Welcome to another Mathologer video. Today I'll start by showing you something absolutely amazing, a really nice animation by mathematician Arnaud Cheritat of an ingenious way to turn a torus inside out. Let me explain. A torus is just a surface of a donut. It has been painted green on the outside and red on the insert. It can stretch, it can contract, it can even pass through itself, just like a ghost. What's important is that all the way throughout the deformation that you'll be seeing the surface is always completely smooth. There are no holes, there are no creases, nothing terrible happens to the surface. Now we've just arrived at a critical stage of this inside-out deformation. The torus is just folded in on itself. Now we're going to take this part which is overlapping and turn it. Ok and you see some of the inside starting to poke out and there's more and more red here and at this stage what you see is half red half green and the rest of the deformation is really running backwards what we've just seen just on the other side of the circle. And there the two kinks and annihilate. Et voilà, at this point in time the torus takes a bow and I hope you as impressed as I was when I first saw this thing. So the technical term for what you've just seen is a torus eversion. This stage I want to have a really close look at. So here I've redrawn the cross-section just after the critical move and does anything here look familiar? Well, you should see two Klein bottles. So what we're really seeing here is a double Klein bottle. If we deform this a little bit more we can actually arrange this in a really nice symmetric way. So a double Klein bottle is a torus in some way. Very strange. Even stranger when you think about that Klein bottles don't really have insides and outsides but somehow when you stick them together in a double Klein bottle they become a surface that has two sides, the torus. Anyway because of all this I'll call our torus eversion "The Double Klein Bottle Trick" and I will use the Double Klein Bottle Trick to do all sorts of ghost gymnastics in the following. Now I've got lots lots of Klein bottles, I even have a triple Klein bottle but actually didn't have a double Klein bottle, so I just ordered one from Cliff Stoll who a lot of you will know from Numberphile and actually Cliff assures me I'm his absolute best customer and in fact at the moment he says you have more Klein bottles, different types of Klein bottles, than me because I just ran out of a couple. So that's something new :) Ok so a torus ghost can turn inside out smoothly. What about a sphere ghost? Can that turn inside out? I should really have done this video on Halloween but somehow I ran out of time. Okay, so a straightforward idea is you just take the North Pole, the South Pole and push them through each other and just keep on going like this but at this stage here you hit a crease and that's not something we want. We don't want creases, we don't want any holes, smooth throughout, right? And, if you try and try and try you actually never really find anything that works straightaway. So is it actually impossible? Well let's just look at something that's a little bit easier, let's just see whether we can turn a circle inside out. So here's a circle colored green and red and try and try and try and try and you actually find that this is completely absolutely impossible and it is actually fairly easy to see why it is impossible. So for that what I do is I put a race car on the circle and actually use it as a racetrack. The left side of the car is always going to be on the green side of the circle. Ok now chase it around. What we're really interested in is what the race car does relative to its center so I'll just highlight this. So it starts out like this and then it does one clockwise turn Now we're starting to deform the circle and for any intermediate stage there we check out when we chase it around what sort of turning does the car do. And it turns out it always does one clockwise turn. So for example here one clockwise turn. That's even true for really complicated tracks like this. So here you might turn clockwise sometimes anti-clockwise sometimes but overall you get one clockwise twist and again it's very easy to see why this would be the case. Maybe if you're unsure discuss this in the comments. Just one hint: the car always starts and ends pointing in one direction and everything kind of changes continuously. OK, now let's have a look at the inside-out turned circle. Remember the racecar always has to have the left side on the green side of the circle. So to make this work here we have to turn it around. Now if you chase the race car around what you see is well we still do one turn but we do it in the counterclockwise direction and so that shows that it's impossible to turn a circle inside out. You know, starting with a circle you can deform, deform, deform but at all stages we basically get one clockwise turn, we can't just jump all of a sudden to a counterclockwise turn, that's impossible. Alright sphere, obviously more complicated and you certainly would think it's not possible, too, but actually in 1958 a very high-powered mathematician Stevens Smale showed that it's possible to turn the sphere inside out and this really counterintuitive result is actually now refered to as Smale's paradox. What may seem even weirder to a non- mathematician audience is that the proof didn't contain any pictures and didn't give any clue whatsoever as to how you would actually do this in practice. Now subsequently quite a few ways have been found and really people have been trying to get really good ones happening for 50 years and here's four different ones that are all very nice and all very ingenious but you can see they're all pretty damn complicated. I actually stop them here at halfway stages and just to wrap your minds around these halfway stages is a killer. The original clips are linked in from the description so check them out especially that one up there. It's called Outside in. It's actually a half-hour documentary on all this. It's absolutely fantastic. So here are the second halves of these eversions. Now this one down here is actually buy Arnaud Cheritat the person who animated the torus. So I had a really really really close look at pretty much everything that had ever been written about this while researching for this video and I stumbled across something that