 So hi Roddy So this is interesting. I've never done this with a back projection if I laser on to the back. Can you see it? Yeah, okay. I guess we'll do it this way. Maybe I'll jump down in front. So so I'm Henry I'm a mathematician. I work at Oklahoma State University, which is a university in the United States I'm also a mathematical artist. I mostly work in 3d printed sculpture and I'm going to tell you some well how to make sculptures of four-dimensional things So I guess before we start What is four-dimensional space now? I don't mean Like if I was a physicist, maybe I'd think of the fourth dimension as time or There's various other ways that people like added dimensions things. I'm talking about four spatial dimensions. So well, what does this mean? So let's go back to two dimensions, right? So everybody's seen How to draw a graph in high school? You've got an x-axis and a y-axis and you describe some point in space Here we go. Let's try this is so cool Except all my text is backwards. It just makes it a bit tricky So so here's the point three three along and two up and I go three along and two up and I follow the axes Okay, and then to do three-dimensional space Let's get rid of the numbers. Here we go. You just add an extra Direction, right? So I added an extra axis here Right angles to the other two and then I can describe numbers Describe points in that space with three numbers So to describe four-dimensional things all you need to do is add another axis that's at right angles to the previous three And then you can describe a point in that space with four numbers w x y and z And maybe I will come in come down in front so You may be unhappy with me at this point saying what is this nonsense? You can't find Another direction that's at right angles to the three you already started with this picture is a lie And you'd be correct This is also a lie Right this vector here this this arrow here is not right angles to the other two Right this angle here is I don't know 60 degrees or something On the screen, which is a two-dimensional thing You're fine with me drawing this third direction and saying it's at right angles to the other two Why are you fine with this? I don't know It's because you grew up in a three-dimensional world and you're happy with this idea that this is some sort of Picture of a three-dimensional thing, but it's not actually a three-dimensional thing And in exactly the same way You should be happy Maybe With this being a picture of four dimensions. Okay, so what's actually happening here? Hello there we go is This is a sort of projection a shadow of something in three dimensions I've crushed three dimensions down to two But we're okay with this picture because we understand three-dimensional things And in the same way, whoops Here i'm crushing a four-dimensional thing down to two dimensions And i'm just sort of drawing in these little things here to show that they're right angles. Okay, so From a mathematical point of view four-dimensional space is just you have four numbers to describe points You can still talk about lengths and angles and all the things you usually talk about It's just harder to see things because we're not evolved in a four-dimensional space and we don't really understand it okay, so so Let's think of some example What's an example thing that we can put in this four-dimensional space that we're going to try and understand I'm going to try and see So so we're going to make something called a hypercube So we're going to start very simple with a point And i'm going to move that point across take a copy and then connect up those two points and I get a line segment Everything great so far. Let's do it again So here's a line segment. I take a copy. I move it across at right angles To the line that the line segment was in And I connect up the corners and I go square Then I do it again I take a copy of this square I move it at right angles to the plane that the square is in already And I connect up the corners and I get a cube And i'm lying again already, right? This is not a cube. This is a flat thing on a two-dimensional screen But we're happy with this fine. Whatever And then we're just going to do it again Take a copy of the cube move it at right angles to the three-dimensional space that the cube was already in Connect up the corners and that's some sort of picture of a hypercube So so this is Maybe kind of useful right so so you can tell some things from this picture You can tell how many corners a hypercube has right you can count One two three four five six seven eight And then there's another one two three four five six seven eight over here So there were 16 corners on a hypercube and you can count how many edges there are and so on So you can get some sense of it, but it's not a great picture. So so The first bit of this talk is how do we get a better picture of what this thing is? okay So How do you see four-dimensional things unfortunately the answer is we we can't really see four-dimensional things There are people who claim that they can see they can actually visualize truly four-dimensional objects Um, I think they're wrong or misguided But there are people who claim it. I don't know that there's all kinds of strange things going on in people's brains Well, how do people actually try and understand four-dimensional things will use various tricks and here's one of the tricks So you use a shadow Or you project down and this is what we were doing just before Here's a here's a cube Um, it's actually a 3d printed cube and it's sitting on a sheet of transparent plastic And I'm shining a light down from somewhere up here And it's casting this shadow down on here. So What's the idea here? This is crushing a three-dimensional thing down to two dimensions And soon we're going to do the same thing going from four dimensions down to three dimensions so so this thing here If you had a two-dimensional friend who lived in the table And your two-dimensional friend wanted to understand this strange three-dimensional thing called a cube What you could do is cast a shadow down here And then your friend would