1 00:00:12,320 --> 00:00:23,840 Well, thank you for the wonderful introduction, and it's great to be back in Oxford, so so yes, so I'm a mathematician and also a mathematical artist. 2 00:00:23,840 --> 00:00:28,400 And maybe those seem like a sort of strange combination of things to be doing. 3 00:00:28,400 --> 00:00:33,560 And hopefully, I'm going to show you how there are more connexions than you might think. 4 00:00:33,560 --> 00:00:40,070 For that matter, what is it the mathematicians do anyway? Maybe you know what artists do. 5 00:00:40,070 --> 00:00:44,870 We're not sitting around doing really big sums all of the time, or at least not all of us. 6 00:00:44,870 --> 00:00:50,180 So let me sort of start off by saying just a little bit about what my mathematics looks like. 7 00:00:50,180 --> 00:01:00,770 So I'm an apologist. This is more or less typical page of my research work, and you might notice that there's not many numbers here at all. 8 00:01:00,770 --> 00:01:08,810 It's mostly about sort of surfaces intersecting in space and some sort of complicated configurations and so on. 9 00:01:08,810 --> 00:01:11,570 So there's obviously a visual component to what I'm doing here. 10 00:01:11,570 --> 00:01:18,560 And maybe you'd think that's a connexion between mathematics and the arts, but I think there's a stronger connexion, 11 00:01:18,560 --> 00:01:27,230 which is that both mathematics and artists are sort of allowed to and have to think about fictional worlds. 12 00:01:27,230 --> 00:01:36,470 So we set up some well that we're going to explore and we see what the consequences are, and we just had nothing to do with the real world. 13 00:01:36,470 --> 00:01:37,400 It doesn't matter. 14 00:01:37,400 --> 00:01:45,590 Unlike, say, a scientist or an engineer, which maybe has to have some passing acquaintance with what's happening outside of in the real world. 15 00:01:45,590 --> 00:01:55,200 So we're allowed to explore these fictional what if questions and then if we come back with something that sort of is useful for the real world? 16 00:01:55,200 --> 00:02:05,850 Great. If not, it's not a big deal. So let me start talking about my work in in mathematical visualisation and artwork. 17 00:02:05,850 --> 00:02:09,210 And so I'm pretty much just going to show you a whole bunch of different things. 18 00:02:09,210 --> 00:02:15,420 And if something doesn't make sense, don't worry, I'll be moving on to something else soon enough. 19 00:02:15,420 --> 00:02:21,360 And lots of these things that I'm I'm doing a joint work with other people. 20 00:02:21,360 --> 00:02:25,920 I won't always point out their names, but they'll always be on the side somewhere. 21 00:02:25,920 --> 00:02:29,830 So, so let's start with this one. This is. Well, what is this? 22 00:02:29,830 --> 00:02:34,500 This is a 3D print. It's a 3D printed sculpture of something from my work in topology. 23 00:02:34,500 --> 00:02:41,220 So what you're supposed to imagine here is that you've got sort of a an apple or a ball of dough or something like this. 24 00:02:41,220 --> 00:02:49,290 And you have an extremely well-trained worm that you've trained to tunnel through this ball and in this sort of knotted fashion. 25 00:02:49,290 --> 00:02:53,130 And and so what you have left is something called the complement of the Figure eight knot, 26 00:02:53,130 --> 00:03:00,630 which is one of these is sort of this key core example in in three dimensional geometry entomology. 27 00:03:00,630 --> 00:03:07,560 So so that's a sort of transition between mathematics and and and this sort of visualisation. 28 00:03:07,560 --> 00:03:09,550 Let's go back a little bit back to basics. 29 00:03:09,550 --> 00:03:15,930 You know, if you just want to start doing sort of three dimensional work, you might start with the platonic solids, 30 00:03:15,930 --> 00:03:21,810 maybe a little bit more complicated than the platonic solids would be the Ahmadi and solids. 31 00:03:21,810 --> 00:03:28,230 So the platonic solids right you you have these solids made out of triangles or squares or pentagons, 32 00:03:28,230 --> 00:03:33,120 and it's all the same thing that you see everywhere. It's all made out of triangles or it's all made out of squares, 33 00:03:33,120 --> 00:03:38,490 and the committee and sides are a little bit more complicated relative to different kinds of face. 34 00:03:38,490 --> 00:03:43,200 And so maybe you start thinking about Paul here and you say, OK, this is all very nice. 35 00:03:43,200 --> 00:03:49,680 I've asked my what if questions, is there something I can bring back to the real world? So here's something you could do with some poly hydra. 36 00:03:49,680 --> 00:03:56,530 So we were the first to produce mass mass produce one hundred and twenty sided dice. 37 00:03:56,530 --> 00:04:01,540 And so I've got a lot of stuff here and I'm going to be passing it all around. So I guess we'll start here. 38 00:04:01,540 --> 00:04:05,760 I think you'll you'll be getting all of my things first. So lucky you. So. 39 00:04:05,760 --> 00:04:12,690 So this beyond the shape of of the dice here, there's sort of an interesting mathematical story with the numbering. 40 00:04:12,690 --> 00:04:16,080 So how do you put the numbers on a die with 120 sites? 41 00:04:16,080 --> 00:04:19,800 Well, obviously opposite numbers have to add up to 121, right? 42 00:04:19,800 --> 00:04:23,460 That's clear, right? Opposite number. 43 00:04:23,460 --> 00:04:28,260 Why is it that opposite numbers add up to seven on an ordinary cubicle dies? 44 00:04:28,260 --> 00:04:32,220 So it's sort of an interesting story. What's the one reason? 45 00:04:32,220 --> 00:04:33,720 It's not really clear, 46 00:04:33,720 --> 00:04:42,690 but one one reason you could come up with is that an easy way for a die to be unfair is that if it's sort of squished down just a little bit, 47 00:04:42,690 --> 00:04:49,830 so it's a little bit more like a coin than a than a die. And if if it is, then it's much more likely to land on the big faces and small faces. 48 00:04:49,830 --> 00:04:57,990 But if you have the opposite sides adding up to the same number for all pairs of opposite sides, then at least it won't roll high or low. 49 00:04:57,990 --> 00:05:02,760 So it's a sort of dealing with manufacturing defects to get a fair result. 50 00:05:02,760 --> 00:05:09,480 And so we wanted to do a similar thing here. Obviously, opposite opposite sides should be adding up to a hundred twenty one. 51 00:05:09,480 --> 00:05:13,420 But that doesn't really cut down the possibilities. Very much so. 52 00:05:13,420 --> 00:05:19,830 So what we said is, OK, let's suppose that it was you knew that was more likely to land on one of these 10 faces around this vertex. 53 00:05:19,830 --> 00:05:22,980 You would still like it not to be rolling high or rolling low. 54 00:05:22,980 --> 00:05:26,490 And the way to make that happen is that you need that the sum of all of the numbers around 55 00:05:26,490 --> 00:05:32,250 each vertex are would be 10 has to be the same and it is for this arrangement of numbers. 