1 00:00:17,900 --> 00:00:21,830 Tiger. Tiger burning bright in the forests of the night. 2 00:00:22,520 --> 00:00:26,780 What immortal handle I could frame the night before symmetry. 3 00:00:27,920 --> 00:00:34,010 As humans, we seem to be incredibly sensitive to the idea of symmetry. 4 00:00:34,700 --> 00:00:37,790 When we were in the jungle and we wanted to survive. 5 00:00:38,510 --> 00:00:44,270 It was those that could spot symmetry that had a chance to realise that there was a tiger out there. 6 00:00:45,050 --> 00:00:51,920 If there's something with symmetry, it's likely to be an animal and either it's going to eat you or perhaps you could eat it. 7 00:00:52,610 --> 00:01:00,709 So nature has kind of programmed us to be incredibly sensitive to notice things with symmetry because they've got a message inside them. 8 00:01:00,710 --> 00:01:08,030 Very often they're things that we should notice. Certainly many animals are very sensitive to symmetry if you're in the garden. 9 00:01:08,480 --> 00:01:14,600 A bumblebee looks out for things with symmetry because it knows that that is likely to be sustenance. 10 00:01:15,440 --> 00:01:22,040 The flower, in its turn needs the bee, and so it needs to draw the bee to its flower. 11 00:01:22,400 --> 00:01:29,690 And so symmetry is acting a little bit like a language communicating between these two, the bee and the flower. 12 00:01:30,380 --> 00:01:34,700 And for us, too, as humans, we use symmetry to communicate information. 13 00:01:35,420 --> 00:01:42,950 I've got two theses here which I'm going to make artificially symmetrical, and I ask you, which of these do you find more beautiful? 14 00:01:43,340 --> 00:01:50,330 The first two or the second two that I've made artificially symmetrical and most people are drawn to the second two. 15 00:01:50,630 --> 00:01:55,790 Why is it that humans find symmetry and a human face so beautiful? 16 00:01:56,360 --> 00:02:01,700 Well, again, this is communicating information. It's hard to make symmetry in nature. 17 00:02:02,420 --> 00:02:11,180 So this is an indication that this person probably has a good genetic background, has been brought up well, is healthy, is going to make a good mate. 18 00:02:11,480 --> 00:02:17,960 So the symmetry is communicating to you information about the person that they probably be good to have children with. 19 00:02:18,500 --> 00:02:26,900 And Galileo certainly highlighted the fact that symmetry is a way of reading the world, of understanding the universe. 20 00:02:27,200 --> 00:02:34,610 He very famously wrote, The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. 21 00:02:35,090 --> 00:02:42,139 It is written in mathematical language, and the letters are triangles, circles and other geometrical figures without, 22 00:02:42,140 --> 00:02:46,160 which means that it's humanly impossible to comprehend a single word. 23 00:02:47,120 --> 00:02:54,979 Certainly, symmetry is very central to how we understand the scientific world, but it's also be very important to artists, too. 24 00:02:54,980 --> 00:02:59,000 And many artists have talked about the the role of symmetry in their work. 25 00:02:59,510 --> 00:03:01,549 This is Pushkin, 26 00:03:01,550 --> 00:03:10,700 who wrote in a letter that symmetry is a characteristic of the human mind and it seems to be that we've been evolved to recognise symmetry. 27 00:03:11,840 --> 00:03:21,410 Valerie French poets wrote The universe is built on a plan, the profound symmetry of which is somehow present in the inner structure of the intellect. 28 00:03:21,950 --> 00:03:30,140 And it's perhaps intriguing that both of these were drawn to a literary form that has a lot of symmetry at work in the creation of poems. 29 00:03:31,070 --> 00:03:34,150 But it's not always such a happy relationship. 30 00:03:34,160 --> 00:03:40,760 Many artists find symmetry a little bit too controlling and a little bit of an uneasy relationship. 31 00:03:40,790 --> 00:03:44,270 Here's Thomas Mann talking about symmetry in the Magic Mountain. 32 00:03:44,450 --> 00:03:48,380 He has a character describing the snowflakes that land on his arm. 33 00:03:48,740 --> 00:03:54,980 The character said he shuddered at its perfect position, found it deftly the very, very marrow of death. 34 00:03:56,000 --> 00:04:00,049 But for us, as mathematicians, symmetry is far from deadly. 35 00:04:00,050 --> 00:04:06,290 It isn't still. It's something very active and something with a lot of energy and movement in it. 36 00:04:06,620 --> 00:04:15,440 And the person who identified that symmetry is something with energy in it in some ways was a very romantic figure in the history of our subject. 37 00:04:15,590 --> 00:04:20,780 Evariste Galois, killed in a duel at the age of 20 over love and politics. 38 00:04:20,900 --> 00:04:23,930 But already when he was still in school at the age of 18, 39 00:04:24,110 --> 00:04:30,740 he'd come up with a new way of looking at symmetry, which revealed that it's full of motion and movement. 40 00:04:31,250 --> 00:04:39,170 So if you take a wall, for example, in the Alhambra and you ask, what are the symmetries of this wall for Galois and for mathematicians, 41 00:04:39,290 --> 00:04:44,810 this is about the movements that you can do to this figure, which makes it look like it was before you moved it. 42 00:04:45,200 --> 00:04:49,850 So we trace an outline on the this particular wall in the Alhambra. 43 00:04:50,030 --> 00:04:55,610 We can move these tiles by 90 degrees and then they fit perfectly back on top of each other. 44 00:04:55,880 --> 00:05:00,230 And this for someone like Galois, is the element of symmetry. 45 00:05:00,230 --> 00:05:07,670 The ways you can move something, transform it, such it has a connection to where it came from, but it's evolved into something new. 46 00:05:08,240 --> 00:05:16,820 And this is an idea that one particular art form uses a lot to create an idea of movement, and that is music in. 47 00:05:16,920 --> 00:05:22,889 Music very often you starting with a theme and you want to somehow mutated and change it into something else, 48 00:05:22,890 --> 00:05:25,080 which has a connection to where it's come from. 49 00:05:25,590 --> 00:05:33,960 And I think probably the musician who captured most the idea of using symmetry to generate interesting new ideas is Bach. 50 00:05:34,290 --> 00:05:40,770 Metzler Bach's student once described Bach's music as the process of sounding mathematics. 51 00:05:41,400 --> 00:05:47,580 Now, as you came in, you were, in fact, listening to what I think is one of the pieces, which is almost like a hymn to symmetry. 52 00:05:47,610 --> 00:05:51,480 It was the opening aria of the Goldberg Variations. 