be able to interact with this shadow and would be able to touch it and say Okay, I can see this thing. I can see. Oh, there's like there's an edge here and there's an edge here And it looks like they're parallel, right? They're they're going in the same direction and you'd say, yeah Yeah, the real cube has Maybe this is this edge up here and this edge over here The real the real cube has these edges and they're parallel to each other And your friend would also say and this edge here crashes through that edge there, right? They they actually intersect each other And you say, well, no, um, that doesn't really happen. That's just some sort of Problem. It's an artifact with the way that we cast the shadow, right? um, because what's happening, I think Maybe it's this edge here Is crossing over the shadow is crossing over this edge here and then you get them to crash them together So this isn't perfect, but this is something Okay, so this is something called a parallel projection of a cube Um, because the light rays come in parallel to each other And this is a 3d print of a parallel projection of a hypercube Um, I should mention this is uh, this is by uh, bathsheba grossman bathsheba grossman has Really been one of the pioneers in 3d printed mathematical artwork So I have this one here. So I'll hand a bunch of things around Um, that hopefully will come back to me at the end of the talk So I want to emphasize something here because this is kind of a little bit confusing. So This is a, well, what is this thing? This is supposed to be a three-dimensional shadow of a four-dimensional thing So back here We had a two-dimensional shadow of a three-dimensional thing, right? When you cast a shadow the dimension goes down one and so in the same way This is a Three-dimensional shadow of a four-dimensional thing We can work out the math and the maths and figure out where things go and then we can 3d print the result Hopefully that makes sense So, okay, so what do we got? So this is a this is another parallel projection Which means that the light rays are coming in parallel to each other. So in this picture back here My torch was very very far away or if you use rays of light from the sun they're coming to come in parallel And the same thing here There we go Um, so again, you see like there's edges that look like they're parallel In three dimensions and they're really parallel in four dimensions as well That bit's accurate of this way of casting a shadow But you get another sort of problem here. You don't get edges crashing into each other. You see this edge here It goes through this square Here rhombus, whatever So you do get this edge sort of crashing into this face of this thing. So it's not great yet. Okay. So what else can we do? Well, we can move the light Um closer to This objects and get a different kind of shadow So this is a called a perspective projection and I haven't really improved things yet, right? You can see I've still got this problem with the edges are crashing into each other And now I don't even have parallel edges. So this isn't so good There is something interesting though here. You notice that the shadow Looks like a perspective drawing of the cube Right, it looks like what you might draw What a cube looks like and so why is that? It's sort of an interesting little aside here So here's sort of a picture. Here's the light and here's the cube and here's the shadow and If you sort of reverse the direction of the light rays And instead of having a torch up here, you put your eye there Then what would you see of this cube what you would see? Well, this is what this little screen is here This little screen is what you you would see if you put your eye where the torch was and so Of course, it's the same picture, right? You've just shifted the screen down from right in front of your face to on the table So that's why you get the same picture Okay Back to this thing. How can we improve things? So here I've put the the torch Right above the square face of the cube And maybe I'll actually do this or attempt to do this Let's see if this works because I don't really have enough hands So so this this is a mini mag light Flashlight torch and it has this nice feature that you can use it as a candle Like this like the top comes off. But for my purposes, what's really good is the led is exposed. It's a really small point light source So maybe I can get this to work if I do this There you go So I've got this little point light source and I can put the light right above the face of the cube and I get that Okay So, well, am I doing better? Yeah, because none of the edges are crashing into each other Is there any crashing going on at all? Yes, no Yeah, okay, some people say yes, okay Well, what's going wrong? So where are the six faces of the cube? Right the six squares this thing down here. That's the bottom face of the cube and then these four Trapeziums trapezoids. I never remember what they're called Those four things around there. Those are the sides of the cube But then the top face is going over all of the other five, right? So the top face of the cube The light from that the the light that goes through there Goes over the other five. So we haven't sort of completely won yet, but we've removed edges crashing with edges Okay, so if you do that one dimension up you get this This is another Sculpture by baths Cuba Grisman And that around as well and this is pretty good, right? So Let's see. There's there's eight cubes in a hyper cube So there are six square faces of a cube. There are four Edges of a square. There are two ends of a line segment. So two Four six eight. It turns out there's a pattern there So there's one in the middle. That's the cube that's on the bottom of the hyper cube And there are six arranged around it these trapeziomy things So one plus six that's seven and then there's one more And that's this big one that covers all the rest. So this is pretty good. Well, we're not quite there, but this is pretty good Okay, so What can you do that's better? Well for that, we're going to need something called stereographic projection so Um, so