Matt Parker: An Attempt to Visualise Minimal Surfaces and Maximum Dimensions

thank you all very much it's absolutely honored to be here thank you for that very kind introduction so as was outlined then I am some combination of a mathematics author and youtuber and most people have difficulty with one of those career titles and which one is dependent entirely on how old you are for the young people and author is someone I by write blog posts but only one every couple of years it's very long and you have to buy a printed out version oh it turns of youtuber I make youtube videos about mathematics and so Robert referenced the negative 1 12 by the way I can see the person with a camera I'm just ignoring them you're like it's not having a stroke he's talking straight through another human no I know I know they're there they don't know I am NOT asking for it but thank you for heckling me live on stage so at YouTube we did a video for a channel called numberphile many years ago where it was I was not involved in the video someone else a physicist showed that if you sum all the Institute's you get negative 1/12 and them because they were a physicist all of the proof involve doing this and the internet got very upset and it's been a fantastic debate ever since and so numberphile we started many years ago and then stand-up maths I've been doing since and so this is whole kind of genre of math videos you can find online would you a huge amount of fun so that that's my career as such and so I very much like to thank the árbol Prize Committee and everyone involved for I can only assume is an admin error in inviting me a glorified math clown to to round out today's incredibly prestigious events and fantastic talks about incredible achievements in mathematics I mean what what karen'll and Beck has achieved for maths is like I'm doing it a disservice like if it was like in medicine because she's worked with all new techniques and tools and fields of mathematics if this was a medical talk it would be like having someone who's developed new machinery and new techniques for surgery that now people can perform surgeries they weren't able to do before and when they're being honored they would bring me out to basically sing the knee bone is connected to the leg bone to end the day on a high note so so what I'm doing is a gross oversimplification of everything that has come before but what I thought I would try and do and this is why the word attempt is very important here is to try and find some ways that you can visualize some of the concepts that you've heard in the three lectures before me and because some of the ideas are very like looking at minimal surfaces looking at them in higher dimensions these are very difficult to visualize things are very important topics of course with you know all sorts of implications applications ramifications but how do you actually visualize them so I'm going to start way way way earlier with it with a very simple very famous mathematical surface because so much of this is discussing surfaces I thought that would be a nice humble starting point and so what I have here is a rectangle very popular shape in mathematics and you can make one of the most famous surfaces you will come across if you put the two ends together like this well initially you get the cylinder which was seen as a few times obviously same Gaussian curvature as that I haven't changed that and once it's up there I can then put a gentle half twist so if I turn that over I get a mathematical surface known as the Mobius loop they were and so named after Mobius the first person to make a twisted piece of paper and then name it after themselves yes no he was so proud he named himself after the loop and it's cut not all of what I'm saying is strictly a fact and it's got all sorts of fantastic properties and some of you may have seen these before so I've got a picture up on the screen if it's a bit harder to see so famously he's only got one one face as such one surf and when I talk about these when talk about surfaces because I'm using paper to visualize this paper has some thickness but with all these surfaces we referring to something with no thickness so this cylinder has is infinitely thin you could say there's no thickness so we often talk about in his gut hasn't got two sides it's more that there's no thickness to the surface there's no similar sense of being on one side or the other it's more fundamentally it's a non orientated wall surface so if you start moving around the surface by the time you get back to where you started you have gone eh you've been flipped over so left and right have no meaning on a non orientate able surface and so this is an arrow the arrows being slid around the surface and when it gets back to where it started its flipped over and that gives this surface all sorts of fantastic properties there's an incredible thing you can do if you have to to Mobius loops of equal length and the easiest way to do that is you've start with one I'm using reasonably wide paper if you very carefully make a snip right in the center so I'm cutting right in the middle of the surface there if I now cut right down the middle and one continuous loop and I get back to where I started I should have two separate equal lengths Mobius loops or as the vast majority of you were expecting I get one single loop right and so the Mobius loop is a surface you cannot cut in half and when I first saw that that really freaked me out this from the early things that what and you do it again and again you get the same result so that's a weird sort of reassuring and then I was like what on earth and eventually you realize and you cut and down the center you're you're basically the edge you got one continuous edge so as well as the one surface or non-orientable surface you've got one edge which goes all the