56 00:05:32,250 --> 00:05:36,360 And it's also true that the sum of the numbers around each vertex are to be six is the same. 57 00:05:36,360 --> 00:05:40,500 And it's also true that the sum of the numbers around each vertex degree four is the same. 58 00:05:40,500 --> 00:05:44,940 So that's a pretty difficult problem to make all of that happen. 59 00:05:44,940 --> 00:05:50,370 There's things like what is an integer linear programming problem? Bob Bosch is at Oberlin College. 60 00:05:50,370 --> 00:05:58,710 He's an operations research, and he spent about a month of computer time and was very lucky to find this numbering more dice. 61 00:05:58,710 --> 00:06:04,950 So here's some more dice. These are strangely shaped dice. 62 00:06:04,950 --> 00:06:12,630 So your ordinary, six sided die is a cube, and that has, it turns out, an excessive amount of symmetry in order to be fair. 63 00:06:12,630 --> 00:06:15,030 So the thing that you need in order for a die, to be fair, 64 00:06:15,030 --> 00:06:20,760 is that all the faces of the same and the symmetries of the die take any face to any other face. 65 00:06:20,760 --> 00:06:28,620 And so this one has less symmetry, but it's just as fact. It also wins people out because it looks like it's melting or something. 66 00:06:28,620 --> 00:06:32,340 You can do the same thing with 12 sided dice. 67 00:06:32,340 --> 00:06:38,130 Somehow, this is less impressive than the six sided dice because people don't really know what their ticket will look like anyway, 68 00:06:38,130 --> 00:06:42,260 so it's hard to tell if they're not the right shape. OK. 69 00:06:42,260 --> 00:06:50,010 Anyway, onto onto other 3D printed things. So I was in Australia for three years as a post-doc and as a university. 70 00:06:50,010 --> 00:06:55,470 I called Swinburne University of Technology and they had a whole bunch of money left over at the end of the financial year, 71 00:06:55,470 --> 00:07:00,310 and so they decided to blow it all on. Some big 3D printed models. 72 00:07:00,310 --> 00:07:05,050 These are the kind of sort of classic examples from multivariable calculus. 73 00:07:05,050 --> 00:07:11,140 And you'll notice next to them these much smaller versions, which are just as good at explaining things but much cheaper. 74 00:07:11,140 --> 00:07:15,220 So I use these in my classes when I teach calculus. And so. 75 00:07:15,220 --> 00:07:20,420 So this one over here you write down the equation. Something like z equals x squared minus y squared. 76 00:07:20,420 --> 00:07:26,980 And if you just see the equation, it's difficult to know what it looks like or you know, anything about it visually. 77 00:07:26,980 --> 00:07:34,950 And this sort of helps helps my students understand what's going on. Let me move on from one sphere to another sphere. 78 00:07:34,950 --> 00:07:44,790 This is a self-referential sphere. So maybe you can see I've highlighted in red, there's an SE here, a p h e r e. 79 00:07:44,790 --> 00:07:48,780 So this is a sphere tiled with the words sphere 20 copies of the sphere. 80 00:07:48,780 --> 00:07:56,040 There's some interesting symmetry here. Here's a bunny tiled with, I think, seventy two copies of the word bunny. 81 00:07:56,040 --> 00:08:02,670 Let's see if I can point this out. There's a bee here. You and and why. 82 00:08:02,670 --> 00:08:08,340 So this is. Well, it's this more of a sort of computer science problem than a mathematics problem. 83 00:08:08,340 --> 00:08:13,170 But anyway, there we go. So I said, I'm with apologist. 84 00:08:13,170 --> 00:08:17,250 So there's a standard joke about its apologists that they can't tell a coffee 85 00:08:17,250 --> 00:08:23,430 mug from a doughnut because you can continuously deform one into the other. 86 00:08:23,430 --> 00:08:31,710 So the real sort of mathematical difficulty here is, OK, I know maybe it's easy enough to decide what shape a doughnut is, right? 87 00:08:31,710 --> 00:08:40,500 You can write down some sort of equations to describe this shape, and maybe you can use maybe a computer modelling programme to make a coffee mug. 88 00:08:40,500 --> 00:08:44,160 But how do you deform continuously in some nice way from one to the other? 89 00:08:44,160 --> 00:08:49,270 So this is joint work with Keenan Crane, who really did most of the work here. 90 00:08:49,270 --> 00:08:53,350 The words conformal Willmore Flow may make sense to some people in the audience. 91 00:08:53,350 --> 00:09:03,280 This is what it's about. So moving on to other topological things, so not very common things that technologists think about. 92 00:09:03,280 --> 00:09:06,640 So I've got this 3D printed, not here. 93 00:09:06,640 --> 00:09:11,560 This is the trefoil. Not just like the knot you'd make in your shoelaces, except that there's no ends. 94 00:09:11,560 --> 00:09:13,780 You sort of join it together. 95 00:09:13,780 --> 00:09:19,990 This note has a very interesting property, which is that it rolls and you have all of these wonderful desks in front of you, 96 00:09:19,990 --> 00:09:25,750 so you'll be able to try this out for yourself. Why does it roll? What's the sort of mathematical thing going on here? 97 00:09:25,750 --> 00:09:30,490 So this shape of knot has no try tangent. 98 00:09:30,490 --> 00:09:37,600 This is the way to say it. So what's the try tangent plane? It's a plane which is tangent to the knot in three different places. 99 00:09:37,600 --> 00:09:42,490 If it was, if you could sort of hold a plane up to the knot and it was tangent in three places, 100 00:09:42,490 --> 00:09:47,580 then it would make a little triangle and it would be stable on that triangle. And this one only has it most too. 101 00:09:47,580 --> 00:09:51,760 So, yes, it knows to, which means that it's always it's always got somewhere that it can. 102 00:09:51,760 --> 00:09:57,760 It can roll. And as far as I know this, the truffle is the only knot which has this property. 103 00:09:57,760 --> 00:10:05,680 If anybody happens to know of another knot which has this property, let me know so I can make a rolling version of it. 104 00:10:05,680 --> 00:10:09,080 Here's another rolling thing at slightly larger scale. 105 00:10:09,080 --> 00:10:17,440 So, so Li Branswell is in the theatre department at my university, and he's also interested in circus apparatus. 106 00:10:17,440 --> 00:10:25,900 So there's this class of circus performance where you get inside of a big metal thing and you roll around on the stage, 107 00:10:25,900 --> 00:10:30,820 and I was asked a question by a performer friend of mine. 108 00:10:30,820 --> 00:10:37,150 So there's these various things that you can roll around inside of, and they all have the property that, I mean, you might wiggle back and forth, 109 00:10:37,150 --> 00:10:41,980 but essentially you go in a straight line and then you hit the edge of the stage and you have to sort of go backwards. 110 00:10:41,980 --> 00:10:49,240 And the question was, can you come up with some design which lets you keep going forwards but somehow curve around and keep on stage? 