1 00:00:01,330 --> 00:00:18,270 So I. When I was at school, I was very frustrated by the choices we were kind of asked to make. 2 00:00:19,890 --> 00:00:26,040 It was a Shakespeare or the second law of thermodynamics Rubens or Relativity. 3 00:00:26,430 --> 00:00:30,960 WC or DNA art or science. 4 00:00:31,950 --> 00:00:38,880 Now, at school, I found this demand to try and make a choice between these two deeply frustrating i. 5 00:00:39,300 --> 00:00:43,890 When I went up to secondary school, I actually went to school, a comprehensive school in Oxfordshire, just local to here. 6 00:00:44,940 --> 00:00:49,090 I started learning the trumpet, started playing in orchestras. 7 00:00:49,110 --> 00:00:51,840 In fact, Peter Norman, who's here in the audience, 8 00:00:51,960 --> 00:00:57,500 rescued me once when I was stranded here in an orchestra and there was no lift home driving all the way back to Henley on Thames. 9 00:00:57,500 --> 00:01:06,360 So and I did a lot of theatre at school and but at the same time as I started learning the trumpet, 10 00:01:06,360 --> 00:01:13,800 it was about the time that I fell in love with the world of science and in particular mathematics. 11 00:01:14,910 --> 00:01:20,010 The wonderful power that mathematics has to tell us where we've come from and where we're going to go next. 12 00:01:20,850 --> 00:01:28,169 And I seem to have to make a choice between these two. And I was better at my mathematics than I was at my scales on the trumpet. 13 00:01:28,170 --> 00:01:35,940 And so I chose the the mathematical route. But actually one of the things I've always enjoyed about Oxford is when I came up here as an undergraduate, 14 00:01:37,140 --> 00:01:39,120 I found that actually I could still do both. 15 00:01:39,510 --> 00:01:45,389 In fact, I think the college system almost encourages you to spend your time with people doing other disciplines. 16 00:01:45,390 --> 00:01:54,570 So I spend a lot of time trying to justify why I thought mathematics was as exciting as Derrida is deconstruction or various other things. 17 00:01:54,570 --> 00:02:00,780 And so I kept on doing my music hair and my theatre and but I went to mathematical roots. 18 00:02:01,140 --> 00:02:06,120 I became a professor here. But I have always kept my interest in the arts. 19 00:02:06,120 --> 00:02:16,710 And as time went on, I began to realise more and more that this idea of the two cultures that C.P. Snow talks about, it's really a false dichotomy. 20 00:02:17,190 --> 00:02:25,649 And in fact, the more and more time I spend studying, spending time in the arts, working with all creative artists, 21 00:02:25,650 --> 00:02:31,620 I realise again and again how often we're both drawn to very similar sort of structures. 22 00:02:32,520 --> 00:02:41,280 So what I want to do in this presentation is to I've chosen five of my favourite 20th century artists from various different disciplines, 23 00:02:41,280 --> 00:02:48,840 and so I've called the talk the secret mathematicians, because in some sense I think that these artists are a bit like secret mathematicians. 24 00:02:49,090 --> 00:02:57,270 They're being drawn sometimes consciously but often unconsciously, to structures that are a fundamental importance to me as a mathematician. 25 00:02:58,650 --> 00:03:06,690 So the first artistic discipline I've chosen is one which has a lot of resonance with mathematics traditionally, which is music. 26 00:03:07,440 --> 00:03:12,840 So I've chosen a composer. One of my favourite composers from the 20th century is Olivier Messiaen, 27 00:03:13,500 --> 00:03:19,020 who is obsessed with mathematical structures as an inspiration for a lot of his work. 28 00:03:19,440 --> 00:03:24,839 And in particular, there's one piece that he wrote while he was a prisoner of war in the Second World War, 29 00:03:24,840 --> 00:03:28,590 he was in a prisoner of war camp and there was a rickety upright piano. 30 00:03:29,160 --> 00:03:33,059 There was a clarinettist, a violinist and a cellist. And he wrote this piece called The Quartette. 31 00:03:33,060 --> 00:03:40,290 For the end of time. The quartette is meant to kind of represent that that incredibly desperate period of history. 32 00:03:40,290 --> 00:03:48,270 And the first piece, the liturgy, the crystal, is meant to capture this sense of unease and a sense of never ending time. 33 00:03:48,660 --> 00:03:55,110 And the way that Messiaen so sort of captures this unease is actually to use a little bit of mathematics. 34 00:03:55,650 --> 00:04:02,730 In fact, he uses two numbers that are very important to my own research prime numbers, so indivisible numbers. 35 00:04:03,600 --> 00:04:06,600 So the prime numbers he uses are 17 and 29. 36 00:04:06,630 --> 00:04:09,990 Now, the way that he uses this is in the piano part. 37 00:04:10,260 --> 00:04:14,850 Now, when a piece opens, you hear the clarinet and the violin exchanging bird themes. 38 00:04:15,330 --> 00:04:19,290 Messiaen was very interested in bird themes, used to notate them as an inspiration as well. 39 00:04:19,560 --> 00:04:24,790 But it's the piano piece where the mathematical structure can be heard. What it is. 40 00:04:24,790 --> 00:04:32,680 It's a very repetitive piano piece. The rhythm sequence is just 17 notes, a rhythm repeated over and over again. 41 00:04:33,670 --> 00:04:39,700 But the harmonic sequence is 29 chords, which themselves are repeated over and over again. 42 00:04:39,730 --> 00:04:43,240 So you have the 29 chords and then you repeat them again and repeat them again. 43 00:04:43,510 --> 00:04:49,749 But the choice of the 17 notes for the rhythm and the 29 for the Homeric sequence means that you get a very strange 44 00:04:49,750 --> 00:04:58,930 effect because the choice of 17 and 29 mean that these numbers don't come in sync until you've heard 17 times 29 chords. 45 00:04:59,080 --> 00:05:01,870 So you get the sense of you can hear something repeating, 46 00:05:01,870 --> 00:05:07,359 but then it's sort of when it's the rhythm starts, the harmonic sequence is still working its way through. 47 00:05:07,360 --> 00:05:10,630 When the harmonic sequence starts, the rhythm is halfway through its pace. 48 00:05:11,380 --> 00:05:17,110 So if you see a score for the piano piece, so the 17 note rhythm sequence, you starts with crotchet, 49 00:05:17,110 --> 00:05:21,130 crotchet, crotchet, and then you get a syncopated rhythm until you get crotchet and then minim. 50 00:05:21,400 --> 00:05:24,610 And then you see the crotchet, crotchet, crotchet repeated again and off it goes. 51 00:05:24,610 --> 00:05:31,629 The rhythm is exactly the same. 17 Notes. The harmonic sequence goes all the way up to this point where you see the same chords repeating, 52 00:05:31,630 --> 00:05:35,620 but you see when the records are repeated again, it's a completely different rhythm, rhythm structure. 53 00:05:36,310 --> 00:05:44,630 So it's very hard to hear what Messina is doing, but you certainly get a sense of there being structure there, but a huge unease as well. 54 00:05:44,650 --> 00:05:49,240 So let's hear the prime 17 at 29 at work in the quartette for the end of time. 55 00:06:07,600 --> 00:06:13,420 So that's the rhythm sequence repeating again. The harmonic sequence is still working its way through the 29 chords. 56 00:06:19,930 --> 00:06:24,670 And now the harmonic sequence starts repeating, but the rhythm sequence is doing something completely different. 57 00:06:25,180 --> 00:06:27,669 And so the choice of 17 and 29, 58 00:06:27,670 --> 00:06:36,490 these two primes are chosen very deliberately by Messina in order to create this sense of unease and the things keeping out of sync. 59 00:06:37,830 --> 00:06:41,070 Now. I think what's interesting is that very often at the core of this, 60 00:06:42,330 --> 00:06:47,430 the fact that artists and mathematicians and scientists seem to be interested in similar structures is that in 61 00:06:47,430 --> 00:06:53,250 some sense we're all reacting to the natural world around us and trying to find patterns in the natural world. 62 00:06:53,430 --> 00:07:02,400 So you actually find this effect at the heart of the survival of a very strange insect that you can find in North America. 63 00:07:02,940 --> 00:07:06,510 It's a cicada which has an incredibly strange lifecycle. 64 00:07:07,560 --> 00:07:10,680 There was a very big brood this summer which appeared on the East Coast. 65 00:07:11,010 --> 00:07:16,260 It was, but they hadn't been seen for 17 years because what these cicadas do is that 66 00:07:16,470 --> 00:07:21,690 there are the eggs in the ground and then the the eggs produce the cicadas. 