way around am i cutting down the center you're basically this is the shape of the edge so it's popping out into that shape which means you can go one step worse with a more twisted loop so what I'm going to do now is a three twist Mobius loop so there's our initial zero twist cylinder now I turn it over and once you get a single twist Mobius loop and then that's a to twist Mobius Loup and then finally a three twist Mobius loop and if that looks familiar it's because it is the recycling logo yeah next time you see a recycling logo you can say hey that's a three twist Mobius loop it only works if there's someone else forget it so so here's the three twist loop which I'm now going to cut in half and so if you have not seen this before see if you can have a guess what will happen if I cut right down the center of a three twist loop if you have seen it before no spoilers keep it to yourself and I find very few people can just guess I mean obviously if you're familiar with this you'll know what's about to happen but guessing this from nothing is very difficult so I'm right back to where I started I've got like a tangled mess of paper and if I cut it I still don't get two pieces of paper so obviously there are other surfaces if I cut all the way through them I'd get two pieces of paper this particular closed loop I've done I don't get two bits I get one bit but now there is a knot tied in that loop right so I went from a piece of paper with no knots tied in it and then by cutting it in half I've got a piece of paper with a knot right when I first saw that that really that really freaked me out and this is the trefoil knot that I think Robert referenced earlier which can be embedded on the surface of a torus so there's the torus knot and before I started cutting the edge of a three twist Mobius loop forms a trifle knot and so by cutting it down the center I end up with the bit of paper now the surface forms that same knot and so I get they're not there and there's a whole area of math where you're looking at the surfaces you can form where you have knots as a boundary which is what I'm going to do in a second but I cannot with them with I can't go that far to Mobius loops without showing you my favorite side fact about cutting bits of paper like this and I skipped early on over the cylinder so if you start with a cylinder there we go and you cut that down the center you get exactly what you expected you can alway down the middle you come back to be started you get two cylinders it's not at all surprising if you make one slight change if you get a second cylinder and you attach it to the first cylinder at right angles actual I'm going to put some tape on the other side if you're going to try this yourself afterwards and a lot of you are make sure you put there's a pro tip put plenty of tape on both sides because I'm going to cut both of these in half while they're attached and your challenge if you're not seeing this before see if you can guess what shape or shapes I'm gonna get if I cut that one in half right down the center and then this one in half right down the center okay here we go I have never done this in front of a more qualified audience to work it out it's so okay I'm cutting right down the first one and I get that when I cut down the second one and two loops stuck together both cut in half give you a square yeah all right after this that's one of the main reasons I wanted to do this is if anyone comes in now or tunes into the live feed right then they go I won't know how the other árbol lectures are going and there's me going now this is a square and a bunch of you are like oh yeah wow that's some complicated math here all right of course the natural the natural follow-on question is what would happen if you stuck to Mobius loops together and then you cut them both in 1/2 and this is one of my absolute favorite things to do with cutting up mathematical pieces of paper so I'm cutting straight down the first one and if you make it back to where you started you get that that's the same shape from before but now it's got the other ones still attached to it I cut down this one this time we do get two pieces of paper but if you then attempt to untangle them you get it's two hearts that go through each other isn't that that's great like 80% of you alike that's quite cute and 20% of you are like this is not why I came to this lecture okay if you want to try that yourself a lot of you will the to Mobius loops have to be different twists one's gonna be right-handed one's gotta be left-handed so there are mirror images of each other then you stick them together then you get those if they're the same twist you get you you end I said of getting too hot you end up with disappointment okay so now you can start to have some fun like the the three twist shape that I showed you where the boundary is a knot you can start to create those surfaces deliberately and so if you do it just the way I did it you'll end up with a non orientate able surface the edge of which till the boundary is the trifle not this is a trifle not however it's been rearranged so that is an orient a table surface which has been put between them and someone Madeleine Smith who live in Edinburgh crocheted a knitted a bunch of mathematical models that I've used various things that she's a great mathematician University of Edinburgh and this is a fabric version of a little crocheted crochet thank you Wow is there any area of expertise this audience does not have oh wow I'm more scared than I was a second ago so so she crocheted this to show that you've got the triple knot in the white and then you've got surface surface an Orient a table surface around the rest of it and there's so many areas of math where having a surface will come in incredibly useful and several times today we've heard about having a minimal surface