain way and that certain way I can actually show you very easily also with arrows, goes like that. So we have to do is, we have to glue these two points together, we have to glue these two points together, and so on and that will give you a Klein bottle and well since we have 3d beings I can actually show you this construction in space. So here I've got a Mobius strip. Now I'm just going to bring corresponding points of the edge together, like this, and there you've got your Klein bottle. Now obviously once you've found one of those mirror reversing paths and a two-sided Mobius strip it's pretty easy to imagine that we might be able to extend this strip into a two-sided Klein bottle and this is exactly what happens. All right, now we've got two pictures of a Klein bottle here and just like QED can use the square to describe a Klein bottle we can use a solid cube to describe a solid counterpart of a Klein bottle, so basically a solid Klein bottle and this is also done by these fancy arrows. What the fancy arrows show you is how opposite faces of the solid cube are supposed to be glued together. For example, these two points get glued together, these two points, those two points, and so on, should be pretty obvious and actually this solid Klein bottle there is one of those mirror universes and if you have a really close look you can see that this here is a two-sided Klein bottle within this 3d mirror universe, very very fancy, very, very pretty. It's an absolutely beautiful Klein bottle much nicer than the one that I showed you before. The one I showed you before has this strange sort of self-intersection which is really annoying. This one doesn't have any of this so so much much nicer in this respect. Ok, now I learned about all the stuff for the first time from this book here, The Shape of Space by Jeffrey Weeks. This is an amazing accessible introduction to two- and three-dimensional universes manifolds. I really recommend it to everybody here. Jeff's also created some amazing pieces of software, totally free that you can download from the website I'll link in from the description and they allow you to you play chess on Klein bottles on tori, play pool, all kinds of other things but he also has pieces of software that allow you to fly around in strange 3d universes. So, for example, here's a view of a very small version of this solid Klein bottle universe which just basically has space for one Earth and as you kind of look around because of the way it kind of connects up to itself you can actually see yourself, see Earth over and over, not only Earth but also the mirror image of Earth and you know I leave it to you to kinda figure out how exactly this works how how exactly the pattern of Earths and mirror Earths comes about. Now there's a lot more to be said about all this, e.g., four dimensional stuff. I may say it's a little bit about this in the description. Also I'll definitely come back to these strange 3d universes, make another video about that but for the moment I just like to say thank you very much for all your support throughout 2016 and Happy New Year to all of you and I'll see you again soon.
Don't know whether you guys are familiar with it but in my opinion Jeff Weeks book is hands down the most accessible introduction to as complicated a topic as 3d manifolds, absolutely brilliant. Have not watched the video yet, but it's clear from the title that it's about the very surprising fact that the sidedness of a surface is not an intrinsic property of the surface. Still remember being blown away by the chapter on two-sided Klein bottles, etc.
woah this is really interesting!