67 00:07:21,690 --> 00:07:26,550 They emerge into the forest, they eat the leaves, they mate, they lay eggs. 68 00:07:27,060 --> 00:07:29,170 So actually, this is the sound of one of these cicadas. 69 00:07:29,190 --> 00:07:35,160 The sound is so loud that residents in general move out for the 17th year because it's totally unbearable. 70 00:07:35,790 --> 00:07:40,290 People make sure they don't plan their weddings while this is happening because you can't hear anything. 71 00:07:40,530 --> 00:07:42,120 But after six weeks of passing, 72 00:07:42,360 --> 00:07:50,460 the cicadas all die and the forest goes quiet again for another 17 years before the next brood next generation emerges from the ground. 73 00:07:50,770 --> 00:08:00,540 That's I had a chance to go and see these the year before last in the it was it was Nashville, Tennessee. 74 00:08:00,540 --> 00:08:05,460 They had a brood there. But this wasn't a 17 year cycle. This was actually a 13 year cycle. 75 00:08:05,670 --> 00:08:12,239 So it's very curious. There are different species across America, but they only have 13 or 17 unicycles. 76 00:08:12,240 --> 00:08:15,900 There are none with 12, 14, 15, 16 or 18. 77 00:08:16,260 --> 00:08:24,940 So two prime numbers, 13 and 17. So it's very curious, what is it about the primes which seems to be helping these cicadas in some way? 78 00:08:25,830 --> 00:08:26,790 Well, we're not too sure, 79 00:08:26,790 --> 00:08:34,499 but one theory is that it's very much the same principle as Mason was using to keep the rhythm and the harmony out of sync in the liturgy, 80 00:08:34,500 --> 00:08:42,240 the crystal, because there is a belief that there might have been a predator around in the forest, which also used to appear periodically, 81 00:08:42,570 --> 00:08:46,950 and the predator would try and time its arrival to coincide with all of this card is appearing. 82 00:08:47,730 --> 00:08:55,190 Now the cicadas, which had a non-prime number, life cycle, they got wiped out because they got in sync too quickly with the predator. 83 00:08:55,200 --> 00:09:02,700 So, for example, I've got a little example here where the Predator appears every six years and the cicada appears every nine years. 84 00:09:02,880 --> 00:09:12,030 So they have a common factor three. So that means that the predator and the cicada will meet each other every 18th year. 85 00:09:12,030 --> 00:09:16,110 So every second time this cicadas emerge, they meet this predator. 86 00:09:16,350 --> 00:09:20,880 And so they're quickly wiped out. But those cicadas, which have a prime life cycle. 87 00:09:20,940 --> 00:09:24,480 So let's shift. So let's make the cicada have a seven year life cycle. 88 00:09:24,660 --> 00:09:31,710 So seven means actually it's appearing more often in the forest. So you might think, well, this is got even more of a chance of getting wiped out. 89 00:09:31,860 --> 00:09:35,549 But no, because now six and seven are co prime. 90 00:09:35,550 --> 00:09:41,370 Seven is a prime number. It means that this cicadas keeps out of sync of this predator. 91 00:09:41,370 --> 00:09:46,160 So the first time they actually meet is a six time seventh year, the 42nd year. 92 00:09:46,410 --> 00:09:52,530 And so this prime number of years gets a better chance of keeping out of sync and so surviving. 93 00:09:53,070 --> 00:09:56,790 So it seems like there was a quite a competition in some of the forests in North America. 94 00:09:56,790 --> 00:10:03,270 The Predator would move its lifecycle, get in sync, but it's the 17 year lifecycle. 95 00:10:03,360 --> 00:10:06,450 So if you know your primes, you kind of survive in this world. So it's a good message. 96 00:10:06,450 --> 00:10:10,529 There's an impact for you. You know, you, Max, you survive. 97 00:10:10,530 --> 00:10:16,060 So so the cicadas knew they survived and they seems like the predator got wiped out. 98 00:10:16,090 --> 00:10:21,270 It didn't it couldn't get into sync with the 17 year lifecycle or a 13 year lifecycle. 99 00:10:21,540 --> 00:10:26,460 But it's kind of curious. This is exactly the same trick as Messina's using that the Predator, 100 00:10:26,520 --> 00:10:32,399 that's a bit like the rhythm sequence and the the cicada, a bit like the harmonic sequence. 101 00:10:32,400 --> 00:10:41,010 And keeping those things out of sync gives a sense of kind of unease and never ending time that well, almost never ending time. 102 00:10:41,340 --> 00:10:43,470 The message I wanted to know. 103 00:10:43,620 --> 00:10:50,700 For me, what's intriguing is that he is missing who's read up on a bit of mathematics and realise that it can be useful for him. 104 00:10:50,940 --> 00:10:58,019 But often artists are being drawn to mathematical structures for without knowing it. 105 00:10:58,020 --> 00:11:06,750 And often you can find examples where the mathematics was discovered first by artists and only subsequently realised were important by mathematicians. 106 00:11:06,930 --> 00:11:10,350 And one very interesting example of this, it's a very famous sequence. 107 00:11:10,350 --> 00:11:13,950 So if you're not a mathematician, what's the next number in this sequence? 108 00:11:14,460 --> 00:11:19,770 You always say, if you haven't read The Da Vinci Code, you're not allowed to have read the definition. But what's the next number in this sequence? 109 00:11:20,610 --> 00:11:26,610 34 It's a very famous sequence that a lot of kids get exposure at school. 110 00:11:26,880 --> 00:11:30,270 It's a lovely sequence because you get the next one by adding the two previous numbers. 111 00:11:30,270 --> 00:11:37,110 In the sequence they call the Fibonacci numbers. Fibonacci discovered them because he realised they were very much. 112 00:11:37,190 --> 00:11:42,260 Relate it to two things in the natural world, again, very much associated with growth. 113 00:11:42,680 --> 00:11:48,350 So the way that things seem to grow, I mean, the numbers themselves have a natural sense of growth in them. 114 00:11:48,500 --> 00:11:51,510 And that seems to be also apparent in the natural world. 115 00:11:51,560 --> 00:11:58,130 So, for example, if you count the number of petals on a flower, invariably it's a number in the Fibonacci sequence. 116 00:11:58,320 --> 00:12:03,050 Sometimes you get two copies of the flower, so you can get double the number in the Fibonacci sequence. 117 00:12:03,320 --> 00:12:05,450 And if it isn't the number in the Fibonacci sequence, 118 00:12:05,630 --> 00:12:10,730 that's because the petals falling off your flower, which is how mathematicians get round exceptions. 119 00:12:12,230 --> 00:12:16,610 But actually Fibonacci wrote about these in relation to another problem, 120 00:12:16,610 --> 00:12:25,930 which is if you look at two generations of rabbits from one season to the next, how many rabbits do you expect with a particular mathematical rule? 121 00:12:25,940 --> 00:12:29,060 So the mathematical rule was you start with one pair of rabbits. 122 00:12:29,660 --> 00:12:36,020 They take a month to mature, at which point that they can give birth to another pair of rabbits who then 123 00:12:36,020 --> 00:12:40,510 themselves take a month to mature before they can give birth to another generation. 124 00:12:40,520 --> 00:12:43,910 And so it's quite a sort of complex problem to get your head around, as you know. 125 00:12:44,000 --> 00:12:47,570 Oh, these ones are all mature, yet these ones are still giving birth. 126 00:12:48,000 --> 00:12:52,970 These are very mathematical rabbits which never die, of course, and which always give birth to a man and a female. 127 00:12:53,390 --> 00:12:59,550 But Fibonacci realised that these numbers were described by the female. 128 00:12:59,720 --> 00:13:04,250 We apply this sequence where you add the two previous generations together to get the next one. 129 00:13:05,000 --> 00:13:08,090 So they've been very much attributed to Fibonacci. 130 00:13:08,090 --> 00:13:14,659 But the intriguing thing is, and I only discovered this quite recently when I was looking into the subject of Indian mathematics, 131 00:13:14,660 --> 00:13:19,190 I did this programme about the history of mathematics and sort of discovered I mean, 132 00:13:19,190 --> 00:13:24,260 we teach our subject very historically and somehow you don't realise where a lot of these things come from. 133 00:13:24,500 --> 00:13:29,629 But actually I discovered these Fibonacci numbers that Fibonacci wasn't the first to discover it. 134 00:13:29,630 --> 00:13:37,820 In fact, they were discovered already in India by poets and musicians who were interested in what sort of rhythms that you can generate. 