53 00:05:51,780 --> 00:06:00,000 And for me, the Goldberg Variations is Bach really exploring the idea of symmetry in sound and music? 54 00:06:00,450 --> 00:06:07,680 So the Goldberg Variations, a piece for solo piano, starts with this aria, which is what you were listening to when you came in. 55 00:06:08,640 --> 00:06:15,870 And then Bach goes through 30 variations on this aria and then concludes with the same aria. 56 00:06:16,290 --> 00:06:22,680 So already you have a sense of a circle that the thing joins up the beginning and the end are exactly the same. 57 00:06:23,610 --> 00:06:30,090 And to really reinforce this idea of a circular structure happening inside the Goldberg Variations, 58 00:06:30,390 --> 00:06:34,500 if you move to the 16th variation, which is halfway through the piece. 59 00:06:35,400 --> 00:06:38,700 Bach intriguingly calls this an overture. 60 00:06:39,120 --> 00:06:43,169 Now, an overture in music is usually something which begins the piece. 61 00:06:43,170 --> 00:06:49,020 So already you're questioning where quite is the beginning of the Goldberg Variations. 62 00:06:49,020 --> 00:06:58,950 Maybe you can start anywhere and join the whole thing up in this circle where Bach really plays with the idea of of symmetry to create an idea, 63 00:06:58,950 --> 00:07:05,400 a variation on the theme that we've heard in the ARIA is in every third variation. 64 00:07:05,580 --> 00:07:12,300 Every third variation in the Goldberg Variations is something called a canon, which you probably all remember from school. 65 00:07:12,510 --> 00:07:15,390 A canon is where one voice starts singing. 66 00:07:15,780 --> 00:07:21,930 And then a little bit later, the second voice comes in and starts singing the same thing, but with a delay in time. 67 00:07:22,110 --> 00:07:26,000 And this creates a beautiful effect if you choose the right the right tune. 68 00:07:26,010 --> 00:07:29,760 So every third variation is an example of a canon. 69 00:07:29,760 --> 00:07:34,350 So here's the first canon, the third variation. 70 00:07:34,950 --> 00:07:42,479 And so I'm going to play a very short snippet of this and you will hear the first voice starting. 71 00:07:42,480 --> 00:07:48,209 It's a start on this B and does a nice little trill up and then you hear the second voice coming in doing exactly the same thing. 72 00:07:48,210 --> 00:07:54,960 So let's hear the first canon, the Bach heads in the Goldberg Variations and listen out for just the repeat of the pattern. 73 00:07:55,110 --> 00:07:59,700 This is like a pattern which is being repeated, like a tile across a wall. 74 00:08:02,310 --> 00:08:05,340 There it is, then tilted. 75 00:08:06,780 --> 00:08:11,700 So over the two bars you had to fill the first voice and the second ball came in with a second voice. 76 00:08:12,270 --> 00:08:19,020 But Bach isn't content with each new canon repeating this idea with just a second voice repeating what the first voice did. 77 00:08:19,350 --> 00:08:23,160 So he decides to take the pianist on a journey. 78 00:08:23,460 --> 00:08:30,660 So in the second canon, the sixth variation, the second voice doesn't just repeat what the first voice does, but starts. 79 00:08:30,660 --> 00:08:35,070 We've seen a shift in tone, the sort of tile moving this way. 80 00:08:35,120 --> 00:08:42,449 Now we're going to hear a shift in pitch because the second voice starts one note higher in the third variation. 81 00:08:42,450 --> 00:08:50,100 So it's little different out today. So you hear the the G cascading down and then you hear one step up in a casting Kafkaesque cascading down. 82 00:08:58,020 --> 00:09:01,710 So each time you hear the first voice, but the second voice is one note higher. 83 00:09:02,040 --> 00:09:04,619 So as each new canon is introduced, 84 00:09:04,620 --> 00:09:13,320 the second voice climbs gradually higher and higher and higher until something quite interesting happens when you get to the eighth variation. 85 00:09:13,680 --> 00:09:22,080 The eighth canon, the 24th variation. Because at this point, this the second voice has been stepping up one step at a time through the scale. 86 00:09:22,320 --> 00:09:25,530 And by eight notes, it actually hit something called the octave. 87 00:09:25,800 --> 00:09:31,230 We actually get to a note, which sounds like the note you started with, but just an octave higher. 88 00:09:31,440 --> 00:09:34,470 So Bach kind of joins up this circle again. 89 00:09:34,920 --> 00:09:42,410 In this case, actually, it isn't higher, but lower. So you'll hear the first voice start on a G and then an octave lower, in fact. 90 00:09:42,420 --> 00:09:45,990 So he varies quite where this second voice is coming in. 91 00:09:46,170 --> 00:09:54,120 But you suddenly hit this point where suddenly the two themes join up again as if they're starting on the same note, but just an octave higher. 92 00:09:59,690 --> 00:10:07,249 So you heard the second boys come in there. But we still got one more cannon left. 93 00:10:07,250 --> 00:10:11,710 So where does this voice go on then, when in fact, it takes that next step upwards? 94 00:10:12,140 --> 00:10:19,280 So for me, when I hear this, these cannons and what I'm hearing, it's actually a kind of geometric shape embedded in here, 95 00:10:19,280 --> 00:10:25,730 because we've already heard the circle where we've had the ARIA starting and it joins up again with the ARIA at the end. 96 00:10:25,850 --> 00:10:33,500 But then these cannons are creating a second worth of circles in this piece, or maybe even a spiral because you're getting this octave effect. 97 00:10:34,130 --> 00:10:40,850 So embedded in this piece is somehow the shape of this Taurus, a circle's worth of circles. 98 00:10:41,240 --> 00:10:48,260 And as I said, the ninth variation continues this step upwards as if you it's telling you where the piece is going to go, 99 00:10:48,260 --> 00:10:52,520 if you want it to it, extend it off to infinity. And indeed, the first voice comes in. 100 00:10:52,520 --> 00:10:56,090 The second voice is an octave and a note. Nine notes higher. 101 00:11:03,390 --> 00:11:07,280 Now, Bach also uses ideas of symmetry to do these variations. 102 00:11:07,280 --> 00:11:14,760 So sometimes the second voice isn't a complete copy of the first voice, but actually has some symmetrical operation acting on it. 103 00:11:15,000 --> 00:11:21,180 So for example, the first voice might shoot upwards and then you'll hear the second voice shooting 104 00:11:21,180 --> 00:11:25,380 downwards as if there's been a reflection in the horizontal line through the music. 105 00:11:26,040 --> 00:11:29,579 So here we see the note going. 