it's been a reoccurring theme and and so what I thought I'd do now is I'm going to try and do some stuff visualizing minimal surfaces and then I'm going to try and visualize higher dimensions and we'll see how we go so the classic example of a minimal surface is using soap film because it naturally contracts to the smallest possible surface area so if you can somehow arrange soap film between whatever the constraints you have are it will take on the minimal surface least a 2d surface in 3-space and so I brought with me a small amount of bubble mixture and the classic one is just if you blow a bubble it's constrained by some volume of air and it will form the minimal possible shape and so we'll see if this works okay there we go and so you get the classic classic sphere you're never gonna blow this it's not because those are round a lot of young kids very rightfully go with that's round right if they were square you would have got a cube bubble good good point I can't I cannot knock your logic but even if you do have a square one or any other shape you're never gonna get a cube bubble when you blow something like this you're always going to get a spherical bubble and that's where it's just constrained by its got a certain volume that has to be inside of it and the sphere is the smallest possible surface much like before we will show if you've got a certain amount of ball hide and you want to get the maximum amount of area so the circle is your biggest possible line you can get around that now you can have much more complicated constraints so this is just me putting some air into a bubble and it becomes a sphere you could have a lot more fun with doing things like putting in other rings which were reference before I could try and find so if this was the throat joining my two holes in 3d space my two my ball holes here and here like mr. on the diagram I could use so film to try and show what that throat would look like I could even get something like a box or a cube in 3d right and I could you soak them to show me some of the minimal possible surfaces which would fill something like this the issue is something like this is not going to fit it's right so if you if you want to try this yourself afterwards you need to get a container big enough that you can completely fill it with bubble solution to get something like this in which I may have done so down here I have a ridiculously big container of bubble solution which I'm going to use to break all of this equipment okay here we go so let's regret this immensely okay here we go whatever you're imagining it's bigger okay okay like before 80% of you are impressed 20% of you were like this is not why I came to the lecture okay right so what let's say what are we going to do first let's do let's do the rings first so so you're able to see this a bit better I have both got a camera that I can put over here on the side and you one second to bring that up so okay so there's it looks like I'm about to do washing the dishes okay so there's that tub there we've also got the overhead camera which we can go to in a second I'm going to try on this one first and then if that works we'll go to the visualizer no promises that this will pair now we'll give it a go okay so I've got paper towel I'm gonna make a huge mess that will stop it from okay okay here we go okay I think I think we're gonna be alright so here we go two rings we'll see if we can get the minimal surface that joins the two rings together if you put one in and bring it out you just get the flat plane there right or over a big bubble I guess oh okay if you put the two in together and this might take me a second to get it to work and then pull them apart you should there we go right so now we've got the the catenary that we mentioned before will join the two of these together and just naturally sometimes it's joined in the middle so if I'm not careful ah there it is there oh no wait so you can that's it okay was it still got the middle bit there we go no - what is I need a volunteer who's gonna come and give me a hand with this who is who's the youngest person who's here there are the young people for whom this is clearly intended that's you're both pointing at each other it's okay do you want to actually let's go you the person leaning forward that's you just the most enthusiastic I've seen a young person in some time okay come on up is this the team with the young people who won the math competition hey all right give them a round applause well huh okay what was your name okay I'm gonna give you a pencil there you go your job is if the bit of bubble forms in the middle you pop it with the pencil can't go wrong it can go wrong if you stab literally anything else with the pencil it has gone wrong okay here we go say it won't even happen this time we'll give it a go okay okay so can you pop that bit in the middle now okay so now you stand there awkwardly okay so you can see wow you done that literally okay so you can see as I move these apart we it forms the minimal surface it's not a cylinder it curls in it's got the catenary what a catenoid I guess shape and actually if we go to the overhead oh my goodness now it's multi-media okay so you can see there but as was mentioned before if you start to move these apart it won't just form an incredibly there goes it won't form go two in a row okay it won't form an incredibly thin tube there's a point where the minimal surface look it flips across and you get the two flat surfaces so if there's a ratio which actually I'm not sure nearly I'm not sure what it is where the this is stable so that is our stable minimal surface if it's perturb slightly it will stay where it is once you go past a certain distance then suddenly you're in that unstable region and you can see on the chalkboard behind me where Robert was explaining that part of