135 00:13:38,060 --> 00:13:43,639 We use long and short beats, so if you allowed long and short beats, I suppose you've got four beats in the bar. 136 00:13:43,640 --> 00:13:47,810 So for short beats and how many different rhythms can you make? 137 00:13:47,820 --> 00:13:56,210 So what you can do for sure beats, or you could do a short, short, long or short, long, short or long, short, short or long, long. 138 00:13:56,660 --> 00:14:00,410 So you've got five different rhythms that you can make out of these long and short beats, 139 00:14:00,410 --> 00:14:05,750 which was of interest to know what sort of different possibilities, where have you doing poetry or creating music? 140 00:14:05,910 --> 00:14:09,680 And so you say, okay, if I had an extra short piece, I've got five short beats in the bar. 141 00:14:10,190 --> 00:14:11,810 How many different rhythms can I make? 142 00:14:12,950 --> 00:14:18,979 Well, they discovered it's the same rule is for the Fibonacci numbers, because you add the two previous numbers together. 143 00:14:18,980 --> 00:14:24,500 So there are eight different rhythms and actually you can see why I think it's much more obvious in there with these rhythms. 144 00:14:24,500 --> 00:14:27,920 And then four for the rabbits in a way, because how do you get the rhythms? 145 00:14:27,920 --> 00:14:32,930 So you can take all the ones with four beats in the bar and add a short beat to those, and that gives you the sum of them. 146 00:14:32,930 --> 00:14:36,380 But you can also take the ones with three beats in the bar and add a long beat to those. 147 00:14:36,620 --> 00:14:39,140 They're all different and they give you all the different possibilities. 148 00:14:39,380 --> 00:14:47,240 So it seems much more obvious why these numbers, these Fibonacci numbers, actually count rhythms rather than sort of rabbits. 149 00:14:48,380 --> 00:14:52,730 So actually it was the musicians and poets who discovered these numbers. 150 00:14:52,730 --> 00:15:01,400 First they had written in Hamish. Sandra writes about these numbers before Fibonacci, and it seems that even before Ramachandra but you know, 151 00:15:01,400 --> 00:15:04,790 they really deserve to be called the Sandro Fibonacci numbers. 152 00:15:05,660 --> 00:15:10,790 And I think, you know, I talked a lot about rhythm and mathematics, and I think that, you know, 153 00:15:11,240 --> 00:15:17,719 there's a kind of obvious connection between counting and and mathematics, mathematical trumpeter. 154 00:15:17,720 --> 00:15:21,590 I spent a lot of time in my orchestra sort of counting balls, rest, you know, seem to be all counting. 155 00:15:21,590 --> 00:15:32,990 90 423, 4952, three, four. Now, I think loudness kind of summed up that sort of connection between the rhythms of music and count and counting. 156 00:15:33,270 --> 00:15:39,350 He said to music is a pleasure, the human mind experiences from counting without being aware that it is counting. 157 00:15:39,770 --> 00:15:43,040 But as you know, I think this connection between art and music goes a lot deeper. 158 00:15:43,040 --> 00:15:51,640 The sort of structures that a musician embeds inside that musical composition uses a lot of sort of mathematical ideas. 159 00:15:51,650 --> 00:15:55,430 After all, in some sense, what distinguishes music from noise? 160 00:15:55,520 --> 00:15:59,720 You're looking for structures and patterns and things being related to something you just heard. 161 00:15:59,990 --> 00:16:04,940 So and Stravinsky wrote very nicely about mathematics being so important to a composer. 162 00:16:04,940 --> 00:16:11,120 He writes, The musician should find in mathematics. His study is useful to him as a learning of another languages to a poet. 163 00:16:11,300 --> 00:16:14,480 Mathematics swims seductively just below the surface. 164 00:16:15,020 --> 00:16:17,929 And if you take a piece like Box Goldberg Variations, 165 00:16:17,930 --> 00:16:22,760 you can see Bach just playing around with combinatorial possibilities for the different variations. 166 00:16:23,000 --> 00:16:31,400 Each of the variations is, in some sense, using ideas of symmetry to create something which is related but different to what you've just heard. 167 00:16:31,760 --> 00:16:37,070 If you move into the 20th century, you have lots of examples. Schoenberg, for example, he said, okay, well, we're going to throw away. 168 00:16:37,120 --> 00:16:41,500 Atonal music. But if you throw away structure, you need new structure to be put in place. 169 00:16:41,710 --> 00:16:46,830 And so he introduced this idea of the 12 tone rose and looked at it very mathematically. 170 00:16:46,840 --> 00:16:52,930 He would take permutations of the 12 notes of the chromatic scale and then apply symmetrical rules to that used to 171 00:16:52,930 --> 00:16:59,800 create a palette of a 4812 tone rose which he would use to compose things with a and used to love this as well. 172 00:17:00,340 --> 00:17:08,090 And actually in one of his pieces, he seemed to discover a very significant mathematical structure by using these 12 tone rose. 173 00:17:08,110 --> 00:17:14,800 Purely from for aesthetic reasons, but from a mathematical perspective is extremely significant in my own subject. 174 00:17:15,490 --> 00:17:23,140 One of the I of it I had this kind of dilemma whether to choose Messiaen or another of my favourite 20th century musicians, which is Xenakis. 175 00:17:23,410 --> 00:17:27,970 And Xenakis is extraordinarily mathematically literate. 176 00:17:28,450 --> 00:17:32,260 In fact, this is a piece Novus Alpha, which is written for solo cello. 177 00:17:33,250 --> 00:17:35,680 He dedicated to every scholar amongst us, 178 00:17:36,640 --> 00:17:45,520 as well as two other mathematicians who was in fact the inventor of the language that I talk every day when I'm doing the subject of symmetry. 179 00:17:46,000 --> 00:17:51,040 And in fact, there's a symmetrical object hiding behind the composition of this piece of music. 180 00:17:51,220 --> 00:17:56,460 Actually, the score I've got here isn't for Name Alpha. It's actually for another piece called Metastasis. 181 00:17:56,470 --> 00:18:00,550 But you can see that the natural this score looks like a piece of geometry, not a piece of music. 182 00:18:01,510 --> 00:18:05,919 But I'm going to play you this an extract from Nomos Alpha for solo cello. 183 00:18:05,920 --> 00:18:10,900 And I want to say it's actually based on a on a three dimensional symmetrical object. 184 00:18:11,020 --> 00:18:17,020 And I want you to try and here what symmetrical object is sum it up in your mind is only by the following piece. 185 00:18:48,950 --> 00:18:52,130 So did anyone get a symmetrical shape preparing in their mind? 186 00:18:53,270 --> 00:18:56,840 A circle. Interesting. And a spiral. 187 00:18:57,200 --> 00:19:04,370 Well, in fact, that was a cube hiding behind there. But I must say, even when I knew that, I find it quite difficult to hear the cube. 188 00:19:04,760 --> 00:19:11,149 But actually it's a little bit unfair because I haven't really played you enough of the piece to really get you a sense of the cube, 189 00:19:11,150 --> 00:19:17,510 because as an artist does, is you want to hear what's an arcane star is is to put. 190 00:19:17,750 --> 00:19:21,530 He uses a cube and he puts certain musical ideas that the cello can play. 191 00:19:21,530 --> 00:19:28,990 So you heard the kind of pizzicato there's an effect where you can turn the bow over and hit the strings with the wooden side of the bow necklace. 192 00:19:29,030 --> 00:19:32,270 Sandy So these are put on the eight corners of the cube. 193 00:19:32,570 --> 00:19:40,790 And then in each new variation, what's in our co-stars is a symmetry of the cube, which then tells him the new order in which these should be done. 194 00:19:40,940 --> 00:19:46,999 There's another cube which is controlling the sort of time spent on each of these items and then using those constraints. 195 00:19:47,000 --> 00:19:52,370 He allows his creativity to take over and then he plays within those particular constraints. 196 00:19:52,370 --> 00:19:56,450 So it's interesting because he he uses the symmetries of the cube there. 197 00:19:56,630 --> 00:20:00,770 There are eight factorial different ways that you can arrange each of these eight things. 198 00:20:00,770 --> 00:20:08,600 But but by constraining it to the symmetries of the cube, it somehow creates some sort of hidden order amongst different variations. 199 00:20:09,560 --> 00:20:19,040 And so so that was the symmetries of the cube, which, again, you can only I don't think as an artist necessarily expects you to hear the the cube. 200 00:20:19,250 --> 00:20:24,530 But. Simon Stravinsky used to write, you know, actually, I can only be creative under huge constraints. 