1 00:00:15,860 --> 00:00:19,790 I would like to welcome you all to the Mathematics Institute. My name is. 2 00:00:19,790 --> 00:00:24,590 I like O'Reilly, and I'm in charge of the public lecture series for the Institute. 3 00:00:25,130 --> 00:00:31,040 Before we start the event, let me remind you that, as usual, the green sites indicate the emergency exit. 4 00:00:31,700 --> 00:00:41,420 I also want to acknowledge a sponsor that you can see the markets there dispense of the whole Oxford Mathematics Lecture series. 5 00:00:41,900 --> 00:00:49,190 X Market is a leading quantitative driven electronic market maker with offices in London, Singapore and New York. 6 00:00:49,310 --> 00:00:50,960 And we're very grateful for the support. 7 00:00:52,190 --> 00:01:01,270 Now onto the main event, as you can see or as you know, you can feel in the air, too, that tonight is a very special event. 8 00:01:01,280 --> 00:01:07,460 It's actually co-organized with the Clay Mathematics Institute and Oxford Mathematics. 9 00:01:08,060 --> 00:01:11,600 As such, it will be a little bit different from the usual format. 10 00:01:12,260 --> 00:01:19,790 The public lecture today will be followed by the presentation of the Clay Award for Dissemination of Mathematical Knowledge. 11 00:01:20,390 --> 00:01:23,490 I call it cat dunk for show. Good. 12 00:01:24,650 --> 00:01:31,250 So you will have the distinct honour to be part of the event and the presentation, the award ceremony for the dunk. 13 00:01:33,230 --> 00:01:39,740 And since we want it to be directly after the talk, we will not take question today. 14 00:01:39,740 --> 00:01:43,219 We can go directly into the awards ceremony. Nick Woodhouse, 15 00:01:43,220 --> 00:01:47,900 which is who is the former head of our department and the current president of the Clay Mathematics 16 00:01:47,900 --> 00:01:53,870 Institute will give us a full description of the awards and of Professor Penrose's achievements. 17 00:01:54,320 --> 00:02:03,500 So I will be very brief in my introduction for tonight's letter, I would just say the following When you work in mathematics and physics, 18 00:02:04,190 --> 00:02:09,200 one sign of becoming famous is that your name may be associated with a theory. 19 00:02:09,200 --> 00:02:19,069 We say, Oh, the theory, yes, that that person made it. Now you realise that Roger is in a completely different category when you see the partial list 20 00:02:19,070 --> 00:02:23,870 of theories that has been named that either this name or that naturally associated with him, 21 00:02:24,440 --> 00:02:32,569 for instance, you'll find a twist. The theory the notion of spin network cosmic censorship while curvature hypothesis 22 00:02:32,570 --> 00:02:38,330 Schrodinger Newton equation andromeda paradox conformal cyclic cosmology just to name a few. 23 00:02:39,290 --> 00:02:44,900 No, it gets even better when you name is use for a famous phenomena or theorem. 24 00:02:45,940 --> 00:02:53,500 As you probably guessed, people don't usually name things after themself where some have tried, but it usually doesn't work. 25 00:02:54,430 --> 00:03:01,240 So for that to happen, for a name to be given to you, a name to be given to something spectacular. 26 00:03:01,540 --> 00:03:04,540 First of all, you have to done something quite remarkable. 27 00:03:04,930 --> 00:03:09,040 But you also must have the admiration and the respect of the entire community. 28 00:03:09,580 --> 00:03:12,430 And again, here, the list is quite long. 29 00:03:12,760 --> 00:03:20,920 And I'm just naming a few people in mathematics and physics know about Penrose inequalities, Penrose interpretation of quantum mechanics. 30 00:03:21,430 --> 00:03:30,580 The more Penrose pseudo inverse, the Newman Penrose formalism, Penrose diagrams, and of course the Penrose Hawking Singularity theorems. 31 00:03:31,480 --> 00:03:37,900 Just to give you an idea. But things get even better and transcend science. 32 00:03:38,290 --> 00:03:44,620 When you name are associated with things for the public at large, beyond physics and mathematics. 33 00:03:45,040 --> 00:03:54,250 And here again, we have Penrose tiling that you might have seen coming in here tonight, the Penrose Cube that you can buy online. 34 00:03:55,810 --> 00:04:05,890 Penrose stairs. Penrose Triangle. And of course, these last things are closely associated with this contact and influence on the artist M.C. Escher. 35 00:04:06,130 --> 00:04:13,960 And this is a topic of tonight lecture. So without further ado, please let me welcome Roger and let's hear directly from him about this topic. 36 00:04:27,780 --> 00:04:38,670 Well, thank you very much. Yeah, well, it's a great pleasure and an honour for me to be able to address this very distinguished audience here. 37 00:04:39,240 --> 00:04:42,240 And I should explain the title. 38 00:04:42,240 --> 00:04:49,410 First of all, it's a book which I'm supposed to be writing, and this is a way to get me started on this book. 39 00:04:49,420 --> 00:04:55,410 I keep telling my publishers that I'm just one way getting through, and it's been nearly finished. 40 00:04:55,770 --> 00:05:05,340 I haven't actually started yet, but this is the start in a certain sense because I had to organise my thoughts while I had done that much already, 41 00:05:05,760 --> 00:05:09,239 but I had to organise my thoughts particularly I should explain. 42 00:05:09,240 --> 00:05:16,440 The idea is to use pictures as an introduction to certain various areas of mathematics. 43 00:05:16,440 --> 00:05:20,489 So it would be so, so that it would be a book for the general public. 44 00:05:20,490 --> 00:05:25,380 And you would introduce the idea just by looking at your pictures. 45 00:05:25,950 --> 00:05:31,550 I found it quite difficult in a way, because one of the problems is it's not as a sitting. 46 00:05:31,590 --> 00:05:35,550 This actually illustrates this this one illustrates that. This illustrates the other thing. 47 00:05:35,940 --> 00:05:39,390 You find that the same picture illustrates about three things all at once. 48 00:05:39,690 --> 00:05:45,060 And then to try and put this into some kind of organisation I found extremely difficult. 49 00:05:45,760 --> 00:05:57,710 This is something I can start with. Escher, I think when he was in Italy he did a lot of prints, lots of pictures in which he okay, 50 00:05:58,080 --> 00:06:04,110 sometimes a little exaggerated but very accurate pictures of of scenery and so on buildings. 51 00:06:04,470 --> 00:06:12,180 And I think it was when he visited the Alhambra in Spain and he realised all these 52 00:06:12,180 --> 00:06:19,440 incredible Moorish patterns which he tried to understand what was going on. 53 00:06:19,440 --> 00:06:28,979 There were different kinds of arrangements and then he learned from a mathematician, I think it was well, actually, I'm not quite sure where it was. 54 00:06:28,980 --> 00:06:37,410 I did not say. But he got this picture, which just shows the 17 different Christian Christian groups, if you like, 55 00:06:37,410 --> 00:06:46,890 symmetry groups of the plane where you have discrete motions and there are 717 different ones illustrated by these pictures you see here. 56 00:06:47,580 --> 00:06:59,100 Well, Escher then developed these pictures in his own unique way, using tessellations of various creatures and so on. 57 00:07:00,660 --> 00:07:02,970 I should explain. You see, he did this by. 58 00:07:03,600 --> 00:07:11,190 In this one, you can see on the outside you've got the ordinary squares or something, and then he will modify them in terms of creatures. 59 00:07:13,380 --> 00:07:16,680 This is the sort of thing you have. There are symmetries. 60 00:07:17,520 --> 00:07:20,550 Either you have a translation which is just a motion. 61 00:07:21,270 --> 00:07:27,489 Without any kind of rotation, you can have glide planes where you translate to note and you also reflect. 62 00:07:27,490 --> 00:07:31,170 At the same time, you can have simple rotations. 63 00:07:31,170 --> 00:07:34,500 That's the one down here. I don't remember. And all the other ones are. 64 00:07:34,920 --> 00:07:36,390 Yes, the reflection. That's the other one. 65 00:07:36,810 --> 00:07:44,130 So all these different motions combined together to give you the various 17 different groups that you can have. 66 00:07:44,670 --> 00:07:54,479 And each one he would illustrate in his own unique way he was able to economise a little bit, although this one was to be in colour, I think. 67 00:07:54,480 --> 00:07:57,030 But I just have a black and white version, 68 00:07:57,930 --> 00:08:04,320 so you have to imagine they're all different colours and the two of these different groups are illustrated with the same picture. 69 00:08:04,680 --> 00:08:10,739 If you take the colours into consideration, then you have one of them and if you don't, then you have the other one. 70 00:08:10,740 --> 00:08:14,910 So, so you can do two at once. That's the sort of economy that he liked. 71 00:08:15,510 --> 00:08:21,030 I shall just show these things rather quickly because I have an awful lot to show you. 72 00:08:21,690 --> 00:08:25,500 There's another one and these groups are illustrated. 