Symmetry: a talk based on his second book, Finding Moonshine - Marcus du Sautoy

Well, the 30th of May 1832, a gunshot was heard ringing out across the building this morning. Paris, a peasant who was walking to the market that morning, ran towards the end. Gunshot was over and on the ground, he found a young man writhing in pain, clearly shot by a shooting. The young man was taken to a local hospital in a country hospital where he to stay in the arms of his brother, Alfred. The lawsuits, he says, were too high. Me, Alfred.

I need all the courage I can muster to die at the age of 20. The young man's name was every Galois. He was a well-known revolutionary in Paris at the time. Actually, this is if you want to see me any private at all. Yes. Yes. Now, this is that is actually happening when I was about ten years before the IDF came out and. The young man. The young man's name was every schoolboy. He was a well-known revolutionary in Paris at the time. Now, it's not quite clear what the jewel that morning was about.

Some say that it was, in fact, a duel with a friend of his over a woman. They both had fallen in love with others. And actually it was the establishment trying to get rid of this troublesome revolutionary. There's even a conjecture that he staged his own dance in order to try and spark a new revolution in Paris.

Well, now it's unclear what that duel was about, but it wasn't revolutionary politics for which this young man became famous, because a few years earlier, while still at school, every school law had solved one of the big mathematical problems at the time. Now, he went to the parents academy trying to explain his great breakthrough. But the academicians in Paris couldn't understand a word of what he was talking about, partly because this is how he wrote most of his mathematics.

Anyway, the night before that duel he realised that probably this was going to be his last chance to try and explain the great breakthrough that he made. So he actually set off the whole night before the duel, writing the line to a friend, trying to articulate this great breakthrough he made. And as a dawn arose and the new day, he went out to meet his destiny. Probably in fact been doing maths all night with the fact that he was such a bad shot that morning.

But. But as you say, there's documents that he left behind on his desk. Many of us regard as some of the most important mathematical documents in the whole history of my subjects, because hidden in so many, that was a new language, a language to help us to explain one of the biggest ideas in mathematics and in science, namely the idea of symmetry. This language that he developed was called group theory.

Now, I, as a mathematician, use this language that got a lot of evidence already, just by the age of 20, every day in my working life, in order to be able to understand and navigate the world of symmetry. It's interesting. Although I became a mathematician on my Galois, I think our Morning Edition went on. When I was younger, I hadn't wanted to be a mathematician at all. In fact, my dream when I was a school was in fact to become a spy.

This is partly fuelled by my mother, who works for the Foreign Office, and when she had children, she had to retire. That's a she told me and my sister that she'd been allowed to keep the black gun of every member of the Foreign Office and that it was hidden somewhere inside our house. So my sister and I used to spend all our face and try to find where this black gun was. Never find it. They obviously told her the answer concealed very well as well, but some.

So I decided that I was going to join the Foreign Office like my mother. And then I thought having becoming a spy. So when I got to secondary school, I went to a safe school here in Oxfordshire. I signed on for every language that my my school, the Foreign Office, found any languages, the French, Latin, German. Actually, at the time when I went to school, there was a course on the BBC teaching Russian and I thought,

Oh, fantastic, that's what you need for a spy. Now you can tell how old I am because I grew up in the Cold War when the Russian fabric. And so my French teacher help me with the Russian course on the BBC. But as I wanted to learn these languages, I more and more frustrated because there were all these kind of strange spellings that you just have to learn. It was never long term at Learn Irregular Verbs, which is S.A. Cotton to the Russian course was actually a disaster.

I couldn't get past the word hello, which has so many consonants and no vowels. My strong suit was the name of the cause. I couldn't even say the name of the school said. So I became very disillusioned. And it was strange to kind of spy and learn all these languages, but it was about that time, about 12 or 13, that my maths teacher in my school in May, Wilson, just like I want to see you after class, I got into trouble. So I went off to the end, of course ended and he said, Follow me.

We went round the back of my small car, the whole dropping out, and then he took out his right time. Cigar. You said you weren't allowed to smoke in the common room. So two guys, right? So I said the same time, I think you should find out what mathematics is really about because it isn't really about all the kind of multiplication, percentages, signs and cosines that we're doing in the classroom. Much more exciting and he recommended a few books to me.

But he's always a rock, this world of mathematics. And so we took this I took this list of books, and I went home and my dad and I came up and we came to Oxford with this wonderful bookshop called Blackwell's, which is going to have, you know, your daughter, who, as you know, I think that was taught, is this kind of small shopfront. Do you think there can't be anything interesting here? You go inside and it just goes on forever.

You go down into the basement and there are all these science books and maths books, and that was so exciting. And so my dad took this list of books and I was just went round sort of wandering, putting books off the shelves. I couldn't understand words. You seem to be in some sort of secret code, but there are undergraduate stands of reading up against the bookshelves, reading these books as if they were novels, little stories and things.

So. So I became very intrigued. It was exciting excitement so much. And we took these books home and I began to read about this exciting world of mathematics, and I began to have a sense of the amazing stories that this world contains. But what intrigued Moses was one of the books that my teacher recommended because it was called The Language of Mathematics. I still have the book that we bought that Saturday morning. It costs £1.75. I defy you to find a book in black holes now that costs £1.79.

But this book I know because first of all, I never really thought of mathematics as a language, but as I began to read this book, I realised, Yeah, it's an amazing language. It's a language which actually helps us to articulate the structures in the world around us, where we come from, where we're going to go next. It was one language because I didn't have the irregular verbs that only make perfect sense.

And so this is the language for me. So that's not to say that they didn't have strange twists and turns and surprises, and that's one of the joys of mathematics that not everything is totally obvious, yet everything seemed to make perfect, logical sense. And this is the book. It's not a very famous it's been really helpful in language.

Mathematics is a language on the language that really I found so intoxicating in this book was this man with power who developed to understand the world of symmetry and language through group theory. And I began to understand when I read these books that in a way, symmetry is its own language. Symmetry is a language that nature uses to communicate information. If something has symmetry, then it's generally got a message inside of it.

For example, if you go to the garden and you look at the bumblebee, a bumblebee has incredibly fine vision and can't judge distances. It sees the world very monochromatic. KLEE But what it can make out is shapes with symmetry because it knows the shape. The symmetry is likely to be a flower, which will have substance inside it. Now, intriguingly, the flower itself needs to be to this day in order to propagate is genetic heritage.

So the symmetry of the flower is little bit like a bill saying, you know, come and visit me. And in fact, research has been done to show that the more symmetrical the flower, the sweeter the nectar inside the bee, inside the flower.

Although we also have become very sensitive to symmetry, I think, you know, we survived in the jungle because of our ability to spot things, especially if you're in the jungle, you see the chaos of the trees and the leaves and things, and suddenly you see something with mirror symmetry. You better take notice because that's an animal. Either it's going to eat you or you're going to it. So symmetry. Those who are sensitive to symmetry somehow survive in this world.

And even now we use symmetry very often when we're choosing a mate. I'm going to show you two pictures of faces I talk to, just think normally, but I made them artificially symmetrical. And the bottom line now, if you ask somebody between these two images, which ones do you find the most beautiful? Most people are drawn to the face, which is most symmetrical. And why is that? Why we associate symmetry with beauty? Well, again, symmetry is communicating information. We're looking for a mate.

We're looking for somebody who has very good genetic heritage, good upbringing, good conditions. And the thing is that symmetry is quite hard to make. It's very easy to break. And so if you find somebody who has a perfectly symmetrical face that's actually communicating information, it's communicating information about their genetic background, that they're their upbringing. And so that's why, again, symmetry is being used in the face to communicate information when you're choosing your mate.

And I rather like quite which comes from Galileo, which I think rather sums up the power of mathematics as a language to to understand the world around us. He writes, The universe cannot be raised until we learn the language, become familiar with the characters, which is written. It is written in mathematical language. And the letters are triangles, circles and other geometric figures without which means it is humanly impossible to comprehend a single without.