I've got this black slide so that I can show you this with the shadows So here's here's a sphere a spherical 3d printer thing. I'm going to put the the microphone down for a second So there is a trick I figured out how to do this one handed If I hold it like this Then uh, yeah, so hidden inside of this sphere is this perfect grid Anyway, hopefully that's clear to everybody um It's not so easy to uh to to do that. So here's a slide Which is a much nicer easier way to do this Um Right. So so what is this stereographic projection thing? So here I've got the flashlight or sorry the torch. I'm in england now Um, and I put it at the north pole of this sphere And what happens is the light rays Come down and they hit the sphere somewhere maybe here and then they carry on and they hit the table And there's and this map this function goes from the sphere to the plane and it's just that map So it's just where does it hit on the sphere? Where does it hit on the plane? And so it maps this Weird sort of curvy grid to this square grid here. Okay So so that's what stereographic projection is just before I move on. Um, I thought I'd just comment. Um So This photo was not so easy to take I mean you could you could probably see I was sort of like lining it up and like my hand was shaking a little bit And if you move the the light just a tiny little bit the shadow it doesn't work anymore So in fact what's going on here the hand is purely decorative What's going on is that the the torch is attached to a rod that's coming down from above And there's a beam across the top and there's a couple of clamp stands and I just put my hand in there to look like I was holding it Um, yeah, yeah, okay. So so okay, great. This is awesome. What does this do it makes a Three-dimensional thing two-dimensional great Um, so how am I going to make a picture of a cube right because I want to show my two-dimensional friend a cube using this thing and Well and the problem with this is that this is a sphere. It's not a cube So first of all I have to get the cube onto the sphere So this is this is supposed to sort of illustrate this. So I got my torch now in the middle of a cube And it's projecting outwards to a sphere that's Centered on the cube this this is sort of I got a cube inside of a sphere and then I just sort of Blow it up Like a like a beach ball cube Hopefully that makes sense um, so so I do that and then Then I 3d print that it's all about 3d printing shadows So so here's that sort of beach ball cube and then I do the same thing again and now I've got another picture of this shadow over cube and This is my favorite way to cast shadow of cube in two dimensions. So so Well, what's so good about this? I mean Sure, it looks kind of similar to the last one that we did there's a square in the middle There's four squares around it. There's squares, right? The edges aren't even straight But here's a really really important thing. Where are the six squares? There's one at the bottom There's four around the outside. Where's the sixth square? It's not the same problem as last time It's outside Right. So so here's here's the light beam. It hits it starts on the On this square face really but just at the north pole and it goes along and it hits outside Here and it hits over here. So the top face actually goes outside So this is better And if you do the so here's another view and if you do the same thing to the hyper cube Then you get this thing So So this is a little bit better. I would say And I'll go into more Yes, it's all curvy which is a problem In some ways, but the fact that you don't have any crashing is really important. That will come up later So, um, let's see. Here's a render of the same thing which shows the square faces as well Unfortunately 3d printing technology isn't there yet with being able to do like the edges and sort of transparent faces When that does happen, that will be great. That will be awesome. But at the moment, it's not so great So here's a computer render instead, but you can see the cube in the middle the Six cubes around it and then we're actually inside of the eighth cube Just like in the previous one the top one goes outside. So it fills up the rest of the plane Back here. Whoops. Come on. Here we go. So the top face fills up the rest of the infinite plane and here Oops too far the the top cube fills up the rest of space and we're inside of it Okay, um, so that's stereographic projection Let me just say a couple more things because it's awesome There are these really pretty Things you can do. So this is like a this one here is a regular pattern on the plane and a weird pattern on the sphere As is this one. This is a hexagonal one These ones are nice regular patterns on the sphere that give pretty regular patterns on the plane as well I'll just mention, uh, maybe One thing about this that's super cool. So you can see here that this there's this sort of radiation symbol right on the front And the shadow down here is also a radiation symbol Um, and the angles are correct. So the lengths are all super distorted But the angles here these are 60 degree angles here and they're also 60 degree angles here Um, and that's true with like there's these, uh, I guess 10th of a circle. So that's 36 36 degree angles here the same thing here Another cool thing is there's all of these circles here which come from circles on anyway I like stereographic projection carrying on Let's take a step back for a second So when we made the the hypercube the sort of stereographic projection hypercube, I went by pretty quickly What did I say I said do the same thing one dimension up and you get that thing which is now going around So what does that mean? So hold on. Let's just go back a few slides So what do we do first? We took the cube Going from three dimensions or two. We took the cube We made a beach ball cube and then we stereographic projected that so beach ball cube is a cube that sits on the sphere So what did I do here? I take a hypercube it's sitting there in four dimensions And then after make a beach ball hypercube that's sitting on the sphere in four dimensions So