111 00:10:49,240 --> 00:10:56,440 And so I'm not a very good acrobat. This guy is, so I'm going to show you what this design actually does. 112 00:10:56,440 --> 00:11:03,280 So this points in the motion. This is the view from the inside. Maybe you can see on here there's a GoPro attached to it. 113 00:11:03,280 --> 00:11:05,650 And this is, of course, the view from the outside. 114 00:11:05,650 --> 00:11:14,180 So the and points in the motion where he can lean one way and switch from rolling in a straight line to sort of shifting from one line to another. 115 00:11:14,180 --> 00:11:20,380 And so this was an attempt to try and fix this problem, which went somewhere else an attempt anyway. 116 00:11:20,380 --> 00:11:26,020 Interesting thing and more cars in space. 117 00:11:26,020 --> 00:11:28,930 So this is a soap film on a frame. 118 00:11:28,930 --> 00:11:36,790 So this is the mathematical area here is minimal surfaces, surfaces which sort of are pulled tight as as much as possible. 119 00:11:36,790 --> 00:11:42,820 One of the nice things about working in 3D printing as opposed to, say, computer graphics, 120 00:11:42,820 --> 00:11:46,180 is you end up with a real world objects and with a real world objects, 121 00:11:46,180 --> 00:11:55,340 you can interact with other real world things, for example, soap, soap, film, and so you can make these these nine surfaces. 122 00:11:55,340 --> 00:12:02,950 The not that I showed you earlier, you can also make a nice, minimal surface on the surface with something called a soft surface. 123 00:12:02,950 --> 00:12:08,750 So a soft surface is the surfaces boundary is a given not. 124 00:12:08,750 --> 00:12:14,670 And you get this this beautiful shape here. 125 00:12:14,670 --> 00:12:21,180 They can get a lot more complicated these type of services, but unfortunately it's difficult to get soap films to do the right thing. 126 00:12:21,180 --> 00:12:24,340 But instead you can 3D print the right thing so. 127 00:12:24,340 --> 00:12:33,030 So these are sort of various examples of complicated services that would be too complicated to to make the the the the soap film do the right thing. 128 00:12:33,030 --> 00:12:38,070 But we can calculate with various scary terms like fractional automotive. 129 00:12:38,070 --> 00:12:44,500 It forms what the shape should be and then we can generate these these things. 130 00:12:44,500 --> 00:12:52,570 So onto another collaboration, this was a collaboration with Mark O'Mara is a he's a an artist who makes mobile 131 00:12:52,570 --> 00:12:58,870 so big sort of things that you put up in a hotel lobby usually made out of metal. 132 00:12:58,870 --> 00:13:04,390 And we looked into making 3D printed versions of these things, and it's sort of nice to be able to. 133 00:13:04,390 --> 00:13:10,600 You can calculate. OK, so so what does Marco do when he's building this kind of mobile out of metal? 134 00:13:10,600 --> 00:13:16,930 He starts from the bottom. So you've you've built up this file here, and then you need to decide where does the everything down here? 135 00:13:16,930 --> 00:13:20,680 Where does it balance? Because that's where you want to put the hook for the next level up. 136 00:13:20,680 --> 00:13:28,360 And so, so we did something similar, except you can calculate with various things where to put the hook and then you print it out and it works, 137 00:13:28,360 --> 00:13:35,860 and he doesn't have to spend ages sort of balancing things. Here are some other examples from that same collaboration. 138 00:13:35,860 --> 00:13:39,400 This is a binary tree. I guess this is more computer science. 139 00:13:39,400 --> 00:13:45,460 This is a ternary tree when it's split into three at each time, each level as it goes down to Quaternary Tree. 140 00:13:45,460 --> 00:13:49,150 If you look at the bottom of the ternary tree, you see this happens to you, triangle. 141 00:13:49,150 --> 00:13:57,730 So to move into fractals, I want to show you some more fractals to explain what the fractal is before I get there. 142 00:13:57,730 --> 00:14:02,590 So we're going to start with a straight line and then we're going to replace the straight line with a zigzag. 143 00:14:02,590 --> 00:14:06,490 And then each straight line here, we're going to replace with the same zigzag, right? 144 00:14:06,490 --> 00:14:10,270 So straight line goes to zigzag. And this one becomes a zigzag. 145 00:14:10,270 --> 00:14:15,610 I smooth it off a little bit. But it's pretty much going on like this. 146 00:14:15,610 --> 00:14:21,100 And so usually you might see this. This construction is sort of an animation like I've shown you here. 147 00:14:21,100 --> 00:14:28,250 And the idea here was to instead of doing an animation through time, you can do an animation through space, which looks something like this. 148 00:14:28,250 --> 00:14:35,740 You get this beautiful, crinkly surface. Here's a big 3D printed version of this. 149 00:14:35,740 --> 00:14:45,100 This was at an art show at the Simon Centre for Geometry and Physics at Stony Brook in New York, and we had this set up with this mirror bellows. 150 00:14:45,100 --> 00:14:49,870 You could see the squiggly thing on the underneath, as well as as seen as sort of the curtain shape. 151 00:14:49,870 --> 00:14:56,410 And here are some other designs the same sort of idea of it based on different space filling curves. 152 00:14:56,410 --> 00:15:05,710 These these these constructions are. This is maybe the most familiar or the most famous one is the Hilbert curve turns into this kind of square. 153 00:15:05,710 --> 00:15:09,700 This one is the dragging curve. Pinsky, our head curve. 154 00:15:09,700 --> 00:15:15,130 One of the interesting things about working in this sort of field is that, you know, why am I doing this? 155 00:15:15,130 --> 00:15:19,450 I'm doing this to say, here's this mathematical thing, this abstract thing. 156 00:15:19,450 --> 00:15:23,710 Let me try and sort of drag it out of the abstract world and make it sort of physical. 157 00:15:23,710 --> 00:15:29,170 And, you know, and as accurate as a way as possible, realise it. 158 00:15:29,170 --> 00:15:36,820 But whenever you put something out into the world, people can interpret it however they want and bring some other interpretation. 159 00:15:36,820 --> 00:15:41,940 So somebody somebody described this one to me as as a skyscraper going for a walk. 160 00:15:41,940 --> 00:15:51,010 It's sort of here's another one that people sometimes have strong reactions to this sort of ask. 161 00:15:51,010 --> 00:15:54,970 Monkeys with these weird, distorted limbs. 162 00:15:54,970 --> 00:16:01,930 But really, this is just a sculpture which is showing a particular kind of symmetry that comes in four dimensional space. 163 00:16:01,930 --> 00:16:07,960 So there is these sort of there's a ring of four monkeys going around here and there's another ring of four monkeys going around here, 164 00:16:07,960 --> 00:16:13,870 and this is symmetric if you're able to sort of see what's going on in four dimensions. 165 00:16:13,870 --> 00:16:18,640 This is based on the geometry of the Hypercube. 