73 00:08:25,770 --> 00:08:31,260 If you remember what those numbers and letters were and so on, and then you can see which one it is. 74 00:08:31,720 --> 00:08:35,760 I'm not going to go through that in detail because I have too many other things to say. 75 00:08:36,180 --> 00:08:44,430 But I'm just giving you examples of the way in which he illustrated these different symmetry groups, either with these birds and fish. 76 00:08:45,270 --> 00:08:53,670 I said, look carefully to see which there are of them fish in that case, or we have in this case dogs. 77 00:08:54,930 --> 00:09:04,850 And he liked the way that the I think they probably have it backwards doesn't make too much difference because this involves the glad plane you see. 78 00:09:05,830 --> 00:09:09,460 So you have to turn it over to get the symmetry just as it is. 79 00:09:09,640 --> 00:09:17,980 It's good, but you just can't because says okay upside down doesn't make too much difference in some the 80 00:09:18,030 --> 00:09:23,820 particularly those ones and here we see another one in the groups is illustrated at the bottom. 81 00:09:24,150 --> 00:09:30,560 I shall run through this rather quickly. Because I've got far more transparencies to show that I normally would have an election like this. 82 00:09:31,430 --> 00:09:35,790 So I think you have to. 83 00:09:36,290 --> 00:09:39,860 So is it not working? No, I suppose it's. It's just. 84 00:09:40,170 --> 00:09:45,370 It's okay. That's okay. Thank you. And tried to see what was happening there. 85 00:09:45,470 --> 00:09:48,650 Never mind. They can get very elaborate. 86 00:09:50,240 --> 00:09:54,709 For example, with this one, you see the. The animals, the creatures. 87 00:09:54,710 --> 00:09:58,430 I have to see what they are there and beetles or something of some sort. 88 00:09:59,090 --> 00:10:04,130 And the arms all interlock and complicated way. But he was good at doing that. 89 00:10:05,900 --> 00:10:10,810 And of course, you can different groups. You have to look carefully to see which group it is. 90 00:10:11,600 --> 00:10:16,310 Did you here have a key at the bottom so you can tell. Anyway, 91 00:10:16,700 --> 00:10:21,290 that that gives you a good way to start because he does illustrate these different symmetry groups 92 00:10:21,290 --> 00:10:25,640 and one can go and talk about what groups are and motions of the plane and that sort of thing. 93 00:10:27,170 --> 00:10:34,910 This one, this last picture here. Has a certain history to it, which I should explain. 94 00:10:35,660 --> 00:10:44,450 I once met Asha. I come back to that story as to how I happened to get in touch with him. 95 00:10:44,690 --> 00:10:58,250 But I did meet him once, and I came with a certain set of puzzle pieces, which in the shape of the ones that are at the top one minute. 96 00:11:01,200 --> 00:11:04,800 I put my pointer down here at the very top. 97 00:11:05,190 --> 00:11:11,640 I had a bunch of wooden pieces. They were I think they're wooden and they were the shape that we have here just. 98 00:11:12,120 --> 00:11:15,360 And the thing was, you have to assemble them so that they cover the whole plane. 99 00:11:15,930 --> 00:11:21,180 It was quite interesting because you have to have 12 different orientations, six one way up and six the other way up. 100 00:11:21,720 --> 00:11:25,700 And they all have to be incorporated to cover the plane. 101 00:11:25,710 --> 00:11:33,510 And then you can I had to he did it and wrote me a letter and said, well, he didn't quite see what the principle was. 102 00:11:33,510 --> 00:11:38,880 And so I showed him this pattern here. Basically have run buses and these little lines. 103 00:11:39,390 --> 00:11:39,810 Mark, 104 00:11:39,810 --> 00:11:49,950 how you might have a this this one has to match the next one and you have to have a modification of the shape which so that one agrees with the other. 105 00:11:50,670 --> 00:11:56,639 And this this is what it was. This one shape was designed so that a lot of false tries as well. 106 00:11:56,640 --> 00:12:04,890 But that's that basically. And then I explained how it was done and I suggested some possible animal. 107 00:12:05,080 --> 00:12:09,930 I think it was a bird of some kind, but rather crude. And I said, I'm sure you can do a lot better than that. 108 00:12:10,350 --> 00:12:14,820 And he produced the picture, which I just showed you previously. 109 00:12:15,750 --> 00:12:22,200 This is actually the only example he had of what you could call a non non Asahi. 110 00:12:22,200 --> 00:12:30,960 Ideal tiling is it's a tiling of one shape, but the different instances of that shape may not be on equal footing with each other. 111 00:12:31,200 --> 00:12:36,030 So although I think the dark ones and the light ones, I'm not sure if it is illustrated that way, 112 00:12:36,270 --> 00:12:39,970 but some of them have a different relation to the passengers as a whole. 113 00:12:40,140 --> 00:12:46,200 So you see two, one and another one. And if you move that one to the other one, the pattern doesn't follow it. 114 00:12:46,680 --> 00:12:51,600 So you have that's what's called the non Asahi general tiling. 115 00:12:52,020 --> 00:13:05,730 It's quite interesting historically because in the famous speech that Hilbert, when he introduced his his famous problems at the turn of the 19th, 116 00:13:06,090 --> 00:13:11,760 20th century, he one of the problems he made was if you have a three dimensional shape, 117 00:13:12,660 --> 00:13:20,879 is it possible that you have a shape which will tile only not so he really I'm not sure if that word was used at that time, 118 00:13:20,880 --> 00:13:22,890 but that's what he was trying to do. 119 00:13:23,070 --> 00:13:28,680 I think he thought the plane shapes are completely trivial and so there was nothing interesting you could do with plain shapes. 120 00:13:29,670 --> 00:13:39,210 But then one of his colleagues or students, I think he I think it was he produced an example which showed that even in the case of plane shapes, 121 00:13:39,690 --> 00:13:45,989 this conjecture was incorrect and that you did have shapes which would only tile the plane that would tell the plane, 122 00:13:45,990 --> 00:13:49,560 but only in a way watch, which was non a hydrogen. 123 00:13:49,800 --> 00:13:59,700 So that was an example. But anyway, the picture that Escher made with, I believe the last with certainly the last watercolour he ever made, 124 00:14:00,840 --> 00:14:10,680 which partly due to my slowness in responding to his letters, but never mind about that. 125 00:14:11,400 --> 00:14:14,310 Anyway, you can do other things with similar sorts of shapes. 126 00:14:15,240 --> 00:14:23,069 Let me just give you an example here, very similar to some of the ones which did have periodic shapes. 127 00:14:23,070 --> 00:14:28,050 But this isn't it's not obviously periodic. It may be it extends in some such way. 128 00:14:28,380 --> 00:14:34,590 They're not all the same size. I'm not even quite sure what's involved in the arrangement here. 129 00:14:34,920 --> 00:14:42,060 Of course, you can do more or less symmetrical things still by just having initials animals and things like that. 130 00:14:42,440 --> 00:14:47,130 There's no symmetry at all. And of course, many of his examples or things like that. 131 00:14:49,080 --> 00:14:54,330 What about three dimensions? Well, of course, that's a bit hard to illustrate, but he certainly did things in three dimensions. 132 00:14:54,990 --> 00:14:58,920 Here's an example of of just cubical arrangement. 133 00:14:58,920 --> 00:15:03,719 This one, I think this is what I wanted to show. 134 00:15:03,720 --> 00:15:08,760 Here is a sketch of this picture. 135 00:15:08,760 --> 00:15:13,950 That's another one, two of the three dimensional things, the things that he then his interest in. 136 00:15:14,880 --> 00:15:20,430 And here we have another three dimensional picture, this fish. 137 00:15:21,840 --> 00:15:25,370 And you can see that again, it illustrates a symmetry. 138 00:15:25,380 --> 00:15:31,290 This is just translation. So it's not so exciting, but it's as a picture is impressive. 139 00:15:31,440 --> 00:15:39,239 Of course, the only one I know of which which illustrates a tiling of three dimensions is this one. 140 00:15:39,240 --> 00:15:48,180 Here you have these flatworms, whatever they are swimming around and the tiling that you see in the background, 141 00:15:48,180 --> 00:15:59,040 this swimming through it and they just pass it of course is the tiling of regular tetrahedra and and octahedron and they fit together. 142 00:15:59,120 --> 00:16:03,830 To form a three dimensional lattice tunnelling three space. 143 00:16:04,760 --> 00:16:10,760 So you see that he was interested in three dimensions of his harder to draw pictures of those things so I don't know that 144 00:16:10,760 --> 00:16:19,480 he did so many of the specific didn't do so many of them he also explored with distorting geometry in interesting ways. 145 00:16:20,630 --> 00:16:28,070 And here you have an example where the building you go up and it twists around and you find a copy of the building up there. 146 00:16:28,670 --> 00:16:33,440 But the geometry is twisted around in some strange way. And this is the sketch for it, I think. 147 00:16:34,430 --> 00:16:46,639 So you see examples of that you rather like these creatures which he invented because they crawl along like a caterpillar or something, 148 00:16:46,640 --> 00:16:52,280 and then they roll up and they can. When they go downhill, they can roll up and roll down the hill. 149 00:16:52,760 --> 00:16:59,360 These are steps. Oh, that thing they do, it is they go out the door, you can see them anyway. 150 00:16:59,370 --> 00:17:03,110 He rather like these creatures and made use of them in various ways. 151 00:17:04,370 --> 00:17:09,350 But the the the geometry of the space is distorted in interesting ways, too. 152 00:17:11,420 --> 00:17:16,580 So anyway, that's other things in three dimensions. 