So where do we start to try and explore the world as soon as we go off to the beginning of the 19th century, developed this language, but actually maybe trying to articulate not a sad world of symmetry for millennia. In fact, if you go back to some of the very first objects that humans carved out. So there's a wonderful game in which we find in the British Museum in London.

This game dates back to 2600 B.C. and it's got for the first time, interestingly, the first ice cubes that we used to play, Monopoly, things like that, but a little tetrahedral ice. So it's a kindred symmetrical object. The first choice that we used in history were the knuckle bones of sheep, which all are irregular. They would land. Several are often on one side rather than another.

And people soon realised that if you wanted to find ice, you want something which is symmetrical, which doesn't on one side over the other. So they were naturally drawn by their claim to produce shapes which had an inbuilt symmetry. And so the four sides of the novel gradually got carved into this tetrahedron, which is made up of four equal outlines. Now, interestingly, I like this one and one of the points upwards.

So what's the the they describe this game as they would colour two corners and then you throw it the tetrahedron and then you count the number of black spots that were pointing out and that would be all new in the game. And so this game of war actually is one of the forerunners of backgammon. It's an early version of backgammon. And even the board itself, you can see some actually the significant squares all have symmetrical shapes attached to them around the same time.

But in Scotland you find other artists playing around with symmetry. In fact, these stone balls you couldn't find here in Oxford in the Ashmolean Museum, they make they match 2500 B.C. Every game you can see the artist taking a ball and carving different patches on the side of the ball, almost like a football, trying to make out of all the different patches. And you can see the art is exploring different possibilities. So something like a cube or the tetrahedron or some more complicated shapes.

And it's interesting, we don't actually know what these stones or whether they had a purpose. They don't seem to be part of the game. Some have suggested they might have been part of a fortune telling, but it isn't the way they had power for the clans. But it's facilitating all these 2500 B.C., exploring one possible symmetries that are out there. And that's somehow what drives mathematics, is that the the the challenge of trying to find what possible symmetries are.

And it really was it wasn't we had to wait until the ancient Greeks for a more analytical analysis of these symmetrical shapes began to emerge. And in fact, you find in Euclid's elements a proof sort of culmination of this great book. Is it a proof that you can actually only have five different shapes which make up dice? It always has a symmetrical face, and you have to have the same symmetrical face about the whole shape.

And they have to arrange such that no faces and this can be distinguished from any other. With those conditions you find there are only five different possible types and actually place. I believe that the symmetry so important is that the associated these five basic shapes with the building blocks of nature we believe in mathematics was the foundation of the whole of the natural world. And so each of these five shapes is associated with one them.

One of the elements now the elements at of time in the ancient Greeks was things like oxygen and hydrogen, but with things like fire, earth, air and water. So fire, the shape of fire was this little tetrahedron. So the tetrahedron is sometimes spiky, is of all the dice. And that was the shape of fire. The cube made up of its six square faces that was stable earth.

And then you have another shape made out of rats and triangles. You could put eight equal out to times together to make an octahedron. That was the shape of this place. As I say to air, you can put 20 faces together, an easy classify move to make an icosahedron. This is the most circled of the shapes and so Plato associated with water.

And then you had this one shape match over, which was the dodecahedron beautiful shape made 212 pentagonal faces and he associated this one with the shape of the universe. At the time, the ancient Greeks believe that somehow the universe was basically just some sort of glass ball with the star painted on it and place. They believe that that's where we were sitting. Inside was basically just a big dodecahedron and you can find examples.

Actually, again, it's 12 faces. There are no examples of called calendars or with Zodiac signs with these 12 phases. So, yes, it's interesting. Now, from a small world perspective, Plato's classification of these shapes associated with the elements as we find what looks kind of ridiculous. But actually, Plato talked to something very deep about symmetry. So there was this symmetry. There is symmetry absolutely at the heart of science and the natural world.

So, for example, in chemistry, chemists learn about different ways that molecules can be put together as way. So the different credit, the way that crystals behave of crystal structures, is very often dependent on proteins or symmetry. The biologies also in biology don't really want to guess because nobody knows. Generally, viruses like herpes virus or the AIDS virus based on symmetrical shapes. In fact, that virus is using the fact that symmetry has a very simple rule to create the object.

It's not a very complicated object. So virus is heart broken has a very short is of RNA or DNA, which is a program to reproduce itself. It needs to reproduce itself as quickly as possible. And so the fact that a symmetrical shape is a very false shape to make with the very simple program is probably at the heart of why the virus is choosing to use this sort of a very stable shape and a very strong shape as well.

And so we've begun to understand the way that viruses work. Understanding online symmetry in physics as well. And what's happening in a Large Hadron Collider, the fact that we're able to predict new particles that combined in there to make sense of all this kind of menagerie of particles that we've discovered so far. Actually, this horse is a simple object in very high dimensions, which somehow makes sense of all of these strange objects that we discovered.

And also that helps us to to make predictions about what's missing. Because if one of the pieces isn't represented, there's likely to be a particle and we can look for it will correspond to that face. So symmetry is all over the scientific world, but essentially symmetry is very much part of the artistic as well, although I think all of these have a slightly more ambiguous relationship with symmetry.

This is Thomas Mann in the Magic Mountain, talking about the symmetry of a snowflake, and they might be a shot at precision. And it definitely the very marrow of death, I think for all this symmetry is something they're a little bit nervous of. It's very restrictive. It tells you what should happen and and just like how it somehow can be filled with a sense of something without life and energy to it. But I think there are some artists who really love revelling in symmetry.

And one of the parts, and particularly taps into symmetry and fantastic I think is music and in particular the music of ball. For example, the Goldberg Variations, I think is almost like a song celebrating symmetry. So here's a little bit of one of the variations from Bach, Soledad. Variations. Now the variations actually start with an aria, very simple aria. And then there are three variations through to that all study, which is the reputation of the aria you saw.

Now, interestingly, so there's sort of a circle going on there already because the two actually got one means to the other. And you go to the 16th variation, Bach pulls out a banjo, which usually is the beginning of a musical piece. So it really got the sense with this piece. So all four ends and there's a sense, a circle already there that everything that variation Bach makes a hand out of the variation.

So a kind of you might remember from singing in school when all this all starts off the song. And then a little bit later, the other half of the class joins in. And you have to tune and the same tune, guys, with the different times and you get this lovely interaction. So Bach loved the idea of canons, but he wanted something slightly more sophisticated.

So each new canon, actually the second voice when it comes in doesn't just repeat with the first, but I think the set up was so soft, higher and higher and higher. So actually, by the eighth variation, what happens is that you guess the second voice comes in an octave away. Now I don't even want to says, I know you started with. So again, you just kind of set the pattern happening in the canon, so almost even has a spiral shape.

And so for me on hearing this, there's a embedded inside the structure. This this is what we call a chorus of circles where the circles inside the structure of this piece and again, you know, the rhythms ball is very careful to to get all the different rhythms that are possible. So in the in the canons, the variations are broken off sometimes reasonable or reasonable or do these in them are the beats and divided into quavers, triplets or quavers?

And what about going through the symmetries that if the two triangles are combination? He makes sure that each hand covers one of the possibilities either three B's and of all divide it up into seven quavers and Bach, the true imputation. Make sure that each and every different possibilities he is covered in brackets. Ball student medicine used to call box music a process, a sound in mathematics. And Bach isn't the only one who has lots of ideas symmetry in order to create variations.

Very often the variations will mirror what is going on before when you were trying to push. And so symmetry is very often something you can hear embedded inside the music of, for example, Bach and another of the also I think, which uses symmetry or lost its architecture. We're in a building which really mixes art and mathematics here, police officers. We're right back to, in ancient times, the pyramids. In Egypt, for example.

I looked we like these all Hadrian's Wall one hall stuck inside of the fence and he squares right through to modern day Paris. I think Paris is one of my favourite cities full of symmetry in fact. So the revolutionaries of the time of Galba had wanted to build a sphere at a hall to Paris because they felt that a sphere was a shape which represented the idea of égalité no direction favoured and over any other sort of perfectly symmetrical building.