106 00:11:29,580 --> 00:11:33,000 It goes down and then up and then down. But the second going it goes up. 107 00:11:33,000 --> 00:11:54,990 Then down, then up. And then. But also likes to use a bit of symmetry at work in the rhythm and structure of these canons as well. 108 00:11:55,010 --> 00:11:59,080 So if we go to, for example, let's go back to the 24th canon. 109 00:11:59,660 --> 00:12:08,150 He's broken up the beats in this bar into three beats, and each beat in turn is divided into three triplets. 110 00:12:08,390 --> 00:12:11,900 So we get a kind of three, three, lots of three in this structure. 111 00:12:12,440 --> 00:12:17,329 So let's hear this one at work again. One, two, three, and one, two, three. 112 00:12:17,330 --> 00:12:20,210 One, two, three. One, two, three. One, two, three. One, two, three, one, two, three. 113 00:12:20,870 --> 00:12:24,330 So Bach has some different combinations that he can do in all of these canons. 114 00:12:24,350 --> 00:12:29,870 Here we see a ball divided up into three and each beat divided up into three. 115 00:12:29,870 --> 00:12:35,810 But there are different possibilities. So we see Bach, the mathematician, at work who says, Well, what are the other things I could do? 116 00:12:35,960 --> 00:12:42,200 So I could divide the bar up into four beats or two beats and each beat I could do into triplets? 117 00:12:42,290 --> 00:12:49,940 Semi quavers or quavers. So he goes through all the different possibilities that you could have with these different ways of breaking up the rhythm. 118 00:12:50,210 --> 00:12:54,950 So this is a bit like having a combination lock where you've got two triangles. 119 00:12:54,950 --> 00:12:58,910 One triangle is controlling how many beats are in the bar? Two, three or four. 120 00:12:59,120 --> 00:13:04,850 The other triangle is controlling how that beats is broken up into quavers, triplets or semi quavers. 121 00:13:05,430 --> 00:13:12,290 Bach makes sure, as if he's trying to break this combination lock, that he covers every particular possibility. 122 00:13:12,320 --> 00:13:18,200 So how many different ways are there to arrange these three. Two triangles on my little combination lock. 123 00:13:18,410 --> 00:13:23,660 There are three times three nine different ways. We had nine canons that we saw already. 124 00:13:23,900 --> 00:13:31,590 So indeed, Bach makes sure that each canon covers one of these different possibilities of the symmetries inside the different rhythm. 125 00:13:31,610 --> 00:13:36,140 So we've seen three broken, three beats broken up into triplets. 126 00:13:36,740 --> 00:13:42,980 For example, variation number 15, we see two beats broken up into semi quavers. 127 00:13:43,430 --> 00:14:03,509 That's. And I think this really illustrates that BARC is not doing this kind of unknowingly. 128 00:14:03,510 --> 00:14:10,319 I think that would be very hard to do this unknowingly. I think Blanck is very aware that he wants to make sure that this structure covers 129 00:14:10,320 --> 00:14:14,100 all of the different possibilities and there is so much structure in here. 130 00:14:14,100 --> 00:14:18,150 You've seen how the canons build up. They climb each time, the rhythms and things. 131 00:14:18,300 --> 00:14:23,730 But we've got one more cannon that we haven't heard, which is the last variation, the 30th variation. 132 00:14:23,730 --> 00:14:30,540 So you might think, well, I know exactly what is going to happen now. It's going to take a step upwards again, perhaps some start the rhythms again. 133 00:14:30,720 --> 00:14:36,500 But no, when Bark comes to this last variation, he destroys the structure completely. 134 00:14:36,720 --> 00:14:40,440 And this last variation is called a quarter limit, a musical joke. 135 00:14:40,680 --> 00:14:48,719 It's in fact, just a variation on two folk themes of the day has nothing to do with the rest of the structure of the piece, which is kind of curious. 136 00:14:48,720 --> 00:14:53,340 But I think it goes to illustrate how much structure has been up to that point 137 00:14:53,490 --> 00:14:58,680 that you find this suddenly the two folk tunes woven together so surprising. 138 00:15:10,280 --> 00:15:18,530 And this is not uncommon with artists. I think art is very often like this idea of setting up symmetrical expectations and then breaking them. 139 00:15:18,890 --> 00:15:23,030 One of my favourite examples I work with a professor in Japan. 140 00:15:24,950 --> 00:15:30,040 Professor Kurokawa. And we went to he took me up to Nikko and I took this photograph. 141 00:15:30,050 --> 00:15:36,470 It's of an arch. As you enter the temples and on the arch there's eight columns. 142 00:15:36,620 --> 00:15:39,800 There's a beautiful pattern, symmetrical pattern on these eight columns. 143 00:15:40,730 --> 00:15:45,170 Seven of them are exactly the same, but the eighth one is turned upside down. 144 00:15:45,500 --> 00:15:50,059 And I said to Professor Kurokawa, wow, the the architect must have been really angry, 145 00:15:50,060 --> 00:15:54,980 the builders, that they got this wrong and turned it upside down. And he said, no, this is a very deliberate act. 146 00:15:55,550 --> 00:16:02,650 And he referred me to this wonderful Japanese essays in Idleness from the 14th century, wonderful book title. 147 00:16:02,660 --> 00:16:08,600 I need to write a book called That, where the essayist wrote in everything, uniformity is undesirable. 148 00:16:08,810 --> 00:16:15,140 Leaving something incomplete makes it interesting and gives one the feeling that there is room for growth. 149 00:16:15,740 --> 00:16:20,180 And I think that's what artists like doing. They like setting up this expectation and then breaking it. 150 00:16:20,990 --> 00:16:30,500 And in the Goldberg Variations, that upside down column is that 30th variation when I guess a ball is probably one of the 151 00:16:30,770 --> 00:16:36,559 the musicians who use symmetry a lot and it's a classical period in the modern period, 152 00:16:36,560 --> 00:16:42,560 probably the person who took up the mantle of using symmetry to do themes and variations is Schoenberg. 153 00:16:42,920 --> 00:16:52,610 Schoenberg system that he introduced has symmetry at its heart to generate a palette of themes that he would then compose with what he would do. 154 00:16:52,820 --> 00:17:02,450 He broke the kind of harmonic structure of the major minor scale and said every note of the chromatic scale 12 note should be considered as equal, 155 00:17:03,020 --> 00:17:08,750 and they will then do some sort of variation of those 12 notes and arrange them in some interesting way. 156 00:17:09,050 --> 00:17:12,440 And then you would do symmetrical operations on these 12 notes. 