I'm going to try and explain to you what the sphere in four dimensions is because it's kind of weird so And again The whole thing is like you go back down to down a dimensions try and understand it and you go back up again So what is a sphere a sphere is the set of points at some fixed distance from the center Right, so the ordinary sphere in three-dimensional space is the set of points which are equidistant from the center um Here it is and so This so we just saw with this stereographic projection thing that this is the same as the whole two-dimensional plane down here We're actually plus one point right, so There's one point on the sphere that doesn't get hit by a shadow And that's the very north pole itself right you don't get the north pole So you have to add one extra point up here. That's sort of infinitely far away Okay, so Plane plus a point Let me just mention while i'm here. So this is a different pattern here I've got like an equator to this sphere and I've got four triangles in the southern hemisphere And I've got four triangles in the northern hemisphere All right, and the shadows look quite different There's these four sort of triangle things down here And then there's these really stretching out things down here and they look really different But they're really the same thing And if you're sort of feeling unhappy about this thing being the same thing as this thing Just look back over here, and then it's obvious that they're the same thing. Yeah, okay Okay, now we do it one dimension up Hopefully there we go So the four dimension sphere in four dimensions space is the same as all of three-dimensional space plus one point Which is the north pole So this is sort of the same picture Of what the sphere in four-dimensional space is as this thing over here So over here, we had an equator. It was a circle here. The equator is a sphere Everything goes one up one dimension The southern hemisphere over here had four triangles The southern hyper hemisphere over here has eight tetrahedra Well, you can see there's a tetrahedron Four-sided a d4. Maybe that's better for this ground So I've got eight tetrahedra in the southern hyper hemisphere And then there's another eight tetrahedra in the northern hyper hemisphere And they look really different, but they're actually the same They look different in the same way that this triangle and this triangle look different This doesn't even look like a triangle, but it's a triangle, right? They're the same thing over here So I can't show you, you know the whole picture of where the light is because it's in four dimensions But I can show you the shadow Okay So right moving on so that's that's the sphere in four dimensions Let's talk about some other things that we can build in four dimensions and then see what they look like So So in two dimensions, we have the regular polygons, right? Triangle, square, pentagon, hexagon, and so on and so on and so on And these really nice regular symmetrical things In three dimensions, these are the only regular Nice symmetrical polyhedra Otherwise known as five of the six standard d and d dice So you've got the d4 down here, which is the tetrahedron You've got the cube the six-sided Dice and you have the octahedron the d8 The dodecahedron the d12 and the icosahedron the d20 and that's it There are no other sort of super symmetrical polyhedra in three dimensions Which is a little bit odd. There's infinitely many in two dimensions, but there's only five in three dimensions Um, in fact, there are other infinite families So we saw the poly the polygons the two dimensional ones is infinitely many of them There are three other infinite sequences, but they all skip dimension So we already saw this sequence in the middle before right here's a point and then a line segment and then a square This was the copy make a copy across connect up the edges So point line segment square cube hyper cube and it goes on there's a hyper hyper cube in five dimensions And six dimensions and so on and so on you just keep doing the same thing Then there are two other things you can do. So so here's another sequence Of these regular things and this is a slightly different Rule it looks the same in zero and one dimension, but then it changes What you do is you you have your thing A line segment you choose a new point that's A waft off the line segment and you connect up the edges and you get a triangle And you choose a new point that's not on the triangle It's not on the plane that contains the triangle and connect up the corners and you get a tetrahedron And then this sequence also continues onwards This is something called the five cell and then there's something else in six dimensions and so on and so on And then there's one other thing you can do here which is instead of making one extra point That's not on the thing you started with you make two So here is a line segment. I make two points one up here one down here I connect up the corners and I get a diamond Also known as a square And then I do the same thing again. I choose a point up up in front and I choose a point behind And I get an octahedron and then there's something called the 16 cell and this goes on and on okay, so four infinite families the polygons and then these three different ways of doing things And then there are these five exceptions These are the only exceptions to this rule. There's nothing else in any other dimension So there's a dodecahedron and the icosahedron which are weird There's the 24 cell which is this thing in Four dimensions, which is really bizarre. There's nothing like it in any other dimension at all And then there were two other things as the 120 cell and the 600 cell and these are both four dimensional polytips And these pictures are just what we did before right You really project onto the sphere in four dimensional space and then you Stereographically project to three dimensional space and you see these pictures whatever Is going on here so What's going on here? Well first of all We can 3d print these or rather half of it. So this is the half 120 cell. So I'll hand that around as well This is the