153 00:17:17,420 --> 00:17:31,639 He like playing with polyhedra and also combinations of polyhedra such as here you have the two tetrahedra and there's a straightforward dodecahedron, 154 00:17:31,640 --> 00:17:37,040 I think, and that's quite a team he played with. 155 00:17:38,060 --> 00:17:42,440 I'll give you another example here, which is a. 156 00:17:46,790 --> 00:17:52,820 I have to see what it is because I'm not sure I notice. See the picture. 157 00:17:53,030 --> 00:17:57,960 It's a. You can probably tell me. 158 00:17:58,110 --> 00:18:01,739 You can see it better from where you're sitting than I can. Anyway, these. 159 00:18:01,740 --> 00:18:06,390 These are two polyhedra interlocking, which you have. I think it's an icosahedron, a dodecahedron. 160 00:18:07,140 --> 00:18:10,950 It'll enter locking and preserving the same symmetry as each other. 161 00:18:13,440 --> 00:18:20,750 And here you have, I think, three octahedron and lizard creatures and various things involved. 162 00:18:20,880 --> 00:18:24,270 So and he's got other ones dotted around like that. 163 00:18:25,410 --> 00:18:32,130 So he was quite keen on that idea of showing into penetrating polyhedra and the symmetry. 164 00:18:32,880 --> 00:18:39,390 This is not the symmetry of the octahedron you have. When you finish that, you see you can. 165 00:18:42,920 --> 00:18:46,610 I should have worked out what the symmetry was before I gave this talk. 166 00:18:46,610 --> 00:18:52,030 But here we have then in. 167 00:18:52,520 --> 00:18:56,719 I always like this one because you can see it's a planet of some sort and the two 168 00:18:56,720 --> 00:19:04,010 planets are both tetrahedra and they interlock and they don't seem to know anything 169 00:19:04,010 --> 00:19:07,219 about each other because one of these is all dinosaurs and things and the others 170 00:19:07,220 --> 00:19:14,810 are some kind of civilised community and interlock in that extraordinary way. 171 00:19:15,770 --> 00:19:25,190 So I'm going through this rather quickly because you really have to look at this pictures in detail to appreciate all the intricate, 172 00:19:26,330 --> 00:19:33,500 the tremendous artistry has and detail which is extraordinary and it needs an examination. 173 00:19:34,100 --> 00:19:40,580 So that one seems to be stuck to the bottom here. That one too. Here we have a this is just delighted. 174 00:19:42,680 --> 00:19:45,860 That's the still isolated dodecahedron these. 175 00:19:48,330 --> 00:19:55,590 A pentagram and passing through each other, and it has the same symmetry as the icosahedron. 176 00:19:55,590 --> 00:20:06,450 But it's more interesting in that the faces go through each other and the polygons also go through each other as well. 177 00:20:07,290 --> 00:20:12,390 And I think this is illustrating sort of various kinds of rubbish around here. 178 00:20:12,690 --> 00:20:14,670 And then you have this perfect shape in the middle. 179 00:20:16,650 --> 00:20:25,290 One of the things he liked doing was exploring the contrast between the image of a three dimensional object and the three dimensional object. 180 00:20:25,830 --> 00:20:32,700 And here you see, this one looks as though it's a three dimensional object, and then one at the bottom is flat. 181 00:20:32,970 --> 00:20:36,870 But nevertheless, it's a picture of a three dimensional object. 182 00:20:37,200 --> 00:20:41,220 I mean, the spheres this one here is, I guess, a hemisphere or something. 183 00:20:41,370 --> 00:20:47,550 But the top, you see a picture. So he's playing around with the picture of the object in the object itself. 184 00:20:48,360 --> 00:20:52,290 And he does much more elaborate things of that nature. 185 00:20:54,790 --> 00:21:02,490 And the striking one is this what you see in the background, these two, I don't know, humanoids or something. 186 00:21:03,010 --> 00:21:13,750 We seem to be forming this interlocking pattern and then they sort of get a bit fed up and walk out of the picture and make friends with each other. 187 00:21:14,230 --> 00:21:19,990 So that's one example. It has other examples of the same thing. 188 00:21:21,910 --> 00:21:23,500 And this I've always liked that one. 189 00:21:24,220 --> 00:21:34,600 You see the tiling here of these lizard creatures or whatever they were, and that could continue as a plain tiling. 190 00:21:35,320 --> 00:21:41,350 But then they decide to come to life and they walk around on top of these polyhedra and other objects. 191 00:21:41,720 --> 00:21:45,910 Then, after having had a stroke, they get back into the into the two dimensional picture. 192 00:21:47,110 --> 00:21:58,780 It's a paradox, of course, but he loved that kind of thing. And then we have a much more elaborate version of the same kind of situation. 193 00:21:59,470 --> 00:22:07,620 Also mirrors involved. So you can see the mirror, I guess the blind symmetry too, you see, because it's reflected. 194 00:22:07,630 --> 00:22:15,560 You have a reflection in the symmetry. When they're black and white and somehow they go around and. 195 00:22:17,130 --> 00:22:20,340 You just have to follow it. It's very, very complicated what's going on, 196 00:22:20,640 --> 00:22:28,290 because they go through and they're also reflected and you see the reflections merging with the original creatures. 197 00:22:33,360 --> 00:22:38,820 And here's another case where he illustrates two kinds of paradox at the same time. 198 00:22:39,690 --> 00:22:45,090 I mean, one of them is this contrast between a picture of something and the thing itself. 199 00:22:45,690 --> 00:22:52,110 Here you have a picture of a hand gets more and more realistic until you think it's a real hand, and that's drawing a picture. 200 00:22:52,530 --> 00:22:54,420 And then, of course, it does the same thing there. 201 00:22:54,780 --> 00:23:01,979 But there's also a kind of paradox in that the thing is the thing being drawn by this hand and therefore, 202 00:23:01,980 --> 00:23:07,320 in some sense, this is a it's a product of this thing. 203 00:23:07,320 --> 00:23:15,899 And then this is itself a product of that. So you have something which is is it's a bit like a Russell paradox where something is the asset 204 00:23:15,900 --> 00:23:21,210 which has itself as a member and you see that kind of thing being illustrated in that picture. 205 00:23:22,290 --> 00:23:26,880 I had this opportunity to visit that Escher, as I mentioned before, 206 00:23:27,480 --> 00:23:33,959 and when I was there he had this long table I was expecting when I visited him that 207 00:23:33,960 --> 00:23:39,360 he would have a very extraordinary house with staircases going over the window, 208 00:23:40,560 --> 00:23:45,810 that sort of thing. And I was rather disappointed to find it was a very neatly organised house with a 209 00:23:45,810 --> 00:23:51,600 lovely picture window of the view outside and a very neat and organised place. 210 00:23:51,900 --> 00:23:58,170 And he had this long table, I don't know about as long as. Yeah, and it's about as wide as that. 211 00:23:58,350 --> 00:24:04,890 And he was sitting at one end and he had two piles of prints and he said well look this size, 212 00:24:05,130 --> 00:24:09,150 this one, I don't have many left but the other side have quite a few left. 213 00:24:09,450 --> 00:24:15,810 You can choose one. So I thought, Wow. And so I brought the pile over and I started going through it. 214 00:24:16,410 --> 00:24:21,480 And it was really difficult because some of the favourite ones he had already gone there were in the other pile, 215 00:24:22,080 --> 00:24:25,290 but there were a lot of things and it was difficult to make the choice. 216 00:24:25,890 --> 00:24:35,010 And eventually I picked this one, though he was very pleased when I picked that one because he said people don't usually understand that one. 217 00:24:36,120 --> 00:24:40,590 And so he felt since I seemed to understand it, I thought at least I thought I did. 218 00:24:42,360 --> 00:24:46,350 I was very flattered that he that he was pleased with my choice. 219 00:24:47,040 --> 00:24:49,980 I should explain this a little bit. Here we have a fish. 220 00:24:51,090 --> 00:24:57,810 And you see, as you move around here, the scales of the fish become a tessellation of other fish. 221 00:24:58,800 --> 00:25:06,540 And as you go round, if I'm going which way around it is, and you find that one of these fish become this one here, and then it scales. 222 00:25:06,690 --> 00:25:11,340 They have bigger and bigger and bigger, and then one of its scales become the original fish. 223 00:25:11,760 --> 00:25:17,310 So it's a bit like that, the two hands drawing each other, but in a somewhat more subtle way. 224 00:25:18,600 --> 00:25:21,960 And so it takes a little while to figure out exactly what's going on here. 225 00:25:22,410 --> 00:25:25,440 But again, you have this sort of Russell Paradox idea. 226 00:25:25,470 --> 00:25:34,290 You see the scales are a subset of the fish, but yet the fish, the scales become a fish and then it scales or one of the subsets, 227 00:25:38,160 --> 00:25:43,590 one of the elements at least of this set of little fish, if you like, becomes the original one. 228 00:25:43,590 --> 00:25:47,399 So you have a Russell Paradox idea involving sets, 229 00:25:47,400 --> 00:25:52,680 which are members of themselves or members of other sets which are members of themselves, that sort of thing. 