157 00:17:12,440 --> 00:17:19,610 So this would be called a 12 tone row, which you would then you would choose a 12 tone row that would be your kind of theme, 158 00:17:19,850 --> 00:17:23,720 and then you would do these variations on this in a very mathematical way. 159 00:17:23,960 --> 00:17:31,550 So the first variation is really to kind of push. If you think of this like a little tile, you would push the tile gradually higher and higher. 160 00:17:31,550 --> 00:17:36,920 So each note would be taken one semitone higher and you might drop a one down if it goes 161 00:17:36,920 --> 00:17:42,049 above the octave and you would generate 12 different shifts of this particular motif, 162 00:17:42,050 --> 00:17:46,100 which all have pretty much the same shape, but it's just a different pitch. 163 00:17:46,670 --> 00:17:52,850 So you've got 12 those. So those would be little translations, but then you would do some more symmetrical moves. 164 00:17:52,850 --> 00:17:58,579 So you would do a reflection in the horizontal line, you would flip the thing upside down, just as Bach did. 165 00:17:58,580 --> 00:18:01,130 Remember that with that theme and variation. 166 00:18:01,310 --> 00:18:07,040 So Shambhu would say, You turn all the 12 upside down, but you can also vary them by playing them backwards. 167 00:18:07,040 --> 00:18:14,030 So he would reverse them and do a reflection in the so that was the horizontal line in the vertical line. 168 00:18:14,180 --> 00:18:19,490 And then he would combine those which would be a combination which would be like a rotation. 169 00:18:19,700 --> 00:18:25,579 So then you would have 48 different themes from which you would then start your composition. 170 00:18:25,580 --> 00:18:31,670 And this was Schoenberg's kind of the way that he changed music at the beginning of 171 00:18:31,670 --> 00:18:37,280 the 20th century by using these symmetrical tools and composers that were followed, 172 00:18:37,280 --> 00:18:47,870 Schoenberg system were drawn to interesting 12 tone rows that when you made these symmetrical moves, perhaps interesting coincidences happened. 173 00:18:48,350 --> 00:18:50,720 One of my favourite is Olivier Messiaen. 174 00:18:50,930 --> 00:18:58,490 Olivier Messiaen was very interested in the ideas of using mathematics quite intuitively, very often in his music. 175 00:18:59,030 --> 00:19:02,720 He used he created some new scale structures based on symmetry. 176 00:19:02,930 --> 00:19:11,810 But he also was very intrigued by this system of Schoenberg's to use symmetry to create themes and then bury them with these symmetry moves. 177 00:19:12,740 --> 00:19:19,309 But I particularly like one piece that he wrote for solo piano called Ildefonso, too, 178 00:19:19,310 --> 00:19:28,790 because in this particular piece he chooses a 12 tone row, which has huge mathematical significance for anybody who studies symmetry. 179 00:19:29,420 --> 00:19:32,420 This 12 tone row, it's a little bit like taking. 180 00:19:32,510 --> 00:19:38,089 So you can think of the notes a bit like 12 cards, a pack of 12 cards, and there's a way. 181 00:19:38,090 --> 00:19:45,860 So the notes will be arranged from, say, the ice through to the the queen and those will be your chromatic notes. 182 00:19:45,860 --> 00:19:50,210 But then you want to make some new variation, some, some permutation of those notes. 183 00:19:50,540 --> 00:19:57,650 So what Marcion was intrigued by was a particular permutation where you take the top and the bottom cards and put them on this hand, 184 00:19:57,980 --> 00:20:01,280 and then you would take the next top and bottom card and put them on top of this hand. 185 00:20:01,490 --> 00:20:07,250 And you would keep on doing this until you cleared this pack and you had rearrange the cards on this side. 186 00:20:07,490 --> 00:20:10,830 This is a particularly. A shuffle called the Mongolian Shuffle. 187 00:20:11,550 --> 00:20:15,840 And he used this. He said, Well, that's interesting. I wonder what sort of music that will create. 188 00:20:16,170 --> 00:20:22,860 And so this is the the Schoenberg 12 tone row that he uses in this piece, Ildefonso, too. 189 00:20:23,250 --> 00:20:31,280 And actually he doesn't use the techniques that Schoenberg used, but he said, okay, well, I can repeat that shuffle so I can keep on it. 190 00:20:31,680 --> 00:20:37,110 I can take that permutation, take it over to here, repeat the particular shuffle I've done, 191 00:20:37,230 --> 00:20:43,560 and that would be the next 12 tone row that you would hear. So I think if I got a piece of you know, I think it's coming up. 192 00:20:43,680 --> 00:20:50,700 So this was the this actually is the piece goes on. You hear the shuffle being applied again and again to these 12 notes. 193 00:20:51,570 --> 00:20:58,979 This particular permutation for a mathematician is one that we recognise because one of the Galois started 194 00:20:58,980 --> 00:21:04,860 this journey that we've been on for 150 years since he first introduced this new way to look at symmetry, 195 00:21:05,100 --> 00:21:11,339 which culminated in us being able to produce what we call kind of a periodic table of symmetry. 196 00:21:11,340 --> 00:21:14,640 It's called the Atlas of Finite Simple Groups. So I brought it along here. 197 00:21:14,970 --> 00:21:21,120 And this thing here is like the periodic table that the chemists have in the labs across the road. 198 00:21:21,300 --> 00:21:26,580 It lists all the kind of atoms of symmetry from which we can make all four symmetries. 199 00:21:27,270 --> 00:21:31,440 So there are some very simple symmetries kind of shapes with the prime number of sides. 200 00:21:31,650 --> 00:21:33,900 There are symmetries which fall into kind of patterns. 201 00:21:34,110 --> 00:21:40,230 But then there are 26 very strange symmetrical objects that seem to have no pattern to them at all. 202 00:21:40,240 --> 00:21:46,200 We call them the sporadic, simple groups, and some of the first to be discovered were by a mathematician called Matthew. 203 00:21:47,340 --> 00:21:52,860 And he he discovered one which uses this permutation that Messiaen was using. 204 00:21:52,860 --> 00:21:57,000 And it it creates this extraordinary symmetrical object. 205 00:21:57,300 --> 00:22:04,440 It's not an object I can show you. The first dimension that you will see this represented in, I think is 11 dimensions. 206 00:22:04,620 --> 00:22:11,610 So I can't show you this symmetrical object, but wonderfully messier has given away as a way to listen to it. 207 00:22:11,970 --> 00:22:16,560 So here's the sound of an 11 dimensional, sporadic, simple group. 208 00:22:49,030 --> 00:22:55,870 And then it goes off into something else. But but that was the first permutation that you were hearing of the set of 12 cards. 209 00:22:55,870 --> 00:23:00,399 The 12 notes. Amazing. The mission was drawn to this for aesthetic reasons. 