I had never seen before, a really original idea for turning a sphere inside out based on the double Klein bottle trick and this is by mathematician Derek Hacon and he never published it and maybe only a handful of people know about this. So my main mission today is to actually let the world know about this amazing idea, you know, this is the way to go. So what does he do? Well he takes the sphere and he pushes in the bottom part. Now let's look what this thing looks like inside. Take off the dome here, another dome sticking up inside. Take that off too and then you see that the bottom part of our shape is actually just half the torus that we looked at before and it's bounded by these two circles and the domes are attached to the circles. Now the idea is to just unleash the double Klein bottle trick on this and have the domes follow the movement of the two circles at the bottom and while they're doing this the contortions of the circles sort of die out towards the top. I just want to show you what that looks like for the dome in the middle. So there's the dome in the middle attached to the circle. At some point in time the circle is folded up like this, so the dome would look roughly like that, so it's got a flap and the flap dies out towards the top of the dome. Later on we've got this double kink. The dome would look like this, roughly, dying out towards the top and stays like this all the time. Now we started with the circle on the inside. At this stage, at the halfway stage the circle is already half in half out and when it now completes the eversion the circle has moved all the way to the outside and so now this dome here is on the outside and of course if everything goes well then the other dome has moved to the inside. And if you now take out the bulge we've actually completed our eversion. Isn't that neat? It's really really nice and actually it sounds a little bit too good to be true and there's a complication but just to emphasize, the main work that's done in this eversion is really done by the double Klein bottle trick. Now the complication comes about when you look at what the outer dome does. The inner dome deforms in a very tame way. So here, for example, very tame, very tame. Let's go back to this folded stage and look at what happens to the outer dome here. It's also completely tame but now after the critical move you see these two twirls appearing and it's a bit harder to imagine what the dome now has to do. If you're fixing your own bikes whenever you're manipulating a chain it happens that you get these two twirls appearing and usually it's a bit of a pain to get rid of them. Now they are a pain here but actually in this sphere eversion they are very, very useful. OK now the twirls, what do they do? So they travel around the torus and then they annihilate on the other side just like a particle antiparticle pair. Really neat, right? So I'll provide a link to a discussion of how exactly the outer dome deforms in the description, so check that out if you're interested. Anyway, this version is my absolute favorite from now on. There's a funny or sad story that goes with this. Now, everybody who knows anything about sphere eversions knows it's very hard to visualize them. So Derek Hacon was obviously very excited when he found his way of turning the sphere inside out and he submitted it to a popular maths magazine and ... they rejected it because they thought it was too trivial. How silly can you get. Derek Hacon actually died a couple of years ago so we can't ask him which magazine he submitted it to but hopefully the people who rejected it will see this video and will kick themselves now. Here's another ghost, a double torus ghost and you can also use the double Klein bottle trick to turn that one inside out. That's also very neat so I'll just show you. So let's just focus on half of it, the other half you have to imagine it's still there, just to show you the inside and the outside. Now we shrink one half and unleash the double Klein bottle trick on the big one. The little one is just going to be carried around like a fly sitting on the big one here. So unleash it on the big one and at the end of it the fly will sit on the inside. Now to complete the eversion what you do is you just reach in through the hole and then pull out and the eversion is finished and I've got a nice animation here that shows what actually happens. So you put your finger inside and pull and as you pull that's what happens and you kind of just rearrange by some more deformation and you've turned to double torus ghost inside out and you can do this with all of these shapes, all of these shapes can be turned inside out smoothly, nice! If you want to know more about all is google Smale's paradox or sphere eversion. Now you may think that all this completely useless. Well I have to tell you that this sort of ghost maths is often really, really good for finding out when certain things are not possible with more solid counterparts of the ghosts and the principle here is if the ghost can't do it, the solid counterpart can definitely not do it either and it's often a lot easier to prove that a ghost can't do something than to show directly that something solid can't do it. So it often comes in handy in this way. Finally the Mathologer inside out challenge. So to take part in the challenge what you need to make yourself is a ring consisting of four paper squares. These paper squares have been creased along the diagonals and the ring is colored differently inside and outside and you're supposed to evert this by just folding along the diagonal creases and the creases at which two squares meet. It's a really nice challenge. If you succeed, send me some video evidence of you performing this feat and i'll include your name in a Mathologer inside-out challenge hall-of-fame limited to 100 participants in the description. At the end of two weeks or something like this I'll publish the best video I got submitted on Mathologer 2. So no submission please in the general comments I'll just delete them. And that's it for today. Well, it's really late here and really the only thing left for me to do is to go to bed. So before I do this I'll take my special Mathologer glasses case which actually nicely turns inside out, put my glasses in, say good night and I'll see you next time.
Does anybody know whether higher-dimensional spheres can be turned inside out in the same way?