bit that's in the southern hyper hemisphere because if you printed the whole thing It would be huge and it would be really super expensive and it wouldn't fit in my luggage Um So I should mention right 120 cell. What is that? Where does that name come from? So it's the same thing with the names of the polyhedra, right? So the dodecahedron why is it called dodecahedron? So dodeck means 12 and hedron means faces. So it's got 12 faces Well, this is one dimension up. It has 120 things has 120 120 cells. It's like a honeycomb So it's got a hundred and twenty three dimensional faces which are called cells And each one of them is a dodecahedron and then the 600 cell has 600 tetrahedra Anyway, so these things are kind of confusing So I wanted to try and understand this And 3d prints and more stuff. So that's what we did And I should say this is joint work with one of my collaborators, Saul Schleimer Who's at the university of warwick Here in england Okay, so it's this super complicated thing. What the heck is going on? Um, it has 120 of these dodecahedral cells You can see this sort of distorted thing out here And and there's a big one here. And again, we're standing inside of the one on top It's got 720 pentagons sort of between those dodecahedral cells It's got 1200 edges and 600 vertices. So this is super complicated. Um, Let's try and sort of break it down. What is this thing? So one way to try and understand it is to look at sort of the layers of these these cells going outwards um, so Here's a picture so In the middle of that 3d print that's going around somewhere There's a little small dodecahedron right in the middle And I've drawn it here in red and this is sort of a schematic picture of where it's sitting on the sphere in four-dimensional space So here's the light at the top and here at the south pole Is this thing here? okay, so then Arranged around that central one our 12th dodecahedra at well, okay It's at a distance, but distances on spheres you can measure with angles. So it's an add and distance quote slash angle of Pi over 5 are these 12 ones here There's a 20 of these at angle pi over 3 Another 12 at 2 pi over 5 building these layers out from the south pole And we get to the equator which remember as a sphere And there's 30 dodecahedra there at pi over 2 so 90 degrees um And if you sort of continue this pattern going further outwards the the pattern mirrors in the final four layers So there's another layer of 12 and then 20 and then 12 and then 1 if you add all these up you get 120 Um, okay, great Um, here's another way to understand Where how these dodecahedra arranged? so What you can do is it's starting one of these dodecahedra That's in this three-dimensional surface of the four-dimensional thing And you can say okay, I'm going to go out through one of the pentagonal faces of this thing And then I'm in another dodecahedra And I can just go straight through that to the opposite side and keep going So I go through go through this dodecahedron into this one and keep going and just keep walking through dodecahedra And after 10 dodecahedra, I get back where I started it turns out if you start again in another dodecahedron next to this one and Start going then you again get back after 10 um, and it sort of wraps you make this second ring that kind of wraps around the first one And there's a third ring a fourth ring And a fifth ring So this gray ring that we started with there are five rings I've not drawn in the last one. There's sort of like a death star trench along here that you can fly down um Let's put it in there. So there's one two three four five rings arranged around the central one So that's each one had 10. So this is six in total. So six times 10 that's 60 This is half of the 120 cell so This is half and then there's the seventh the eighth come on the eighth the ninth the tenth The 11th ring and as usual we're inside of last one So the last ring goes down through a hole in the middle of this and there's your 12 rings 12 times 10 is 120 okay so, um, we wanted to 3d print this because that's what we do Um, we wanted to 3d print. Okay, not everything because it would fill up the universe with plastic But we wanted to do the ring around here and then this the the five ones around it Um, so there's a problem here is because we wanted these things to be loose You wanted to be able to take it apart and put them back together again But when you put it in the 3d printer Your rings have to be separate from each other if they're touching then they'll just be stuck together when you're done Um, so so we had to sort of arrange them somehow I could only actually manage to get two rings to surround this central one I couldn't work out a way to to put in um, a third one Um, but we did that where is it here it is and you get this kind of uh, I don't know Kids rattle toy with the the the central ring and then the two rings around that and that around as well Um, and so this is them fitting together and they're slightly apart as well There we go um, so Okay, that's great, but we really wanted to have all five of them so Well, so what did we do we cheated so the problem the problem with 3d printing the two rings that are linked as you have to print them in one Um You can't print them separately and put them together because they're linked together So well, maybe we can just get rid of these big four things on the outside that make it really expensive anyway So let's just get rid of them And instead of a ring of 10 dodecahedra. We have a rib of Six dodecahedra And why do we call it a rib? Isis gently curving And it's made out of dodecahedra Anyway, um, so okay, so here's a rib. It's it's six of the original ring And then I can put in The other five right this the the sort of What would be rings except I've cut out the bits at the end Let's get rid of the Um central ring because we don't actually need it And there's this sort of cool ring thing Thanks And here it is And uh, so so we'd inadvertently made a puzzle because you print them separately And then you have to fit them together Um So just to prove that this is Really a puzzle. Let's