230 00:25:53,700 --> 00:26:02,880 It's very much like a perhaps better known picture, the picture gallery where you see there's a boy. 231 00:26:04,130 --> 00:26:10,540 I've got it the right way around. Yes, I think so. A boy over here somewhere who was in the picture gallery had to cover up the rest of it. 232 00:26:10,540 --> 00:26:13,630 Really? To see that he's in a picture gallery and he's looking at the picture. 233 00:26:14,020 --> 00:26:19,690 And then you follow the picture around and you see eventually that the boy is in in the picture. 234 00:26:19,690 --> 00:26:28,479 So he's again, it's a sort of Russell paradox kind of thing where the boy is in the picture itself. 235 00:26:28,480 --> 00:26:39,370 But it illustrates something else which was actually explored more by a mathematician, Lindstrom, who is a Dutch mathematician in London. 236 00:26:40,000 --> 00:26:46,989 And he and a colleague decided, you see in the middle of the picture, Escher didn't quite know what to do. 237 00:26:46,990 --> 00:26:50,020 Well, I don't know if that was it or what. He didn't quite know what to do with it. 238 00:26:50,020 --> 00:26:57,340 He decides at least that was a place to put his signature. So it seems as though he gave up on trying to fill that thing in the middle. 239 00:26:57,610 --> 00:26:59,980 He sort of filled it up in the middle with the other picture, 240 00:27:02,770 --> 00:27:11,170 but it's done in a way which you don't quite see that there's anything strange about it, but I suppose it is a sort of singularity in the middle. 241 00:27:12,730 --> 00:27:16,390 But here he gave up on the singularity and decided just to have a whole. 242 00:27:17,230 --> 00:27:25,420 But what Lindstrom and his colleague decided to do was to find a transformation of this picture which would fit inside that hole. 243 00:27:26,140 --> 00:27:31,900 And then, of course, there's a hole inside that picture. And so the whole picture goes into that as well. 244 00:27:32,380 --> 00:27:40,959 And he made a very nice film of this where you sort of zoom in on it and you see the pictures inside itself all the time. 245 00:27:40,960 --> 00:27:46,000 And it gives you a very eerie feeling. This just goes on and on and on with a slight rotation at the same time. 246 00:27:47,530 --> 00:27:54,429 But to do this and I think Escher also used this sort of thing in it to use of bit of 247 00:27:54,430 --> 00:28:02,140 complex analysis here because it's a conformal mixture and you use the conformal. 248 00:28:03,550 --> 00:28:15,370 This is of course where one can talk about the mathematics in developing the idea of complex analysis and how the mapping of the complex plane, 249 00:28:16,360 --> 00:28:19,000 polymorphic mapping to another part of a complex plane. 250 00:28:19,300 --> 00:28:28,180 You always have these things which are these things which are conformal so that angles are preserved. 251 00:28:28,840 --> 00:28:32,590 Sizes are not preserved. What was that? 252 00:28:33,670 --> 00:28:36,729 Oh, I better start that again. 253 00:28:36,730 --> 00:28:43,610 Yeah. So yeah, I tend to not my microphones off by waving my arms around but um, 254 00:28:44,680 --> 00:28:50,979 so this is the sort of mathematics kind of lead into because it relates to the conformal 255 00:28:50,980 --> 00:28:57,880 geometry that is closely certainly to the one dimensional complex analysis which is complex. 256 00:28:59,290 --> 00:29:04,390 You have to have real parameters to describe your complex number. And so that's the normal, complex plane. 257 00:29:04,630 --> 00:29:08,020 And then when you transform with a complex analytic function, 258 00:29:08,560 --> 00:29:15,310 the or differentiable complex differentiable function, then the mapping is one which is conformal. 259 00:29:15,400 --> 00:29:19,600 And the Dutch mathematicians don't make great deal of use of this. 260 00:29:19,910 --> 00:29:28,270 In order to make this picture map into itself, you have to do something about making the square outside into a circle to make it fit. 261 00:29:28,270 --> 00:29:39,130 But they managed to do this very well. Okay, well, conformal transformations are also interesting. 262 00:29:40,480 --> 00:29:43,270 Escher made use of them here. I hope you can see this one. 263 00:29:44,230 --> 00:29:50,500 It's a bit like the tessellations of a plane we had before, but as you work your way down, you see the sizes change. 264 00:29:51,070 --> 00:29:57,729 And so it really isn't quite a Euclidean motion, but it's a little motion where you squash it down. 265 00:29:57,730 --> 00:30:08,590 So it's one of these conformal motions which related to complex analysis and that would be very nice area to discuss, certainly in more detail. 266 00:30:10,480 --> 00:30:14,860 One of the more famous of Escher's pictures is this one here. 267 00:30:18,590 --> 00:30:21,050 And here we see one of his circle limits. 268 00:30:23,060 --> 00:30:35,630 It's quite curious because it's very I find it very useful to use this picture to illustrate the different cosmologies in cosmology. 269 00:30:35,660 --> 00:30:44,840 You usually make an assumption that the universe especially isotropic and homogeneous, 270 00:30:45,050 --> 00:30:51,740 and these things were a limited number of geometries you can use, basically characterised by the curvature. 271 00:30:51,920 --> 00:30:55,819 The curvature can be positive, there can be more complicated topologies, 272 00:30:55,820 --> 00:31:03,320 but there are basically the three types the positive curve, zero curvature and the negative curvature. 273 00:31:03,830 --> 00:31:10,250 And Escher was very ingenious in showing that he could use more or less the same shapes. 274 00:31:10,820 --> 00:31:16,850 Illustrate each of these three. If you go carefully, you see the number of the devil's feet coming together as three. 275 00:31:16,850 --> 00:31:22,880 Here was it, whereas it's only two there and there are different numbers of things that the the wings and so on. 276 00:31:23,720 --> 00:31:27,170 And so you can get this without changing the shapes very much. 277 00:31:27,680 --> 00:31:30,710 You can get these three different geometries illustrated. 278 00:31:31,610 --> 00:31:41,990 This geometry illustrates various things, which I've often use it in the lectures on cosmology. 279 00:31:44,000 --> 00:31:48,500 As I said before, it's an illustration of the. 280 00:31:49,250 --> 00:31:52,430 But this is what's called hyperbolic geometry. That's the Euclidean geometry. 281 00:31:52,430 --> 00:32:02,719 That's the geometry of the sphere. Hyperbolic geometry is negative curvature and is often eschewed in several of these as called circle limits. 282 00:32:02,720 --> 00:32:14,570 He learned about it from Cox Sitter, who was at the same meeting of the International Congress of Mathematicians in 1950. 283 00:32:14,570 --> 00:32:21,770 I can't remember exactly 53, 54 or something where I went to this in my second year as a graduate student, 284 00:32:22,280 --> 00:32:25,880 and this is where I first became acquainted with Escher's work. 285 00:32:26,540 --> 00:32:34,340 Escher didn't at that time have these examples here, but he let me go back to the one I just showed you. 286 00:32:35,180 --> 00:32:39,229 You didn't have that then, but he learned from Cox. 287 00:32:39,230 --> 00:32:43,670 Sitter two is attending this meeting and Cox is to explain to Escher. 288 00:32:44,120 --> 00:32:50,510 He said, look, would be interesting if you could illustrate the hyperbolic plane. 289 00:32:52,850 --> 00:32:57,110 This is the plane that people usually mathematicians usually refer to as the Poincaré disk. 290 00:32:57,680 --> 00:33:06,170 I always worry about that because it was initially due to Beltrami and I guess people don't seem to remember it was Beltrami. 291 00:33:06,590 --> 00:33:10,669 He had all these different transformations. He had the Kline representation. 292 00:33:10,670 --> 00:33:16,220 They put incorrect half plane in the park or a disk before these other people did. 293 00:33:16,850 --> 00:33:20,089 I'm not quite sure about Riemann because Raymond already had the half plane, 294 00:33:20,090 --> 00:33:23,959 apparently, but I don't know whether he realised it was a non-Euclidean geometry. 295 00:33:23,960 --> 00:33:28,310 I'm not sure. But anyway, this is this was the first one that Escher made. 296 00:33:28,310 --> 00:33:39,620 It's actually quite useful to use that one because then you can see how this the you see these are the straight lines are the 297 00:33:39,620 --> 00:33:48,589 straight lines diameters or else you have Oxford circles which make the boundary orthogonal and the since it's conformal, 298 00:33:48,590 --> 00:33:54,710 the angles are correctly represented and you can see how you can have lines which never meet each other. 299 00:33:55,040 --> 00:34:00,439 I mean, these would be parallels which which actually never meet had an ethical these parallels because 300 00:34:00,440 --> 00:34:08,269 the parallel to infinity and then triangles you could see the angles don't add up to 180 degrees, 301 00:34:08,270 --> 00:34:14,210 they add up to something less. And all that is well understood as a piece of mathematics. 302 00:34:14,690 --> 00:34:24,380 But Escher had this very nice representation of it and you can see the way you you know, nothing figures if I get it right and. 303 00:34:26,850 --> 00:34:31,680 Something about that anyway. And then this is a little bit about that. 304 00:34:32,130 --> 00:34:46,320 As it does now, I've also used this where let's say this one as a nice illustration of a kind of geometry where infinity is represented. 