201 00:20:24,710 --> 00:20:30,060 And I think a lot of composers love the constraints that mathematics gave them in which they confine. 202 00:20:30,090 --> 00:20:34,560 Then they can play and find a new sort of creative structures. 203 00:20:34,910 --> 00:20:42,140 Now, the interesting thing about Xenakis as well is that not only was he interested in mathematics and also a composer, 204 00:20:42,860 --> 00:20:47,630 but he's also an example of my my second art that I want to turn to, which is architecture. 205 00:20:47,960 --> 00:20:50,630 And he actually worked on a pavilion in Brussels. 206 00:20:50,630 --> 00:20:57,890 And if you look at the plans for the pavilion, they look very much like that score for metastasis and fighting roots. 207 00:20:58,160 --> 00:21:06,500 He constructed this pavilion alongside my second choice of a secret mathematician from the world of architecture, which is Le Corbusier. 208 00:21:06,500 --> 00:21:11,840 You look obviously I worked with Xenakis on this this pavilion. 209 00:21:11,840 --> 00:21:15,590 But look, APJ was obsessed again with mathematics in particular. 210 00:21:15,770 --> 00:21:22,940 He liked this idea of the Fibonacci numbers. The Fibonacci numbers were obviously embedded in the way that things grow naturally. 211 00:21:23,570 --> 00:21:27,500 So he believed that you should sort of mimic that in a building. 212 00:21:27,770 --> 00:21:31,429 And so he produced the series of numbers, his rouge and still his blue, 213 00:21:31,430 --> 00:21:37,070 which were used as his kind of numbers to the proportions that you should find within the building. 214 00:21:37,080 --> 00:21:41,270 So you can see after a while they settle down to this. They have exactly the same rule. 215 00:21:41,720 --> 00:21:46,070 You add 0.86 to 1.40 and you get to 2.26. 216 00:21:46,070 --> 00:21:49,219 So this and he he thought that these are actually related. 217 00:21:49,220 --> 00:21:56,240 Something that goes back, of course, to Leonardo as well, that some of these Fibonacci numbers are related to proportions in the body. 218 00:21:56,840 --> 00:22:04,370 And this is a an idea that goes back to Vitruvius, the Roman architect thought that a building should somehow mimic also the proportions in a body. 219 00:22:04,370 --> 00:22:08,120 And that's the sort of buildings that we all love. And actually, if you use these Fibonacci numbers, 220 00:22:08,120 --> 00:22:14,120 one of the reasons they seem to appear naturally in things that are growing in the 221 00:22:14,120 --> 00:22:18,049 natural world is the term you can use them very nicely to build up a structure. 222 00:22:18,050 --> 00:22:24,770 So if I take, say, a one by one room and add another one by room, one next to that, I'll have a one by two room. 223 00:22:25,190 --> 00:22:29,750 Now I can add a two by two room because I know about the number two now on the side of that, 224 00:22:29,900 --> 00:22:34,700 but now I've got the dimension three so I can add a three by three room and you keep on adding these rooms. 225 00:22:34,700 --> 00:22:41,810 The Fibonacci of course one of the Fibonacci numbers and you get this natural spiral appearing and in fact this 226 00:22:42,170 --> 00:22:49,670 rectangle that's emerging is getting closer and closer to a rectangle that very many artists have found appealing, 227 00:22:50,180 --> 00:22:53,510 which is the rectangle which is in the proportions of the golden ratio. 228 00:22:54,170 --> 00:23:02,209 So in the limits, this would be a rectangle which has this very special property that if you look at the, the ratio of a long side to the short side, 229 00:23:02,210 --> 00:23:11,240 if that's the same as the sum of the two sides to the long side, then this some rectangle is one that we seem to find most aesthetically pleasing. 230 00:23:11,630 --> 00:23:17,780 If I cut a square out of that, so then what I get is a rectangle which is also in the golden ratio. 231 00:23:17,780 --> 00:23:23,719 So it has a and a lot of canvases will often have these proportions. 232 00:23:23,720 --> 00:23:30,140 Leonardo especially love these kind of portions. Also in architecture, if you take something like the Parthenon, 233 00:23:30,410 --> 00:23:34,370 the the ancient Greeks certainly knew about the golden ratio and it's believed that 234 00:23:34,820 --> 00:23:40,730 you can find golden ratios kind of hidden inside the proportions of the Parthenon. 235 00:23:41,690 --> 00:23:44,900 And in fact it's not only architects is. 236 00:23:45,350 --> 00:23:49,460 I did an extraordinary thing at the Royal Opera House. 237 00:23:49,700 --> 00:23:57,200 Looking at the mass behind the Magic Flute. And I discovered that the overture to the Magic Flute. 238 00:23:58,040 --> 00:23:58,970 I don't know whether you know that. 239 00:23:59,000 --> 00:24:07,280 So there's something called the triple chord, which is Mozart's way of embedding kind of the idea of the Masons inside there. 240 00:24:07,280 --> 00:24:13,040 And when this triple chord occurs in the overture, is it exactly the moments where the golden ratio is? 241 00:24:13,640 --> 00:24:16,760 And Messiaen, Mozart must have known about this. 242 00:24:16,760 --> 00:24:22,460 It's too deliberate. So it's kind of 83 bars and you get this triple chord and then 130 bars after that. 243 00:24:23,000 --> 00:24:28,910 So there's a lot of examples of composers also putting key moments in a piece. 244 00:24:29,730 --> 00:24:33,290 As this kind of proportion of the golden ratio. Debussy also did it, for example. 245 00:24:34,040 --> 00:24:35,809 But look at blues. You love this as well. 246 00:24:35,810 --> 00:24:43,520 And he felt that these movies in the series Blue gave rise to buildings, which had this sense of aesthetics inside it. 247 00:24:43,880 --> 00:24:49,610 One of the great examples is this one, which doesn't look a fantastically wonderful building from the outside, 248 00:24:49,610 --> 00:24:56,300 but if you look at the layout of the rooms on the inside, they're constructed according to these series rooms in this series Blue. 249 00:24:56,630 --> 00:24:59,180 And I was talking to somebody recently, you know, 250 00:24:59,180 --> 00:25:05,090 somebody who lives here and says it's actually a beautiful building to live inside because of the way these rooms are laid out. 251 00:25:06,120 --> 00:25:12,200 Of course, look, appreciate in the first use the idea of proportions being important in the way that a building is built. 252 00:25:12,590 --> 00:25:19,010 Palladio, for example. PALLADIO Villas are so perfect in a way, because he was very mathematically sensitive. 253 00:25:19,220 --> 00:25:23,510 He liked kind of whole number ratios rather than these Fibonacci relationships. 254 00:25:24,200 --> 00:25:31,070 And actually, it's very much related to music because if you take whole number ratios in music, you actually get notes with harmony on them. 255 00:25:31,080 --> 00:25:36,320 So here's a Palladio building, but I could put strings on the side and pluck them with give them to the links, the rooms. 256 00:25:43,600 --> 00:25:48,819 So in some way plant is buildings, as they are often called frozen music, 257 00:25:48,820 --> 00:25:55,959 because the proportions inside those villas are actually the proportions that we respond to musically as notes with harmony, 258 00:25:55,960 --> 00:26:01,270 octaves and the perfect fit. It's intriguing that if you look at Paris, Le Corbusier, 259 00:26:01,270 --> 00:26:07,140 his sketchbooks and Palladio sketchbooks, they look very much like a mathematician sketchbooks. 260 00:26:07,150 --> 00:26:12,070 He is trying to find out all the different possibilities for the way a piece of geometry might work. 261 00:26:12,430 --> 00:26:15,669 Look, obviously, I love the idea of things with asymmetry. 262 00:26:15,670 --> 00:26:26,409 Palladio loved symmetry, but he's kind of the mathematical spirit at heart, at work, trying to see what different possibilities. 263 00:26:26,410 --> 00:26:35,260 With these constraints, you can construct buildings. I mean, there are other wonderful examples of I mean, the Guggenheim in Bilbao, 264 00:26:35,260 --> 00:26:42,840 which is a Frank Gehry building, is almost like a piece of Riemannian geometry, which is a Corbusier chapel. 265 00:26:43,270 --> 00:26:50,559 It looks like a bit of hyperbolic geometry. And, you know, I think the modern skyline is full of mathematics these days. 266 00:26:50,560 --> 00:26:55,030 And we have a project here in Oxford called Max in the City dot com where we've tried 267 00:26:55,030 --> 00:26:59,589 to record interesting examples of buildings with mathematics hidden inside them. 268 00:26:59,590 --> 00:27:04,809 And if you have any examples in your own city that you would like to add, it's a very interactive thing. 