210 00:23:00,400 --> 00:23:08,950 He had no idea that this had a connection to a significant object for us as mathematicians with extraordinary symmetrical implications. 211 00:23:09,490 --> 00:23:18,310 Another composer that one of my favourites from the 20th century who was drawn to a symmetrical object that you can see was Yanis Xenakis. 212 00:23:18,430 --> 00:23:25,330 He's a Greek composer who worked with Le Corbusier. He was an architect, but he was also fascinated in mathematics. 213 00:23:25,570 --> 00:23:32,890 He wrote a piece called Nomos Alpha, which he actually dedicates to three mathematicians, Galois included in them. 214 00:23:33,220 --> 00:23:43,000 And this is a piece for solo cello. And he uses, again, the idea of symmetry to do a variations on a particular theme. 215 00:23:43,030 --> 00:23:47,439 So this is I think I got the yeah, so so I'm going to play you the piece. 216 00:23:47,440 --> 00:23:50,230 A little excerpt from Nomos Alpha. 217 00:23:50,500 --> 00:23:56,800 I want you to try and see what sort of symmetrical shape is conjured up in your mind's eye by this particular piece. 218 00:24:09,620 --> 00:24:19,540 And then, you know. 219 00:24:28,010 --> 00:24:31,459 You can open your eyes now. What did you see? Did you see a symmetrical shape? 220 00:24:31,460 --> 00:24:36,440 Anybody? It's very asymmetrical, I think. In fact, that was in fact, a cube. 221 00:24:37,220 --> 00:24:44,570 Although I find it quite hard to hear the cube and I. And it's a little bit unfair to only play you that amount of music because you only start 222 00:24:44,570 --> 00:24:50,180 to hear the cube and its symmetries and work actually as the variations begin to build. 223 00:24:50,180 --> 00:24:57,320 Because then you start to see some connection between this the, the, the piece of music you've just heard and as it's varied. 224 00:24:57,620 --> 00:25:09,259 What's going on here is that Xenakis took the 24 rotations of the cube, and what he did was to put musical ideas on the corners of that cube. 225 00:25:09,260 --> 00:25:12,950 So I think we've got them here. So so the eight corners on the cube. 226 00:25:12,950 --> 00:25:16,040 So I actually had several cubes at work at the same time. 227 00:25:16,760 --> 00:25:22,910 So you see here s12s8 so this would describe the quality of the sound. 228 00:25:22,920 --> 00:25:31,490 So you heard lots of different things that the cello could do, a sort of pizzicato glissando, and these are described by these particular shapes. 229 00:25:32,160 --> 00:25:40,460 And then there'll be another cube keeping track of how long was spent on those particular elements and also how loud or soft they were being played. 230 00:25:40,910 --> 00:25:49,129 So he would read the corners of the cube and then compose with those restrictions, and then he would play apply a symmetry. 231 00:25:49,130 --> 00:25:55,730 So one of the symmetries that we've seen here to the cube and then the next variation would read off in a different order, 232 00:25:56,960 --> 00:26:00,830 the things that the cello would have to do. So you'd hear these, these things varying. 233 00:26:00,950 --> 00:26:06,950 So by actually the end of the piece, you've heard a sequence of permutations at work rotation, 234 00:26:06,980 --> 00:26:12,740 symmetries of the cube, and you hear begin to hear some sort of interconnection between them. 235 00:26:14,210 --> 00:26:19,820 There are 24 variations. But what I discovered just recently is that he isn't going through all the rotations. 236 00:26:20,390 --> 00:26:23,270 He's actually produced a new structure, which I've never seen at work. 237 00:26:23,270 --> 00:26:26,930 And since there are some group theorists here, they might tell me that this has been studied, 238 00:26:27,320 --> 00:26:34,700 but he actually uses a kind of Fibonacci rule to generate which symmetry is going to be used next in the next variation. 239 00:26:34,970 --> 00:26:43,670 So he explored, he took two symmetries of the cube, and then the third symmetry would be used to be the combination of those two symmetries. 240 00:26:43,850 --> 00:26:48,620 Then the fourth symmetry that would be used with the combination of the two previous ones. 241 00:26:48,920 --> 00:26:55,489 And what he did was to find which are the two symmetries which have the longest chain before repeats itself. 242 00:26:55,490 --> 00:27:01,130 So he found a chain of 18 kind of Fibonacci things, which then return to the beginning again. 243 00:27:01,760 --> 00:27:06,620 So you had 18 variations and then he would pit between every third one another variation. 244 00:27:06,620 --> 00:27:16,609 So to bring us up to 24. So this is one of the beautiful things I find sometimes with working with collaborations, 245 00:27:16,610 --> 00:27:22,790 but across disciplines between art and science that out of this perhaps comes a new question is that 246 00:27:22,790 --> 00:27:30,109 an interesting structure for a mathematician to look at the idea of a Fibonacci chain of symmetries, 247 00:27:30,110 --> 00:27:39,439 which I'm not sure I've ever seen studied. So it's an arc is using the cube as a way of generating those variations. 248 00:27:39,440 --> 00:27:45,229 Again, we see symmetry as a way of restricting the possibilities that he could do. 249 00:27:45,230 --> 00:27:51,050 He could have just done eight factorial different ways of arranging those kind of sounds for the cello. 250 00:27:51,230 --> 00:27:56,180 But no, the cube restricts and constrains his creativity. 251 00:27:56,480 --> 00:28:04,790 And this is often why a composer will use mathematical structures in their work because they re too much freedom and you can't create. 252 00:28:05,030 --> 00:28:11,930 This is Stravinsky talking about the importance for him of constraints coming from the things like mathematics. 253 00:28:12,110 --> 00:28:19,010 My freedom consists in my moving within the narrow frame that I have assigned myself for each one of my undertakings. 254 00:28:19,250 --> 00:28:23,569 I shall go even further. My freedom will be so much the greater, a more meaningful, 255 00:28:23,570 --> 00:28:30,170 the more narrowly I limit my field of action, and the more I surround myself with obstacles. 256 00:28:32,150 --> 00:28:40,550 So we've heard a lot about the way that composers have used symmetry as a way to create this framework within which to compose. 257 00:28:40,940 --> 00:28:43,370 But there's also something interesting working the other way, 258 00:28:43,370 --> 00:28:51,170 because we've also discovered that within sound we can discover symmetry at work and creating that sounds. 259 00:28:51,830 --> 00:29:01,070 And this is what we've got set up here is a demonstration of how symmetry is embedded in just the nature of sound itself. 