But since quite difficult to build architecturally on this one, eventually you'll be able to school Agios in Metz in Paris. I think the revolutionaries of debate just wanted to point out that this is in fact an all maximal revolutionary idea. But if you go to Israel, you actually find amazing blocks of flats which are based on the dodecahedron shape.

The place you associate with the shape universe and solar panels is in Iran, not in Israel, although I'm told that these are hopeless furniture that I see. I went very closely with a mathematician in Japan called Professor Corncob, and I went out to all my studies and symmetry to go and do this work with him.

And Professor Kurokawa took me up to his beautiful city, which I don't know whether any of you need to measure, which is full of all these beautiful Shinto shrines and Buddhist temples, and they're full of symmetry all over the place, the patterns inside the building. And as you all know, we took this photograph. We we climb the stairs to this archway and the Australia had these beautiful columns in these symmetrical designs.

So eight columns exactly the same, except for one, which is turned upside down. And I said to Professor Kurokawa, Oh my gosh, that the architect must be very angry. The builders, I mean, go that one wrong. It's upside down. And this is not it. It was a very deliberate decision because in Japanese they love setting all these expectations and then embracing. And he referred me to this lovely book called The Japanese Essays in Idleness. What a wonderful title for a book from the 14th century.

And he revealed this particular requirement on these. And then one thing uniformity is undesirable, leaving something incomplete. Interesting, given the feeling there is room for growth. Even when building the Imperial Palace, they always leave one place unfinished. Nancy even if you go back to the ball. Ball. GOLDBERG Variations seem to be full of symmetry. You reach the third variation. You sort of think you know exactly what's going to happen next.

And then back to exactly the same thing turns the whole thing upside down like that. It's going to call it a musical check that this variation has absolutely nothing to do with the rest of the structure, too. And it's such surprise because you write it and then suddenly realise how much symmetry there being on that point that you appreciate that break in symmetry involved in that.

But I think actually if I was going to be cast out, I had to leave the rest of my life in one building across the well. Being somebody who studies symmetry and love symmetry, I probably would choose the Alhambra in Granada because, you know, Hungary was not a palace. There was really a past century symmetry, this symmetry all over the place. There's lots of water throwing up reflections of the buildings in the water.

They came to realise how, how, how special symmetry is because it was a little breath of wind for that symmetry suddenly to be shattered. So it's something very special. But I think it's funny, the walls where I really see symmetry being celebrated. So because all of these beautiful symmetrical tiles they use across the walls here in the offices exploring all the different possibilities that they have, think of the two candles.

And actually, I think you see in the Alhambra, which is really where you can ask the really big questions about symmetry. Reporter What is a tree? What water? What does it mean to say that something is symmetrical? How can you say two things have the same symmetry. How do you understand if you actually around all the different symmetries that are possible? And as you go on to body language, you want more symmetry is is really what what Galileo's young revolutionary French mathematician did.

I think there's a sort of feel that symmetry is all about that fine, rational symmetry, but there is so much more to this language. And it's what the guy, along with the one who was able to articulate where the symmetry and actually how I like to describe it is that symmetry is kind of something that you can do to an object such that when you finish doing something that will, it looks kind of like the tape we started with in a way, I kind of call it the magic trick.

So if I give you, for example, if I show you this tetrahedron, what are the symmetries of this tetrahedron? Well, there's some moves that I can make to this tetrahedron. Sasha, when I made the move, it looks like it did before I saw it. So, for example, I could pick it up and I could do that. Now it looks like it did before I saw it. So that is a symmetrical move. So I like school symmetry, but I mean, I and you show your eyes, which are your eyes?

Now I'm going to do something to this and open them. Now the tetrahedron looks like it did before you shut your eyes. But I did something to that which has this is one of these symmetries. So that's what I want you to think is symmetry is something you do to an object which brings it back to look like it did before. But actually it has changed something. So let's go to the walls in the Alhambra and try to understand these what symmetry really means.

So, for example, this wall here, where's the symmetry here? Well, how we picked up the tiles and put them back down here. So if they look like for I move them. So I fix the tiles at this red point, pick them up and turn them by 90 degrees, pull them back down again. The tiles look like they were before I move them and that is a symmetry. So symmetry or all the structure is something that I can do to the objects, which means in some way makes it look like it did before I moved it.

So I can take two very simple objects and we can do a small. So what are the symmetries that are made of these tools? So I'm going to take the object, a triangle and a 6.2. So I put a little twist off. So let's start with the six point itself. So for me, symmetry things. I can do movies, but it looks like you did before I knew this. So I put it together so I can keep track of what we're doing. So I can take the ten, put it back down again. And it looks like it did before we saw it.

So that's one symmetry. Okay, well, I can move it by a third so I can take it around to see. So that's two symmetries. I can rotate it by half for 10 seconds, around two days as a third symmetry, I can do it by two thirds a turn. As for cemeteries and finally I can tell you do away rounds to five six to the fence of a565 moves that I'm going to make to this object such that it fits back down in size. Now there is a cemetery. Does anyone have an idea of six cemetery?

I miss that. Yes. Second around the back to it. Yes, exactly. In fact, that's just leaving it where it is, isn't it? Because I could take it all the way round and then it hasn't moved at all. It's not so. So the same symmetry, he's actually just picking it up and put it back down here. Okay. You might say that's awesome, but actually it's a bit like zero zero. It's a very difficult concept as a number for mathematicians to get hold of because what's it counting?

It isn't counting anything. So I want you to think of this as a like zero six, but everything has a symmetry. So zero is you just leave it all in. Even if you go to this of the irregular face, you see some symmetry which is just leaving the face levels, which is good. So I we say there are six symmetries there is going to break through for quite a while to think of just leaving it as a six symmetry and you can't pick the seven,

for example. That might have been one of your other thoughts that it has. It hasn't got a reflection of symmetry, because if I flipped over, it's pointing the other way. So it hasn't got another symmetry coming from reflection, however, I won't take the triangle instead. This does have what is called two rotation symmetries. I'm going to rotate my third clockwise.

I want to rotate my third anticlockwise, but now I've got refraction so I can take the triangle flipped over and I'm going to go three ways I can do that. I can flip it over through the line, through X or get it over through the line through y. So the X is in point swap or I could fix it and put it over there. So that's five different symmetries. Again, two rotations for three reflections and then I six symmetry, which is just leave me where it is.

I pick it up, put it back down again. So it's got six entries. So this is quite interesting because these two objects both have six symmetry. So perhaps we should say they have the same symmetry because they have the same amount of symmetry. But what I want is to explore why actually these two how we articulate really that no, these, these objects are two genuinely different symmetrical objects.

But before I did I say a little challenge, which I want you to think about through the rest of the lecture, not, not see that this is the other stuff, but now you've got some idea of magic for me. So I want you to, to consider another object. I want you to consider the Rubik's Cube. Rubik's Cube. So what does symmetries of the Rubik's Cube? Well, think about this magic trick. So no colours. I don't care, of course, but a symmetry in the cube.

It's really just anything I can do to the cube, which makes it look like still cube. So I could do that fancy symmetry. Or I could do that. That's. That's. And that's enough. So what I want you to think about, there'll be a prize for the person who gets places. How many symmetries do you think this thing has? How many different, genuinely different moves could I make to this cue? Such it still looks like a cue. We'll come back to that later.

And I've got a price with this in this case. But so let's come back to these two shapes, because the real breakthrough got along. First of all, to think of of of symmetry is something about motion as you went very much against what Thomas Mann but for Thomas Mann, symmetry was something deathly something which didn't move but actually got lost. And now you should think about symmetry as something with lots of movements in it.

It's all about it's all about life. It's all about the things that you can do as an object, which it looks like it did before I knew it. But the second great breakthrough was that he realised it isn't just the individual symmetries that are important, but how they interact with each other. So what do I mean by that? Well, if you think about it, if I do a magic trick, move to shape, then I can do another move to it. And then the combined effect is actually a third magic trick.