305 00:34:46,740 --> 00:34:55,350 You see the angels and devils actually, for that in the West is as well as this one, because you can see the fish here. 306 00:34:57,030 --> 00:35:04,290 The black ones are supposed to be all congruent in the geometry. So you have to imagine that this fish and that fish that think they're the same. 307 00:35:04,740 --> 00:35:10,980 The geometry is such that we represent it in this way, so they get squashed up towards the edge. 308 00:35:11,340 --> 00:35:19,260 But as far as the fish are concerned, the white the white ones are the black ones because it is completely homogeneous. 309 00:35:19,260 --> 00:35:24,330 This geometry that you see also in this example, the eyes of the fish are exact circles, 310 00:35:24,900 --> 00:35:28,740 and these circles remain exact circles no matter how close you are to the boundary. 311 00:35:29,370 --> 00:35:34,880 His line is extraordinarily accurately, right down to the edge. You can look at look at it with a microscope. 312 00:35:35,010 --> 00:35:38,250 You can still see it's very, very precise, right down to the edge. 313 00:35:41,670 --> 00:35:49,860 Now, you see this is useful. I find it useful in my discussions of cosmology because you can use the same trick for space times. 314 00:35:49,860 --> 00:35:59,070 This is this is a two dimensional space. But in cosmology, you talk about space and time together and three dimensions of space, an amount of time. 315 00:35:59,490 --> 00:36:02,820 And you could talk about conformal representations just the same way. 316 00:36:03,300 --> 00:36:06,180 And so I find it very handy here. 317 00:36:06,180 --> 00:36:14,310 We see a picture this time going up the page and this is meant to be, well, you see, can only represent two dimensions of space. 318 00:36:14,850 --> 00:36:24,989 But I've got this wiggly part in the back because I don't want to prejudice the whether or not the space is actually closed or not. 319 00:36:24,990 --> 00:36:29,790 It might be open or closed, so I've allowed for it to wiggle around at the back. 320 00:36:29,790 --> 00:36:38,580 But to have it look as that starting to be closed is useful because otherwise you can't convey the the the size of the space very easily. 321 00:36:38,970 --> 00:36:43,980 But anyway, there is the big bang here. The universe expands, slows down a bit, 322 00:36:43,980 --> 00:36:52,560 and then starts to expand again with this exponential expansion which was recently observed and got the Nobel Prize and all that. 323 00:36:53,460 --> 00:36:59,310 And you see, here's the trick. You squashed that down so it becomes a finite boundary. 324 00:36:59,310 --> 00:37:04,230 This is a trick which has been useful to use talking about gravitational radiation. 325 00:37:04,530 --> 00:37:11,310 And certainly I was in the in the sixties and seventies playing with that. 326 00:37:11,730 --> 00:37:15,680 You can also do the opposite trick, which is to expand out the big bang here. 327 00:37:16,560 --> 00:37:20,400 Escher. I was trying to find issue representations of this. 328 00:37:20,730 --> 00:37:25,920 We did things rather like that. Here we have a picture as you guys to the middle and again, 329 00:37:25,920 --> 00:37:30,270 it's conformal and they go right down to I think you have to cut a hole and stop 330 00:37:30,270 --> 00:37:35,910 it for a bit because the big bang is really a surface rather than the point. 331 00:37:36,390 --> 00:37:44,100 So you have to stop it off somewhere. But it is very another an Escher representation of this idea. 332 00:37:44,940 --> 00:37:48,740 And here we have another illustration. So you could think of the big bang as somewhere in the middle there. 333 00:37:49,560 --> 00:37:55,709 And it's control may represent it, but it's sort of opposite trick here. 334 00:37:55,710 --> 00:37:58,710 You have infinity squashed on in there you have the big bang stretched out. 335 00:37:59,790 --> 00:38:02,640 This is a perfectly respectable thing to do in cosmology. 336 00:38:03,120 --> 00:38:12,360 What's not quite so respectable is this model, which is a model which which I've been playing with for a long time. 337 00:38:12,870 --> 00:38:18,059 I think there's some evidence for it, which I won't go into here because that's not the right talk. 338 00:38:18,060 --> 00:38:26,130 But that said, I just want to show you this picture that you can do something very Escher like, which is join the the big bang. 339 00:38:26,940 --> 00:38:29,970 It's the stretching out of the big bang to the squashing down. 340 00:38:30,090 --> 00:38:34,500 This is the this is our Ian, I should say big bang to remote future. 341 00:38:34,950 --> 00:38:38,550 And then this is the next Ian after us, this is the one before ours. 342 00:38:38,850 --> 00:38:46,920 And they form this nice continuous stretch with all these aeons going right back to infinity in both directions. 343 00:38:47,220 --> 00:38:53,550 Okay, that's enough of that. I haven't played with the topology of the universe. 344 00:38:53,940 --> 00:38:58,679 You can certainly have these homogeneous isotropic models and pull them up to various 345 00:38:58,680 --> 00:39:03,510 ways to make several logical things that you did in a straight topology in various ways. 346 00:39:04,020 --> 00:39:12,600 They were just some examples. See, this is a bit like a mobius strip that you have to go around four times I think before you get back to you started. 347 00:39:14,280 --> 00:39:18,780 He did play with Mobius strips as well and you're going to see these ants or whatever they are 348 00:39:18,780 --> 00:39:25,080 crawling over the Mobius strip and they can be one way or the other way on top of each other. 349 00:39:26,460 --> 00:39:32,520 Nice things to examine. Okay, now let me talk about something else. 350 00:39:33,270 --> 00:39:36,330 I did talk about tessellations of the plane. 351 00:39:38,460 --> 00:39:45,120 And unfortunately, what I'm going to show you came just too late for Escher. 352 00:39:45,240 --> 00:39:48,299 He died before I was able to. 353 00:39:48,300 --> 00:39:55,010 I think I wasn't quick enough off the mark. If I say he died too early, I wasn't quick enough to mark this. 354 00:39:55,020 --> 00:40:01,050 I want to show you this because it has some relation to the tiling that you see as you come in the building here. 355 00:40:01,440 --> 00:40:06,660 And I think I want to explain how that idea is behind that tiling. 356 00:40:07,170 --> 00:40:14,010 Well, this is the Italian I came across not long after she had actually died. 357 00:40:14,310 --> 00:40:17,730 I think he would have appreciated this picture. I should explain. 358 00:40:18,210 --> 00:40:26,010 It's a well-known theorem of crystallography, or, if you like, of the types of motions of the plane, 359 00:40:26,010 --> 00:40:31,260 which I've been talking about, that the only symmetries you can have when you get translational symmetries. 360 00:40:31,800 --> 00:40:37,980 But if you have a translational symmetry that's sliding along without any rotation together with a rotational symmetry, 361 00:40:38,370 --> 00:40:45,060 then the only rotational symmetries you can have two fold, three fold, four fold and six fold. 362 00:40:45,150 --> 00:40:53,100 And it's quite easy to prove that. But then here you have a pattern which seems to be very uniform. 363 00:40:53,100 --> 00:40:58,380 It is indeed very uniform. I've expanding, but it seems to have a five fold symmetry. 364 00:40:58,410 --> 00:41:03,870 You see, it's made out of pentagons, but actually four different shapes tend to consist of around pieces. 365 00:41:04,560 --> 00:41:12,210 You have these tentacles, a five pointed star, and just as caps as I call them, sort of half stars. 366 00:41:14,280 --> 00:41:17,400 It's quite interesting to look at even without knowing how it's built. 367 00:41:17,910 --> 00:41:27,810 If you set your eye along where I'm standing better than where you are and you can see that things line up, every little line continues. 368 00:41:29,130 --> 00:41:37,080 So with infinite number of line segments, whichever line you choose and the density of those line segments all the way along is the same. 369 00:41:38,190 --> 00:41:44,040 You also have this interesting decadence, or the place you see regular decadence. 370 00:41:44,040 --> 00:41:52,790 Regular decadence. Whenever you see a regular decadence like this one here where I live, I say it is not what it used to be. 371 00:41:52,800 --> 00:41:58,440 So I have trouble finding these things. So let me try one by looking at it first and then come back to you. 372 00:41:59,640 --> 00:42:03,380 Here's one there. The decorations are always the same. 373 00:42:03,420 --> 00:42:09,330 Three pentagons, one just this cap and tour offices, wherever they are. 374 00:42:09,840 --> 00:42:14,820 And whenever you find one of these decadence, it's surrounded by a ring of ten pentagons. 375 00:42:15,750 --> 00:42:18,810 So I always rather like those rings. I'll come back to them in a minute. 376 00:42:20,400 --> 00:42:24,330 The pattern is very uniform. In fact, although it never repeats itself, it can't. 377 00:42:24,330 --> 00:42:27,660 That would violate the theorem. It almost repeats itself. 378 00:42:27,900 --> 00:42:37,800 That is to say, if you give me, say, any percentage, less than 100%, say 99.9%, then I could give you a motion of translational motion of this, 379 00:42:38,160 --> 00:42:52,230 such that the pattern agrees to 99.9% and also the rotation of the five fold rotation to 99.9%. 