260 00:29:01,340 --> 00:29:04,549 And this is an experiment that was done at the beginning of the 18th, 261 00:29:04,550 --> 00:29:13,640 19th century by scientists in Scotney who discovered that symmetry is actually hiding in the sounds that we hear. 262 00:29:14,060 --> 00:29:21,260 He became very famous. He was kind of the Brian Cox of his day going on tours around Europe, giving lectures about physics. 263 00:29:21,740 --> 00:29:28,219 And he discovered that if you take a metal plate and you take a violin bow and you vibrate 264 00:29:28,220 --> 00:29:36,560 the bow and you put sand or salt or pollen on top of the bow with careful vibrations, 265 00:29:36,560 --> 00:29:40,490 you can get some interesting kind of symmetrical patterns appearing in it. 266 00:29:40,490 --> 00:29:46,549 And. And he went around the courts of Europe. He actually gave up his position university. 267 00:29:46,550 --> 00:29:49,070 He managed to make enough money doing this tour. 268 00:29:49,880 --> 00:30:00,500 He went to Paris, where he demonstrated this for Napoleon, who thought it was so extraordinary, gave him 6000 francs for his demonstration. 269 00:30:00,740 --> 00:30:08,630 So what we thought we would do is to give you a kind of insight into the way symmetry appears in the vibrations of a plane. 270 00:30:08,660 --> 00:30:16,190 So we set up here on the front here an example of a flat in his experiment at work. 271 00:30:16,460 --> 00:30:23,030 And what we've done is a slight variation because we've chosen we put four plates together. 272 00:30:23,030 --> 00:30:27,559 So we're going to see how each plate will produce some symmetry and how these symmetries 273 00:30:27,560 --> 00:30:32,750 then build up to make perhaps something a little bit like the the wall in the Alhambra. 274 00:30:32,930 --> 00:30:40,580 So now you've been given these earplugs, which are not, because in case you find the lecture so boring, you wanted to go to sleep. 275 00:30:40,880 --> 00:30:45,460 They are, in fact, for this moment here, because what what we're going to do now, Aaron, 276 00:30:45,530 --> 00:30:51,380 the frontier is going to be operating and picking out particular frequencies of the vibrations of this plate. 277 00:30:51,590 --> 00:30:56,450 And we'll see the different shapes that are emerging in the salt that I'll put on the plate. 278 00:30:56,480 --> 00:31:03,650 So you can now put these in and I will put mine in because I'm the closest, probably good the only to it. 279 00:31:15,150 --> 00:31:19,110 Aaron got his fancy ones. He's got like, I need to get some of those on this. 280 00:31:25,250 --> 00:31:31,129 Okay. So we're going to put the salt on and then as the frequencies. 281 00:31:31,130 --> 00:31:35,510 So what we've got underneath is amplifiers which are going to make the plates vibrate. 282 00:31:36,320 --> 00:31:41,570 So. Maybe. 283 00:32:24,170 --> 00:32:27,790 Fantastic. You can take them out now. 284 00:32:28,120 --> 00:32:35,879 Yeah. Thank. Now. 285 00:32:35,880 --> 00:32:39,630 I'm not saying I found that magical the first time I saw this actually at work. 286 00:32:40,350 --> 00:32:45,810 And if you want, you can find it. This is this was a YouTube video just in case that didn't work. 287 00:32:46,290 --> 00:32:55,080 But but he documented all of these extraordinary different shapes, some quite simple, but as the frequency gets larger. 288 00:32:55,230 --> 00:33:03,960 So what we have here is a little speaker with a roll to connect you to the speaker, and the road is vibrating the plate. 289 00:33:04,200 --> 00:33:07,770 And we see these extraordinary patterns at work. So what is going on here? 290 00:33:08,610 --> 00:33:12,900 Well, this is very similar to this kind of two dimensional version, really, 291 00:33:13,290 --> 00:33:22,320 of the discovery that Pythagoras made thousands of years ago that music and mathematics are intimately related. 292 00:33:22,770 --> 00:33:30,180 He discovered that the notes that we find harmonic and nice combined together have mathematics hiding behind them. 293 00:33:30,420 --> 00:33:33,440 And you can demonstrate this with a string. 294 00:33:33,450 --> 00:33:37,349 So if you vibrate a string, then you can vibrate it twice as fast. 295 00:33:37,350 --> 00:33:42,630 And we see a point in the middle which doesn't move, something we call the note. 296 00:33:43,050 --> 00:33:46,830 So this point is not moving at all. This is a note which is an octave higher. 297 00:33:47,060 --> 00:33:51,090 If we vibrate three times as fast, we see two points which don't move. 298 00:33:51,270 --> 00:33:59,820 And this is a note which is one perfect fifth and this is the basis of musical harmony, is the combination of these second two notes at work. 299 00:34:00,060 --> 00:34:06,990 But the interesting thing for our flattening plates is that what we're seeing here, the two points which don't move at all. 300 00:34:08,250 --> 00:34:15,990 The what we're seeing with the sand stabilising is these the plate, the lines in the plate which are not vibrating at all. 301 00:34:16,000 --> 00:34:18,690 They're the nos. They're a bit like these stable points here. 302 00:34:19,290 --> 00:34:25,020 So when we see all of these patterns at particular frequencies, there's a complete shift and change. 303 00:34:25,110 --> 00:34:30,780 We've picked out the particular frequencies. If we done it as a continuous frequency increasing, 304 00:34:31,170 --> 00:34:37,290 you would have seen a complete shift of the pattern from one to another as we hit each resonant frequency. 305 00:34:37,530 --> 00:34:42,210 So the patterns you are seeing are the places where the plate is not vibrating. 306 00:34:42,600 --> 00:34:52,530 The challenge was, can you mathematically explain why you're seeing these and even predict for different shapes what sort of patterns will emerge? 307 00:34:52,580 --> 00:34:55,500 What sort of frequencies will you hear? Those patterns. 308 00:34:56,070 --> 00:35:03,270 So as I said, Napoleon heard this lecture, was so excited about it, he said, this man makes sound visible. 309 00:35:03,480 --> 00:35:08,670 And it was at this time that at the Paris Academy, they used to like to set prizes. 310 00:35:09,060 --> 00:35:15,000 Challenge is to try and stimulate the mathematicians at the time and they choose different sorts of challenges. 311 00:35:15,240 --> 00:35:24,780 And the one that was chosen, I think, in 1808 was to explain the patterns that were appearing at these points, 312 00:35:24,960 --> 00:35:30,960 highlighting the experiment had found these. But how can you predict what you will see in these plates? 313 00:35:32,270 --> 00:35:36,500 The prize didn't get many entries. The first entry, the two people injured. 