269 00:27:04,810 --> 00:27:12,100 We're trying to build up a kind of a sort of walking tours of the whole of the world, which viewing the world mathematically. 270 00:27:12,730 --> 00:27:21,190 I mean, for one example, I think Zaha Hadid, who's been done so many buildings recently, the Olympics in particular, 271 00:27:21,550 --> 00:27:26,530 but Zaha Hadid studied mathematics in Iraq before she became an architect here in London. 272 00:27:27,980 --> 00:27:33,730 But it's interesting that I think this tension of the symmetry of the asymmetry Modular Man, I think, 273 00:27:33,730 --> 00:27:39,490 is kind of like a 20th century asymmetrical version of the Vitruvian Man that Leonardo drew. 274 00:27:39,640 --> 00:27:43,800 And in fact, the Vitruvian Man is a solution to an architecture problem. 275 00:27:43,810 --> 00:27:46,420 Vitruvius left in those notes. 276 00:27:46,420 --> 00:27:55,120 He wrote about architecture, this kind of challenge that he believed that a building could encapsulate a human body, both in a square and in a circle. 277 00:27:55,130 --> 00:27:59,950 And a lot of artists try to solve this, and they would often put the square in the circle with a common centre, 278 00:27:59,950 --> 00:28:01,660 but they could never put a person inside it. 279 00:28:01,840 --> 00:28:09,100 You've got these very sort of disproportion bodies when you stuck this in and it was Leonardo who's kind of solved this, 280 00:28:09,100 --> 00:28:13,929 realise you had to shift the centre of the square down off the centre of the circle. 281 00:28:13,930 --> 00:28:18,340 So you have different centres. The circle is centred on the bellybutton, the square somewhere else. 282 00:28:19,840 --> 00:28:23,530 But it interesting, I was kind of wanted in a way to choose. 283 00:28:23,530 --> 00:28:28,419 Leonardo is my choice for my third secret mathematician from the world of the visual arts, 284 00:28:28,420 --> 00:28:33,780 because he so captures somebody who is a bridge between the sciences and the arts. 285 00:28:33,790 --> 00:28:38,919 He did so much sort of with these wonderful inventions that he made and fantastic art as well. 286 00:28:38,920 --> 00:28:41,950 But I went for a 20th century one as well. I sort of kept to that theme. 287 00:28:42,220 --> 00:28:47,379 And so my choice was visual artist for my secret mathematician, actually Salvador Dali, 288 00:28:47,380 --> 00:28:56,890 because Salvador Dali's work just seems to be just so obsessed with ideas of science and mathematics threaded through them. 289 00:28:57,070 --> 00:29:03,550 In fact, he once wrote, I, I'm a carnivorous fish swimming in to waters, the cold water of art and the hot water of science. 290 00:29:03,880 --> 00:29:07,180 And it said that you always used to invite scientists round to his house, 291 00:29:07,180 --> 00:29:11,620 and he found them much more stimulating than artists for for generating new ideas. 292 00:29:11,980 --> 00:29:15,040 And so you can see with lots of scientific ideas, 293 00:29:15,220 --> 00:29:20,860 there's DNA examples of DNA drawn inside the spiral of DNA relativity clearly 294 00:29:21,730 --> 00:29:25,990 inspired him with all his sort of watches falling off and things like that. There's always a lot of mathematics. 295 00:29:25,990 --> 00:29:33,129 So, for example, if you take the Sacrament of the Last Supper, he embeds that inside a dodecahedron. 296 00:29:33,130 --> 00:29:37,420 The symmetrical shape actually, that Plato believed was the shape of the universe. 297 00:29:38,020 --> 00:29:44,770 And it's intriguing because this again harks back to artists who were using symmetrical shapes in the Renaissance. 298 00:29:45,130 --> 00:29:50,590 And in fact, the artists of the Renaissance were very helpful for mathematicians, 299 00:29:50,590 --> 00:29:56,560 because you'll see in this picture here, this is a portrait of a mathematician, Luca Pacioli. 300 00:29:57,100 --> 00:30:00,130 And you can see the dodecahedron again on the table here. 301 00:30:00,220 --> 00:30:08,290 But then there's another extraordinary symmetrical object in the top left hand corner, which is a it's a sort of glass structure. 302 00:30:08,290 --> 00:30:11,649 And I think it's got a sort of field half filled with water as well. 303 00:30:11,650 --> 00:30:15,639 And this is a structure. It's made out of squares and triangles. 304 00:30:15,640 --> 00:30:22,150 It's actually called a round bead cube octahedron, and it's example of an all committee in solid. 305 00:30:22,750 --> 00:30:31,300 Now, you might have heard of platonic solids. There are these five platonic solids which make good dice the cube, the dodecahedron and three others. 306 00:30:31,780 --> 00:30:37,659 But Archimedes just discovered that actually, if you don't stick to the face, it's all being the same symmetrical shape. 307 00:30:37,660 --> 00:30:41,170 So square a cube obviously has six squares, but. 308 00:30:41,390 --> 00:30:47,450 If you take something like a football, the classic football that you kick around on a sun that's made out of pentagons and hexagons. 309 00:30:47,690 --> 00:30:53,000 They all have the same length of edge, but you've got different shapes, but they're all arranged symmetrically on the side of that shape. 310 00:30:53,210 --> 00:30:59,330 So the challenge was how many are the different shapes are there? And Archimedes discovered that there were 14 different shapes and. 311 00:30:59,330 --> 00:31:05,990 Q Including this wrong be cube octahedron. But his description of them was lost in antiquity. 312 00:31:06,260 --> 00:31:11,840 And it wasn't until the Renaissance that we kind of recovered what those 14 shapes were. 313 00:31:12,050 --> 00:31:18,310 And I think it is in part thanks to the artists who were working at the time with perspective, that this was the real challenge for them. 314 00:31:18,320 --> 00:31:21,890 You know, can they draw these objects and sort of bring them alive? 315 00:31:22,070 --> 00:31:28,830 And Leonardo in particular, he illustrated a book by Pat Chorley, and it gives rise to lots of different examples of these. 316 00:31:28,850 --> 00:31:36,500 So. So I think here you see the artist and the scientist sort of helping each other in recovering what these 14 actually were. 317 00:31:37,190 --> 00:31:39,739 But Tony wasn't interested just in these very classical shapes. 318 00:31:39,740 --> 00:31:43,580 He was interested in shapes, these kind of modern shapes appearing in the 20th century. 319 00:31:44,150 --> 00:31:48,440 So it's an amazing picture, the design of war, which pictures a skull. 320 00:31:48,440 --> 00:31:52,490 And then inside each of the sockets, the only sockets in the mouth, you have another skull. 321 00:31:52,760 --> 00:31:56,840 And inside there the hole was in there you've got another skull. And so you have this infinite regress. 322 00:31:57,410 --> 00:32:04,700 This is actually an example of a Lipinski gasket, a fractal with its infinite sums of complexity, 323 00:32:04,700 --> 00:32:08,810 and never get simpler as you go further and further inside the building, inside the picture. 324 00:32:09,560 --> 00:32:12,920 And then there was another artist, Dali was doing this very deliberately. 325 00:32:12,920 --> 00:32:16,729 There was another very famous 20th century artist who was drawn to fractals, 326 00:32:16,730 --> 00:32:23,780 but totally intuitively and this is Jackson Pollock and now Jackson Pollock's paintings. 327 00:32:24,410 --> 00:32:31,190 I mean, he I think he's now been beaten now, but he held the record for the painting, which sold for the most amount of money in history. 328 00:32:31,910 --> 00:32:34,490 But a lot of people said at the time, well, come on, my, my, 329 00:32:34,580 --> 00:32:39,320 my 210 year old twins could make this and just scatter the painting and you can make a Pollock. 330 00:32:39,860 --> 00:32:47,720 So what is it that Pollock was doing that was so special compared to what, you know, your kids might do with a pot of paint? 331 00:32:48,050 --> 00:32:51,200 Well, it turns out that what he was doing with something was something very special. 332 00:32:51,530 --> 00:32:58,729 In fact, you can use a bit of mathematics to discover people started trying to fake Pollock because if they with so much money and you just 333 00:32:58,730 --> 00:33:05,990 need to scatter a bit of paint around but actually is quite difficult to fake Pollock because a Pollock has a very special property. 