166 00:16:18,640 --> 00:16:27,250 Here's a similar sort of structure built on top of a different four dimensional poly headroom, which is built on the twenty fourth cell. 167 00:16:27,250 --> 00:16:31,150 Oh yeah, this one's called more fun, more fun than a hypercube of monkeys. 168 00:16:31,150 --> 00:16:38,500 This one's called more fun than a 24 set of monkeys, and this one's called more fun than a hundred and twenty twelve monkeys. 169 00:16:38,500 --> 00:16:44,290 There's a series, and there's also an interactive virtual reality version of this. 170 00:16:44,290 --> 00:16:47,980 This is by heart who you may recognise from her work on YouTube. 171 00:16:47,980 --> 00:16:53,050 If you go to monkeys, don't happen on the come on your laptops afterwards, not during the lecture. 172 00:16:53,050 --> 00:16:58,090 Then you will see this, which shows the the symmetry. 173 00:16:58,090 --> 00:17:06,700 So every few seconds this monkey here is becoming this monkey is becoming this monkey that's doing the symmetry sort of twisting screw motion, 174 00:17:06,700 --> 00:17:11,970 which is the symmetry that doesn't exist in three dimensional space but does exist in four dimensions. 175 00:17:11,970 --> 00:17:16,720 If I press various keys and I had any internet, then I would see other things. 176 00:17:16,720 --> 00:17:20,500 Oh, there it is. Yes, so there's the there's the 24 cell version. 177 00:17:20,500 --> 00:17:28,900 It turns out with 24 cell is self dual, which means that you can fit another twenty four monkeys in the gaps left between the first set of 24 monkeys. 178 00:17:28,900 --> 00:17:35,980 And then this is the one hundred and twenty cell version with lots and lots of colourful monkeys. 179 00:17:35,980 --> 00:17:42,190 And again, it's a different cemetery group in four dimensions, but you can see monkeys become moving to be over the monkey. 180 00:17:42,190 --> 00:17:47,940 And so on and so forth. OK, so what next? 181 00:17:47,940 --> 00:17:52,730 I guess what makes a good visualisation, so I'm going to turn down the lights for a second. 182 00:17:52,730 --> 00:17:56,330 And let's see, does this do it? 183 00:17:56,330 --> 00:18:00,890 Here we go. No, that doesn't do it at all. How about like that? 184 00:18:00,890 --> 00:18:11,780 There we go. So. So I'm going to show you one one of, I think, maybe my most successful examples of visualisation here. 185 00:18:11,780 --> 00:18:15,890 So I've got this 3D printed sphere with a strange pattern on it. 186 00:18:15,890 --> 00:18:23,240 And the trick here is that if I put the light in exactly the right place. 187 00:18:23,240 --> 00:18:32,000 Then the curvy grid becomes this beautiful regular, if I could get it in the right place grid. 188 00:18:32,000 --> 00:18:39,170 Now let me turn the lights back on again and show you a much better photograph of the same thing. 189 00:18:39,170 --> 00:18:44,780 So, so, so so again, this is using this, this idea, you know, it's a 3D print. 190 00:18:44,780 --> 00:18:49,850 You can combine it with other real world things and get interesting effects. So. 191 00:18:49,850 --> 00:18:58,100 So first of all, this sort of a wow effect, right? There was a surprise. How did this curvy thing become this straight grid? 192 00:18:58,100 --> 00:19:03,170 And then maybe that draws the viewer in, and maybe you get to teach them some mathematics, which is the real point. 193 00:19:03,170 --> 00:19:10,520 What is the mathematics, by the way? I have to tell you that. So so this is showing something called stereo graphic projection. 194 00:19:10,520 --> 00:19:14,030 Stereo graphic projection is a map from the sphere to the plane. 195 00:19:14,030 --> 00:19:20,030 And the way that it works is you trace the light rail from the north pole of the sphere it hits. 196 00:19:20,030 --> 00:19:24,290 It goes inside the sphere, it hits the sphere somewhere and it continues on down to the plane. 197 00:19:24,290 --> 00:19:29,010 And the map is just where does it hit on the sphere to where the where does it hit on the plane? 198 00:19:29,010 --> 00:19:34,190 So I could write down a formula for this, and the formula is not even that complicated, 199 00:19:34,190 --> 00:19:37,520 but you don't need to know the formula to understand what this map is. 200 00:19:37,520 --> 00:19:44,750 You can just sort of understand it visually so that the mathematics here is is correct. 201 00:19:44,750 --> 00:19:50,810 Of course, I mean, there's no point in doing a visualisation if the mathematics isn't correct. 202 00:19:50,810 --> 00:19:54,920 And there's something else that I really like about this, which is to do with the medium here. 203 00:19:54,920 --> 00:19:59,180 So, so again, it's a it's a it's a 3D prints and a torch. 204 00:19:59,180 --> 00:20:03,830 And you know, you could make the same sort of image using computer graphics. 205 00:20:03,830 --> 00:20:07,430 But if you make it with computer graphics, then people are sort of rightly suspicious, right? 206 00:20:07,430 --> 00:20:11,480 You can make anything look like anything with with enough computer graphics. 207 00:20:11,480 --> 00:20:17,850 But this is a piece of plastic in a torch like this is really happening with something actually going on here. 208 00:20:17,850 --> 00:20:21,690 Last thing to say about this. This was not an easy photograph to take. All right. 209 00:20:21,690 --> 00:20:28,410 So you might have seen as I was sort of trying to lighten things up. A tiny movement in the in the flashlight produces a big change in the shadow. 210 00:20:28,410 --> 00:20:35,970 So how did I make this photograph right? I always had my hand here very carefully moving that, and I'm also taking the picture with my other hand. 211 00:20:35,970 --> 00:20:41,160 What's actually going on here is that the the torch was taped onto a rod and there's a cross beam 212 00:20:41,160 --> 00:20:46,830 going across the top and there's a clamps down either side and the hand is purely decorative. 213 00:20:46,830 --> 00:20:53,100 It's just sitting there trying to not move the flashlight and trying to convince you that it isn't a computer render right, 214 00:20:53,100 --> 00:20:57,580 that this is real because this is a problem that I run into. 215 00:20:57,580 --> 00:21:05,040 The 3-D printing is so accurate. And of course, the light rays travel in pretty much straight line, so it just looks like a computer. 216 00:21:05,040 --> 00:21:14,220 So you can do lots of different designs, so all kinds of different things that you can project. 217 00:21:14,220 --> 00:21:23,490 These were a fun project to illustrate different relationships between the different models of the hyperbolic plane. 218 00:21:23,490 --> 00:21:26,580 So the 3D print here is this hemisphere. 219 00:21:26,580 --> 00:21:32,310 And if you sort of do the stereo graphic projection down here, you get the Poincaré disc model, which is maybe the most familiar of these. 220 00:21:32,310 --> 00:21:34,680 You've seen it in actual prints and so on. 221 00:21:34,680 --> 00:21:40,350 If you raise the light up very high, so the light rays coming down parallel, you get the Klein model of the hyperbolic plane. 