take it apart Okay, let me try and Solve this live on stage somebody hold my mic for a second. Thank you. Uh, I've done this many times before if you break it Solve it Or don't worry about it because I can solve it. I've done it many times before Um, let's see. So That is one puzzle. So we ended up with um, let's see So here's another way to cut it up into dodecahedra. So Here's this sort of central Straight rib which we call a spine and then you can wrap things around that whoops Five and then another five of these ones and you get this um symmetrical kind of dodecahedral thing made out of these Uh 11 pieces here um And it turns out um, let's see so so We sort of saddle on like different ways to cut them up into into these ribs because you don't want them to get too big and go around the outside Um, but with these six things you can make this sort of the wieldering away array of stuff Um, so here's a little puzzle for you right now Uh, two of these are photographs of the same thing seen from different directions Any guesses as to which two they might be This is basically impossible. It's The last two these two. No, sadly. No Top left middle right. Yeah, this has got a hole through the middle whereas this one doesn't Yeah, this is essentially impossible. I've had I think one person ever guess it right and given how many times I've given this talk That's probably statistically what you would expect. In fact, there are two pairs that are the same Say again That one and that one that one and that one. Yeah, also wrong. Sorry Um This is the same as this And this is the same as this um So, okay So this so I'll try and show you the the second from the from the start one as this hexagonal whole thing And if I just turn it like that Then there's the cross So, I mean, what's the moral of this? Um, the moral is that photographs are useless And uh, I guess there won't be much time just after this, but I'm at the maths village So you should come by at some point and check them out. Um, okay So Next project Uh, so this was a joint work with vye hearts who you may have heard of from youtube and my brother will segment um, and this is Um, a sculpture called more fun than a hyper cube of monkeys Um, and this is about four dimensional symmetry So so before I get into the details of what earth is going on here Um, I need to tell you about symmetry So so there's lots of different ways of thinking about symmetry The sort of modern mathematical way to think about symmetry Is that a symmetry is a way to move something move an object so that when you're done it looks the same Right, so here's this sort of symmetrical thing And so I can move this so that it looks the same after I've done moving Right, I can turn it so There are five symmetries that this thing has I can rotate by a fifth of a turn Let's say this way Or two fifths of a turn or three fifths of a turn or so on and I'm also going to count the do nothing Like there's a there's as as often as the case in in maths There's usually like some trivial case where yeah, okay Yeah, fine. You could do nothing and that's also a thing that doesn't change it. Okay, and um I mean, this is sort of one of the ways into group theory abstract algebra Um, these symmetries these things you can do you can add them together by doing one movement and then another one So like I could rotate by one fifth of a turn and then I could rotate by two fifths of a turn And then doing those together is the same as rotating by three fifths of a turn Try saying that five times quickly So anyway, you can add these things together and then you get groups and so on. Okay, whatever Um, here's another example. So how many symmetries does this have? How many ways are there of moving this? So it looks the same afterwards four six eight Okay, so so the answer is eight So you can rotate it By one fifth what sorry one quarter two quarters three quarters Or you also also can't do nothing and then there are four different axes of Reflection symmetry. Okay So to describe what's going on with this monkey thing I have to go and talk about monkey blocks And those of you who know of vikart may see her influence here in the in the picture here So so so here's a cube That we've drawn things on top of and here's the unfolded cube. So you can see how it works So um, so there's uh two tail faces on this cube There's a a question mark tail And a sort of unquestion mark tail that goes the other way There's a right paw and a left paw the two Paw faces and then there are two face faces This one the tongue sticking out to the left and this one the tongue sticking out to the right and they're sort of Twisted somehow so when you put it together this thing has no symmetries at all Well, other than they do nothing symmetry. There's nothing you can do to this that leaves it looking the same As it did when you started other than just leave it the same thing Um, and I'll just mention one other thing. So you see there's like a a paw a right paw here and there's a sort of like a A sort of twisting motion to go from one side to the other and that's the same on the other ones as well This question mark here goes to a question mark here and then the faces as well Okay So how can you fit these things together so that the faces match up and by match up I mean that they should come together and be exactly the same. So like I would need a right paw to go to a left paw Okay, so here's one way you can fit them together You can stack them together in a line like this right, so I got a cube and then Right, so there's another paw on the other side that is So this this one is like this and the next one's going to be like this Which means it glues on to the one that's up there and then it sort of goes along like this Okay, so I can stack them together like this And then I can make an infinite line of blocks because I'm a mathematician And I don't have to practically produce an infinite number of blocks. I just have to say dot dot dot But anyway, okay, so I've got an infinite number of blocks What are the symmetries of this thing? What are the ways of moving this thing? So that looks the