his talk to me in advance where if you've moved outside those two lines you're suddenly in the unstable region and blip it'll go back to the two sides like this okay that was the end of your involvement you know at the pencil down you get a round of applause and there's all sorts of other things we can solve with bubble mixture so this is a representation of what's called the the four towns road problem so here I have acrylic with four poles in it and this is a challenge where if you had four towns what is the shortest possible combination of roads which will join all four towns to each other of course you could just have like we tow every single road could link every single town but that's longer than you need you could just have like one road coming down here one going over there if you want oh actually one over there one over there one over there and so there's different ways you can try and work out what is the minimal length of road to join all four towns if we use the slope film it will work it out for us automatically so if I pop this in here and then pull it out and then bring it over here to the OHP you can see there the soap film has formed what I can probably do is what that is a bit clearer the soap film has formed what is the minimal possible series of roads that will join all four of those towns together so they all go in towards the middle and then you have one little joining bit in the center and that is it it is found that well there's the global minimum that is the best you can possibly do and a couple times today we've heard reference to different local minima and you may not be in the global minimum if I was to take it out of the bubble mixture a different way it'll still solve the problem it'll still find a minimal solution but it's not the global minimal solution so now it's found that solution which locally is the minimum solution it's joined those ones are joined those are joined those are joined and if you perturb that slightly if you move it away from that solution it'll spring back to that solution but globally it's not the best what we have to do you see if I can get this to work we have to do where's that in the shot okay is if I breathed on it very gently there we go there you are you weren't impressed by the hearts you lose it for this you're all dead inside okay so the last one actually know what let's do how much you know the speakers keep saying how much time have I got and they look over here there's just someone's written you're out of time and that's it you know what let's do why not we'll do one more another demo this doesn't always work I was hesitant to do this we'll try it because people have mentioned very similar things today I want to see if it's gonna work could I have more young people with the two of you like to give me a hand is that acceptable okay you had a little you checked with each other first that's good teamwork that's why you won the competition okay you get a round of applause once you get to the nuts yet not yet and you both of you both you this is the to two human tasks okay right so who would like to be on pencil Duty there you are you're on bitumen do you want to come over this side over here so here's what we're going to do if it works so I have here a tiny tiny loop of thread and so what you're going to do is take the loop of thread you're going to dip it in the mixture and you're going to put it onto the bubble surface that I have here and then you're going to pop the bubble film inside the thread and we'll see what happens so there's your thread over there and also if like the kind of secret to this is if your hands if we go to the other head one if your hands are covered in bubble mixture you're not going to burst the bubble right you can go right in and out so if you dip it in and make sure your fingers get I have not thought this through okay and then can you pop that on the surface of there so we guys will close loop yep has it gone give me a second can you sing okay yeah oops do you want to dip your fingers further in so they're covered in bubble mixture I know this was not part of the deal when you volunteered okay there you go put it on there okay and okay okay that's it perfect okay all right so you can see there's the closed loop the Meuse make sure it's under the camera okay very carefully burst inside that loop that's okay that now the pencil has too much bubble mixture on it which is why I brought a spare pencil it's the yellow one right under the clock under the sign that says I'm out of time okay Leah Hey all right so by bursting that is the same as the ball hide we had earlier by bursting the oops trying to get in the middle here we go so now I mean we're minimizing the area outside which means we've maximized the area inside and so that has popped out into a perfect circle is able to solve the same problem and give us the minimal area outside maximum area inside and we get the circle like before so give these two a huge round of applause thank you very much and finally the cube so let's let's see if this is going to behave itself so I have the full cube here I'm gonna try and put it in the mixture and then what I'll do is I'll probably just hold it above the tub so you can see it clearly I'm not gonna try and get it and okay okay so depending on which of the surfaces it's Oh which one's that on okay there we are so now I've got a minimal surface joining those six edges and so it's given me that shape there what I really want is one that covers all of them now there we are and so you end up with a there's a square suspended in the middle of all these other surfaces that go out to all twelve edges if you want to have some extra fun you can inflate that square so let me see can I hold it into the camera please there you are hang on now let me see if I can get that oh wow this that's the best oh