222 00:21:40,350 --> 00:21:46,940 And if you put the light on the equator of this hemisphere, then you get the upper half plane model of this. 223 00:21:46,940 --> 00:21:53,840 So we took some of these ideas and ran an exhibition in Edinburgh in 2017, I think it was. 224 00:21:53,840 --> 00:22:02,270 So we had a globe. It was all very interactive. You could rotate the the globe around to put your favourite country in the middle of the projection. 225 00:22:02,270 --> 00:22:11,210 This is sort of a full media version of the grid. We had a room full of various timings and this was the sort of finale. 226 00:22:11,210 --> 00:22:18,890 This is a zoetrope. How many people know what a zoetrope is? Some people zoetrope is sort of the first version of of movies. 227 00:22:18,890 --> 00:22:25,760 So what's going on here is that there's a disk that's spinning around very quickly and there's a strobe light going on, 228 00:22:25,760 --> 00:22:31,310 going off every time one of these 30 models I hear and it's spinning once a second. 229 00:22:31,310 --> 00:22:37,070 So every time one of the models comes to the right place, it flashes the stripes. You always see the right, the right frame. 230 00:22:37,070 --> 00:22:43,430 And this is some rotating hypercube, very similar to the monkey sculpture that we saw before. 231 00:22:43,430 --> 00:22:49,130 So. So let's go from right. This is not, you know, the 3D print doesn't actually do this. 232 00:22:49,130 --> 00:22:53,060 It's an illusion. This is something which looks like it's moving, but it's actually not. 233 00:22:53,060 --> 00:23:01,700 Let's go to something that should move, but actually doesn't. So maybe you've seen this kind of logo or graphical motif. 234 00:23:01,700 --> 00:23:09,080 I grew up in Manchester, and I think this this bus shelter ad was about, well, making three different transportation systems work well together. 235 00:23:09,080 --> 00:23:13,310 But do you think about this for a second if this one's going this way? 236 00:23:13,310 --> 00:23:17,960 I mean, this was going this way and this one can't move at all. 237 00:23:17,960 --> 00:23:22,940 And over here, the teachers, the students, the parents come together and nothing is achieved. 238 00:23:22,940 --> 00:23:31,640 No movement is possible. So here's the challenge. How can you make three gears which are meshing with each other, right? 239 00:23:31,640 --> 00:23:36,590 They have to. Each power of the gears has to be interacting with each other, and yet they can all move. 240 00:23:36,590 --> 00:23:43,490 And so here is one of our solutions. There are these three sort of interlinked rings. 241 00:23:43,490 --> 00:23:50,480 Here's an animation, and I'll pass this one around the the stick through the middle with sort of an afterthought. 242 00:23:50,480 --> 00:23:58,340 It just makes it easier to to work. But the real thing is the three years he has oh, there's lots of these. 243 00:23:58,340 --> 00:24:08,000 OK, so he has a sort of linear version of the same idea or three racks rather than gears that that move, you're still working on that. 244 00:24:08,000 --> 00:24:11,810 So you missed that one story. Here's a version with four. 245 00:24:11,810 --> 00:24:14,720 This one is very satisfying. 246 00:24:14,720 --> 00:24:24,560 You have this sort of tetrahedral symmetry, which goes in and out in the current record is five, and the geometries have to keep changing. 247 00:24:24,560 --> 00:24:29,330 So the question of how to do this and also how to do six yet have any ideas? 248 00:24:29,330 --> 00:24:37,400 Let me know. This is another example of the medium being important, right, with a 3D printed mechanism. 249 00:24:37,400 --> 00:24:44,180 You can make it work and you can see that it actually works. If it was a computer animation, you'd be like, Yeah, sure, whatever. 250 00:24:44,180 --> 00:24:49,580 Pass things around, here's a more mathematical example. 251 00:24:49,580 --> 00:24:58,190 So if you talk to any geometry or topology, they will tell you that the correct geometry for a Taurus is Euclidean geometry. 252 00:24:58,190 --> 00:25:03,350 It's flat, and this Taurus really is flat because you can open it out flat on the plane. 253 00:25:03,350 --> 00:25:08,000 And there it is other sorts of hinged geometry. 254 00:25:08,000 --> 00:25:14,390 This is a very fun one. This is a sort of hyperbolic doily. 255 00:25:14,390 --> 00:25:21,500 So this is a model of the hyperbolic plane made out of little hinged triangles. 256 00:25:21,500 --> 00:25:26,090 I did not prints out three tiny little triangles, then piece them together by hand. 257 00:25:26,090 --> 00:25:32,810 It had to go in the printer with all of the hinges already in place, and it just comes out of the printer and you shake out. 258 00:25:32,810 --> 00:25:38,570 The dust and dirt is sort of amazing technology. So there's an interesting mathematical question how do you do that? 259 00:25:38,570 --> 00:25:43,190 How do you decide where to put the triangles to put it in the printer? 260 00:25:43,190 --> 00:25:48,830 So this is a sort of simple simulation that shows what actually happened to make the shape so. 261 00:25:48,830 --> 00:25:59,390 So what we do is, OK, so we need all of these triangles to be, you know, linked to each other and they're not allowed to collide with each other. 262 00:25:59,390 --> 00:26:08,000 And we start out with them drawing on the Poincaré desk model where we know we can, we can sort of set up where to put them and connect them together. 263 00:26:08,000 --> 00:26:12,680 So the combinatorics is correct was not corrected at the start is the length. 264 00:26:12,680 --> 00:26:16,250 As you go further on in the park, create this model. Lengths get shorter. 265 00:26:16,250 --> 00:26:22,580 And so what we do is we put some springs on each of the edges and shake computationally shake, 266 00:26:22,580 --> 00:26:30,050 and it eventually converges to some final position where all of the lengths are correct and you 267 00:26:30,050 --> 00:26:37,370 get this thing here and then another little bit of programming to put sort of hand designs, 268 00:26:37,370 --> 00:26:45,120 pieces with hinges into the right place. And then this is what we send off to the printer. More sort of moving things, 269 00:26:45,120 --> 00:27:00,180 this is a puzzle based on a combinatorial version of the vibration 120 cell are getting on to mechanisms that are all exotic, that expand so. 270 00:27:00,180 --> 00:27:07,260 So this is an interest of mine. A little bit like people know the Hoberman sphere, 271 00:27:07,260 --> 00:27:13,320 the sphere that sort of you got this little ball and you open it out and you get this big, big ball. 272 00:27:13,320 --> 00:27:18,810 And this is a similar sort of idea, except that we really wanted to do things in three dimensions. 273 00:27:18,810 --> 00:27:22,380 So actually, this one I have here alive. 274 00:27:22,380 --> 00:27:29,790 So rather than right, so the Hoberman sphere, if you think about it, the mechanism is sort of spread out over the two dimensional surface of a sphere. 