334 00:33:05,990 --> 00:33:13,220 This, this idea of a fractal that if you zoom in on a fractal you use the it doesn't get simpler, the complexity remains. 335 00:33:13,250 --> 00:33:17,030 So I've actually taken four different regions from one of Jackson Pollock's paintings. 336 00:33:17,720 --> 00:33:23,540 One of them is the painting itself, and the other three are zoomed in portions of that painting. 337 00:33:23,870 --> 00:33:29,899 Now, I think you can probably tell that the top right hand corner is the most zoomed in one, but amongst the other three, 338 00:33:29,900 --> 00:33:35,120 I think it's pretty difficult to tell which is the original and which is a zoomed in region of it. 339 00:33:35,480 --> 00:33:42,260 And so this is being used. The fact that if you look at the way the paints put on the canvas, 340 00:33:42,500 --> 00:33:47,900 it's possible to distinguish quite a lot of fakes that don't have this particular property. 341 00:33:48,740 --> 00:33:54,500 The reason Jackson Pollock was able to do this because is because the fractal is kind of the geometry of chaos. 342 00:33:54,740 --> 00:33:59,780 And the way that Pollock painted was very chaotic, I mean, in a mathematical sense. 343 00:34:00,350 --> 00:34:05,900 So he was apparently had incredibly bad balance and he used to often paint when he was drunk. 344 00:34:06,110 --> 00:34:14,149 So the combination was such that, in fact, he created a chaotic pendulum, because if we if only was splattering paint, 345 00:34:14,150 --> 00:34:18,229 actually, I would have this is a fixed point and I would create a lot of regularity. 346 00:34:18,230 --> 00:34:23,660 But because Pollock was sort of not able to balance, particularly the pivot here was moving all the time. 347 00:34:23,660 --> 00:34:30,230 And he created this kind of chaotic pendulum, which when you look at the geometry that's associated with that, is this kind of fractal shape. 348 00:34:30,470 --> 00:34:32,300 So there is a way to fake a Pollock, 349 00:34:32,420 --> 00:34:39,170 which is to set up a pot of paint on a string and then push it to make sure the top of the string is being pushed. 350 00:34:39,170 --> 00:34:43,970 Every now and again, you create a chaotic pendulum. So we actually did this in one of the BBC programs that I made. 351 00:34:44,180 --> 00:34:49,669 So this is so De Soto. Number one, we haven't quite perfected it. 352 00:34:49,670 --> 00:34:53,329 I didn't get anything on eBay for this, so I'm still working on it. 353 00:34:53,330 --> 00:34:58,010 So. But Pollock also. So Pollock was interesting. 354 00:34:58,100 --> 00:35:01,819 We went to his studio and his studio is in Long Island. 355 00:35:01,820 --> 00:35:06,080 It's surrounded by all of these very fractal like trees. 356 00:35:06,080 --> 00:35:13,430 We went in the winter and you could see that he was sort of responding to the natural world around him because those trees have this 357 00:35:13,430 --> 00:35:21,169 fractal branching property and you can see you can even measure Pollock's kind of periods because of their sort of a fractal dimension. 358 00:35:21,170 --> 00:35:29,720 And he he focuses in on a particular sort of fractal that is the one that is most resonant with the fractals in nature. 359 00:35:31,400 --> 00:35:36,320 Now, I'm going to come back to Dali because there are other examples of Dali's pictures where he was interested in geometry. 360 00:35:36,320 --> 00:35:40,830 Not that you can see around us in the physical world, but actually sort of beyond the physical world. 361 00:35:40,850 --> 00:35:46,770 So. He was very interested in four dimensional geometry, and here's an example of a crucifixion. 362 00:35:46,810 --> 00:35:54,170 So he was a very spiritual man as well. So the idea of the fourth dimension was great for him, something that transcended the physical world. 363 00:35:54,260 --> 00:35:56,930 Somehow the fourth dimension had a spiritual side to it. 364 00:35:57,380 --> 00:36:04,160 So he did this crucifixion on a so this is actually a four dimensional cube unwrapped into three dimensions. 365 00:36:04,670 --> 00:36:08,480 So if you think about how you would make a three dimensional cube out of a piece of paper, 366 00:36:08,960 --> 00:36:14,480 you would have six squares that you would cut out in cross shape, and then you would follow them up and you could make your cube. 367 00:36:14,630 --> 00:36:21,950 Well, it's the same principle at work here, because this is eight three dimensional cube stacked four on top of each other and four on the side. 368 00:36:22,160 --> 00:36:28,910 If you were living in four dimensions, you'd be able to wrap this net, this three dimensional net up to make a four dimensional cube. 369 00:36:29,120 --> 00:36:33,649 And obviously, we're not in four dimensions, but you can still see the unwrapped four dimensional cube, 370 00:36:33,650 --> 00:36:37,490 which is this two inch locking sort of cross shapes. 371 00:36:37,760 --> 00:36:38,990 So for Dani, this is wonderful, 372 00:36:39,200 --> 00:36:46,820 this idea of the Christ being crucified on this four dimensional cross unwrapped if cube unwrapped into three dimensions. 373 00:36:47,570 --> 00:36:54,140 And the idea of the fourth dimension was also very fascinating for my fourth secret mathematician who comes from the world of literature. 374 00:36:54,530 --> 00:37:00,409 Now, I think literature is a little harder to find sort of connections to in mathematics and and sort of works of literature, 375 00:37:00,410 --> 00:37:02,840 although I suppose poetry is a very obvious place to look. 376 00:37:03,410 --> 00:37:13,730 But Borges, I think, is a great example of somebody who is, again, stimulated by the ideas of mathematics in the sort of short stories that he wrote. 377 00:37:13,940 --> 00:37:18,200 And there's one in particular that I really love, which is called the Library of Babel. 378 00:37:18,590 --> 00:37:22,120 And so if you haven't read it, I really recommend it's only ten pages long. 379 00:37:22,130 --> 00:37:28,340 But in this story, he's really sort of exploring the ideas of paradox of infinity, the shape of space. 380 00:37:29,030 --> 00:37:35,330 And actually he's asking questions that were really intriguing scientists in the 20th century at the same time. 381 00:37:35,540 --> 00:37:39,410 So this is how the piece opens. He describes this library. 382 00:37:39,470 --> 00:37:44,900 There's a librarian who's in this library. He's trying to find out what the shape of his library is. 383 00:37:45,500 --> 00:37:49,760 He describes the universe, which others call the library is composed of an infinite, 384 00:37:50,270 --> 00:37:55,910 of an indefinite and perhaps infinite number of hexagonal galleries from any one of the galleries. 385 00:37:57,020 --> 00:37:59,630 One can see internally the upper and lower floors. 386 00:38:00,080 --> 00:38:06,140 So it's actually looks like a beehive, all of these rooms, and they're kind of layers of the beehive, one on top of the other. 387 00:38:06,320 --> 00:38:10,670 And the librarian, through the short story, sort of starts to explore and tries to understand, 388 00:38:10,670 --> 00:38:16,640 well, does this library go on forever or is it infinite? Could he ever know that or is it finite? 389 00:38:16,880 --> 00:38:20,750 But how would that work? And by the end, he actually comes up with a solution. 390 00:38:20,750 --> 00:38:26,900 I venture to suggest this solution to the ancient problem. The library is unlimited and cyclical. 391 00:38:27,050 --> 00:38:34,760 If an eternal traveller were to cross it in any direction after centuries, he would see the same volumes repeated in the same disorder. 392 00:38:35,270 --> 00:38:42,739 And it's very intriguing because his solution is actually one of the solutions that we think maybe the shape of our universe. 393 00:38:42,740 --> 00:38:45,799 If you think about our universe, well, it's the universe infinite. 394 00:38:45,800 --> 00:38:50,270 Does it have a shape? Is it finite? Well, it's kind of funny. 395 00:38:50,270 --> 00:38:52,000 If it were finite, how does that work? 396 00:38:52,010 --> 00:38:58,249 Because, well, the ancient Greeks used to think it was somehow enclosed in some sort of glass ball of stars on that. 397 00:38:58,250 --> 00:39:04,340 But then what's on the other side of that? You know, are we living in The Truman Show with a camera crew sort of looking in on us? 398 00:39:04,830 --> 00:39:10,280 Often I do feel like that's but but but the interesting thing is that story is a came 399 00:39:10,280 --> 00:39:14,839 up with a solution that mathematicians came up with as well because here's a universe. 400 00:39:14,840 --> 00:39:18,110 This is a smaller universe than our universe. It's a two dimensional universe. 