hang on hang on the rest of the talk is just this okay so we've got that square in the middle again it's changed orientation that's fine though I should have a school a cube based puzzles bubble suspended within the outer cube frame so there are you can see on there right so there we have a cube bubble which is the minimum a given I've put the air in at the minimum I'm gonna get if we go to the overhead camera you oh it's gone no we're doing this again oh okay this time oh we know we have got the middle bit okay so to if I put under the visualizer I'll see if I can blow on it this way is that in the shot and there we go so another I mean most audiences would applaud now but that's fine I'm actually curious and I was wondering this on my way over and I haven't done the working out but if any um younger people want to give it a go whenever you do this you always seem to before I turn it into a bubble you always seem to end up with that square in the middle and I don't know what the ratio the size of that square is it like should it be zero and it's just the physical nature of the bubble mix which means that you end up with a square right in the center or is there some optimal ratio for the size of that square imagine if you've done a bit of calculus you can probably set that up I haven't done it so maybe this is a terrible suggestion don't do it it's a world of pain but it feels like you should rather setup some equations reasonably easy and to work out what size square gives you the minimal surface area of those but I don't recommend giving it a go I'll give it a go give it a go but don't blame me if it ends up being either a tedious or difficult problem so that's I think I'm gonna leave minimal surfaces there obviously what I'm doing here these are just you know two-dimensional surfaces in 3-space and every single speaker before me has referenced three space or n space or you know infinite dimensional space and so I thought for the last bit of my talk is have a look at what mathematicians mean or at least how you could visualize the concept of having higher dimensional space and oh and actually just show up before we do that this was me I spent absolutely ages I made a trefoil not out of a piece of wire such that you could suspend the surfer Orion total surface of the knot by using bubble mixture I'm not doing that live because it took a very very long time because one kid like the wire can't touch itself the whole way around so that's one bit of wire very carefully bent suggest I realized why I prefer abstract mathematics to physical problems but eventually it got to work and again the soap film too that minimal surface there which was deeply pleasing and of course you can have a lot of fun with the cube that's the cube in a cube classic classic shapes out of cube in a cube and then you've got the surfaces joining all the outside edges to the inside edges but now high dimensional space so a lot of people have put out variations on something like it looks a bit like r-squared so you will have seen various things like I haven't broken it good break that right and so we saw we use R to represent the reals and if you got a square then you've got to lots of the real some very simplistic terms which is what we normally use when referring to our to space or two-dimensional space or space with two dimensions as we now need to be very careful to define and that's that's the classic XY plot and in both cases these should be the classic number line as a nice way to keep all the rules in order specifically though I'm only going to color in the whole number coordinates using only zeros and ones so they're all the combinations of zeros and ones if you want to pick points in this two space and if you join them together you get our friend the square if we now want to go one dimension high if we want that to to become a three where now we've got our classic three different directions we can move in the three ignoring the three rotations the three spatial dimensions you can go backwards and forwards in and if you wanted to you could go through and mark in all the combinations of zeros and ones as coordinates in 3d space and if you join them together you get the classic cube you very briefly get the classic cube the question now is what happens if you go up to r4 and several times we've had the need to go into higher dimensional space so even looking at like a plane in flight as Robert pointed out it's got three degrees of freedom in terms of rotations and it's got three degrees of freedom in terms of moving around so there's actually six numbers which define it and if you want to plot all six of those numbers simultaneously you would need six orthogonal axes because they're all independent and if you do a lot of science you will know that you're doing science school you will often do two dimensional plots you'll have two things that vary and you won't be able to plot them both at the same time or you'll often come across three dimensional plots if you do more advanced science we want to plot three things at once and then a lot of the time it kind of stops there and that's because we as humans cannot easily visualize what something would look like if we went one or more steps further what were they for the plot look like or following the examples we've already had what would a 4d cube look like now a lot of the time mathematicians have the solution of not caring and the simple I mean a lot of problems in math are fixed by have you considered not worrying about it and it's true because being able to visualize it is not you know necessary to be able to then go and do mathematics with it a lot of mathematics happens at levels where we cannot intuitively visualize it in fact that's kind of the point of mathematics it allows us to do things beyond our normal intuition we can use logic to deduce