275 00:27:29,790 --> 00:27:36,240 And what we wanted to do was make a truly three dimensional version of an expanding mechanism. 276 00:27:36,240 --> 00:27:42,660 And so there you go. This is based on the crystal structure of diamonds. 277 00:27:42,660 --> 00:27:48,990 So the way that the carbon atoms are arranged is these sort of, uh, 278 00:27:48,990 --> 00:27:52,950 I won't hand this around is kind of big, but you can come up afterwards and play with it. 279 00:27:52,950 --> 00:28:00,300 But the the the way that the the carbon atoms are arranged with each other is sort of beautiful crystal structure, 280 00:28:00,300 --> 00:28:07,110 and we can exploit that to make some complicated shape like this. 281 00:28:07,110 --> 00:28:12,060 We can also do not this diamond geometry. 282 00:28:12,060 --> 00:28:30,560 We can also do a different geometry. So this is cubicle geometry here, I've got a small one here, which I can hand around. 283 00:28:30,560 --> 00:28:36,110 And so, or again, you go out and explore and maybe you come back with something useful, 284 00:28:36,110 --> 00:28:40,880 so potentially this has applications in, for example, space architecture. 285 00:28:40,880 --> 00:28:48,620 You want to some sort of space station or or whatever it is, you want it to be small in the rocket when it goes up. 286 00:28:48,620 --> 00:28:55,320 But then once it's in space, you want it to expand to a much larger objects. 287 00:28:55,320 --> 00:28:59,000 So a few other kind of exotic things. 288 00:28:59,000 --> 00:29:04,220 This is Buckminster Fuller jitterbug. 289 00:29:04,220 --> 00:29:09,380 So so what's going on here is. 290 00:29:09,380 --> 00:29:20,930 Well, so when it's small, it's an octahedron. And the four the eight triangles of the octahedron were represented by these sort of y shapes. 291 00:29:20,930 --> 00:29:30,500 And they're linked together at corners so that they can sort of hinge outwards to a larger figure, which is called a cube octahedron. 292 00:29:30,500 --> 00:29:33,350 And then it sort of goes back down again. So. 293 00:29:33,350 --> 00:29:40,040 So Buckminster Fuller was maybe most famous for geodesic domes and things like that, but he did a lot of interesting things. 294 00:29:40,040 --> 00:29:43,970 Our continued contribution here with adding these gears. 295 00:29:43,970 --> 00:29:53,510 So it turns out that this structure, mathematically at least, does this kind of motion without the gears. 296 00:29:53,510 --> 00:29:59,960 But it turns out that the real world sort of doesn't quite behave, and it's sort of jammed sometimes, 297 00:29:59,960 --> 00:30:07,430 and the gears kind of help it to keep aligned and in the right place. 298 00:30:07,430 --> 00:30:15,290 And just for fun, here's a larger version, so rather than the previous one one from an octahedron to a Cuban octahedron, 299 00:30:15,290 --> 00:30:22,340 this goes from a Cuban to eat into a rumba, Cuba to Iran, and just to give a sense of the horrors that await. 300 00:30:22,340 --> 00:30:30,230 When you try and do this kind of thing here, you'll notice the gearing here is between a square piece and a triangular piece. 301 00:30:30,230 --> 00:30:35,120 And it turns out that the amount that those pieces rotate is not the same, right? 302 00:30:35,120 --> 00:30:40,460 In the octahedron, everything was triangle a triangle and everything was rotating the same amount here. 303 00:30:40,460 --> 00:30:49,430 The the square rotates 90 degrees, whereas the triangle rotates, I think, 120, and they don't rotate a linear rates relative to each other. 304 00:30:49,430 --> 00:30:53,700 So you have some horrible relationship between the two. Why do I care about all this? 305 00:30:53,700 --> 00:30:59,900 Well, I need to make gears that rotate at different rates, depending on where they are in their motion. 306 00:30:59,900 --> 00:31:08,870 And so this is sort of a tricky problem. And here are some more recent work this is. 307 00:31:08,870 --> 00:31:15,440 So these are two ideas, which in some sense not really connected to each other. 308 00:31:15,440 --> 00:31:21,920 I mean, they are connected to each other, then fall apart. But there's no axle that goes through these and there's no frame holding it together. 309 00:31:21,920 --> 00:31:27,410 He has a very similar one, which is showing you that they're really not actually connected together. 310 00:31:27,410 --> 00:31:35,430 Uh oh. So if you happen to have a screwdriver on you, then you can. 311 00:31:35,430 --> 00:31:40,500 It's unfortunately just a little bit too too small to fit a pen. 312 00:31:40,500 --> 00:31:47,010 And this is this is very recent work using similar sorts of ideas. 313 00:31:47,010 --> 00:31:52,690 So, so there were three guys that are linked together with the red one in the middle. 314 00:31:52,690 --> 00:31:58,730 I guess you can look. OK. Well, there it keeps changing. Which ones in the middle groups? 315 00:31:58,730 --> 00:32:06,080 We just play that again. But then you can switch which one is in the middle still holds together, 316 00:32:06,080 --> 00:32:11,760 but somehow the one that's in the middle, it's actually holding things together is changing. 317 00:32:11,760 --> 00:32:15,440 So this is sort of a bit like the bombing ring somehow. 318 00:32:15,440 --> 00:32:24,470 So any teacher will tell you that it's all very well to see somebody do something, but you're not really learning until you do it yourself. 319 00:32:24,470 --> 00:32:33,920 So I certainly lots of things. What about doing this yourself? Well, maybe not you yet, but my students. 320 00:32:33,920 --> 00:32:38,690 So I have a 3-D printing lab that's in the mathematics permanent Oklahoma State 321 00:32:38,690 --> 00:32:45,170 University and I run a class where students learn how to 3D print things. 322 00:32:45,170 --> 00:32:53,060 So this is the homework for me. I think the fourth week they learn how to use the software and use the 3-D printers. 323 00:32:53,060 --> 00:32:59,060 And then the homework is, you know, go off and find some cool parametric cabin space and then print one. 324 00:32:59,060 --> 00:33:11,630 And it's a project based class. So they end up producing beautiful things at the for the midterms and their final fractals and poly hydra. 325 00:33:11,630 --> 00:33:18,680 And this is to do with sort of pine cone geometry, more things along these lines. 326 00:33:18,680 --> 00:33:27,170 This has to do with Kepler's laws of planetary motion that these little sections here are filled with water. 327 00:33:27,170 --> 00:33:31,670 And you can sort of use this to show that they all have the same area, which is Kepler's. 328 00:33:31,670 --> 00:33:36,920 Third, I think and this is a beautiful example from the last time I saw this. 329 00:33:36,920 --> 00:33:38,900 So this is showing various areas. 330 00:33:38,900 --> 00:33:50,170 So this is the different levels of this is showing different times of the Forest Series for the sawtooth wave and the square way. 331 00:33:50,170 --> 00:33:58,830 And this is just a great, great quote from from George Horn. The more maths you know, the more stuff you can make, which is absolutely true. 332 00:33:58,830 --> 00:34:09,450 Let me say a little bit about my work in virtual reality. So this is a recent example of non-nuclear and virtual reality. 