314 00:35:36,980 --> 00:35:42,500 One was became a member of the academy and wasn't allowed to enter any more because he became a judge. 315 00:35:42,800 --> 00:35:52,880 The other one was a certain Antoine, the blonde. And this was quite intriguing for the professors at the time because this student, Antoine the blue, 316 00:35:53,180 --> 00:35:59,899 have been particularly bad students and then suddenly was performing extremely well and they were kind of intrigued, 317 00:35:59,900 --> 00:36:02,960 well, how come the student is suddenly become incredibly good? 318 00:36:04,460 --> 00:36:10,370 The solution that was submitted was not really worthy of the prize. 319 00:36:10,400 --> 00:36:15,350 But then a second solution came in. At which point they decided to find out who this was. 320 00:36:15,350 --> 00:36:21,560 It turned out not to be Antoine Loblaw, who had left the University of the École Polytechnique some time ago. 321 00:36:22,280 --> 00:36:30,350 But it was in fact a woman, Sophie Germain, who was using Antoine de Bloom's name because she could study at the École Polytechnique, 322 00:36:30,950 --> 00:36:37,550 and so she had assumed a man's name in order to be able to try and get access to the university. 323 00:36:37,850 --> 00:36:44,390 Her third submission for this prize was under her own name, by which time she'd been outed, 324 00:36:45,080 --> 00:36:49,240 and it was a worthy enough contribution to the understanding of these plates. 325 00:36:49,430 --> 00:36:58,310 Now that she was awarded the prize, it was the first time that a woman had won this prize given by the Academy in Paris. 326 00:36:58,820 --> 00:37:02,720 The solution was not complete, and it seems it might have had some mistakes in it. 327 00:37:03,260 --> 00:37:06,290 But Sophie, I mean, it's one of the things that she contributed. 328 00:37:06,560 --> 00:37:10,820 The other one is very relevant to this building here because she was one of the 329 00:37:10,820 --> 00:37:17,210 people who made a significant contribution to the solution of Fermat's Last Theorem. 330 00:37:17,840 --> 00:37:25,010 She was able to show particular equations of the form X to the p plus winds appliqués into the p do not have solutions, 331 00:37:25,580 --> 00:37:32,030 but the P had to be a special p prime such that also two times p plus one was also prime. 332 00:37:32,330 --> 00:37:37,030 These are now. So for example, five, two times five plus one is 11. 333 00:37:37,040 --> 00:37:41,899 So that's a prime which with you with this method would work. These are now called the Sophie's. 334 00:37:41,900 --> 00:37:49,100 You mean primes. But she also made a contribution to the first attempt to understand the mathematics behind these plates. 335 00:37:49,430 --> 00:37:57,919 But it wasn't until the beginning of the 20th century when Valter Ritz finally came up with a way to solve what was going on, 336 00:37:57,920 --> 00:38:01,219 to be able to make predictions. He worked out what the frequencies were, 337 00:38:01,220 --> 00:38:07,220 and they comparing very well to the experimental frequencies at which these particular vibrations were appearing. 338 00:38:08,630 --> 00:38:16,340 His work wasn't really recognised at the time. As I understand it, Russians took his work and realised that it was very powerful. 339 00:38:16,340 --> 00:38:21,230 It now is, I think, the basis of the finite element method which is used by many people solving pdes. 340 00:38:22,910 --> 00:38:26,390 Lord Raleigh kind of claimed that he'd stolen his ideas. 341 00:38:26,900 --> 00:38:34,940 So I think it's it's right that we celebrate somebody who perhaps name isn't so much attached with these plates as it should be. 342 00:38:35,120 --> 00:38:42,890 And he came up with this way of producing the particular vibrations that gave a prediction for the particular plates. 343 00:38:43,730 --> 00:38:49,610 But this particular mathematics and the physics behind these plates isn't just relevant to understanding 344 00:38:49,610 --> 00:38:55,459 the vibrations that HLAD was producing in that wonderful lecture that he gave across the corks. 345 00:38:55,460 --> 00:39:04,190 In Europe, we now use the same sort of ideas across so many different disciplines that this has an incredibly rich heritage. 346 00:39:05,120 --> 00:39:07,910 The ideas that came out of these plates. 347 00:39:08,480 --> 00:39:16,070 So, for example, musical instruments, it's very important to understand the particular resonances that will happen in particular shapes. 348 00:39:16,580 --> 00:39:24,469 Why is a Stradivarius violin so much more valuable than a Yamaha manufactured factory? 349 00:39:24,470 --> 00:39:33,110 Manufactured violin? Some people believe it's the nature of the woods that's made that the violin is made from. 350 00:39:33,620 --> 00:39:40,250 But one of the things which is particularly relevant is the shape, the perfection of the shape of that particular violin. 351 00:39:40,490 --> 00:39:47,330 And you can analyse the particular craton, the patterns that appear in the violin box. 352 00:39:47,360 --> 00:39:51,769 Of course, you know, you've just got a, a string vibrating. So you might say, you know what's so important? 353 00:39:51,770 --> 00:39:53,620 Surely the string is the most important thing, 354 00:39:53,780 --> 00:39:59,420 but it's these resonances that are produced in the violin which are actually the characteristic of the violin. 355 00:39:59,770 --> 00:40:05,540 A comes from these these patterns that you see if you take a circular plate, 356 00:40:05,930 --> 00:40:10,910 which is one that we analysed and it's actually mathematically the easy one to analyse, 357 00:40:11,960 --> 00:40:20,000 what you discover is that the first frequencies where these patterns appear are actually very related to each other. 358 00:40:20,000 --> 00:40:26,659 They're in the first frequency and the second frequency are in a 2 to 3 relationship, 359 00:40:26,660 --> 00:40:31,370 which is the perfect fifth from which we make musical harmony the next frequency. 360 00:40:31,400 --> 00:40:37,010 Is it? So 3 to 4, 4 to 5, five to about 5.9 to almost six. 361 00:40:37,280 --> 00:40:40,730 5.92 to 6.9, I think almost seven. 362 00:40:40,940 --> 00:40:49,160 So the first six frequencies that you hear of the of a circular plate vibrating are almost in whole number ratios to each other. 363 00:40:49,430 --> 00:40:55,610 And this is why if you have a timpani, which has a circular shape to it, it has a note to it. 364 00:40:55,610 --> 00:41:02,150 You can hear a note because you're actually hearing the harmonics that you are hearing when a violin string is vibrating. 365 00:41:03,280 --> 00:41:07,269 While intriguingly the note that you hear is not one of the notes that it's 366 00:41:07,270 --> 00:41:11,649 vibrating at because your ear is tricked into putting the bottom fundamental. 