patterns and structure beyond what we can just you know naturally imagine but wouldn't it be nice if we gave it a go so we're gonna see if we can work out what on earth it would actually look like if you did try to put together one of these 4d cubes and you could already did you Squire bit about it because you've already seen the 3d cube and the 2d cube the square all right and you can work out how many corners there are just by the coordinates like I showed you you could very quickly calculate how many corners there should be on a 4d cube you could very quickly calculate how many edges there would be and you're already working out properties of this shape even though you've never seen it you've never held it you have no idea what it looks like but you're already discovering things you know must be true about it but to actually visualize it there actually there are a couple of different ways you can cheat and bring a higher dimensional shape down a notch so you can try and actually see what it looks like so if you want to visualize a 3d cube so this this is a 2d claw BC I'm doing everything is on a 2d screen here this is our 2d representation of a 3d cube so you that middle cube is not square is not smaller this middle square is not smaller than the biggest questions further away and one way you can cheat and take things down a dimension is by unfolding them into their nets so 3d shapes have a 2d net and so what I've done here is I've taken the 3d cube and unfolded it into its 2d now it's made of six squares and then I can fold that back up again into the 3d cube so we go up and down a dimension by folding and unfolding the net and just before I go any further are we all happy I've got a video here of a 2d net becoming a 3d queue okay it's it's key because actually that's not that's not a video of a 2d net becoming a 3d cube there is a video video of the shadow of a 2d net becoming a 3d cube so before I was only showing you this bit at the bottom above it is the actual there we go actual cube unfolding into its net and then folding back up again that's the other way you can cheat to go down a dimension you can project it so you can see there's a dot at the top which is representing the light source it's casting its shadow down in that 3d shape has a 2d shadow which is cast over load these animations are done by answering code David curve on Union College in New York has put together these fantastic animation showing some of these ways you can take shapes down a dimension and we can do the same thing one dimension higher so for D shapes can have 3d shadows if you project them down and they can unfold into 3d nets so on the Left we have our 2d net of a 3d cube on the right is the 3d net of a 4d cube and so you can see some similarities there it's made of eight cubes in a very similar arrangement and just like we can watch the projection of that 2d net folding into a 3d cube we can watch the projection well yep the 3d projection of the 3d cube folding 3d net folding into a 4d cube there we go and over that's not so bad so you can see a lot of similarities with a dimension lower down and you know naturally looking at them on the left that top square when that squares turning over it's not actually stretching around the projection makes it look like it's stretching around but I'll see a dimension higher we know it's just turning over same deal on the right here that cube that the magenta cube looks like it's being distorted and stretched around in fact it's nice just being turned over a dimension higher and setting the whole thing off and the actual the finished cube looks like that so again it's the classic cube inside a cube all joined together it looks very similar to the bubble cube inside a cube which I showed you previously and one of the wonderful things about mathematics is when you're doing what seem to be different areas of math and suddenly the same pattern appears and when that happens you can go oh my goodness I didn't realize that there is some deep connection between these two areas and so in this case here it's pure coincidence anyone know no meaningful link whatsoever as far as I've concerned if anyone wants to prove me long wrong the the next árbol prize is yours so I don't think I had the authority to promise that and again this is a prospective trick so that blue cube on the inside is the same size as the red cube and the answer is just further away in perspective right and so we're using a weird little perspective to better see this a dimension higher and just like we can mess around and we can take a 2d projection of a 3d cube which you're looking at here and we can set it rotating which to our human brains is fine we look at that and we're like oh it's fine it's just a cube rotating we're not thinking about the fact the actual projection the squares the red and the blue square taking turns being bigger and smaller they're going through each other so the red and the blue are projected at the same point on the screen occasionally so if we weren't aware of 3d space and we just looked at that we'll be thinking that's weird there's the squares are getting bigger and smaller they're all getting distorted just thing at the top square looks like this just this weird shape that that's rotating and distorting around on the spot we're like wow that's that is strange but because we understand 3d space we're fine if we now look at this one here we can do the same thing we can set that road and it's exactly the same process the cubes are taking turns being bigger or smaller they look like they're pushing through each other when in fact they're going in front and behind a dimension higher and of course there's many other ways that you could I could have rotated a 4d cube so when you're dealing with 2d shapes you've only got one way to rotate you can clockwise or