333 00:34:09,450 --> 00:34:14,640 This is a three dimensional, hyperbolic virtual reality. 334 00:34:14,640 --> 00:34:20,460 So I started over here and I walked around the sides of a square in the real world. 335 00:34:20,460 --> 00:34:29,310 But it turns out that after going round back to the same place in the real world and not back in the same place in the hyperbolic world, 336 00:34:29,310 --> 00:34:39,990 I have to in fact turn an extra corner to make a right angle Pentagon in hyperbolic space, because that's how hyperbolic space works. 337 00:34:39,990 --> 00:34:44,070 And this is sort of a good sort of visceral way of of seeing the difference between these things. 338 00:34:44,070 --> 00:34:52,590 And then I'm going to talk about some other things, but the sounds of so ignore them. So, so taking these sorts of ideas. 339 00:34:52,590 --> 00:34:56,880 This is using a graphical technique called re-watching. 340 00:34:56,880 --> 00:35:05,580 Last semester, I was one of the organisers for a programme at a mathematical institute at Brown University in the States. 341 00:35:05,580 --> 00:35:10,320 And the programme was about illustrating mathematics. So all these sorts of things. 342 00:35:10,320 --> 00:35:18,660 And with a team of people, we were trying to extend this from three dimensional, hyperbolic geometry to other strange, three dimensional geometries. 343 00:35:18,660 --> 00:35:22,910 We were particularly going for the eight first in geometry, 344 00:35:22,910 --> 00:35:29,820 so you may have heard of Bill First and Field's medallist and his famous geometric zation now theorem, 345 00:35:29,820 --> 00:35:37,290 which says that more or less three dimensional spaces can be classified into into three, sorry, eight different geometries. 346 00:35:37,290 --> 00:35:40,560 And this is one of them, and it looks very strange. So what is this space? 347 00:35:40,560 --> 00:35:45,180 Here is the sort of x y plane is a sphere. 348 00:35:45,180 --> 00:35:49,260 And then the other direction is just a sort of Euclidean translation. 349 00:35:49,260 --> 00:35:53,640 You get these very weird graphical effects and things get even weirder. 350 00:35:53,640 --> 00:36:02,280 This is something called nail geometry, which is one of the sort of and the sort of last three really strange geometries that there are. 351 00:36:02,280 --> 00:36:06,810 And we're still working on this and trying to bring these into again. 352 00:36:06,810 --> 00:36:09,480 So to make it more accessible and available, 353 00:36:09,480 --> 00:36:15,180 this you probably won't want to do in virtual reality because the stereo vision just isn't going to work at all. 354 00:36:15,180 --> 00:36:17,880 So in hyperbolic geometry and three dimensional, 355 00:36:17,880 --> 00:36:23,430 hyperbolic and three dimensional spherical and a few of the others, stereo vision does work more or less, OK? 356 00:36:23,430 --> 00:36:31,410 And here it's just completely terrible. It'll just make you sick in no time flat. 357 00:36:31,410 --> 00:36:40,920 This is very recent work on three dimensional software for investigating three dimensional manifolds called snappy. 358 00:36:40,920 --> 00:36:47,520 So this has been around. I want to say, for 30 or so years, people researchers have been using the software, 359 00:36:47,520 --> 00:36:53,340 and now we have the ability to actually visualise what it looks like inside of these spaces that we've been working with. 360 00:36:53,340 --> 00:37:01,710 This is again, hyperbolic geometry. And this also allows us to does a there's a sense you can deform the space that you're sitting inside of. 361 00:37:01,710 --> 00:37:05,640 And it gets faintly terrifying when you do. 362 00:37:05,640 --> 00:37:12,960 So this is again sitting inside of a three dimensional manifolds of the geometry is being squished around in some crazy way, 363 00:37:12,960 --> 00:37:18,000 which I could get into given another few hours of talking about it. 364 00:37:18,000 --> 00:37:25,650 But I won't. And one of the fun VR simulation thing, this is another very recent idea. 365 00:37:25,650 --> 00:37:31,500 Let me quote out of this and actually show you what it is. These things called Camozzi Fractals. 366 00:37:31,500 --> 00:37:37,650 You can drive through these fractals, which are again based on the hyperbolic geometry of three dimensional manifolds. 367 00:37:37,650 --> 00:37:43,920 But maybe I won't say much more about what they actually are. You can come and ask me later if you interested. 368 00:37:43,920 --> 00:37:53,550 And the last topic I want to tell you about is radical video. So, so this is taking video that was taken by a special kind of camera. 369 00:37:53,550 --> 00:38:01,290 It's a spherical camera. So so here's here's the output from this camera as viewed in the view that comes with the camera. 370 00:38:01,290 --> 00:38:09,570 So, well, here we are on a tripod, and this camera has lenses on both sides that allow you to see everything around you, 371 00:38:09,570 --> 00:38:17,170 so it's capturing the entire sphere of data around you. And here we are juggling at Stanford, it turns out. 372 00:38:17,170 --> 00:38:25,180 So that's what it's supposed to look like when you when you experience it as it's intended, 373 00:38:25,180 --> 00:38:29,560 but of course, any kind of image or video stored, it's transmitted as a rectangle. 374 00:38:29,560 --> 00:38:34,750 And so this is what it looks like as it's stored. 375 00:38:34,750 --> 00:38:41,200 There's sort of the equator around the camera here. This is the South Pole where the tripod is and there's a light up at the North Pole. 376 00:38:41,200 --> 00:38:44,680 So we're sort of unwrapping it's called the rectangular projection. 377 00:38:44,680 --> 00:38:52,900 So I got interested in these cameras thinking about the transformations of space that you should do so with flat video, 378 00:38:52,900 --> 00:38:58,990 for example, you know how to transform it to zoom into something, right? You just make the picture bigger and then crop. 379 00:38:58,990 --> 00:39:04,630 What are you supposed to do with zooming in spherical video? How do you make it bigger? 380 00:39:04,630 --> 00:39:09,280 You can crop. There is nothing to crop. You see everything. What is Zoom even mean? 381 00:39:09,280 --> 00:39:16,180 And thinking about this led me to think about. Well, it's a sphere of data, and you should be viewing this sphere as the Remon sphere. 382 00:39:16,180 --> 00:39:20,470 And then you should be doing the transformations that are natural to do on the Roman sphere. 383 00:39:20,470 --> 00:39:30,340 Complex numbers. So soon, this video is going to switch to the same thing as before, but modified slightly. 384 00:39:30,340 --> 00:39:38,040 Yeah, yeah. So this video is sort of doubled up. 385 00:39:38,040 --> 00:39:45,360 So what we did here is we took the very briefly we take the image and then we apply the Z squared map z squared. 