367 00:41:11,650 --> 00:41:19,690 So we've heard two to 3 to 4 to 5 to 6, but actually your ear fills in and you hear the bottom one, which isn't vibrating at all. 368 00:41:20,440 --> 00:41:25,840 So certainly these finding plays will be important for understanding the creation of musical instruments. 369 00:41:25,990 --> 00:41:31,480 But the same sort of idea what is at halt here is discovering something called the organ values. 370 00:41:31,990 --> 00:41:36,070 These are the frequencies of an operator controlling these vibrations. 371 00:41:36,310 --> 00:41:41,440 And the idea of an eigenvalue. Some of you students here from school may be studying matrices. 372 00:41:41,680 --> 00:41:44,800 And matrices have these things called eigenvalues attached to them. 373 00:41:45,070 --> 00:41:49,250 And they're really at the heart of understanding many things across the physical world. 374 00:41:49,270 --> 00:41:57,429 So, for example, in quantum physics, the idea of these particular vibrations of a particular operator, just as in here, 375 00:41:57,430 --> 00:42:04,660 we went through a set number of variation frequencies which produce these patterns in quantum physics as well. 376 00:42:04,870 --> 00:42:11,200 The particular frequencies which you get these energy levels at is how we understand how an atom is working. 377 00:42:11,530 --> 00:42:16,900 In particular, if, you know, computers are getting smaller and smaller until we're hitting the quantum world. 378 00:42:17,170 --> 00:42:22,750 And what very often what we're having to understand is a kind of small area where an electron is trapped. 379 00:42:22,930 --> 00:42:26,799 And we have to understand the resonances of that electron. And, again, 380 00:42:26,800 --> 00:42:30,940 those particular waveforms that will control the quantum behaviour of that 381 00:42:30,940 --> 00:42:35,590 electron in this little stadium will be very similar to the current NI plates. 382 00:42:36,400 --> 00:42:41,680 And it's understanding the different frequencies, which will give us an understanding of how that electron is behaving. 383 00:42:42,100 --> 00:42:51,940 But it's not just very small things like quantum physics. I discovered that the understanding of earthquakes in Mexico, a particular lake, 384 00:42:52,390 --> 00:43:00,280 if you look at this particular shape of that lake and understand the flattening patterns that will appear in something of that shape, 385 00:43:00,610 --> 00:43:04,060 helps you to understand the fault lines that are lying beneath that lake. 386 00:43:04,840 --> 00:43:12,520 And in an area very close to my own, I study symmetry, but I also very interested in number theory in prime numbers. 387 00:43:13,000 --> 00:43:18,670 One of the ways we believe we can understand prime numbers is using something called the Riemann Zeta function. 388 00:43:18,910 --> 00:43:26,350 And we believe the way the places where this function outputs zero are going to give us the understanding of how the primes are laid out. 389 00:43:26,950 --> 00:43:33,100 It's a kind of landscape and the point it points at sea level in this landscape of the DNA, which will explain the primes. 390 00:43:33,850 --> 00:43:39,280 We believe that these are also eigenvalues of something, their resonances there, 391 00:43:39,280 --> 00:43:44,110 like the moments in this flattening play where suddenly the shapes change from one to the other. 392 00:43:44,410 --> 00:43:51,850 If we can understand what that that kind of thing is, whose frequencies are controlling the these points at sea level? 393 00:43:51,850 --> 00:43:56,140 It might tell us about one of our biggest mysteries, namely understanding prime numbers. 394 00:43:57,400 --> 00:44:01,150 But we're using these particular plates, the idea of putting several of these together, 395 00:44:01,150 --> 00:44:06,100 because we were intrigued to use this as a way of perhaps generating some new patterns. 396 00:44:07,420 --> 00:44:12,490 Traditionally, crab NI plates are shown one at a time, and you see these wonderful symmetries appearing. 397 00:44:12,820 --> 00:44:20,170 But we were wondered what would happen if you put quantity plates together with different frequencies operating on different plates. 398 00:44:20,350 --> 00:44:25,120 Sometimes the pattern seemed to be asking to be propagated across the plates. 399 00:44:25,450 --> 00:44:33,430 And so our project this is our first experiment, your ask kind of soft launch and kind of experimenting with this idea. 400 00:44:34,000 --> 00:44:37,810 We're going to you might have seen as you, some of you when you came in, 401 00:44:38,080 --> 00:44:44,650 we spent the afternoon with 16 of these plates connected together in a four by four grid. 402 00:44:44,830 --> 00:44:49,660 And we've been altering the frequencies and just seeing whether we can generate some new patterns. 403 00:44:50,050 --> 00:44:57,730 And this is a collaboration with an artist, Richard RIESS, who's here in the audience who runs something called A Pattern Foundry. 404 00:44:57,940 --> 00:45:03,250 And he's been curating patterns by different artists and patterns of his own. 405 00:45:03,520 --> 00:45:09,550 And we've created a pattern together, this second pattern along something we call the ghost tile. 406 00:45:09,760 --> 00:45:18,280 It's a little variation on let's see whether I guess I'm God, it's a variation on one of the Alhambra tiles. 407 00:45:18,280 --> 00:45:25,330 And the idea is actually the the three sides of this tile are musical notes, which are the octave and the perfect faith. 408 00:45:26,020 --> 00:45:29,880 But now we're looking at these plates to see whether we can create interesting patterns. 409 00:45:29,890 --> 00:45:35,920 So here's one we just started with, which is the same frequency on each plate, 410 00:45:36,100 --> 00:45:39,670 but already is creating an interesting kind of tessellation across the plain. 411 00:45:40,270 --> 00:45:46,680 But our challenge is to kind of use this as a kind of tool to create a sort of musical version of the Alhambra, 412 00:45:46,690 --> 00:45:51,400 something which absolutely will be living in bearing almost like a musical instrument. 413 00:45:51,730 --> 00:45:57,730 So thank you for those who came this afternoon little earlier to try and help us to explore these plates. 414 00:45:58,360 --> 00:46:02,890 We're going to be taking this to the Cheltenham Science Festival. So having any of you around in Cheltenham. 415 00:46:03,050 --> 00:46:09,710 We're going to be there for the whole weekend just playing with our plates and seeing what symmetries we can make out of sound. 416 00:46:10,220 --> 00:46:10,640 Thank you.