anticlockwise that's it so 2d has one rotation trigger your freedom in 3d space we suddenly get three right that's that's nice thanks if you got a 4d you get six there are six different ways you can rotate a four-dimensional shape if you go up to 5d there are 10 someone will definitely correct me if I'm wrong all right and so there's loads of different ways you this is one particular way that it's rotating that we're now visualizing and so when people are talking about 3-space they're talking about our normal reo they often talk about normal like Euclidian nothing ridiculous going on space right it just means everything there's nothing hyperbolic happening it's all very straightforward and simple and but the bulk of the time like I said we're talking about surfaces in those spaces so when I was making the Mobius loop earlier let me take that camera feed out we're not making the Mobius loop I was waving it around in a 2d surface I was waving it around in 3-space it's embedded in 3-space and the word embedded has come up a couple times today and that means you can just you can take a surface and you can put it into that space with the required number of dimensions without any major problems so you can take a 2d Mobius loop and you can embed it in 3-space which I was doing when I made it but you couldn't have embedded it in to space so if you tried to make a Mobius loop and you tried to do it without taking it off a flat surface you wouldn't have much luck so here are the matching instructions for a Mobius loop so I've got two arrows and to make the Mobius loop you've got to join the arrows together so they match if you try and do that keeping it flat and just stretching measures rubber stretching it around the arrows will never match the only way you can get this work is if you lift it up into the third domain and turn it over and now when you stretch it all the way around oops we get to enjoy all these again now when you there we go turn over now when you stretch it all the way around the arrows will match but to do that we had to go up a dimension we had to go to three space to get that to work and in this picture I'm showing you on this to space screen you can see this outside edge looks like it goes through the other surface so this edge here if you followed this edge all the way around here it goes well from this image if you just look at the picture and naively as drawn that line goes through the middle this surface we of course know that's a representation of it going up and over and down the other side right but if we weren't aware a three space if we were just looking at this from a two space point of view would be like now this is no good that doesn't count right this is no longer embedded this is just it would say it's immersed we've immersed it into space there it's there but we've had to make some compromises now there are other shapes we can do so this here this is the range of all the different fantastic surfaces you can get when you identify the various edges of a plane we've become across a cylinder several times we have some tourists the donor has popped up and the climb bottle has been briefly mentioned a Mobius loop is what I did before a projective plane we haven't come across and that is a whole world of fun which I will leave to the reader okay we are however going to do the Klein bottle which Robert very briefly put up in his talk and that's the same idea but the different matching instructions so this time you've got two pairs of arrows red arrows have to match blue arrows have to match the red arrows no problem you're joined together you get yourself a cylinder very straightforward in 3-space in fact that could be embedded in to space without any major issues we now stretch them around to join the blue arrows together but they don't match ones going well depend on how you look at it one's going one way one's going to like what way 10 o'clock eyes they don't they don't mesh up when you put the two ends together if they did that was the other shape from before the other Taurus or as physicists call it the donut and the although mathematicians eye there's a strictly tree you'll see methods to be very careful they'll say circle and disk to mean different things but the circle is the perimeter and the disk is that the whole area or ball and sphere to mean the the shell or everything inside it I tend to use torus and donut torus as the outside surface doughnut is the solid object and it's more delicious but the problem here now with this shape the Klein bottle is they're not going to move they're not going to match and they won't work in 3-space the only way to get it to work in our humble three dimensions is to shove it through the side then fold it back and then join it up right that gives you your your twisted doughnut that'll do and in 3-space like as our three dimensional beings we're like no what is that doesn't work you can't just shove this through this there's gonna be a hole there and but as Robert mentioned we're still imagining it being this perfectly smooth continuous surface right you can't suddenly have this so this only works in four space if we add an extra dimension it would work fine a four dimensional creatures like that's fine it goes around don't worry about it have you considered not worrying about it and so sadly we will never see this work properly in our humble reality but if you add a fourth dimension it does work out nicely I should flag up a common question or comment is I'm not using the fourth dimension here to refer to time so a lot of physicists will repurpose four-dimensional mathematics and one of the dimensions instead of being a spatial dimension will be time or via temporal dimension and that's kind of that's kind of unfamiliar with how physics works there's lots of experiments and they work up the theory image they get