386 00:39:45,360 --> 00:39:53,610 So I am in the UK and that's it's to to preserves angles, which makes means that people aren't sort of squished in some crazy way. 387 00:39:53,610 --> 00:39:56,280 But it means that there's now two copies of everybody. 388 00:39:56,280 --> 00:40:02,640 So if you show up to joggling and you really want to do a pattern, then of all six people, but only three people are here. 389 00:40:02,640 --> 00:40:13,520 If it has the right symmetry, you can cheat. Here's another version or another VIDEO This is using a particular. 390 00:40:13,520 --> 00:40:17,820 Well, let me just play it and I'll come back in the second version of me. 391 00:40:17,820 --> 00:40:22,800 So I will say my experiment before the start of DR. 392 00:40:22,800 --> 00:40:32,700 So slowly zooming this way along the frame gives you this way, and that is a sort of weird pattern also. 393 00:40:32,700 --> 00:40:39,790 And then what happens? This is the future for the future versions of the year for that and this is the past. 394 00:40:39,790 --> 00:40:47,070 So I want to ask me to explain again what's going on a past version of me. 395 00:40:47,070 --> 00:40:53,340 So I want to say what I can say there. 396 00:40:53,340 --> 00:40:58,740 So, right, so this is a looping video, as you can see. 397 00:40:58,740 --> 00:41:03,840 So this is actually using this kind of zoom that that that is sort of the right thing to do. 398 00:41:03,840 --> 00:41:10,110 Again, it's coming from things called Mobius Transformations and with a bit of, 399 00:41:10,110 --> 00:41:15,960 well, it's not actually green screen, but it's sort of fake green screen. You can get this kind of effect. 400 00:41:15,960 --> 00:41:20,610 It turns out the zoom in in this spherical video feels very much like motion. 401 00:41:20,610 --> 00:41:25,230 It's not quite the same thing, but it feels like like motion. So the camera is not moving at all. 402 00:41:25,230 --> 00:41:31,320 You can see the tripod down here. It's not moving at all. It's just sort of this transformation is giving the impression of movement. 403 00:41:31,320 --> 00:41:40,400 And then I put that the the offset by 30 seconds versions through these these different portals. 404 00:41:40,400 --> 00:41:46,220 OK, so I have one more video to show you. 405 00:41:46,220 --> 00:41:53,320 I'm just going to let this this play, and then maybe I'll come back at the end and say something about this. 406 00:41:53,320 --> 00:42:32,960 This is worth my heart. Oh. 407 00:42:32,960 --> 00:42:43,500 Oh, oh, wow. 408 00:42:43,500 --> 00:43:01,710 And for me. 409 00:43:01,710 --> 00:43:54,320 Oh, oh, oh, oh, oh, oh, oh, oh, oh, oh, oh, oh, oh oh, oh oh oh. 410 00:43:54,320 --> 00:44:13,560 Ow, ow, ow, ow, ow, ow. 411 00:44:13,560 --> 00:44:22,740 Last thing, let me say a few words about what's going on here. So this is around. 412 00:44:22,740 --> 00:44:32,160 There are three different singers. Well, actually, there's only one singer VI, who is offset in time by 20 seconds from copies of herself. 413 00:44:32,160 --> 00:44:43,260 But we've also offset copies of the room. Well, so we all of that offset the video not by 20 seconds, but by 120 degrees. 414 00:44:43,260 --> 00:44:51,420 The mathematics here in how the transformation of the video happens here, we had tripled space rather than doubled space. 415 00:44:51,420 --> 00:44:54,900 So like the jugglers had a there was a z squared back here. 416 00:44:54,900 --> 00:45:01,560 It's a z cubed. So that sort of explains how we got this image here. 417 00:45:01,560 --> 00:45:08,420 But there's a lot of other strange things going on here. Let's see. 418 00:45:08,420 --> 00:45:14,010 And there are three copies of my website. I've let it go too long. 419 00:45:14,010 --> 00:45:19,260 There were three copies of my. There's only one copy of me. There were moves previously, only two copies of me. 420 00:45:19,260 --> 00:45:26,760 Maybe I'll go back a little bit. 421 00:45:26,760 --> 00:45:49,340 Maybe I'll just let it play out, and then I'll say something you. 422 00:45:49,340 --> 00:45:56,840 OK, so as I was saying, somewhere in the middle. Right. 423 00:45:56,840 --> 00:46:01,040 So where am I? I'm over here and I'm also over here. 424 00:46:01,040 --> 00:46:05,270 Why are there only two of me? First question, second question. 425 00:46:05,270 --> 00:46:11,060 Here's the sheet music. There's only one copy of the sheet music. 426 00:46:11,060 --> 00:46:18,260 So I picked it up. She plays a couple of notes. She passes it to me. I pass it back to her through the piano. 427 00:46:18,260 --> 00:46:25,110 She comes around to put it on the music stand. She plays a couple of notes on the piano. 428 00:46:25,110 --> 00:46:31,200 She picks up a hammer for some reason. Just in time for her to pick up the music again. 429 00:46:31,200 --> 00:46:35,830 So, so there's a lot of sort of interesting choreography going on here. So this video is on YouTube. 430 00:46:35,830 --> 00:46:41,020 Also, there's an explanation video that goes into a lot of detail about what's actually going on. 431 00:46:41,020 --> 00:46:48,690 I encourage you to check that out of your system to see more. I am pretty much done. 432 00:46:48,690 --> 00:46:56,100 I am going to take just a few minutes to say something about my book visualising mathematics with 3D printing. 433 00:46:56,100 --> 00:47:10,110 So the what's the trick with this book is that most of the most of the figures in the book are photographs of 3D printed objects. 434 00:47:10,110 --> 00:47:17,850 And there's a website associated with the book, and you can go, for example, to the chapter on probably Hydra. 435 00:47:17,850 --> 00:47:29,580 And you can click on the page for a figure, and there's a sort of virtual 3D model which you can look around and sort of explore. 436 00:47:29,580 --> 00:47:35,850 And the idea here is there's lots of very interesting, accessible mathematics that people don't really write about. 437 00:47:35,850 --> 00:47:42,270 I think because it's too hard to make the figures that in order to really understand it, you really need three dimensional figures. 438 00:47:42,270 --> 00:47:46,620 And so with this, you can have actual three dimensional figures. 439 00:47:46,620 --> 00:47:53,550 You can buy these things on shape ways. You can download the files for yourself to print on your own 3D printer. 440 00:47:53,550 --> 00:47:56,610 There's also a link to think about if people are familiar with thing of us. 441 00:47:56,610 --> 00:48:04,740 This is a sort of repository for 3D files that if the internet worked for me, I'd be able to go there and show you. 442 00:48:04,740 --> 00:48:11,760 But you can download the files in this sort of community there that's looking at the files and printing them out and saying, 443 00:48:11,760 --> 00:48:18,120 here are the settings to make it work on my on my printer. But that's my talk. 444 00:48:18,120 --> 00:48:22,710 I'm Henry Saige on Twitter and Instagram. And that's my website. 445 00:48:22,710 --> 00:48:43,646 Thanks for your attention.