So actually, all of these things are related. If I do one move followed by another. The combined effect is is divided in a third symmetrical move. So in fact, Galois began to explore, okay, what was the relationship between the 66 symmetries of the sixth point you saw? What's the relationship between all of those symmetries? If I do one followed by another. It's got to be a third symmetry, so it must be one on that list. So I can two names to all of these symmetrical moves.

So the name Amy goes to just leaving the thing where it is correct to be moving it by a sixth, but to see a third of a and so on. So now let's see what happens if I combine a six of inside followed by ten. So I'm going to do I'm going to do count will be first is equal to ten and I'm going to do capital C a third. So we do a combined effect and this is what this table records, sixth of a term followed by a third of a time. Well, the combined effect is actually about the same half term in one go.

So actually, this kind of algebra, a language hiding behind these symmetries, which says, if I can combine two sentences, it's going to be a third symmetry. And this table records for you how these symmetries interact. So the combination of being followed by C is actually the same as just in the half step. Now I want to look do it in the other order with a three mass. I could have done the third return first, followed by the same for the turn.

And still I get half as it really doesn't matter. What do you do? And in fact, you can see that reflected in this table. The table records all of the different pairs of symmetries. And what the effect is, is the symmetry. Whatever you. Doesn't matter what order you do. However, if we come to the triangle, it does matter. What will you do?

So I'm going to do a third button clockwise that's going to be you go and you follow by flip and the reflection in X, the combined effect is actually as if the triangle was relaxing it. But if we do it in a different order, let's do the reflection in experts and then the rotation by third anticlockwise.

Actually, that's the triangles ended up in a different place. It's a table like reflected in Y. If I. If it is if it goes to the part later, I want you to take a b-max and look at the picture on the bottom. And then what I want you to do is do a 90 degree turn clockwise and then a reflection in a line away from you, and then see when the picture is pointing, put it back, back to where it was, and then do the reflection first and then a rotation.

You find that the picture is pointing in the other direction. So here it matters what order you do the symmetries in. And so it's the way the symmetries interact that we're able to start to distinguish why the symmetries of the triangle that genuinely difference, the symmetries, all the six points are. In fact, what I want about was a set of rules for the way the symmetries interact. And he shows that it's an object with six symmetries.

Either the six symmetry must behave like the 6.2 starfish, or they must behave like the triangle. And there isn't a third symmetrical object. It will either. The symmetry will interact in the same way as one of these two objects. So suddenly this language now you've enabled him to say, Yeah, actually there are only two symmetrical jets with six symmetries. If I could use this language, it's a bit like a set of rules for Sudoku, for example.

Let me do it. Inside the body of this table, you never see a symmetry twice in a line or in a corner. And so Galileo developed some set of rules to the way symmetries must interact, which helps him to then explore, for example, symmetries in the Alhambra. So it took actually till the end of the 19th century for us to use Galois language. But for example, these tools here, these two always look very different.

One is made up of these kind of shimmering triangles, the other six point to solve these shapes. But actually using Galileo's language, we can say that these two tilings actually represent the same group of symmetries. So where all the symmetry decided, let's take a guess and measure moves. So there's one point where I can rotate by six and ignore the colours. So think of a turn to the point where all the triangles meet. That's one symmetry.

There's another symmetrical move I can make, which is to take the points in the middle of the triangles, rotate one third of the time, the sun will look down on top of each other, and then an isolated corner place is fine, but halfway along the edge you can flip these by half a turn. And so what happens now is that the triangles sort of swap places on top of each other. So the white triangle went to the blue and blue went and said once and that's a third move.

So basically these are the ingredients and all the symmetries of this particular set of toys. But if we go to the other wall, which I showed you, well, the same moves work here. There's a six out of a ten you can make at the centre of a six pointed star. There's a fair return. If you look where the pieces meet, you can rotate by a third return there. So that's point there. And then again, after you take the halfway point between six.

Point in stores. That's another place where you can make these rotations. So although these pictures look completely different, actually using our language, we can articulate the symmetries of these two walls or exactly the same facts. In this case, we can put one tiling on top of another, and any move that we made to one will actually represent the symmetry of the other.

So these two, I think you're going to think of Gauloises work a little bit like the concept of number and the concept of number is quite an abstract idea. For example, level three people in the front row here sitting in three chairs. Now the people of right difference the chairs, but the abstract idea of the number three is common to both. So symmetry, I want you to think of it in a similar way. It's not about geometry in pictures.

These two pictures that I showed you, they have the same underlying symmetries, but they're different representations of those symmetries just in the same way as we got different representations of the number three. So it enables us to take, for example, these three different images. There's one is the tilings, the entrance to the Alhambra, one's the floor, one's the ceiling.

Actually, these will have the same symmetrical moves that can be made, all of them, although the pictures look very different. This is a symmetry for 14. It's nothing to do with football but to do with the fact that there's a place where you can do a quarter of a turn. Another place? A quarter of a ten and a half of ten. So this language suddenly gave us a way to articulate when two different things might look very different,

actually have the same underlying symmetries. By the end of the 19th century and allowed us to actually articulate the fact that in the Alhambra, there are only 17 different sorts of underlying sentences that you can make on a two dimensional wall. And if you think about those pictures of Escher, as he was very inspired by his trips to the Alhambra and he made lots of different pitches with devils and angels,

a little selfish and efficient garden. But actually we can prove mathematically that however, on Escher drawings or anyone else, they will never be able to find anything more than the 17 that are on this list. And interestingly, what I do these in community holidays is my family, which they hate. But I took them to the Alhambra and I said, okay, we can see where they actually found 17 different symmetries in the Alhambra and they almost stayed there.

The last one is Sol three, three, three, and those blue tiles, we need to paint those black. And then we would have had an indication that we didn't actually see this abstractly. But but it's amazing. That's actually even the artist was kind of ahead of the mathematicians in being able to find all of these. But it was only with mathematics that we're able to articulate the sort of limits of this world of symmetry.

The only thing Gabor did was to actually realise that symmetry also has a kind of atomic symmetries, a bit like the possibility the periodic table of symmetry. Now, I think it's actually extraordinary and I studied Galileo to realise this. This man, aged 20, had come up with so many amazing ideas and this is perhaps the most savvy one because he showed why a symmetrical object can actually be broken down into the symmetries of smaller atomic symmetrical objects.

And this gave us a way in somehow really get to grips with the world of symmetry mathematically. So, for example, here's a 15 sided coin. The symmetries in that 15 sided point, you now know how to do their sort of rotations of a 15 sum to four things and things like that. But actually, you can make the symmetry of this 15 sided coin at the centrepiece of two options, namely a pendulum and applying so and so on.

So actually symmetries of the paintings. I do think that I can do all of them by doing Symmetries of the Pencil, a triangle. For example, I want to rotate the Pentagon by 50 return. So the key here is the fact that 15 can be broken down into the Primes three and five. So let's say I want to rotate by 5610. So I want to get let's say we're going to get six green dots. 515 is the ten. I want to get it to yellow. I'm going to use the symmetry, symmetries, the triangle, the principle.

So I'm going to take over the pendulum. I'm going to rotate by 2/5 to the 10th. So my brain is going to go down to the blue dot, but now I'm going to let the triangle take over and I'm going to do a rotation, my third return and equalise. And then also the combination of those two, hey presto, I've got a 15 for this year and this basically is at the half of all of the symmetry.

You say any symmetrical object, actually, you'll be able to find smaller symmetrical objects inside of it whose symmetries can be used to build out the symmetries of the large objects. If you just like two dimensional flat shapes like this, basically it's the primes which are the atomic shapes. So any shape of 105 sides, you can make that out of triangle Pentagon 675 theta. So it may. Daniel's discovery that he could do this gave us the idea that you could produce a periodic table of symmetry.

Now we can try to classify what all of these building blocks are symmetry on to understand the whole of the symmetrical world. So. So already we found the primes of their prime sided shapes of some of the building blocks. It turns out they're not the only ones. In fact, there are some much more exotic, little more complicated ones. And intriguingly, the first sort of interesting prime shape in all this is the symmetries of the icosahedron.