stuck and like what are we going to do now now let's go see what the math department I've been up to recently hey what are you doing I was going to borrow some of these matrices thank you notice theorem we'll have that Oh eigenvalues cheers and then they go back and this will work for at least 50 years and so and so you'll see a lot of theories of every time and and I was putting that bit of thread together before for that demo so I was like are you doing string theory I'm like a very rough approximation of a joke is what you just said and so and so you will see time for 4d I'm using it as a spatial dimension and that's what you'll often see people when they're talking about R to the N up to you know infinitely infinite dimensional space we've gone long beyond attempting to visualize things right so we can try and visualize things low down it's kind of fun it's very useful there's some cool graphics and there's a huge intersection obviously with math and art and all these ways you can visualize stuff which is great but a lot of it is just driven by the need for more variables we have more variables they're independent we've got to have them all at once we don't want them to get in each other's way and you just go up you increase n until it works right and that's as Karen said we're ambitious in mathematics will increase n as far as we can go see when it breaks that's the bulk of maths in a nutshell and so actually the client bottle I mean I it has a special it's probably my favorite shape a special place in my heart because you can you can kind of make one so I've got here this is a glass version you can get a guy called cliff Stoll in the US has also done videos on numberphile will sell glass versions of these things Acme Klein bottle page right and so this does actually have the bit with the tube goes through that bits missing it's not continuous there so you can get water and things in and out and of course famously it makes for a mighty fine hat so you can knit these alright and when I realize you can knit these I got so excited and I went to my mum right because I can't knit I was like one you've got to knit me one of these you've got to knit me that the three-dimensional shadow of a four-dimensional twisted doughnut it's classic mat unbelievably work together right because she's super nerdy as well she's like we're going to do this and eventually we ended up with this right there's there's my climb bottle hat sadly I don't take it out anymore but a we've since made a spare traveling version in brighter colors so if you want to see what it looks like when I wear it it looks a bit like that picture there now I'm it looks like there's a handle sticking out of your head that's what you get and it's a continuous tube of knitting right so the Hat becomes a tube the tube goes around and through this I'd if I reach in here I can then pull that tube through and here's where it folds back on itself right so it's obviously because of the constraints of physical reality it gets bigger and smaller but same idea it's a continuous tube that goes through itself sadly it's got to have the whole day because we're only a motion immersing it in three dimensions oh and you have it's striping the first version my mum made for me which I called the prototype which she called the perfectly good gift was was all the same color sounds like mum could you make me another one with stripes of different colors she's like yeah of course I just changed the colour as I go like ah could you make each stripe different thicknesses based on a long list of numbers I give you and she's like yes and so these are the digits of pi miss it into my 4d hat and this is officially the world's nerdiest hat if you would like the missing pattern this is not a joke send me an email I have visited as a PDF I will very happily share it with you so drop me a line I'll send it over to you you can ever play with it and as always it's it's good to try it like I mean I cannot overemphasize how much what I'm doing is a I know I keep saying it's a gross simplification of the achievements of Professor Erlin Beck but that's it's so true like what what cutting edge mathematicians are doing is absolutely phenomenal right and as I like you talking about it's well beyond what we can do with our normal intuition and understanding of a reality absolutely incredible I just mess around at a very low level having fun trying to visualize things and so it's been an absolute honor to be here desperately out of my depth I will be around so they've got as it was very kindly mentioned I have got copies of my book so they're out there and they're loads of other books by other authors and I will sign any of them it's not a problem so after this I'll be out there I also sign calculators not it was a joke and now I sign a lot of calculators happy to take selfies and everything else I'll be around if you want to say hi I think most of the speakers some of them have other commitments but some of us will be around if you want to say hi and chat has been absolutely fantastic to see so many people here enjoying such incredibly advanced and occasionally trivial bits of mathematics a huge thanks to everyone the árbol prize for making both these lectures possible and the prize everything else so thank you to everyone I'm gonna wrap up there thank you very much [Applause]

Channel: The Abel Prize

Views: 84,555

Rating: 4.9503517 out of 5

Keywords: Matt Parker, Standup, math, mathematician, mathematics, Karen Uhlenbeck, Karen Keskulla Uhlenbeck, Minimal Surfaces, Maximum Dimensions, Abel Prize, The Abel Prize, Abel, The University of Oslo, An Attempt to Visualise Minimal Surfaces and Maximum Dimensions, Science lecture, Popular lecture, Popular science, Popular science lecture

Id: Iip8VNrHK_8

Length: 50min 2sec (3002 seconds)

Published: Fri Dec 20 2019