380 00:42:52,980 --> 00:42:57,550 And then you say, well, maybe what, about 99.99%? 381 00:42:57,600 --> 00:43:01,740 This is yeah, you could do that to any percentage, less than 100%. 382 00:43:02,010 --> 00:43:07,770 You can find a motion which was sent this pattern unto itself, but it's never quite symmetric. 383 00:43:07,770 --> 00:43:12,510 And that's how it wriggles through the theorem. Let me explain how this is constructed. 384 00:43:13,860 --> 00:43:17,220 Here we have, I should say, is a sort of hierarchical arrangement. 385 00:43:18,840 --> 00:43:22,770 And here we have a bigger version of the same thing. 386 00:43:26,490 --> 00:43:32,790 Here we have a again, this is hierarchically arranged. 387 00:43:32,790 --> 00:43:41,490 So now the bigger version of the same thing. And here we have another version of the same thing. 388 00:43:42,060 --> 00:43:48,270 So that is the sort of organisation. It's quite useful to show that and then ask somebody, can you find these pentagons? 389 00:43:49,200 --> 00:43:54,390 And it's not so easy, which is a way of illustrating where, well, how uniform it really is. 390 00:43:55,770 --> 00:44:05,339 Okay. Now, this pattern, I think I'm going to use the second set of hierarchy. 391 00:44:05,340 --> 00:44:08,820 This one here is a little easier to explain what I'm going to say. 392 00:44:10,230 --> 00:44:15,900 Yeah, I can see the patterns more easily to here. Here you have the deck again and the rings. 393 00:44:16,110 --> 00:44:20,130 I always like these rings, you see. I'll come back to that in a minute. 394 00:44:21,300 --> 00:44:25,100 But it's. This is made out of four. 395 00:44:25,440 --> 00:44:30,089 It shapes where the Pentagon is actually. Well, that's not great. 396 00:44:30,090 --> 00:44:36,570 But there's another way of doing the same sort of thing. 397 00:44:36,900 --> 00:44:43,410 These are cuts in DART and the relationship between the cuts in DART and the original pattern is this. 398 00:44:44,280 --> 00:44:46,890 And you, if you look carefully enough, done it right. 399 00:44:47,340 --> 00:44:56,400 You see each cut has the same set of lines in it and each as each other cut and each dart has the same set of lines. 400 00:44:57,180 --> 00:45:00,510 And if you match the lines, then you get this arrangement here. 401 00:45:00,900 --> 00:45:06,540 In fact, the only way of putting the lines together so that you match them is to make that pattern. 402 00:45:06,900 --> 00:45:14,580 So these cuts in dart marks with the lines that you see here can only be assembled in a non periodic way. 403 00:45:15,210 --> 00:45:24,060 And so this example was the first known example where you have two shapes which will only tile the plane in a non periodic way. 404 00:45:25,920 --> 00:45:31,620 You can also Escher ises and this is the sort of things that came as she wasn't able to see this. 405 00:45:31,620 --> 00:45:38,310 I'm sure he would have done wonderful things. I shall try and find the right spot here, which is difficult. 406 00:45:40,050 --> 00:45:44,570 And I think to do it. That's not bad. 407 00:45:44,750 --> 00:45:56,319 Yeah. I don't know how well I've done it, but you can see that this these are cats and dogs, but modified to look like birds. 408 00:45:56,320 --> 00:46:06,910 So this is what I mean by ization. This is my poor attempt at trying to do something that Escher might have done as he lived long enough. 409 00:46:07,660 --> 00:46:11,319 But other people have done other things like this too. 410 00:46:11,320 --> 00:46:18,670 So it's interesting that you can do something which I'm sure you would've revelled in if you ever had the chance. 411 00:46:20,140 --> 00:46:24,250 Right. Let me go back to the cats and dogs and so on. 412 00:46:24,970 --> 00:46:33,250 What? To the initial. This one here. There's another way of doing it, which is these promises. 413 00:46:34,570 --> 00:46:44,530 And again, they are related to the cuts and cuts with lines drawn on the promises. 414 00:46:44,770 --> 00:46:49,900 And if you match the lines, then you get the original pattern back again. 415 00:46:51,670 --> 00:46:57,070 And you can do it directly between cats, dogs and their bosses. 416 00:46:58,330 --> 00:47:01,750 And it shows you can mark one of them to get to the other one and so on. 417 00:47:02,980 --> 00:47:11,560 I want to try and describe the tiling outside the front office building because those are two dots, but what are the markings on those tiles? 418 00:47:12,130 --> 00:47:17,320 Well, that came from the following, and I better make sure I don't get these things upside down when I pull them to the side. 419 00:47:18,280 --> 00:47:26,610 Here we have. The original Pentagon pattern. 420 00:47:27,240 --> 00:47:33,780 And as I say, I rather like these rings of Pentagon's. So it was nice to mark them. 421 00:47:34,110 --> 00:47:39,180 We have these green things go round the through those rings at Pentagon's. 422 00:47:40,950 --> 00:47:49,070 Now, that doesn't quite fill up the regions of the Pentagon. 423 00:47:49,080 --> 00:47:55,410 So let's do the more to that. I'm going to add some more green lives. 424 00:47:56,460 --> 00:48:06,330 And the virtue of doing this is that now I have a pattern which is really the same as that pattern, but it's a distortion of it. 425 00:48:06,690 --> 00:48:12,299 So you see here we have the Pentagon's you've got three different versions of the Pentagon's. 426 00:48:12,300 --> 00:48:23,640 They're very much distorted, but three different Pentagon's curvilinear Pentagon's now and you see the justice caps here is one. 427 00:48:24,870 --> 00:48:31,200 And they put it this way, there's just this gap and there's just this justice gap and so on. 428 00:48:32,130 --> 00:48:35,650 And so that is the same pattern, but somewhat distorted. 429 00:48:36,640 --> 00:48:47,370 Now, can you get these onto a rhombus tiling where you can, but the trouble is that the rhombus isn't quite the same. 430 00:48:48,240 --> 00:48:54,360 So you need to add a few more lines. Having edges, a few lines. 431 00:48:55,920 --> 00:48:59,910 You now see that every fat rhombus has the same markings on it, 432 00:49:00,570 --> 00:49:06,120 every thin rhombus has the same markings on it, and that is the tiling that we have outside the front. 433 00:49:06,930 --> 00:49:15,810 So if you want to know what it is, that's what it is. It's, it's the rhombus tiling with all these green things on it. 434 00:49:16,710 --> 00:49:28,170 So the markings there, I wondered for quite a long time whether the you have places where these rings close up and there are four 435 00:49:28,170 --> 00:49:35,430 different ones where they close up and you can see all four of them with different patterns on the outside. 436 00:49:35,880 --> 00:49:41,940 But there was. But are there any others? Well, the others would have to be ones which actually cross each other. 437 00:49:41,940 --> 00:49:45,390 Cross them. And are there any which close up and cross each other? 438 00:49:45,570 --> 00:49:50,220 I don't think there are. And the pattern there and I sort of conjectured that none of them did. 439 00:49:51,300 --> 00:49:59,280 Realisation of that conjecture is completely wrong. I did find one which wiggles all over the place much too big for the pattern outside 440 00:49:59,280 --> 00:50:03,390 the building here because all over the place and finally gets back to itself. 441 00:50:04,740 --> 00:50:08,970 So that conjecture was wrong to have a new conjecture and I was that that almost all of them do close. 442 00:50:11,460 --> 00:50:16,170 So you can certainly have patterns where where one or two of them don't close. 443 00:50:16,170 --> 00:50:20,340 But if it's more than two, I think probably, you know, closer. That's a conjecture. 444 00:50:20,610 --> 00:50:23,850 I don't know if it's true or not. Okay. That's that. 445 00:50:24,840 --> 00:50:30,360 And this, in fact, is the pattern that I gave to people so that they would get it right. 446 00:50:32,100 --> 00:50:40,580 Now, I want to give you something else. I've been playing around these things quite a bit, and there are other things you can do, too. 447 00:50:41,340 --> 00:50:50,550 There is a set of three shapes, not just two, and they're quite simple shapes the square, regular hexagon and the regular dodecahedron. 448 00:50:51,210 --> 00:50:55,170 And these markings on them are the things you have to match. 449 00:50:55,440 --> 00:51:02,340 And I was interested to know if when you match them, what you see, do you get some feeling for the pattern as a whole? 450 00:51:02,730 --> 00:51:05,670 I tried this with something before and I got no feeling whatsoever. 451 00:51:06,540 --> 00:51:12,420 But Joshua Sokolow, I gave a talk when I mentioned this at the end, and Joshua Sokoloff, 452 00:51:12,900 --> 00:51:19,140 the computer program, which was related to this, and you can see there is the pattern over there. 453 00:51:21,060 --> 00:51:25,380 I'll show you the next picture, which is the next stage of the hierarchy. 454 00:51:26,250 --> 00:51:32,970 And I was really quite surprised how it is really a very attractive pattern. 455 00:51:33,840 --> 00:51:37,080 It's one of these things which never actually quite repeats. 456 00:51:38,040 --> 00:51:48,090 It's not quite right in that those little wiggly, tiny rings ought to be exact circles, but that's the only thing that's not quite right about it. 457 00:51:50,730 --> 00:51:56,440 They should be. The little tiny one should be exact circles. They look they actually look falling. 