Well, actually, not to say that usually the only considering you need to put a slight twist on it. So this is actually a chocolate box that Escher designed for a manufacturer in Holland of celebrating an anniversary. When I retire, I want one of these boxes asking you to go to the SWC and Den Haag. They've got an example, I presume, that makes many of these and the little tin boxes, but they're the most beautiful things.

And but the symmetries of this box are actually one of the first sort of non obvious point symmetries that you can have, because actually, how many symmetries is this object have? Well, actually, it's the same as the centuries old football. So the football his football made out of pentagons and and hexagons. But actually, the symmetries of this are the same as the symmetries of the icosahedron. So how many different moons on that? I'll call with Max Lucas.

Exercise. Quite a twist. So he's twisted this starfish, so there isn't the reflection, like my 620 starfish. So how many different moves on there that I can make? You know, I can I can do that. So I'm just looking to read things all like you find it when you do this, you can do 60 different moons, 60 different symmetries of this shape. Now, come on, 60. That's an incredibly divisible number. The reason the Babylonians use it for the base of the nervous system, because it's so obvious.

It's the reason we have 60 minutes in the air. So but when you consider the symmetries of this object, it turns out that number 60 is very invisible. Actually, the symmetries of this can't be divided into smaller symmetrical objects. You might say, well, hold on. I mean, there's a painting in here. Surely the symmetries of the Pentagon. I just see that that's a kind of symmetry, which is a smaller hiding inside.

But if I tried to divide by the Pentagon and trying to symmetrical objects, it doesn't make any sense. Turns out the mathematics that go on to balance that there's no way to break this down into smaller objects. This turns out to be one of the first kind of interesting atoms in the periodic table of symmetry. Now, it's got a lot of studies on this one, but for 150 years, it's got a long way to go.

We were trying to classify more and more these natural objects, trying to find the acids which make up symmetry. And it culminated in the 1980s with what we believe was the completion of operating a table of US extraordinary objects, including this amazing object, which is a and I'll show it to you because it's an object is a symmetrical object, which is a 996,893 dimensional size. Even in that sense, you can call it altered for you physically, but amazingly, isn't it?

If you want to think about is this kind of weird movement which exists in an unusual size and symmetries. So the more symmetries that they are acting as the sun gets, we have to suppose we couldn't write this object and just lose interest and we just didn't have any pauses, just weird objects that suddenly appeared at this point in space. But how does the mathematicians say that's what I mean by 196,883 objects? I mean, how do mathematicians play around with objects and stuff point on it?

Well, I think this is another example of how fantastic language is in mathematics and language. In a way, you have got lost the physical world symmetry of algebra. Suddenly the pictures disappear. I have access to the table which allows us to sort of see mathematically shapes. And I mentioned school and school of easy geometry, Cartesian geometry space to be a wonderful dictionary, a language which translates a lot of shapes into the world of numbers.

So, for example, if I wants to describe a square in numbers, I can locate a school in a sort of east, west, North-South south location. So the square consists of a point at the origin 001 set along the horizontal axis, 101701 and that is it. One line where I find something that. So in a way I can translate a square into the. Seeing results pairs among school passengers identify for you this way.

If I want to go off the Dimension Cube where I can draw you key, but I can similarly use the dictionary to translate it into numbers of Americans. AIDS triples in numbers. Which car? As long as you know how many moves you make in each direction. No, no, no. One, nought, nought, nought, nought, nought, one. All the way through to the stream point while online. Okay. What about a four dimensional cube?

Now, the point is, this dictionary is wonderfully powerful, because although the visual side runs out of the three, I don't show you all of it. Still, the numerical by the numbers carries on. I can tell you in numbers, one four dimensional cube is it's a it's a shape which has points light occasions 0000. And then from that point, you got an extra now two points at 1000010000100001.

So for edges that I just keep on going. Maybe six in each of these directions it's going at some point at one, one, one, one. Now, how many corners has a four dimensional cubicles? 16 central. You able to work out how many different ways there are putting zeros and ones inside? 16. You could also use the same trick to look at how many edges there are. Now you might have to explore very quickly using this language the geometry of a shape you cannot see.

And little thought should be able to get around the edges that are all there. And that's the amazing thing that we can all see. Move to symmetries of this automation cube by using the same coordinates. So now I kind of have to show you a shadow of this shape. So, in fact, if you go back to Paris, there is a shadow of a four dimensional cube. Actually, the financial area in Paris. The all points is a shadow of information. Q In three dimensions. Do you think about a three dimensional cube?

Actually, then all this is really three dimensional cube on a two dimensional capitals. What they do is make a square and they put a smaller square inside and then they don't see what it says. Because you think the same cube, because it's just a black thing. This is the same sort of concept a project organising cube has in three dimensions. What you get is a small view inside an object view and the size will join up.

So I have an example here. Here's a four dimensional cube projected down into a three dimensional universe. I can see you correctly articulated that there are 16 corners and you can see them nice little whites balls and actually you'll be able to count all the edges. Now. So yeah, I do the maths. So the 12 inch cube on there. So that's 24, 26, 31, 32, 32 edges. But actually, you could have done that. Now, this picture, if I tried, was if I want an edge in terms of these.

But it's going to be around 196,883 are actually shadows. And what's good for you. So I'm trying to make small symmetries of this shape. You really do have to use a mathematical language which translates geometry into numbers, and amazingly, you navigate out of it. Even measure is pretty an amazing pleased to be able to explore the symmetries and all of this number branches. And one of the most is of the will of symmetry.

You have to complete this kind of classification over the centuries to incorporate the masses of all symmetry. In fact, when I finished my degree area in offices, I mean, I've fallen in love with world symmetry. It was now 1985, and I just heard about this sort of completion of this project. The Cambridge seemed to be the place to go to study symmetry, and in fact, they produced this amazing piece that has been around is the real thing.

The Atlas five is this basically inside here are all the building blocks. This is the periodic table of symmetry was produced in about 1985, just when I was finishing my three. And I went up to Cambridge because John Conway and his group to try and find out, you know, what are the next steps after having done this. So I was very keen to join that group. And he sat down and he said, well, you know, we are all very obsessed with symmetry, so let's make out at the same time.

He said, Oh. We have to drop the do something against Rangers, though, because I know the authors in this area have six letters on each word. I don't like two schools in the same place when I say, you know why, but nobody's going to do that. I was like going agree. So as bad as it is to and others. Yeah. And if you drop one of those letters. Walter, we only have two games on the air, so I'm not an MP. So my argument is, if that's not all, that's the first to join the group with John Conway.

And then he got his Ph.D. student, Ralf Hedges, that it was this guy hanging around and Cambridge just kept running around. It seemed to be quite helpful. Simon Norton So he was the third all go joined. He was really good at computing. The things he wore them all through joined with Wilson. They all joined in alphabetical order. So if I was going to join the group, I was going to have to change my name off.

The same for but since. So now about that point I realise well I know that's going a little bit too far though, and so I'm a little bit concerned that I came back to work since I wanted to start my research into the world of surgery. And I, I want to go maybe. Maybe surgery finished. Maybe we all understand everything there is to know about, but that turns out not to be true is never true.

So one problem that opens up so many more interesting columns in particular, particularly single who monster. It was this strange thing, right? In 196,883 inches right. Here is why you suddenly suddenly emerge in this financial space and it has this number or any significance, but it turns out it does this number you find in a completely different area of mathematics called number theory.

This number comes up in some modular forms, which is very much related to the work, what I wanted on Fermat's Last Theorem and even conflate the other dimensions with somebody, this monster appears. All the numbers of the dimensions have something to do with this completely different area of mathematics. And so one of the challenges has been we made some progress, but I still think there's a lot to unlock.

Why has this strange metrological anything to do with the world of number theory in school? Monstrous moonshine. Because John Connolly thinks that there's somehow a third object which is shining lights on both of these things, which in the moonlight we see is these strange numbers appearing. In fact, Richard Fortune's Won the Fields Medal or Nobel Prize for making a lot of discoveries about what? That the sun might be shining a light on a clump.