401 00:39:18,260 --> 00:39:21,660 Some of you may have played and you won't play asteroids in there. Yes. 402 00:39:21,680 --> 00:39:27,560 All the old people sitting around saying, my my son would not be seen dead playing this game. 403 00:39:28,220 --> 00:39:34,100 But it's a very beautiful illustration of how a finite universe. So the universe is just on the computer screen. 404 00:39:34,100 --> 00:39:37,969 It's finite, but it's unlimited. It doesn't have any walls. 405 00:39:37,970 --> 00:39:43,700 It's not The Truman Show. When you go off the left hand side of the screen, you reappear on the right hand side. 406 00:39:43,700 --> 00:39:49,040 And if you go off the top of the screen, you reappear at the bottom. So it feels like this thing is just going on and on and on. 407 00:39:49,040 --> 00:39:57,380 But of course, it's finite. So this is rather like the description that the librarian arrives at at the end of the Library of Babel. 408 00:39:57,710 --> 00:40:05,600 And the reason is that this does have a shape. It's in fact, the universe is in the shape of a Taurus or a bagel or a doughnut. 409 00:40:06,050 --> 00:40:11,720 So when you go off the top of the screen, what you're doing is actually going around the Taurus inside and background again. 410 00:40:11,940 --> 00:40:17,569 If you go off to the left, you're going round the outside. And in fact, we can illustrate the top and the bottom of the screen, essentially the same. 411 00:40:17,570 --> 00:40:21,700 You join them up, you go round, and the left and the right hand side of the screens are also the same. 412 00:40:21,710 --> 00:40:25,730 So you can join those up. What you get is this bagel shaped universe. 413 00:40:26,990 --> 00:40:31,600 Well, we live in a three dimensional universe. The librarian in the Library of Babel is in a three dimensional universe. 414 00:40:31,640 --> 00:40:38,209 So. So what's happening there? Well, it could be the same sort of thing, actually, because, you know, suppose this is our universe. 415 00:40:38,210 --> 00:40:42,650 We've had the big bang. And it's got to this size and. There's nothing outside this lecture theatre. 416 00:40:43,160 --> 00:40:46,850 In fact, the rooms of this lecture theatre are rather similar to this. 417 00:40:46,850 --> 00:40:52,160 So if you go out to the right hand side of the lecture theatre, you repair the left. 418 00:40:52,430 --> 00:40:56,780 So when you get out there and just come out here, when you go out the top of the building, 419 00:40:57,080 --> 00:41:00,559 top of the ceiling, you come in through the floor and then we go in another direction. 420 00:41:00,560 --> 00:41:05,900 So that's like the game of asteroids. So when you go out the screen here, you repair the back of the lecture theatre. 421 00:41:06,800 --> 00:41:11,850 Sam said that we've embedded mathematics in this building. You might find that I get out of this lecture theatre. 422 00:41:11,930 --> 00:41:18,169 So that's what we want for our undergraduate. Yes. But so actually, what does this universe look like? 423 00:41:18,170 --> 00:41:20,299 Because it's very strange, because the light switch is going out, 424 00:41:20,300 --> 00:41:24,590 the back of my head is going through the screen here and then reappearing at the back of the lecture theatre. 425 00:41:24,740 --> 00:41:29,720 So actually I can see the back of my head over there. And then another copy of me and another copy of me. 426 00:41:29,930 --> 00:41:34,670 So actually, this universe is finite, but it doesn't have any walls that you bounce off. 427 00:41:34,880 --> 00:41:41,300 And this is what the universe looks like. And this is a potential shape for all universe. 428 00:41:41,300 --> 00:41:46,460 If the universe is finite, but with no walls, then it could be what I mean. 429 00:41:46,760 --> 00:41:53,600 Again, if I put this in four dimensions, I could wrap it up and make a Taurus in four dimensions or a bagel in four dimensions. 430 00:41:53,840 --> 00:41:56,209 And in some sense, that's the solution that Gabor has, 431 00:41:56,210 --> 00:42:02,540 comes to the library made out of these hexagons going on internally, up and down, left and right. 432 00:42:02,960 --> 00:42:06,830 But it's you come back whenever you go off, you come back to where you started at. 433 00:42:07,340 --> 00:42:14,959 And so this is one solution, but it's actually the heart of one of the great mathematical theorems that was proved in the last decade. 434 00:42:14,960 --> 00:42:22,220 The Poincaré conjecture, which some of you might have read about in the newspaper of proved a few years ago by a Russian mathematician, 435 00:42:22,220 --> 00:42:28,280 Grigori Perlman, who was awarded our version of the Nobel Prize, the Fields Medal. 436 00:42:28,280 --> 00:42:31,819 And he also it was one of the millennium problems. 437 00:42:31,820 --> 00:42:33,320 It was one of the million dollar problems. 438 00:42:34,220 --> 00:42:40,549 What he did was actually to to make a list of all the possible shapes that the universe could be wrapped up in. 439 00:42:40,550 --> 00:42:44,060 I've given you one, but there are other possibilities. What are those possibilities? 440 00:42:44,450 --> 00:42:51,169 So that question that Boneheads Hands was asking, okay, what is the shape of the Library of Babel is actually at the heart of one of the problems. 441 00:42:51,170 --> 00:42:55,940 That is one of the great problems have been solved in the last century, really. 442 00:42:56,540 --> 00:43:05,959 In fact, the library Babel was an inspiration for a project that I did where I because of this connection between mathematics and the arts, 443 00:43:05,960 --> 00:43:12,710 I get asked a lot to go and work with composers or choreographers to try and give them interesting ideas, 444 00:43:12,710 --> 00:43:15,800 some new structures that might might help them in their creative process. 445 00:43:15,800 --> 00:43:23,030 And so I ended up actually working on it's called the 19th step, actually after another Borges story. 446 00:43:23,900 --> 00:43:30,980 But we were inspired by this idea of the Library of Babel. And so it's a piece of choreography which I ended up actually performing in. 447 00:43:31,490 --> 00:43:36,560 So this is a I'll show you a little bit of this. This is me dancing. 448 00:43:36,680 --> 00:43:45,230 The router encompassed construction of a hexagon, followed by a proof of the irrationality of the square root of three. 449 00:43:45,590 --> 00:43:59,390 So. I think that's enough of that. 450 00:44:02,030 --> 00:44:04,490 I think I must be a first for mathematics and dance, actually. 451 00:44:05,570 --> 00:44:10,730 But actually, it was during that project that I learned about my fifth and final secret mathematician who comes from the world of choreography. 452 00:44:11,480 --> 00:44:15,200 Because I think choreography is a great example of sort of geometry in motion. 453 00:44:15,200 --> 00:44:19,700 And very often when a choreographer is trying to, it's quite an abstract world as well. 454 00:44:19,910 --> 00:44:24,770 And so, for example, Rudolf Laban, who's my choice of a secret mathematician from the world of choreography, 455 00:44:24,770 --> 00:44:31,430 really developed a very mathematical language in order to be able to articulate what was happening in a piece of choreography. 456 00:44:31,670 --> 00:44:36,799 And he also used to make his dancers try and give them a sense of the geometry around them. 457 00:44:36,800 --> 00:44:40,760 So he would always ask the dancer to think of the three dimensional shape that was surrounding them, 458 00:44:40,940 --> 00:44:44,550 rather like sort of three dimensional version of Vitruvian Man, in a way. 459 00:44:44,570 --> 00:44:51,110 So I said, Man is inclined to follow the connecting lines of the 12 corner points of an icosahedron with its movements travelling, 460 00:44:51,110 --> 00:44:57,620 as it were, along an invisible network of paths. And so you can really tell when somebody is being trained in a lab and start a dance because 461 00:44:57,620 --> 00:45:02,000 they have this very you can see almost the shape emerging as they move that their limbs. 462 00:45:03,620 --> 00:45:09,290 Now actually the project that I teach, that piece of choreography has grown and actually become a piece of theatre. 463 00:45:09,470 --> 00:45:14,270 So if you want to see the current version of this thing inspired by the Library of Babel, 464 00:45:14,270 --> 00:45:18,950 I'm actually something I'm starting work on next week, but it will be on at the Science Museum. 