same when you're done What could I do? I could shift it four along to the side, right? So this guy is the same as this guy. Yeah, okay. I could do that. Is there anything else I can do? I could shift it backwards. Yes. Yes, this is true I can see some sort of rotation Yeah, yeah, so so that's what I'm after right So if I take this one up here and I go forward and rotate one click do this sort of screw motion Then it's the same question mark comes question mark And the the paw becomes the paw and the face becomes the face and so on. So this is sort of like Screw motion symmetry of this thing So I don't really like this go forward by four things So I'm going to get rid of it by just sort of curling them around into a circle of four of them Also helpful because I don't have to make infinitely many blocks And then I've still got this twisty thing, right? I can still go go forward one and twist one And that works So this one has only the the twist thing because I've sort of got rid of the go forward four by sort of wrapping it around Okay, um, I mean this seems like cheating, but it will make sense in a second What else can I do is the only is that can I so I've made a line of these things Can I sort of stack up and make a wall? Well, if you try doing this you want into trouble So so here's the line of three of them across here and if you put one on the top Then you get these two paws facing each other And there's no block that goes in there right because the paws on a on a block or on opposite sides I can't fit anything in there Although it does look I mean this poor here and this poor here if only I could close up those two faces Then they would fit together But then I would need something with one two three Cubes around this edge and what has three cubes fitting around an edge The hypercube has three cubes that fit around an edge so There's this cube in the middle and there's one on the top on the one on the side And then you see this three of them around this this edge here And so in fact you can glue these eight monkey blocks together to make the cells of a hypercube the eight cells of a hypercube fit together So you get this sort of decorated hypercube that has these things on it Um, this is what it looks like sort of exploded Um, so the next question is uh, what are the symmetries of this thing? Um, I guess I didn't I didn't put it down. But yeah, so the question is what are the symmetries of this thing? So I mean this is kind of drawn on here already. So there's this sort of Twisting thing here and there's a twist. Oh, well anyway Um, it turns out that the same twisting thing going around here works again With these other whoops going the other direction with these four going up vertically And then there's two other directions where you can do this whoops There's the one going through here and there's the one going through here Anyway, there's a lot of stuff going on It turns out that these Symmetries the symmetries of this decorated hypercube Corresponds to the eight elements of something called the quaternion group If people have maybe done some computer graphics, you may have heard of quaternions, which is sort of fancier versions of complex numbers They are not quite as scary as they're made out to be they're not that bad But anyway, so so if you take the eight quaternions one ijk minus one minus i minus j minus k Then these correspond to the eight different ways you can sort of move this thing around and leave it looking the same So one is to do nothing Symmetry ij and k are these screw motions where you go forward and you twist The negatives of those are the reverse screw motions and then there's minus one which sends every cube to its opposite cube Um, and these satisfy the relations of the quaternion group and so you get the quaternions out of this um Okay, great How do I make a sculpture it has this symmetry so well all I have to do is Put something in each cell of the hypercube which has no symmetry So so the monkey block had no symmetry and I ended up with something that had this weird eight element group symmetry thing and I should mention by the way Whatever this group is it doesn't exist in three dimensions at all There's no object you can hold in your hand so that when you sort of rotate it around using ordinary three dimensional stuff You you can do anything so that you get that group So as far as we know This is the first time that anybody's sort of made a picture of something that has this kind of symmetry And you had to go to four dimensions. Anyway, we're getting that So so the block the monkey block has no symmetry and I want to put some design inside of this block of the hypercube Okay, okay, it has to do nothing symmetry. Everything has to do nothing symmetry, but other than that To make a sculpture with this symmetry type You put something which has no symmetry inside of this cube and there is only one choice It has to be a monkey I'm actually mostly serious there. So here's a monkey sitting inside of a cube And one of the big advantages of monkeys is that they have six limbs if you include the tail and the head Which means that you can connect this guy Onto his neighbors in the other eight. Sorry the other seven cubes of the hypercube. Here's this guy. He's surrounded by One two three four five six of his neighbors and they all connect onto each other and you need limbs to do that Um, why are we doing something with limbs? Well, the thing itself can't have any symmetry if I put something symmetrical inside of this cube Then the whole thing would have too much symmetry. There'd be extra symmetries there So I don't want that. So if it's got to be something non-symmetrical, it might as well be a figurative thing So I guess that means we're going to use a monkey. So there we go. Okay, so it's a monkey and we put his neighbors into all the Other cubes and then we do exactly the same thing as before right, so When I was making this picture of a hypercube, what did I do? I started with The edges of a hypercube I Made a beach ball version of the hypercube and then I stereographically projected to three-dimensional