I still believe that there's a lot we still do not understand about this strange symmetrical object, which is sort of one of the strange things. This happens on my symptoms, but actually my own research I might be interested in. If this is the periodic table, then what are the molecules of symmetry that you can make? What can you build? What are the new sort of trees that you can make out of this list in the spirit table?

So as you you know, I spend more time here analysis and doing research and symmetry, trying to build new symmetry out of the the building blocks that we got a in the act as a simple so and so I discovered a strange using that for objects with connections to things called related curves. And in fact I got one of these Newsmax.com objects which hasn't got a name yet, and this is going to be a price for you that I'm going to name the symmetrical object.

Also the person who can work out how many symmetries that Rubik's Cube has. So I got a little certificate here. It's not a name yet, so that is something. Group. This could be your name. No, no. Modern science. Science is a kind of a process of evolution as that one theory gets overturned by another. You know, the science the ancient Greeks proved in law school of business. The amazing thing about mathematics is it's eternal. This group will be there forever and species will die.

Stars will up. But this group may give you a little bit of immortality if you can claim some. So the challenge is, okay, how many symmetries does this Rubik's Cube have? Okay, so I've put it back there, the Rubik's Cube here now. So how does I'm going to do this? Okay. What I want you to do is if you go to my guess, I just want you to make a guess at how many digits this number has. So. So just if you count I did a number of women make a guess how many digits do you also think it's got?

A hundred symmetry. So it's got three digits. Okay, now you want to play you please stand up. I'm going to try and sort it out. So make your estimates using all three symmetries or three in our series. Right. So we've got some sort of players or we need to guess and you will get to be able to see if you get close to the right. So you only have one estimate. I don't need a precise formula. I just want you to estimate how many digits you think there are.

Okay, so I'm going to sort it out. I'm used to calculating the but very impressive. So if you go to ten or fewer digits, I want you to sit down. Underestimated the number. So more than that. So if you now if you got 30 or more digits, you also sit down because you overestimate. And so it's the numbers. It's between ten and 30. Okay. So if you've got some, uh, more than 20 digits in your number. I want you to keep standing. All right.

So it's between. Right. So you go there. Sometimes you got to sit down. So. Okay. If you've got more than 28 tickets and you have a you want to sit down. 27. 23. Sit down. Okay. Right. Right. So it's between. It's a it's more than 23. Okay. It's alright. So how many of you. Tony. So you got to sit down to sit down. 24. 24, 24, 25, 25. This is great news. So what is your name, sir? It's actually has 26 inches 25. Exactly. That's amazing. So I have work here. No, no.

Okay. What's your name, sir? Of course. Yes. See you. Won't see you. Can you, auntie? Do you want to say I think. Bruce. Which Hamilton? Hamilton. Paul Hamilton. Okay. That's cheating because he's a very famous scientist. You go out to the White House, so so let me do this. Okay, so k you all t. So the Kurt Hamilton group has just been born. It's a new symmetrical object built out of the symmetries in the periodic table curtain, so we can come collect this afterwards.

So there are in fact, it's a novel with 25 different surfaces. Occasionally, it's not one of the atomic symmetries. You can break down two centuries and smaller objects, but it's amazing that number of metal moves you can do that object which make it sort look like a cube, which is genuinely, distinctly different. Interestingly, the company that manufactures this Rubik's Cube, the ideal toy company say did all the packaging, the original Rubik's Cubes.

There are more than 3 billion possible states that you can go down now. Such an underestimate is analogous to McDonald's proudly announcing that they sold more than 120 hamburgers. So it is obvious that large numbers have now, as you have discovered, many more of these symmetrical objects. And so you feel that it's a myth that Kurt Hamilton has beaten you to get a bit of immortality. Actually, we have a project here out of my Muscle Institute where it's for donation to charity that I support.

And in Guatemala, an educational charity which gets kids off the streets and into education. I will name a smart girl. I'll take one of these new objects that I discovered off. You are a for a not why you calling but a present for somebody. Then they're going to own all the money goes to the charity. But more than just writing a. You know, it was interesting that, you know, we make a lot of progress in my money. So there's still all of these problems.

And, you know, I think for me, it is those things that, you know, we still don't know, always putting these symmetries together. We still don't really understand what it is we all get. The monster is in there 196,883 dimensions. And but for me, that's that's what makes me want to carry on doing mathematics you know broke out killed at the age of 20. I'm not going to make it off at first, but I often come back to that wonderful quote this Professor Kurokawa told me about a Nico,

which I think sums up mathematics as well as art. You know, in everything, uniformity is undesirable. There is something interesting and gives. One is the feeling that there is room for growth. And he says kind of silly on social things which keep us going here. Study mathematics, I think. So there's an opportunity to ask some questions about symmetry or any other mathematical problems. You only go on the line what might see.

And so yes. Question in back here we're going to like on the assignments to. We're saying. All right. We're going to come back to you. We are going to sit right with that right high. Is it significant? A maximum number of images there are is a prime number. Oh, you mean for the Alhambra? Yeah. For example. Yes. So we were looking at the walls of the Alhambra as as a two dimensional shapes, how they put together. Interesting that it's 17, which is the final. Yeah, it's a very good question.

If that's the sort of thing that we need to be a mathematician, is that in this case, I think it's not significant actually. You know, it's my favourite number 17. I mean, I'm obsessed with it. It's number eight for my football team. And there's the cicadas have a 17 year life cycle. But actually, I think in this case, actually, 17 is more of a military solution so that you find in the number of ways that you can do things.

It just turns out that 17 is the way because for example, if you go, you can consider sort of Alhambra in higher dimensions as well. So for example, the number of crystals that are possible is essentially the same question as the crystal structures, and that isn't so that's the way it has a you know, it's a good question to ask, but an anxiety, what am I going to prove is it should be climb if I go back and it turns out not to be right. So it's more trees.

But I see you know that's what arises in mathematicians is asking those looking for those kind of coincidences. I take the most the most 196,883. That's the number. And they're saying we got the same number somewhere else. And I could have just been a coincidence, but when we started to see more numbers appearing, the next time you see the monster is also hiding inside the ceiling number, there is an explanation. Then you thought, okay, I'm onto something now. So good question, but that's it.

But after a while we see actually doesn't seem to be significant is. Who questions is it? We had a question in the middle here and we could get the mike to the guy in the middle, in the blue shirt. But I'll take this on first. So you just come down with it? Yes. So you spend a lot of time explaining science and mathematics to people and sharing that knowledge. Two little questions.

Firstly, there know a push, it seems when we're funded by the public at doing research and that sort of thing, that what we do is useful. How do you what are your favourite arguments for maintaining mathematical research? I guess from the very purist level, right through all the scales. And also, do you think as publicly funded research as should we be doing more of what you're doing, which is explaining what we do to people and helping them to explain or understand mathematics and its beauty?

Yeah, well, as to the question, I think it's. We're in a kind of rocky situation with mathematics that very often the research that we do with most mathematicians, I don't think do mathematics because of utility. We don't do that because we want to create a new piece of technology. We do mathematics because we feel like we're getting access to internal trees that nature is. I think aesthetics drives a lot of the directions that we choose in mathematics.

So there's very much more similarity between the arts or very often know the choices that we make in science, very often driven by trying to explain the physical world around, you know, these symmetries on discovering things like to see them or why not studying them. Because I think how we use the stories that I can tell, we are all magical and full of surprises and excite me. And that's and a sense of your second question, which is why why do I do this?

I do this partly because I want the next generation of mathematicians to and I want to excite them. My I'm paying back my teachers to Alison. He took me out of the class in the middle of a lesson and got me excited about mathematics. And I sort of want to pay my little way for saying, okay, I want to tell people as well so they can get excited and become mathematicians as well as she's selfish.

The reason I do this because I love my subject so much, I love talking about it and then communicate, communicating. And for me, I really when I make a mathematical discovery, there are two things about being in a position. One is making a new discovery, but the second is communicating it, and that's when it starts to come online. When I tell you about something I've done or I'm in a seminar here in the department, or I write a paper and I send it to a journal and somebody else reads it.