458 00:51:56,660 --> 00:51:57,070 Falling. And. 459 00:51:58,250 --> 00:52:07,170 Anyway, somebody perhaps could make a building and would have these three tiles and they can they'd need a lot of them to see that pattern there. 460 00:52:08,040 --> 00:52:22,680 Okay, anyway, that's enough of this. Now, when I told you that I did go and see the an exhibition when I was in Amsterdam. 461 00:52:23,190 --> 00:52:27,750 At the International Congress of Mathematicians. And it was in the Dunkirk Museum. 462 00:52:30,120 --> 00:52:42,910 One of my lecturers, Sean Wiley, who taught me algebraic topology, and he had a picture of it on the catalogue which had the Escher picture. 463 00:52:42,930 --> 00:52:44,880 Night and day I'm showing you that one, 464 00:52:45,540 --> 00:52:51,000 with birds flying off in different directions and spaces between the birds picking the birds going the other way. 465 00:52:51,360 --> 00:52:54,520 One of the tonight's scene of the day scene on the other side. 466 00:52:55,170 --> 00:53:03,380 And so I went into this exhibition, and I remember I remember being particularly struck by this picture called relativity, 467 00:53:04,770 --> 00:53:11,610 but all these different staircases and creatures going up the staircases of gravity being in different directions. 468 00:53:12,270 --> 00:53:17,910 And I was really very struck by that. And, of course, many of the other things he had there. 469 00:53:18,660 --> 00:53:22,890 So I went away trying to think I would try and do something impossible myself, 470 00:53:24,450 --> 00:53:30,000 not with the skill that you had, of course, but at least with an idea that I hadn't actually seen. 471 00:53:30,630 --> 00:53:35,580 And I produced things with bridges and rivers and that going off in different directions. 472 00:53:36,120 --> 00:53:46,770 And I whittle it down to something which I thought illustrated the impossibility in its simplest sort of reduced it to its simplest form. 473 00:53:47,310 --> 00:53:54,360 And I showed this picture to my father, and then he started producing staircases, buildings and all that sort of thing. 474 00:53:54,360 --> 00:54:00,510 And we wrote a paper where we showed some of his pictures. 475 00:54:00,960 --> 00:54:07,380 We didn't know what the subject was, but my father happened to know the editor of the British Journal of Psychology. 476 00:54:07,710 --> 00:54:17,350 So we decided it was psychology. Probably they could get it published because he was good friends with the editor and indeed that's what happened. 477 00:54:17,860 --> 00:54:27,640 So it was published and we then sent the copies to Esher, giving reference to his catalogue and saying appreciation of his work. 478 00:54:28,210 --> 00:54:33,130 And in the meantime, Asher had actually produced something of a similar nature, 479 00:54:33,610 --> 00:54:40,570 which is this picture of a dairy where you see the top part of the building is joined up in an impossible way to the bottom half. 480 00:54:42,130 --> 00:54:52,960 He, after seeing the paper we had in the British Journal of Psychology, Ezra then produce his own version of The Staircase, 481 00:54:53,590 --> 00:54:57,820 which is a very famous picture now that the monk's going around in both directions. 482 00:54:58,240 --> 00:55:06,270 Very beautifully done. And Escher was very generous and giving credit to the paper. 483 00:55:08,980 --> 00:55:13,560 This was the occasion when I said, I must meet Asher. 484 00:55:13,570 --> 00:55:19,690 My father and Asher had a quite a correspondence backwards and forwards, and I had Escher's telephone number. 485 00:55:20,110 --> 00:55:28,120 I was travelling in, in the Netherlands, I think in Appleton, near to Bonn, where I lived. 486 00:55:28,120 --> 00:55:34,780 And I phoned him up and he said, Just come along. And he said, Come along for tea, which is what happened. 487 00:55:35,230 --> 00:55:36,430 So that's how I got to know him. 488 00:55:37,870 --> 00:55:47,879 But the triangle, which I'm going to show you, I'm going to say something about the mathematics that this is involved with. 489 00:55:47,880 --> 00:55:50,290 But before getting to that, let me show you. 490 00:55:50,290 --> 00:55:59,830 I was a psychology conference and I thought I'd try and go one step better than some of these pictures and produce an invisible, impossible object. 491 00:56:00,670 --> 00:56:04,240 So this this is an invisible, impossible triangle. 492 00:56:05,470 --> 00:56:10,510 This is also an invisible possible triangle and not quite sure which one is better. 493 00:56:11,380 --> 00:56:16,030 And you can put the two together and maybe that works best. 494 00:56:16,030 --> 00:56:19,900 I don't know. Anyway, you can probably see the invisible triangle. 495 00:56:20,560 --> 00:56:23,620 It's not really there, but it's impossible, even if it's not there. 496 00:56:26,800 --> 00:56:34,000 I was at a there was a film being made for some reason about trusses and things like that. 497 00:56:34,330 --> 00:56:44,680 And the people it was shown, I think it was on the BBC, Montel a long time ago, and they were starting to make this film and talking about twisters. 498 00:56:45,190 --> 00:56:49,180 And at one point they finally asked me, What's it good for you see? 499 00:56:49,720 --> 00:56:53,860 So I said, Well, one thing you can use, Twist, is for solving Maxwell's equations. 500 00:56:54,640 --> 00:56:59,680 And then I said, Well, well, you had this. You have to use a certain idea that I can't really explain it. 501 00:57:00,190 --> 00:57:06,360 It's cosmology. I can't explain that to you, but it's. So they got very disappointed about that. 502 00:57:06,370 --> 00:57:11,560 So then I went home and I thought, gosh, I can come on this course and they can explain it. 503 00:57:12,400 --> 00:57:18,580 Here we are. You see, this is a nice way of producing that. 504 00:57:18,580 --> 00:57:23,710 You see your mark. Imagine that you have some pieces which you have to assemble to make that object. 505 00:57:24,310 --> 00:57:31,990 And these pieces consist of a lot of corners like that made they made out of wood, you say, 506 00:57:31,990 --> 00:57:35,620 and you make these out of wood and you have instructions about how to build them together. 507 00:57:35,950 --> 00:57:39,430 That's supposed to glue to that one, and this is to glue to this one in that glue to that one. 508 00:57:39,940 --> 00:57:43,780 And you have to look at these instructions and see whether the instructions 509 00:57:44,230 --> 00:57:49,120 allow you to build the objects or you're going to have a little problem with it. 510 00:57:49,660 --> 00:57:52,390 And then I realise that actually is an element of cosmology. 511 00:57:52,690 --> 00:58:00,490 If the cosmology element vanishes, you're just put and you see these instructions give you a check representation of an object, 512 00:58:00,880 --> 00:58:07,000 and this object you're using the freedom you have of the distance from the eye. 513 00:58:07,480 --> 00:58:13,150 And so that's what enables you to construct the current multi and nontrivial cosmology element. 514 00:58:13,630 --> 00:58:21,730 And if the instructions have a non vanishing coloured cosmology element, then that's something you can't build. 515 00:58:22,240 --> 00:58:25,690 If come on, the element does vanish, then you can build it. 516 00:58:26,170 --> 00:58:30,450 So I thought this was a nice way into that notion of cosmology. 517 00:58:31,630 --> 00:58:35,530 I see the measure. 518 00:58:35,800 --> 00:58:48,970 It's a Czech representation of cosmology. I was talking about this too, at a conference in I think it was in Rome. 519 00:58:49,780 --> 00:58:53,050 It was on that mathematics and asked and I can't remember something like that. 520 00:58:53,320 --> 00:58:57,010 And I was talking to an American mathematician, unfortunately, regarding who it was. 521 00:58:57,610 --> 00:59:04,839 And I was explaining that my talk the next day was going to be about cosmology and how this represents that. 522 00:59:04,840 --> 00:59:08,980 And he said, Oh, well, is there any other group you can represent? I said, Well, I hadn't really thought about that. 523 00:59:09,460 --> 00:59:17,620 So this is the sort of. You can take the distance from the eye and you take the logarithm of it, and that gives you a an additive group. 524 00:59:17,620 --> 00:59:21,700 So you can then talk about the come on in the ordinary check way. 525 00:59:23,140 --> 00:59:25,420 That has to be explained in more detail in the book, of course, 526 00:59:27,010 --> 00:59:32,290 but then in the other groups I thought, well, yeah, there is, you know, I could think of that too. 527 00:59:33,220 --> 00:59:37,150 And so, in fact, I drew this picture out and I gave it and I took the next day. 528 00:59:37,450 --> 00:59:42,580 I think it's a little easier if I don't show you that one. But something which I developed a bit later. 529 00:59:43,600 --> 00:59:49,540 It's the same idea. And what is this you see? 530 00:59:49,780 --> 00:59:55,970 Well, this is an idea that you use. And let me show you this picture here. 531 00:59:56,320 --> 01:00:00,129 It's the only one I know of his, which I'm not sure. 532 01:00:00,130 --> 01:00:05,710 It's quite cosmology. I haven't quite figured it out yet that you see on one side of the picture. 533 01:00:06,940 --> 01:00:11,980 I mean, Escher used this idea for other things where you have this sort of like a cube paradox. 