space Now I have a design inside of each cell of the hypercube and I make a beach ball version of that design by projecting it onto the sphere in four-dimensional space And then I stereographically project that down to three-dimensional space and I get this sculpture Um And there you go. So there's a web link here we'll get to in a second We didn't stop there, right? So we've sort of done this with the hypercube, but there were other four-dimensional versions of the polyhedra there. So we can also do More fun than a 24 cell of monkeys, which has 24 monkeys and we can also do More fun than 120 120 cell monkeys again This is only half of it because again, it would be way too expensive. Let's go check out what this website is about So this is a Animated version So if you go to monkeys.hypernome.com This is web web vr. So if you have an oculus rift headset or probably a vive You can see this in glorious four-dimensional stereo vision Um, so the the controls are not documented, but w ast as you might imagine and some other things to do with the arrow keys Okay, so Let's see what's going on here So so this is showing that symmetry, right? So every few seconds These lighter colored monkeys are rotating around and this one becomes this one And there's this sort of twisting motion and everything is doing exactly the right thing Um Maybe if I look upwards You can see that there's actually monkeys that go around infinity So there's sort of a monkey god That sort of gets smeared out around everything There we go, okay Let's see. So here's More fun than a 24 cell of monkeys same sort of thing this time you get a ring of six Uh, you get rings of six monkeys. So the six lighter colored monkeys that are twisting around You may be aware that the 24 cell is self-dual what that means Well, um, what it means is I can fit another 24 monkeys around The first so there's the first one and then there's another set of 24 monkeys that fit in between those ones And let's go all the way to the 120 cell of monkeys. There we go This is sort of monkey cosmology um, again, there's um This time this rings of 10 monkeys going around here Um, if you're going to do this at home, you just press the number keys to switch between the different kinds of symmetries and the different number of monkeys Okay, oh and that ring of 10 monkeys is exactly the same as the rings of 10 dodecahedron the puzzle from before Of course um, okay So I'm almost done. I've got a couple of things to say. So I have a book coming out It's called visualizing mathematics with 3d printing available soon I'll just mention one cool thing about this so um So the book has all these figures and the figures are photographs of 3d printed things And you can go to the website 3d print math.com and you can download the 3d files You can print them on your own printer. You can rotate them around on screen. You can order them from shapeways Um, why does the biology textbook not come with a 3d printed model of dna or the shape of the heart? Right, if you're going to represent 3d content Why are you using photographs like 3d print something? um, okay There's um a few things. There's one other thing that I wanted to show you which is another app um Which is on an iPad. So it will take me a second to switch over So talk amongst yourselves I see is this working? Do I have Do we have input? No Not yet. It's not on the shall I unplug and replug? try again Let me unplug it and replug it and see if that makes a difference I didn't warn them that I was going to do this. So I'll try unplugging and replugging Something's happening No, let me try unplugging and replugging. Otherwise, I'll just hold it up so you can see what's going on Okay, it's plugged in This usually works Okay, well, I'll just hold this up so you can see it. No nothing Okay All right, so this is an iPad app Or android or again, um, if you prefer You can use a vr headset And as I say this and the the monkeys app was joint work with my heart and uh, andrea hawksley and mark 10 bosh So what is going on here? So you can see as I move the iPad around There's dodecaedra sort of Wandering around these are the same dodecaedra is all along. These are the dodecaedra of the 120 song But you can see that I'm interacting with this by moving the iPad if you want to go to um So if you go to hypernome.com on your smartphone or Otherwise gyroscopically enhanced device then you can check this out for yourself. It's just it's not a download It's just actually on the website So what's happening here is that we're navigating through the 120 cell by moving the iPad so This is a little hard to um comprehend but roughly speaking what's happening is The set of possible orientations of the iPad is some three-dimensional space right? I can sort of roll or I can your or I can tilt up and down so there's three three dimensions of ways you can move this thing and It turns out that that space is very closely related to the sphere in four-dimensional space, which is also a three-dimensional thing And so you can use one to navigate the other. So that's what's going on here. Let me just uh, oh wow this is uh Unusual Let me just try to read we see here we go That's better. Okay, so So as I twist this thing to the right you can see that I'm sort of going forwards And when I get close to these Dota key drive eat them So this is why the name is hypernome. This is hyper as in hypersphere and nom as in nom nom nom So this is four-dimensional pac-man um The aim of the game is to eat all the cells of the 120 cell and you do this by getting the iPad into every orientation possible In fact twice because there's a double cover going on um, so maybe I'll just finish by Giving you this image. Um, this is originally developed for oculus rift so You have to imagine not doing this, but it being strapped to your face and so instead you have to do this kind of thing And you try and eat all the cells of the 120 cell and it's timed as well. So if you want to compete on Getting very sick from motion sickness as quickly as possible. You can do that as well Okay, I think I'm probably out of time. So thank you very much