That's the moment when suddenly the thing has life and it begins to breathe. And and so I think that's, you know, we all do this in some sort of level. And the more we communicate, the more that subject lives and breathes. But I think we all have these opportunities, although a lot of these symmetrical objects we discovered purely for the excitement of discovering something new, actually, we can very quickly connect it with a utility.

And the fact that this can create new technology, for example in the is the Acme supply and I simple said was an object is just discovered in use imagery which actually that became the basis of a new Eric Watson code that was used by Voyager to take pictures of Saturn and these pictures, right. Corrupted when they were delivered back to Earth. But using the symmetry of this code, they were able to correct these errors and produce some crystal clear images.

That is a lot like if I give an article and experience in the future, but this is my error in one corner. Rushing Roshi know how to correct error because of the symmetry and the rest of the target. So symmetry actually is at the heart of many of the photos which encode a picture or a voice in the digital data, which tend to get to be corrupted in on the way back. So I think we can very easily justify why nobody discovered that symmetry because they wanted to create the perfect prime numbers.

I used the Internet cryptography, but the things we discovered by our numbers that I use were discovered by Fama and Weiner. They to know about the Internet. But I think it's a balance. And, you know, I recognise that governments have limited resources, but you must always have a balance for value. And so yes, sure, there are some things which we can see have short term goals and there's always other. But there are many things that's that's look at this.

I will have another thing that might be useful very much later. Turned out to be just the thing you need to do to to Google, for example. Google isn't like Google elves just sitting there looking at the websites that you've just asked about. It's a clever bit of mathematics, eigenvalues of matrices, which helps you to set up exactly what you're looking for.

So it's, I think, through the way that we should justify why still resources research our brains about symmetry should be funded, even though we may not know where it's going to end. Is that huge number of examples of mathematical research that has been done which which has been subsequently catalysed. But I think we shouldn't miss the fact that actually still worth as a as a culture, you know, why should we come to music?

You know, we we are for our society. If we don't have these different strands, each one helps the other. I think, you know, I spend a lot of time talking with artists. I'm doing a project of the moments with the theatre company. The House mathematics. Actually, that helps me because they make you look at a question in a completely different way and have a different way of looking at things. And that's, you know, I really enjoy it because creating a piece of paper about maths.

But actually it's, I think it benefits all of these things enriching each other. And if you count one else, you are going to have a much poorer society. And so the question here. I probably some of the other people noticed as they came into this building outside one of the doors there's that creating a tiled surface. Penrose tiles based on an Oxford discovery. And they're marked with lines, curving lines. And some of those lines make a neat circle, about six feet in diameter.

But while I was waiting to come in, I tried following the lines like a labyrinth. Some of the more complex patterns, like the outline of a five bedroom floor. But one line went on and on and on. And I was wondering, is that potentially an infinite line in that part of the layout of the tiles isn't complete yet, but I'm afraid it's going to be fine. It's not like that.

Well, this is the beautiful thing about that design, is that it's increasing its symmetry, like five pointed star or something, but it has no repeating symmetry at all. So now I'm I would have to do a little bit of research to see whether there is an infinite line inside there. But I'm absolutely sure that there will be an unbounded. So the even if they're will find it will definitely be an unbounded number, I suspect, and I won't.

But the interesting, interesting question and, you know, if given this answer and with all their actual policies parsing, which must be infinite and what what you discovered with this tiny is a very interesting time because in some sense it contradicts what I said about the Alhambra because it looks like an 18 mean you taught it but it doesn't have the symmetry.

It has no move you can make inside those tiles such that if you move in some way, they would sit back on top of each other because of what is known periodicity to it so dramatically. They're objects that have symmetry that by force, symmetry or other symmetries, you were able to prove the global act on water treaties that nature actually discovered these before Roger. Roger actually famously and his family discovered this pattern of been used on some toilet paper.

And this is very useful for toilet paper because because of the Non-repeating passage, you don't get the kind of build up if you wrap your mind around, around. So you have some experience, you can get a nasty sort of build. So this company had used it had actually cleverly patented the design. And I think they settled out of court for some time. But actually it's be discovered before Roger because we found crystals, quality crystals.

One of the recent Nobel Prizes in chemistry was for the investigation of these crystals, which is quite remarkable because it is when you start to make the tiles. I mean, they have been very careful outside because there's you sort of have to have global knowledge for the whole structure to be able to lay it down.

And so when a crystal is forming these quasicrystals, it's quite an interesting question about how does out how does this kind of crystal known global properties, although it's to down as it grows. So there are a lot of interesting questions around the this norm here only can I say around including on the US.

Do you believe mathematics is discovered and created? And because I know Girdle said that axiom to something we believe which is true, but nowadays people call things which are axioms, which we don't necessarily know whether they're true or not. I do believe the truth lies in mathematics all the time. Or maybe the hypothesis is that we haven't breathed, which I actually don't have any proof, like the continuum hypothesis.

Yeah, well, I think that it's it's a kind of eternal tension which exists in a mathematician's life about these two words about discovery and creation. And I find myself using both of them at different times. For example, there is new symmetrical objects now. I felt I created that. I felt that it was an act of creation to somehow piece together these new set of ideas to create this shape which has lovely connections and with electric cars.

And there was a time I had this feeling of clashing that was all that was there all along, somewhat easier to put together and discover. So then you start to it like discovery. And I don't think there is this continual tension about how much and you see this in a lot of mathematical discoveries gets discovered. At the same time, non-Euclidean geometry was discovered by three different people. So it's a field like it was it was a continent out there for mathematicians to discover.

And it was Gauss Mollaei who who both came close together. And, you know, you want to say that about and I did The Magic Flute by Mozart. You wouldn't think that anybody else could create the Magic Flute. It really isn't that isn't a discovery. But actually, if you talk to a lot of composers, you find them actually countering that and saying, well, you'd be surprised how often actually a new musical idea actually has been zeitgeisty.

And people come across the same musical idea almost simultaneously. They would write the same piece of music. But then even in mathematics, we ratti we might have similar ideas. We rarely write exactly the same mathematics. And so I think actually even in the arts, one can see a little bit more of a tension between this creativity and discovery. There's interesting story.

I was at a festival with Michael Nyman and I asked him a question during the session about whether what did he think as a composer, whether he thought music was just created, or was there an element of. Do you feel like you ever discovered music? And he said, Well, yes, I can give you an example of this. And he said, I found this amazing sequence, of course, where you can get a soprano to just hold a note all the way through.

The chords just changed around her, but she just kept this one note and it was a very beautiful sort of property. So the sequence, of course, and he was very proud of it. And then a few years later, he bought an album by the Scissor Sisters, and it was an extra material. It was an encore from an article or something. And you listen to some of his material and he heard the same sequence, of course, what, running through one of the songs.

And he said he couldn't believe that they'd found it independently. So yeah, she took them to court for ripping off his music. And so there was a whole debate within the court case, and they got musicologists to try and see whether the sequence of chords and ever be found before. And eventually Scissor Sisters had to settle and email that they actually had written off because there was no evidence that.

Michael Nyman take that as an example. Actually, music, you can make discoveries of structures in a way that you think is just creative. Now coming to a question about Google is very interesting because that fact gave us a little bit of room for creativity, because you mentioned the continuum hypothesis. You know, we which I will actually come to in one of my other lectures later on.

But actually it shows that we got this question about whether there's a set of numbers which is strictly bigger than the whole numbers and strictly smaller than the rational numbers of real numbers. And it turns out that we can make a choice. It's a bit like the answer is yes and no. I think it's what students would love in an exam. All right.

So I think it's a very deep philosophical question you are answering, which I think we we constantly wrangle with in a way, how many stick our head in the round about because we just go on doing what we do. That's. You sound a. I'll try and talk loudly as it's coming through. Brilliant. Excellent. I love to see the pictures that have burned already. Scale of the Alhambra. And it seems absolutely incredible that they got all 17 symmetry elements in two days in the Alhambra.