534 01:00:12,280 --> 01:00:17,889 The thing could be one way around or the other. You could see a cube edge on and you may see the back of it as well. 535 01:00:17,890 --> 01:00:22,030 You don't know quite whether you see the cube internally, whether you're looking inside or outside. 536 01:00:22,360 --> 01:00:28,240 And if you see this, you get it one way around and then that tells you the rest of the picture which way around it is. 537 01:00:28,420 --> 01:00:33,940 So Escher has this here and this is clearly one way around and this is clearly the other way around. 538 01:00:33,940 --> 01:00:39,370 But in the middle it's ambiguous so that the whole thing is impossible because you get the ambiguity in the middle, 539 01:00:39,370 --> 01:00:45,040 which if you make one choice, which is this choice that contradicts that, and the other choice contradicts this. 540 01:00:45,640 --> 01:00:51,820 But the thing I had here, I think as you would have done wonderful things with this, too. 541 01:00:52,480 --> 01:00:57,820 But you see the staircase. You could imagine walking around the staircase and well, it's impossible. 542 01:00:58,300 --> 01:01:02,050 I did show it to a but there's another version of it here. 543 01:01:02,380 --> 01:01:05,800 I did show it to an expert on impossible objects who was at the conference, 544 01:01:06,100 --> 01:01:09,640 and I showed him this picture and he said, Can you see what's wrong with this? It's nothing on that picture. 545 01:01:10,630 --> 01:01:15,160 Could you make that out, would you think? Yeah, sure, if I could make that out. What are you really sure you could? 546 01:01:16,000 --> 01:01:21,399 You have to work it because the trouble is your eyes, you go round it, it flips to the other one without you noticing. 547 01:01:21,400 --> 01:01:26,790 That's what's happening. So you certainly couldn't make that out of a piece of wood anyway. 548 01:01:26,800 --> 01:01:30,970 So that's that. Can you make these impossible objects? 549 01:01:31,030 --> 01:01:34,420 Well, that's a good question. You could certainly make the pieces. 550 01:01:35,230 --> 01:01:42,010 One thing you can do is this. In fact, this is a model of something which I believe somewhere was in college. 551 01:01:42,010 --> 01:01:45,520 It's still there. I'm not quite sure where there is a bigger version of this. 552 01:01:46,030 --> 01:01:49,989 I can't show it to you all at once, everywhere, because you see, this is the impossibility. 553 01:01:49,990 --> 01:02:01,120 It's not a Czech version. This is more like a like aa1 of the continuous versions of Come On You Can have and that's one way of doing it. 554 01:02:01,480 --> 01:02:14,780 But let me show you something else. Ah, no, that's the picture we just had. 555 01:02:14,780 --> 01:02:19,850 Isn't this we've done that. Yes. Simple screen. 556 01:02:22,480 --> 01:02:27,120 Wow. And wow, look, it still works. 557 01:02:28,350 --> 01:02:32,850 That was lined up just right. So you have it's a possible impossible triangle. 558 01:02:33,810 --> 01:02:39,060 And of course, if you want to see what it really is, you give us a little twist. 559 01:02:39,450 --> 01:02:42,689 Well, first of all, I think the best expert for this we had. 560 01:02:42,690 --> 01:02:47,490 There we are. Oh, we give us a little twist and then you can see what it is. 561 01:02:53,770 --> 01:03:18,960 Okay. Well, thank you very much. Thank you very much for this wonderful lecture. 562 01:03:19,410 --> 01:03:26,639 And now for the second part of the event, I'd like to please, I'd be welcoming a Nyquist house. 563 01:03:26,640 --> 01:03:30,750 Who is going to take and present the awards, please. 564 01:03:31,080 --> 01:03:42,819 Should I come back or. Okay. 565 01:03:42,820 --> 01:03:53,950 So in a moment, I'm going to call on Lavinia Clay to present the Clay Award for the dissemination of mathematical knowledge to Roger Penrose. 566 01:03:54,850 --> 01:03:59,680 So dissemination of mathematical knowledge is a very important part of the Clay Institute's mission, 567 01:04:00,130 --> 01:04:08,020 alongside supporting mathematical research at the very highest level and of course, giving very large prizes. 568 01:04:10,920 --> 01:04:19,600 The this award recognises the achievements of people who have made exceptional contributions under both headings. 569 01:04:20,740 --> 01:04:30,550 In Roger's case, the award is in recognition of his outstanding contributions to geometry, relativity and other branches of mathematics, 570 01:04:31,000 --> 01:04:38,230 and of his tireless work in explaining mathematical ideas to the public through popular books, public lectures and broadcasts. 571 01:04:39,520 --> 01:04:49,420 But of course, Alan at the beginning gave a long list of his contributions within relativity, within geometry. 572 01:04:49,750 --> 01:04:55,420 And that is takes very the first books very, very clearly. 573 01:04:56,860 --> 01:05:02,770 But he is also, as you seen, a brilliant expositor of mathematical ideas. 574 01:05:03,220 --> 01:05:06,940 And it's the combination of these two which leads to the award. 575 01:05:07,750 --> 01:05:15,760 So we've seen his some of his contributions to geometry and the Penrose tiling. 576 01:05:16,840 --> 01:05:30,190 But there's another aspect of his work which is really very important and which bridges the to the two, which is the passing from one area to another. 577 01:05:30,550 --> 01:05:34,690 So he begins by thinking about something impossible. 578 01:05:35,620 --> 01:05:42,040 So you start thinking about the idea of tiling the plane with Pentagons. 579 01:05:42,880 --> 01:05:50,650 Well, most people might have occurred to earlier to think whether that's possible, and they decide very rapidly that it's not possible. 580 01:05:51,070 --> 01:05:55,090 Well, Roger thinks about that. And what does it lead to? 581 01:05:55,150 --> 01:06:02,620 It leads to the Penrose tiling. And that has an impact, a huge impact outside the area in which he thought about it. 582 01:06:02,890 --> 01:06:07,850 And in that case, an impact in the visual arts and and beyond. 583 01:06:08,500 --> 01:06:12,490 Or he starts thinking about impossible objects. 584 01:06:12,790 --> 01:06:22,209 And where does that lead to? Leads to a paper in the British Journal of Psychology, which is there are other examples of this. 585 01:06:22,210 --> 01:06:25,270 I mentioned briefly the more of Penrose inverse. 586 01:06:26,140 --> 01:06:31,030 Well, again, you you learn at school about taking the inverse of a square matrix. 587 01:06:32,170 --> 01:06:37,210 Well, what about a matrix that isn't square? Well, you give that a few seconds thought. 588 01:06:37,330 --> 01:06:42,160 That's impossible. So you go on. Not if you're Roger. You think about it some more. 589 01:06:42,220 --> 01:06:45,790 And that leads to this generalised inverse. 590 01:06:46,120 --> 01:06:52,630 And again, that has a huge impact. In fact, his paper on the generalised inverse is his most cited paper. 591 01:06:53,350 --> 01:06:56,980 It's a hugely useful tool in numerical analysis. 592 01:06:57,040 --> 01:07:03,909 It's had a big impact on computation and other things. 593 01:07:03,910 --> 01:07:10,210 Where he's crossed the boundary, perhaps is it was Roger who. 594 01:07:10,720 --> 01:07:16,480 This one's for the computer scientist who first made the suggestion that the church's 595 01:07:16,480 --> 01:07:22,270 lambda calculus might be a useful tool in thinking about programming languages. 596 01:07:22,300 --> 01:07:24,370 And that, again, had a huge impact. 597 01:07:25,570 --> 01:07:35,080 But he's best known outside the scientific community for his popular books, which are wide ranging, highly original, 598 01:07:36,220 --> 01:07:44,530 highly controversial, in the very best sense of being controversial in that they provoke debate and get people talking. 599 01:07:45,940 --> 01:07:53,019 So they have a very unusual feature for popular books in science in that they're 600 01:07:53,020 --> 01:07:58,450 addressed not only to the general public in explaining mathematical ideas, 601 01:07:59,200 --> 01:08:04,810 but also in engaging the public in the development ideas of ideas. 602 01:08:04,840 --> 01:08:13,780 So he's he's putting forward his own scientific ideas, original ideas in his books in a way that many other people do not. 603 01:08:14,620 --> 01:08:18,249 His books have been translated into many different languages. 604 01:08:18,250 --> 01:08:26,200 They've sold in the hundreds of thousands, and they are a remarkable achievement. 605 01:08:27,370 --> 01:08:31,750 Well, you may have noticed that Roger doesn't use the most up to date of the technology. 606 01:08:33,370 --> 01:08:39,850 It is in his public lectures. But notwithstanding that, he does have a very significant online presence. 607 01:08:40,540 --> 01:08:48,700 If you look. On YouTube, you will find some 30 expository videos, recording lectures and interviews. 608 01:08:49,180 --> 01:08:55,579 Many of them are been viewed tens of thousands of times, or in one case, over 300,000 times. 609 01:08:55,580 --> 01:08:58,870 So that's a huge impact there. 610 01:08:59,410 --> 01:09:04,300 So I think it's clear that he very amply fulfils the criteria for our award, 611 01:09:04,600 --> 01:09:08,049 both in the contributions that he's made in the development of mathematical 612 01:09:08,050 --> 01:09:13,120 ideas and the incredible work he's done in the dissemination of mathematics. 613 01:09:13,720 --> 01:09:17,530 So I'll now call on Lavinia Clay to present the award to him.