465 00:45:18,950 --> 00:45:23,990 It's become a piece of theatre working with an actress from complicity called X and Y. 466 00:45:24,500 --> 00:45:32,210 So I play X and my Victoria Google plays Y, but that's wanted at the Science Museum from the 10th of October to the 16th of October, 467 00:45:32,420 --> 00:45:35,510 and then it's going to the Manchester Science Festival after that. 468 00:45:35,510 --> 00:45:42,200 So if you want to see the current state of this collaboration between art and mathematics, then head along to the Science Museum. 469 00:45:42,920 --> 00:45:52,730 Now I talk a lot about the way that artists use mathematical structures in their work, sometimes deliberately, sometimes drawn to it intuitively. 470 00:45:53,180 --> 00:45:55,669 But I think it works the other way round as well, 471 00:45:55,670 --> 00:46:04,040 because I believe that mathematics is equally as creative a process as the act of creating something in these artistic disciplines. 472 00:46:04,310 --> 00:46:08,840 One of the books that my teacher when I was at school recommended that I read that made me fall in love 473 00:46:08,840 --> 00:46:13,970 with mathematics and realised that it was a bridge in a way between these two worlds of art and science. 474 00:46:14,570 --> 00:46:21,490 Was a book by G.H. Hardy called A Mathematician's Apology. And in that book, he really describes what it's like to be a mathematician. 475 00:46:21,520 --> 00:46:25,740 He says a mathematician, like a painter or a poet is a maker of patterns. 476 00:46:25,790 --> 00:46:28,910 I'm only interested in mathematics as a creative art. 477 00:46:29,360 --> 00:46:39,100 And actually, Henry James Graham GREENE wrote that this book was the best description of being a creative artist after Henry James's diaries. 478 00:46:40,160 --> 00:46:44,840 And and I think that's very often what motivates us as a mathematician. 479 00:46:45,320 --> 00:46:53,660 It's not the utility of our subject. Symes describes how we're having to at the moment, tell the government what the impact of our work is. 480 00:46:53,660 --> 00:46:57,820 And certainly we have extraordinary impact on the world around us. 481 00:46:57,830 --> 00:47:03,170 But I don't think that's the motivation for for most mathematicians in in this department. 482 00:47:03,350 --> 00:47:08,420 I think that we're just intrigued by discovering new, interesting structures, things that move us. 483 00:47:09,770 --> 00:47:13,910 When I try and create a new piece of mathematics, I want to, 484 00:47:14,290 --> 00:47:19,939 to and I present it at a lecture here to my fellow mathematicians, I want to surprise them. 485 00:47:19,940 --> 00:47:24,500 I want to take them on a journey to tell a story and to to show them something they've never seen before. 486 00:47:25,220 --> 00:47:31,370 And I think, you know, if you take here's a particular example, one of my favourite little theorems from number theory due to ferma. 487 00:47:32,210 --> 00:47:35,480 This is like a wonderful piece of music, the proof of this fact. 488 00:47:35,490 --> 00:47:41,270 So I'm so firmly on proof that if you take a prime and you divide it by four and it has remained a one. 489 00:47:41,810 --> 00:47:46,100 So for example, 41 is a prime. I divide it by four, I get remain to one. 490 00:47:47,060 --> 00:47:51,410 Fermat proved. You can always write that as two square numbers added together. 491 00:47:51,680 --> 00:47:58,080 So 41 is four squared plus five squared. Now they're infinitely many of these primes which have remained two one on division by four. 492 00:47:58,100 --> 00:48:02,390 An absolutely extraordinarily you can always prove that they can be written in square numbers. 493 00:48:02,720 --> 00:48:08,810 Now, that for me is absolutely extraordinary. What a surprise, what an earth of a prime number has got to do with these two square numbers. 494 00:48:09,080 --> 00:48:14,540 And so as you read the proof, it's like two sort of musical themes which seem to have nothing to do with each other or two characters. 495 00:48:14,750 --> 00:48:19,010 And then you see them gradually through the story or through the proof, changing, 496 00:48:19,010 --> 00:48:22,340 mutating, until you suddenly realise, no, the two sides of the same equation. 497 00:48:22,580 --> 00:48:28,640 And and it's that sense of surprise that these two things are connected, which makes this such a beautiful kind of mathematical result. 498 00:48:28,970 --> 00:48:33,950 And I think it's that which motivates us when we're when we're trying to choose what mathematics. 499 00:48:33,950 --> 00:48:41,300 I think there is a lot of choice. I here's one of my mathematical theorems which we're talking about in my impact report to the government. 500 00:48:41,720 --> 00:48:45,230 I don't think this will be particularly useful for anything. 501 00:48:45,260 --> 00:48:48,710 I don't know. That wasn't my motivation. What motivated me about this was a discovery. 502 00:48:48,860 --> 00:48:55,640 I mean, I don't expect you to understand everything is up here. But what I discovered was a new symmetrical object. 503 00:48:56,480 --> 00:48:58,190 Which had very surprising properties, 504 00:48:58,190 --> 00:49:04,999 some surprising properties connected to something completely different to to where the symmetrical object was somehow created. 505 00:49:05,000 --> 00:49:12,920 It's the inside this symmetry were questions about solving equations called elliptic curves are actually one of the other millennium problems. 506 00:49:12,920 --> 00:49:16,010 These million dollar problems is about solving equations like y squared. 507 00:49:16,010 --> 00:49:22,160 It was executed minus x. Can you find two numbers which which satisfy that equation? 508 00:49:22,940 --> 00:49:29,479 And now I can. I could get a computer to churn out new descriptions of symmetrical objects and just run the thing. 509 00:49:29,480 --> 00:49:33,110 I can get computed to churn out new mathematics. But that's not mathematics. 510 00:49:33,110 --> 00:49:36,620 That's like the monkey at the typewriter writing, you know, just. 511 00:49:36,770 --> 00:49:39,890 Just random words. I'm combining them. 512 00:49:40,090 --> 00:49:44,059 The act of a mathematician is to to be involved in the creative process. 513 00:49:44,060 --> 00:49:51,560 So make a choice about what is exciting to talk about the drama involved in actually showing that these two things which look totally different, 514 00:49:51,560 --> 00:49:59,870 are actually related. And so I think that's the motivation for the mathematician and often those things will have an impact then on the physical world 515 00:49:59,870 --> 00:50:06,530 around us because what we respond to are things which are those structures which are sort of hidden in the in the natural world. 516 00:50:07,220 --> 00:50:08,300 I'm going to end with a quote. 517 00:50:08,630 --> 00:50:18,860 I want you to think, is this a quote by an artist or is it a quote by a scientist to create consensus precisely and not making useless combinations? 518 00:50:19,220 --> 00:50:26,480 Invention is discernment choice. The sterile combinations do not even present themselves the mind of the inventor. 519 00:50:27,410 --> 00:50:31,309 Now put your hand up if you think that that's an artist talking about their creative process. 520 00:50:31,310 --> 00:50:35,540 How many people think as an artist talking out? A few votes. 521 00:50:35,550 --> 00:50:42,140 Yeah. How many things the scientists talking about what they do that female scientists. 522 00:50:42,300 --> 00:50:46,560 How many people are not really sure. It could be either. Yes, I see. 523 00:50:46,560 --> 00:50:50,850 I have to put you in that sort of category now because it's sort of like, okay, I'm not quite sure now what? 524 00:50:51,720 --> 00:50:57,930 Maybe the word inventor gave it away because that's not usually a word that's associated with the creative arts. 525 00:50:57,990 --> 00:51:03,150 It was, in fact, only punk who was asked this question about what the shape of the universe could be, in a way. 526 00:51:03,660 --> 00:51:10,140 But actually, Stravinsky used to call himself an inventor. He used to think he was inventing his pieces of music. 527 00:51:11,040 --> 00:51:16,439 But I think it really is that connection, that mathematics is this beautiful bridge because it has that creativity, 528 00:51:16,440 --> 00:51:19,440 yet it is trying to describe the physical world. 529 00:51:19,620 --> 00:51:27,599 And I think in the end I sort of did end up being both on the artistic side and the scientific side. 530 00:51:27,600 --> 00:51:31,650 And I found that it was mathematics that was my way, too, to unite these two. 531 00:51:32,430 --> 00:51:32,790 Thank you.