I've been that. I'm afraid I read as much about it as I would have liked to. But do you know? Well, so did the people who designed the Alhambra. Have they done a lot of I guess that they looked around in nature and kind of looked at the structures and nature? I mean, does it it can't.

It doesn't it feels like it can't be a coincidence or is it something about the sort of symmetry elements are almost wired into us in some way, I guess is a slightly psychological question to put to you rather mathematical one, but I'm really interested to hear your thoughts on that. I think that's a that I think we all very hard line to be extremely sensitive to symmetry.

I mean I think at that which is why we find it again and again, it's something we play with or withdraw and there's lots of evidence of the brain is be so addicted to symmetry, the desire to see symmetry where there isn't a relation to those inkblots, you know, there's able to there's a basically is just a passive which is the god symmetry symmetry around it and use those because. He knew that we would try and interpret. And if you're in a psychological session, you will try and tell a story.

It is called symmetry. It's me telling you something. And so you started to unleash your inner mind. You know that it causes no. Got anything particularly special about it. But these different rules of images just seem to tease out of you as stories because you try and interpret it well. You used to get all his patients to draw mandibles, these symmetrical kind of Buddhist shapes because he believed in they would express their inner workings of their mind.

So as soon as Freud said, I see blank, I believe in a lot of psychological problems based on the fact that we try to make logic symmetrical logic is very asymmetrical. If the if A and B, that's what he does not be repeating that you want to reverse these things. But actually the mindset is always to reverse the thing that actually that could be responsible for strange psychological disorders,

for example, is was a patient of his. That's when when blood was being taken from the arm was that the arm was being removed from the bodies. And just to this reversal, and if you look at a lot of things that you can find the same sort of explanations that this sort of rape is going on. So I think going to the psychological side, I think that, you know, the brain is so online that sometimes it can confuse itself and, you know, hungers.

I think because I think this is a good example. I think all of those images were things that they would have naturally seen around them. I think this is a great example of, first of all, being inspired by nature and a lot of the questions of science and the like is stalled in nature. But then we lose our imaginations, run riots, and and we start to try and push things which are beyond the natural world.

And a lot of the mathematics, I think you can find its source in the natural world, but then it's about trying to find something and the other possibilities. So I would say that some of the Moorish artists who came up within the 14th century, those designs, and it's interesting that I showed you those patterns with the same symmetries, although they look very different. Interestingly, you find those in the same bit of the palace.

So the entrance to the palace is full of lots of rotations of the floor. Two reflections and the second floor of the House is where I was and people visiting the palace would be hosting it and it's quite clients and you could see the symmetry, right. Some of the symmetries. Well the different patterns as you go through into the more intimate parts of the palace to the harem, for example, then you suddenly see reflections disappearing. You have triangles, hexagons, you get rotations.

So a different sort of symmetry. But it seems like they always had a sensitivity to the fact that they they felt like these had some connection, although they never had the language until the 19th century. For us to be able to say, why are these? Why do you feel like these should be in the same bit of the past? But I think it is. That's why it comes back to the other question.

So I think having this combination of ways of looking at the world, why having an artistic sentiments as a mathematician is a very helpful one in creating new possibilities. And in a way you see the artistic side running ahead of the mathematical side in the Alhambra. Is there another question here? I just think, yeah, I just study fractals for school.

And I was wondering, does it make any sense and is it worth studying symmetry in a real value dimension of that's great to do because you want say, well, fractals are kind of the most magical thing because of all that's kind of strange. But actually there's a symmetry, a different sort of symmetry happening here, which is a symmetry scale. So that's another move that you could make use of.

Zoom in and then suddenly you see the same thing repeating itself and then you get this infinite repetition of the fractal. So that's actually symmetry is a very powerful tool in seeing in areas which doesn't look like it should have any sort of impact for, say, symmetry in the world. Or perhaps the geometry is still very powerful. So, yes. How do people go about attacking problems like.

Trying to classify all of the potential building blocks for symmetries and discovering things like this monster group, which is certainly not an object that you just trip on out in the street or walking around. Yeah, that's a that is one of the eternal mysteries. And when you're a research mathematician, I wish I knew the answer what the solution is.

And it is about playing around with these and any kind of and it's kind of intriguing that there were periods in trying to understand the classification of the building blocks of symmetry. What I think people just had the impression this is never going to end, we're never going to find all of the things. And we you know, perhaps this is just beyond the capacity of the human brain, even when flying across the globe of the whole community.

But as you put more and more restrictions on I mean, the monster. Exactly. It's it's a little bit like fundamental particles. The monsters was predicted because of conditions that it had before it was ever constructed. So there was a thing like that should be an object with these particular properties because we can't find any reason why they shouldn't be.

And so the conjecture, they were such that somebody conjectured there should be an object with this number of symmetries and certain properties. And the challenge was, okay, well, let's try and build this thing. And if we can use a very complex object, yet at its heart is what do you need when you're doing these properties to break down into smaller pieces? And the smaller piece in the case of the monster was what you might say is not smaller.

That and there's a way to pack 24 dimensional oranges, something called the beach lattice, which we knew about. And the centrepiece of that we started is it gave us the way to build. So you sort of these smaller jigsaw bits very often to help you to piece together to make a larger story.

But in the case of the story of symmetry, it's a wonderful example out of many hundreds of mathematicians were involved in sort of exploring this, it was a bit like going out and exploring, finding new continents for archipelago of islands, which seem to have nothing to do with anything that your home actually has a vague thought that maybe we we didn't think we found. But he says we could easily have some or will have on these estimates.

Also, he thinks that there might be a chance, there might be something out there that we that we miss because we thought our conditions that it couldn't be satisfied, which actually they could. So he has a theory, which is that I mean, you know, it's a strange thing, this sort of tension between wanting to prove the conjecture you're working on. But when he proved it, it's a sort of melancholy because, you know, you've been with that in conjecture for so long.

And I think when Fermat's Last Theorem proved and we all love. And was he finished? No, but there was another guy. And by this I only realise that that's not such a motivating problem for us. It was kind of a know, I think it was a moment of sadness in life, wonderful cracks. And there was the same symmetry. It was like, Oh gosh, let's do more questions that. I think we should finish there. It's not what you wants to do tomorrow, so thank you for doing great. Sorry. You only had one.

Yeah, it was just. Hello? I'm not sure where to start. Where we got me, actually. Uh, yeah. Just seeing Charlie. Yeah. Yeah, I just. I just took our. Oh, that is sound. No. Because you. It's. And I tried. On the 30th of May 1832, a gunshot was heard ringing out across the 30 on this one in Paris. Apparently it was walking to market that morning, ran towards where the gunshot came from and on the ground and found a young man arriving in a garden, clearly killed by a shooting range.

The young man's name was every single gunman was taken to the local hospital, the coaching hospital where he died the next morning in the arms of his brother Alfred. And he said to his brother on his death, he says, Don't cry from me, Alfred. I need all the courage I can muster to die at the age of 20. I hope likely will at you was about that morning. It must have been between Valois and a friend of his over a woman that they were both in love with them.

Others suggested that, in fact, it was the establishment trying to get rid of this troublesome revolutionary. There's even a conjecture that Galois staged his own dance in order to try and spark a new revolution in Paris. What happened to Clay? What statue was about that morning? This actually wasn't revolutionary politics, which Galois was famous because a few years earlier, while still at school.

That was so one of the great unsolved problems in mathematics versions of the academicians in Paris trying to explain his great breakthrough. But unfortunately, the academicians couldn't understand what on earth he was writing about, only because this is how he wrote most of his mathematics. But anyway, the night before that June, I realised that this probably was the last opportunity he was going to have to try and explain his great breakthrough.

So he stayed up all night writing in a letter to a friend trying to articulate this great breakthrough as the Sun appeared over the new day and he went out to meet his destiny. Maybe the fact that he was doing maths all night, the fact that he was such a bad shot that morning.

But actually, those documents that he left behind on the desk, which many of us regard as some of the most important documents in the history of mathematics, because hidden inside, there was a new language, a language to understand one of the most important concepts in mathematics and science, namely the subject of symmetry. This new language was called group theory.